AE4M33RZN Fuzzy logicTutorial examples

Radomiacuter Černochradomircernochfelcvutcz

Faculty of Electrical Engineering CTU in Prague

2015

Task 2

AssignmentOn the universeΔ = a b c d there is a fuzzy set

120583A = 1114106(a 03) (b 1) (c 05)1114109

Find its horizontal representation

120449A(120572) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

X 120572 = 0a b c 120572 isin (0 03⟩b c 120572 isin (03 05⟩b 120572 isin (05 1⟩

Task 3

AssignmentThe fuzzy set A has a horizontal representation

120449A(120572) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

a b c d 120572 isin ⟨0 13⟩a d 120572 isin (13 12⟩d 120572 isin (12 23⟩empty 120572 isin (23 1⟩

Find the vertical representation

Solution

120583A(a) = sup1114106120572 isin ⟨0 1⟩ ∶ a isin 120449A(120572)1114109 = sup⟨0 12⟩ = 12120583A(b) = 13 120583A(c) = 13 120583A(d) = 23 therefore

120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109

Task 5

AssignmentOn the universeΔ = IR there is a fuzzy set A

120583A(x) =

⎧⎪⎪⎨⎪⎪⎩

x x isin ⟨0 1⟩2 minus x x isin (1 15⟩0 otherwise

Find its horizontal represenation

Solution

120449A(120572) =

⎧⎪⎪⎨⎪⎪⎩

IR 120572 = 0

⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩

Task 7

AssignmentThe fuzzy set A has a horizontal representation

120449A(120572) =⎧⎪⎨⎪⎩

IR 120572 = 0

⟨1205722 1) otherwise

Find the vertical representation

120583A(x) =⎧⎪⎨⎪⎩

radicx x isin (0 1)0 otherwise

Task 8

AssignmentDecide if the following function is a fuzzy conjunction

120572 and∘ 120573 =

⎧⎪⎪⎨⎪⎪⎩

120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise

Solution

The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from

120572 and∘ 1114102120573 and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise

We get the same for 1114102120572 and∘ 1205731114105 and∘ 120574

It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)

Task 10

AssignmentProve that the Sugeno class of negations are all fuzzy negations

notS120582120572 = 1 minus 120572

1 + 120582120572 for 120582 isin (minus1infin )

Solution

When canceling out terms in a fraction we use 1 + 120582120572 gt 0

bull Involutivity Assuming 120582 gt minus1

notS120582notS120582120572 = 1minus 1minus120572

1+1205821205721+120582 1minus120572

1+120582120572= 1 + 120582120572 minus 1 + 120572

1 + 120582120572 + 120582 minus 120582120572 =120572 (120582 + 1)120582 + 1

= 120572

bull Non-increasing If 120572 le 120573 then

1 minus 1205721 + 120582120572 ge

1 minus 1205731 + 120582120573

(1 + 120582120573)(1 minus 120572) ge (1 + 120582120572)(1 minus 120573)120573 ge 120572

Task 12

AssignmentDecide if the following function is a fuzzy conjunction

120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise

Task 13

AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 14

AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573) and1113693 (120572

∘or not1113700120573) = 120572 where the

disjunction∘or is

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Solution

1 No Counterexample 120572 = 05 120573 = 012 Yes max(0 (120572 + 120573 minus 120572120573) + (120572 + (1 minus 120573) minus 120572(1 minus 120573)) minus 1) =

= max(0 120572) = 120572

3 No Counterexample 120572 = 09 120573 = 08

Task 16

AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise

is a fuzzy conjunction

Task 17

AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Task 18

AssignmentDecide which equalities hold

1 (120572 and1113700120572) 1113693or (120572 and

1113700120573) = 120572 and

1113700(120572 1113693or 120573)

2 (120572 and1113693120572) 1113700or (120572 and

1113693120573) = 120572 and

1113693(120572 1113700or 120573)

3 120572 1113700or (120572 and1113693120573) = 120572 and

1113693(120572 1113700or 120573)

Justify your conclusions

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Task 2

AssignmentOn the universeΔ = a b c d there is a fuzzy set

120583A = 1114106(a 03) (b 1) (c 05)1114109

Find its horizontal representation

120449A(120572) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

X 120572 = 0a b c 120572 isin (0 03⟩b c 120572 isin (03 05⟩b 120572 isin (05 1⟩

Task 3

AssignmentThe fuzzy set A has a horizontal representation

120449A(120572) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

a b c d 120572 isin ⟨0 13⟩a d 120572 isin (13 12⟩d 120572 isin (12 23⟩empty 120572 isin (23 1⟩

Find the vertical representation

Solution

120583A(a) = sup1114106120572 isin ⟨0 1⟩ ∶ a isin 120449A(120572)1114109 = sup⟨0 12⟩ = 12120583A(b) = 13 120583A(c) = 13 120583A(d) = 23 therefore

120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109

Task 5

AssignmentOn the universeΔ = IR there is a fuzzy set A

120583A(x) =

⎧⎪⎪⎨⎪⎪⎩

x x isin ⟨0 1⟩2 minus x x isin (1 15⟩0 otherwise

Find its horizontal represenation

Solution

120449A(120572) =

⎧⎪⎪⎨⎪⎪⎩

IR 120572 = 0

⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩

Task 7

AssignmentThe fuzzy set A has a horizontal representation

120449A(120572) =⎧⎪⎨⎪⎩

IR 120572 = 0

⟨1205722 1) otherwise

Find the vertical representation

120583A(x) =⎧⎪⎨⎪⎩

radicx x isin (0 1)0 otherwise

Task 8

AssignmentDecide if the following function is a fuzzy conjunction

120572 and∘ 120573 =

⎧⎪⎪⎨⎪⎪⎩

120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise

Solution

The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from

120572 and∘ 1114102120573 and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise

We get the same for 1114102120572 and∘ 1205731114105 and∘ 120574

It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)

Task 10

AssignmentProve that the Sugeno class of negations are all fuzzy negations

notS120582120572 = 1 minus 120572

1 + 120582120572 for 120582 isin (minus1infin )

Solution

When canceling out terms in a fraction we use 1 + 120582120572 gt 0

bull Involutivity Assuming 120582 gt minus1

notS120582notS120582120572 = 1minus 1minus120572

1+1205821205721+120582 1minus120572

1+120582120572= 1 + 120582120572 minus 1 + 120572

1 + 120582120572 + 120582 minus 120582120572 =120572 (120582 + 1)120582 + 1

= 120572

bull Non-increasing If 120572 le 120573 then

1 minus 1205721 + 120582120572 ge

1 minus 1205731 + 120582120573

(1 + 120582120573)(1 minus 120572) ge (1 + 120582120572)(1 minus 120573)120573 ge 120572

Task 12

AssignmentDecide if the following function is a fuzzy conjunction

120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise

Task 13

AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 14

AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573) and1113693 (120572

∘or not1113700120573) = 120572 where the

disjunction∘or is

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Solution

1 No Counterexample 120572 = 05 120573 = 012 Yes max(0 (120572 + 120573 minus 120572120573) + (120572 + (1 minus 120573) minus 120572(1 minus 120573)) minus 1) =

= max(0 120572) = 120572

3 No Counterexample 120572 = 09 120573 = 08

Task 16

AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise

is a fuzzy conjunction

Task 17

AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Task 18

AssignmentDecide which equalities hold

1 (120572 and1113700120572) 1113693or (120572 and

1113700120573) = 120572 and

1113700(120572 1113693or 120573)

2 (120572 and1113693120572) 1113700or (120572 and

1113693120573) = 120572 and

1113693(120572 1113700or 120573)

3 120572 1113700or (120572 and1113693120573) = 120572 and

1113693(120572 1113700or 120573)

Justify your conclusions

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Task 3

AssignmentThe fuzzy set A has a horizontal representation

120449A(120572) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

a b c d 120572 isin ⟨0 13⟩a d 120572 isin (13 12⟩d 120572 isin (12 23⟩empty 120572 isin (23 1⟩

Find the vertical representation

Solution

120583A(a) = sup1114106120572 isin ⟨0 1⟩ ∶ a isin 120449A(120572)1114109 = sup⟨0 12⟩ = 12120583A(b) = 13 120583A(c) = 13 120583A(d) = 23 therefore

120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109

Task 5

AssignmentOn the universeΔ = IR there is a fuzzy set A

120583A(x) =

⎧⎪⎪⎨⎪⎪⎩

x x isin ⟨0 1⟩2 minus x x isin (1 15⟩0 otherwise

Find its horizontal represenation

Solution

120449A(120572) =

⎧⎪⎪⎨⎪⎪⎩

IR 120572 = 0

⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩

Task 7

AssignmentThe fuzzy set A has a horizontal representation

120449A(120572) =⎧⎪⎨⎪⎩

IR 120572 = 0

⟨1205722 1) otherwise

Find the vertical representation

120583A(x) =⎧⎪⎨⎪⎩

radicx x isin (0 1)0 otherwise

Task 8

AssignmentDecide if the following function is a fuzzy conjunction

120572 and∘ 120573 =

⎧⎪⎪⎨⎪⎪⎩

120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise

Solution

The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from

120572 and∘ 1114102120573 and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise

We get the same for 1114102120572 and∘ 1205731114105 and∘ 120574

It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)

Task 10

AssignmentProve that the Sugeno class of negations are all fuzzy negations

notS120582120572 = 1 minus 120572

1 + 120582120572 for 120582 isin (minus1infin )

Solution

When canceling out terms in a fraction we use 1 + 120582120572 gt 0

bull Involutivity Assuming 120582 gt minus1

notS120582notS120582120572 = 1minus 1minus120572

1+1205821205721+120582 1minus120572

1+120582120572= 1 + 120582120572 minus 1 + 120572

1 + 120582120572 + 120582 minus 120582120572 =120572 (120582 + 1)120582 + 1

= 120572

bull Non-increasing If 120572 le 120573 then

1 minus 1205721 + 120582120572 ge

1 minus 1205731 + 120582120573

(1 + 120582120573)(1 minus 120572) ge (1 + 120582120572)(1 minus 120573)120573 ge 120572

Task 12

AssignmentDecide if the following function is a fuzzy conjunction

120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise

Task 13

AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 14

AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573) and1113693 (120572

∘or not1113700120573) = 120572 where the

disjunction∘or is

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Solution

1 No Counterexample 120572 = 05 120573 = 012 Yes max(0 (120572 + 120573 minus 120572120573) + (120572 + (1 minus 120573) minus 120572(1 minus 120573)) minus 1) =

= max(0 120572) = 120572

3 No Counterexample 120572 = 09 120573 = 08

Task 16

AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise

is a fuzzy conjunction

Task 17

AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Task 18

AssignmentDecide which equalities hold

1 (120572 and1113700120572) 1113693or (120572 and

1113700120573) = 120572 and

1113700(120572 1113693or 120573)

2 (120572 and1113693120572) 1113700or (120572 and

1113693120573) = 120572 and

1113693(120572 1113700or 120573)

3 120572 1113700or (120572 and1113693120573) = 120572 and

1113693(120572 1113700or 120573)

Justify your conclusions

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Solution

120583A(a) = sup1114106120572 isin ⟨0 1⟩ ∶ a isin 120449A(120572)1114109 = sup⟨0 12⟩ = 12120583A(b) = 13 120583A(c) = 13 120583A(d) = 23 therefore

120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109

Task 5

AssignmentOn the universeΔ = IR there is a fuzzy set A

120583A(x) =

⎧⎪⎪⎨⎪⎪⎩

x x isin ⟨0 1⟩2 minus x x isin (1 15⟩0 otherwise

Find its horizontal represenation

Solution

120449A(120572) =

⎧⎪⎪⎨⎪⎪⎩

IR 120572 = 0

⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩

Task 7

AssignmentThe fuzzy set A has a horizontal representation

120449A(120572) =⎧⎪⎨⎪⎩

IR 120572 = 0

⟨1205722 1) otherwise

Find the vertical representation

120583A(x) =⎧⎪⎨⎪⎩

radicx x isin (0 1)0 otherwise

Task 8

AssignmentDecide if the following function is a fuzzy conjunction

120572 and∘ 120573 =

⎧⎪⎪⎨⎪⎪⎩

120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise

Solution

The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from

120572 and∘ 1114102120573 and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise

We get the same for 1114102120572 and∘ 1205731114105 and∘ 120574

It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)

Task 10

AssignmentProve that the Sugeno class of negations are all fuzzy negations

notS120582120572 = 1 minus 120572

1 + 120582120572 for 120582 isin (minus1infin )

Solution

When canceling out terms in a fraction we use 1 + 120582120572 gt 0

bull Involutivity Assuming 120582 gt minus1

notS120582notS120582120572 = 1minus 1minus120572

1+1205821205721+120582 1minus120572

1+120582120572= 1 + 120582120572 minus 1 + 120572

1 + 120582120572 + 120582 minus 120582120572 =120572 (120582 + 1)120582 + 1

= 120572

bull Non-increasing If 120572 le 120573 then

1 minus 1205721 + 120582120572 ge

1 minus 1205731 + 120582120573

(1 + 120582120573)(1 minus 120572) ge (1 + 120582120572)(1 minus 120573)120573 ge 120572

Task 12

AssignmentDecide if the following function is a fuzzy conjunction

120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise

Task 13

AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 14

AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573) and1113693 (120572

∘or not1113700120573) = 120572 where the

disjunction∘or is

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Solution

1 No Counterexample 120572 = 05 120573 = 012 Yes max(0 (120572 + 120573 minus 120572120573) + (120572 + (1 minus 120573) minus 120572(1 minus 120573)) minus 1) =

= max(0 120572) = 120572

3 No Counterexample 120572 = 09 120573 = 08

Task 16

AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise

is a fuzzy conjunction

Task 17

AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Task 18

AssignmentDecide which equalities hold

1 (120572 and1113700120572) 1113693or (120572 and

1113700120573) = 120572 and

1113700(120572 1113693or 120573)

2 (120572 and1113693120572) 1113700or (120572 and

1113693120573) = 120572 and

1113693(120572 1113700or 120573)

3 120572 1113700or (120572 and1113693120573) = 120572 and

1113693(120572 1113700or 120573)

Justify your conclusions

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Task 5

AssignmentOn the universeΔ = IR there is a fuzzy set A

120583A(x) =

⎧⎪⎪⎨⎪⎪⎩

x x isin ⟨0 1⟩2 minus x x isin (1 15⟩0 otherwise

Find its horizontal represenation

Solution

120449A(120572) =

⎧⎪⎪⎨⎪⎪⎩

IR 120572 = 0

⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩

Task 7

AssignmentThe fuzzy set A has a horizontal representation

120449A(120572) =⎧⎪⎨⎪⎩

IR 120572 = 0

⟨1205722 1) otherwise

Find the vertical representation

120583A(x) =⎧⎪⎨⎪⎩

radicx x isin (0 1)0 otherwise

Task 8

AssignmentDecide if the following function is a fuzzy conjunction

120572 and∘ 120573 =

⎧⎪⎪⎨⎪⎪⎩

120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise

Solution

The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from

120572 and∘ 1114102120573 and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise

We get the same for 1114102120572 and∘ 1205731114105 and∘ 120574

It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)

Task 10

AssignmentProve that the Sugeno class of negations are all fuzzy negations

notS120582120572 = 1 minus 120572

1 + 120582120572 for 120582 isin (minus1infin )

Solution

When canceling out terms in a fraction we use 1 + 120582120572 gt 0

bull Involutivity Assuming 120582 gt minus1

notS120582notS120582120572 = 1minus 1minus120572

1+1205821205721+120582 1minus120572

1+120582120572= 1 + 120582120572 minus 1 + 120572

1 + 120582120572 + 120582 minus 120582120572 =120572 (120582 + 1)120582 + 1

= 120572

bull Non-increasing If 120572 le 120573 then

1 minus 1205721 + 120582120572 ge

1 minus 1205731 + 120582120573

(1 + 120582120573)(1 minus 120572) ge (1 + 120582120572)(1 minus 120573)120573 ge 120572

Task 12

AssignmentDecide if the following function is a fuzzy conjunction

120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise

Task 13

AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 14

AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573) and1113693 (120572

∘or not1113700120573) = 120572 where the

disjunction∘or is

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Solution

1 No Counterexample 120572 = 05 120573 = 012 Yes max(0 (120572 + 120573 minus 120572120573) + (120572 + (1 minus 120573) minus 120572(1 minus 120573)) minus 1) =

= max(0 120572) = 120572

3 No Counterexample 120572 = 09 120573 = 08

Task 16

AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise

is a fuzzy conjunction

Task 17

AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Task 18

AssignmentDecide which equalities hold

1 (120572 and1113700120572) 1113693or (120572 and

1113700120573) = 120572 and

1113700(120572 1113693or 120573)

2 (120572 and1113693120572) 1113700or (120572 and

1113693120573) = 120572 and

1113693(120572 1113700or 120573)

3 120572 1113700or (120572 and1113693120573) = 120572 and

1113693(120572 1113700or 120573)

Justify your conclusions

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Solution

120449A(120572) =

⎧⎪⎪⎨⎪⎪⎩

IR 120572 = 0

⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩

Task 7

AssignmentThe fuzzy set A has a horizontal representation

120449A(120572) =⎧⎪⎨⎪⎩

IR 120572 = 0

⟨1205722 1) otherwise

Find the vertical representation

120583A(x) =⎧⎪⎨⎪⎩

radicx x isin (0 1)0 otherwise

Task 8

AssignmentDecide if the following function is a fuzzy conjunction

120572 and∘ 120573 =

⎧⎪⎪⎨⎪⎪⎩

120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise

Solution

The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from

120572 and∘ 1114102120573 and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise

We get the same for 1114102120572 and∘ 1205731114105 and∘ 120574

It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)

Task 10

AssignmentProve that the Sugeno class of negations are all fuzzy negations

notS120582120572 = 1 minus 120572

1 + 120582120572 for 120582 isin (minus1infin )

Solution

When canceling out terms in a fraction we use 1 + 120582120572 gt 0

bull Involutivity Assuming 120582 gt minus1

notS120582notS120582120572 = 1minus 1minus120572

1+1205821205721+120582 1minus120572

1+120582120572= 1 + 120582120572 minus 1 + 120572

1 + 120582120572 + 120582 minus 120582120572 =120572 (120582 + 1)120582 + 1

= 120572

bull Non-increasing If 120572 le 120573 then

1 minus 1205721 + 120582120572 ge

1 minus 1205731 + 120582120573

(1 + 120582120573)(1 minus 120572) ge (1 + 120582120572)(1 minus 120573)120573 ge 120572

Task 12

AssignmentDecide if the following function is a fuzzy conjunction

120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise

Task 13

AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 14

AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573) and1113693 (120572

∘or not1113700120573) = 120572 where the

disjunction∘or is

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Solution

1 No Counterexample 120572 = 05 120573 = 012 Yes max(0 (120572 + 120573 minus 120572120573) + (120572 + (1 minus 120573) minus 120572(1 minus 120573)) minus 1) =

= max(0 120572) = 120572

3 No Counterexample 120572 = 09 120573 = 08

Task 16

AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise

is a fuzzy conjunction

Task 17

AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Task 18

AssignmentDecide which equalities hold

1 (120572 and1113700120572) 1113693or (120572 and

1113700120573) = 120572 and

1113700(120572 1113693or 120573)

2 (120572 and1113693120572) 1113700or (120572 and

1113693120573) = 120572 and

1113693(120572 1113700or 120573)

3 120572 1113700or (120572 and1113693120573) = 120572 and

1113693(120572 1113700or 120573)

Justify your conclusions

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Task 7

AssignmentThe fuzzy set A has a horizontal representation

120449A(120572) =⎧⎪⎨⎪⎩

IR 120572 = 0

⟨1205722 1) otherwise

Find the vertical representation

120583A(x) =⎧⎪⎨⎪⎩

radicx x isin (0 1)0 otherwise

Task 8

AssignmentDecide if the following function is a fuzzy conjunction

120572 and∘ 120573 =

⎧⎪⎪⎨⎪⎪⎩

120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise

Solution

The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from

120572 and∘ 1114102120573 and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise

We get the same for 1114102120572 and∘ 1205731114105 and∘ 120574

It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)

Task 10

AssignmentProve that the Sugeno class of negations are all fuzzy negations

notS120582120572 = 1 minus 120572

1 + 120582120572 for 120582 isin (minus1infin )

Solution

When canceling out terms in a fraction we use 1 + 120582120572 gt 0

bull Involutivity Assuming 120582 gt minus1

notS120582notS120582120572 = 1minus 1minus120572

1+1205821205721+120582 1minus120572

1+120582120572= 1 + 120582120572 minus 1 + 120572

1 + 120582120572 + 120582 minus 120582120572 =120572 (120582 + 1)120582 + 1

= 120572

bull Non-increasing If 120572 le 120573 then

1 minus 1205721 + 120582120572 ge

1 minus 1205731 + 120582120573

(1 + 120582120573)(1 minus 120572) ge (1 + 120582120572)(1 minus 120573)120573 ge 120572

Task 12

AssignmentDecide if the following function is a fuzzy conjunction

120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise

Task 13

AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 14

AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573) and1113693 (120572

∘or not1113700120573) = 120572 where the

disjunction∘or is

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Solution

1 No Counterexample 120572 = 05 120573 = 012 Yes max(0 (120572 + 120573 minus 120572120573) + (120572 + (1 minus 120573) minus 120572(1 minus 120573)) minus 1) =

= max(0 120572) = 120572

3 No Counterexample 120572 = 09 120573 = 08

Task 16

AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise

is a fuzzy conjunction

Task 17

AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Task 18

AssignmentDecide which equalities hold

1 (120572 and1113700120572) 1113693or (120572 and

1113700120573) = 120572 and

1113700(120572 1113693or 120573)

2 (120572 and1113693120572) 1113700or (120572 and

1113693120573) = 120572 and

1113693(120572 1113700or 120573)

3 120572 1113700or (120572 and1113693120573) = 120572 and

1113693(120572 1113700or 120573)

Justify your conclusions

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Task 8

AssignmentDecide if the following function is a fuzzy conjunction

120572 and∘ 120573 =

⎧⎪⎪⎨⎪⎪⎩

120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise

Solution

The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from

120572 and∘ 1114102120573 and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise

We get the same for 1114102120572 and∘ 1205731114105 and∘ 120574

It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)

Task 10

AssignmentProve that the Sugeno class of negations are all fuzzy negations

notS120582120572 = 1 minus 120572

1 + 120582120572 for 120582 isin (minus1infin )

Solution

When canceling out terms in a fraction we use 1 + 120582120572 gt 0

bull Involutivity Assuming 120582 gt minus1

notS120582notS120582120572 = 1minus 1minus120572

1+1205821205721+120582 1minus120572

1+120582120572= 1 + 120582120572 minus 1 + 120572

1 + 120582120572 + 120582 minus 120582120572 =120572 (120582 + 1)120582 + 1

= 120572

bull Non-increasing If 120572 le 120573 then

1 minus 1205721 + 120582120572 ge

1 minus 1205731 + 120582120573

(1 + 120582120573)(1 minus 120572) ge (1 + 120582120572)(1 minus 120573)120573 ge 120572

Task 12

AssignmentDecide if the following function is a fuzzy conjunction

120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise

Task 13

AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 14

AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573) and1113693 (120572

∘or not1113700120573) = 120572 where the

disjunction∘or is

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Solution

1 No Counterexample 120572 = 05 120573 = 012 Yes max(0 (120572 + 120573 minus 120572120573) + (120572 + (1 minus 120573) minus 120572(1 minus 120573)) minus 1) =

= max(0 120572) = 120572

3 No Counterexample 120572 = 09 120573 = 08

Task 16

AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise

is a fuzzy conjunction

Task 17

AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Task 18

AssignmentDecide which equalities hold

1 (120572 and1113700120572) 1113693or (120572 and

1113700120573) = 120572 and

1113700(120572 1113693or 120573)

2 (120572 and1113693120572) 1113700or (120572 and

1113693120573) = 120572 and

1113693(120572 1113700or 120573)

3 120572 1113700or (120572 and1113693120573) = 120572 and

1113693(120572 1113700or 120573)

Justify your conclusions

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Solution

The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from

120572 and∘ 1114102120573 and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise

We get the same for 1114102120572 and∘ 1205731114105 and∘ 120574

It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)

Task 10

AssignmentProve that the Sugeno class of negations are all fuzzy negations

notS120582120572 = 1 minus 120572

1 + 120582120572 for 120582 isin (minus1infin )

Solution

When canceling out terms in a fraction we use 1 + 120582120572 gt 0

bull Involutivity Assuming 120582 gt minus1

notS120582notS120582120572 = 1minus 1minus120572

1+1205821205721+120582 1minus120572

1+120582120572= 1 + 120582120572 minus 1 + 120572

1 + 120582120572 + 120582 minus 120582120572 =120572 (120582 + 1)120582 + 1

= 120572

bull Non-increasing If 120572 le 120573 then

1 minus 1205721 + 120582120572 ge

1 minus 1205731 + 120582120573

(1 + 120582120573)(1 minus 120572) ge (1 + 120582120572)(1 minus 120573)120573 ge 120572

Task 12

AssignmentDecide if the following function is a fuzzy conjunction

120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise

Task 13

AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 14

AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573) and1113693 (120572

∘or not1113700120573) = 120572 where the

disjunction∘or is

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Solution

1 No Counterexample 120572 = 05 120573 = 012 Yes max(0 (120572 + 120573 minus 120572120573) + (120572 + (1 minus 120573) minus 120572(1 minus 120573)) minus 1) =

= max(0 120572) = 120572

3 No Counterexample 120572 = 09 120573 = 08

Task 16

AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise

is a fuzzy conjunction

Task 17

AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Task 18

AssignmentDecide which equalities hold

1 (120572 and1113700120572) 1113693or (120572 and

1113700120573) = 120572 and

1113700(120572 1113693or 120573)

2 (120572 and1113693120572) 1113700or (120572 and

1113693120573) = 120572 and

1113693(120572 1113700or 120573)

3 120572 1113700or (120572 and1113693120573) = 120572 and

1113693(120572 1113700or 120573)

Justify your conclusions

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Task 10

AssignmentProve that the Sugeno class of negations are all fuzzy negations

notS120582120572 = 1 minus 120572

1 + 120582120572 for 120582 isin (minus1infin )

Solution

When canceling out terms in a fraction we use 1 + 120582120572 gt 0

bull Involutivity Assuming 120582 gt minus1

notS120582notS120582120572 = 1minus 1minus120572

1+1205821205721+120582 1minus120572

1+120582120572= 1 + 120582120572 minus 1 + 120572

1 + 120582120572 + 120582 minus 120582120572 =120572 (120582 + 1)120582 + 1

= 120572

bull Non-increasing If 120572 le 120573 then

1 minus 1205721 + 120582120572 ge

1 minus 1205731 + 120582120573

(1 + 120582120573)(1 minus 120572) ge (1 + 120582120572)(1 minus 120573)120573 ge 120572

Task 12

AssignmentDecide if the following function is a fuzzy conjunction

120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise

Task 13

AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 14

AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573) and1113693 (120572

∘or not1113700120573) = 120572 where the

disjunction∘or is

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Solution

1 No Counterexample 120572 = 05 120573 = 012 Yes max(0 (120572 + 120573 minus 120572120573) + (120572 + (1 minus 120573) minus 120572(1 minus 120573)) minus 1) =

= max(0 120572) = 120572

3 No Counterexample 120572 = 09 120573 = 08

Task 16

AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise

is a fuzzy conjunction

Task 17

AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Task 18

AssignmentDecide which equalities hold

1 (120572 and1113700120572) 1113693or (120572 and

1113700120573) = 120572 and

1113700(120572 1113693or 120573)

2 (120572 and1113693120572) 1113700or (120572 and

1113693120573) = 120572 and

1113693(120572 1113700or 120573)

3 120572 1113700or (120572 and1113693120573) = 120572 and

1113693(120572 1113700or 120573)

Justify your conclusions

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Solution

When canceling out terms in a fraction we use 1 + 120582120572 gt 0

bull Involutivity Assuming 120582 gt minus1

notS120582notS120582120572 = 1minus 1minus120572

1+1205821205721+120582 1minus120572

1+120582120572= 1 + 120582120572 minus 1 + 120572

1 + 120582120572 + 120582 minus 120582120572 =120572 (120582 + 1)120582 + 1

= 120572

bull Non-increasing If 120572 le 120573 then

1 minus 1205721 + 120582120572 ge

1 minus 1205731 + 120582120573

(1 + 120582120573)(1 minus 120572) ge (1 + 120582120572)(1 minus 120573)120573 ge 120572

Task 12

AssignmentDecide if the following function is a fuzzy conjunction

120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise

Task 13

AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 14

AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573) and1113693 (120572

∘or not1113700120573) = 120572 where the

disjunction∘or is

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Solution

1 No Counterexample 120572 = 05 120573 = 012 Yes max(0 (120572 + 120573 minus 120572120573) + (120572 + (1 minus 120573) minus 120572(1 minus 120573)) minus 1) =

= max(0 120572) = 120572

3 No Counterexample 120572 = 09 120573 = 08

Task 16

AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise

is a fuzzy conjunction

Task 17

AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Task 18

AssignmentDecide which equalities hold

1 (120572 and1113700120572) 1113693or (120572 and

1113700120573) = 120572 and

1113700(120572 1113693or 120573)

2 (120572 and1113693120572) 1113700or (120572 and

1113693120573) = 120572 and

1113693(120572 1113700or 120573)

3 120572 1113700or (120572 and1113693120573) = 120572 and

1113693(120572 1113700or 120573)

Justify your conclusions

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Task 12

AssignmentDecide if the following function is a fuzzy conjunction

120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise

Task 13

AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 14

AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573) and1113693 (120572

∘or not1113700120573) = 120572 where the

disjunction∘or is

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Solution

1 No Counterexample 120572 = 05 120573 = 012 Yes max(0 (120572 + 120573 minus 120572120573) + (120572 + (1 minus 120573) minus 120572(1 minus 120573)) minus 1) =

= max(0 120572) = 120572

3 No Counterexample 120572 = 09 120573 = 08

Task 16

AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise

is a fuzzy conjunction

Task 17

AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Task 18

AssignmentDecide which equalities hold

1 (120572 and1113700120572) 1113693or (120572 and

1113700120573) = 120572 and

1113700(120572 1113693or 120573)

2 (120572 and1113693120572) 1113700or (120572 and

1113693120573) = 120572 and

1113693(120572 1113700or 120573)

3 120572 1113700or (120572 and1113693120573) = 120572 and

1113693(120572 1113700or 120573)

Justify your conclusions

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Task 13

AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

min(120572 120573) 120572 + 120573 ge 1

0 otherwise

Task 14

AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573) and1113693 (120572

∘or not1113700120573) = 120572 where the

disjunction∘or is

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Solution

1 No Counterexample 120572 = 05 120573 = 012 Yes max(0 (120572 + 120573 minus 120572120573) + (120572 + (1 minus 120573) minus 120572(1 minus 120573)) minus 1) =

= max(0 120572) = 120572

3 No Counterexample 120572 = 09 120573 = 08

Task 16

AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise

is a fuzzy conjunction

Task 17

AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Task 18

AssignmentDecide which equalities hold

1 (120572 and1113700120572) 1113693or (120572 and

1113700120573) = 120572 and

1113700(120572 1113693or 120573)

2 (120572 and1113693120572) 1113700or (120572 and

1113693120573) = 120572 and

1113693(120572 1113700or 120573)

3 120572 1113700or (120572 and1113693120573) = 120572 and

1113693(120572 1113700or 120573)

Justify your conclusions

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Task 14

AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573) and1113693 (120572

∘or not1113700120573) = 120572 where the

disjunction∘or is

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Solution

1 No Counterexample 120572 = 05 120573 = 012 Yes max(0 (120572 + 120573 minus 120572120573) + (120572 + (1 minus 120573) minus 120572(1 minus 120573)) minus 1) =

= max(0 120572) = 120572

3 No Counterexample 120572 = 09 120573 = 08

Task 16

AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise

is a fuzzy conjunction

Task 17

AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Task 18

AssignmentDecide which equalities hold

1 (120572 and1113700120572) 1113693or (120572 and

1113700120573) = 120572 and

1113700(120572 1113693or 120573)

2 (120572 and1113693120572) 1113700or (120572 and

1113693120573) = 120572 and

1113693(120572 1113700or 120573)

3 120572 1113700or (120572 and1113693120573) = 120572 and

1113693(120572 1113700or 120573)

Justify your conclusions

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Solution

1 No Counterexample 120572 = 05 120573 = 012 Yes max(0 (120572 + 120573 minus 120572120573) + (120572 + (1 minus 120573) minus 120572(1 minus 120573)) minus 1) =

= max(0 120572) = 120572

3 No Counterexample 120572 = 09 120573 = 08

Task 16

AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise

is a fuzzy conjunction

Task 17

AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Task 18

AssignmentDecide which equalities hold

1 (120572 and1113700120572) 1113693or (120572 and

1113700120573) = 120572 and

1113700(120572 1113693or 120573)

2 (120572 and1113693120572) 1113700or (120572 and

1113693120573) = 120572 and

1113693(120572 1113700or 120573)

3 120572 1113700or (120572 and1113693120573) = 120572 and

1113693(120572 1113700or 120573)

Justify your conclusions

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Task 16

AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩

2 rarr ⟨0 1⟩

120572 and∘ 120573 =⎧⎪⎨⎪⎩

120572120573 120572 + 120573 ge 1

0 otherwise

is a fuzzy conjunction

Task 17

AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Task 18

AssignmentDecide which equalities hold

1 (120572 and1113700120572) 1113693or (120572 and

1113700120573) = 120572 and

1113700(120572 1113693or 120573)

2 (120572 and1113693120572) 1113700or (120572 and

1113693120573) = 120572 and

1113693(120572 1113700or 120573)

3 120572 1113700or (120572 and1113693120573) = 120572 and

1113693(120572 1113700or 120573)

Justify your conclusions

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Task 17

AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for

1 standard 1113700or2 algebraic 1113682or3 Łukasiewicz 1113693or

Task 18

AssignmentDecide which equalities hold

1 (120572 and1113700120572) 1113693or (120572 and

1113700120573) = 120572 and

1113700(120572 1113693or 120573)

2 (120572 and1113693120572) 1113700or (120572 and

1113693120573) = 120572 and

1113693(120572 1113700or 120573)

3 120572 1113700or (120572 and1113693120573) = 120572 and

1113693(120572 1113700or 120573)

Justify your conclusions

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Task 18

AssignmentDecide which equalities hold

1 (120572 and1113700120572) 1113693or (120572 and

1113700120573) = 120572 and

1113700(120572 1113693or 120573)

2 (120572 and1113693120572) 1113700or (120572 and

1113693120573) = 120572 and

1113693(120572 1113700or 120573)

3 120572 1113700or (120572 and1113693120573) = 120572 and

1113693(120572 1113700or 120573)

Justify your conclusions

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Task 19

AssignmentDecide which equalities hold

1 (120572 and1113700120573) and

1113693120574 = 120572 and

1113700(120573 and

1113693120574)

2 not1113700(120572 1113682or 120573) = not

1113700120572 and

1113693not1113700120573

3 (120572 and1113693120572) 1113693or not

1113700120572 = (not

1113700120572 and

1113693not1113700120572) 1113693or 120572

Justify your conclusions

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Task 20

AssignmentVerify that 120572 and∘ (120572

1113699rArr∘ 120573) = 120572 and1113700 120573 holds for

1 algebraic ops

2 standard ops

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Solution

We will show the solution for algebraic operations The other ones aresimilar

120572 and1113682(120572 1113699rArr

1113682120573) =

⎧⎪⎨⎪⎩

120572 sdot 1 = 120572 for 120572 le 120573

120572 sdot 120573120572 = 120573 for 120572 gt 120573

⎫⎪⎬⎪⎭= min(120572 120573) = 120572 and

1113700120573

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Task 22

AssignmentComplete the table so that R is a S-partial order

R a b c d

ab 05c 03d 02

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Solution

Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal

R a b c d

a 1 0 xprime yprime

b 05 1 0 0c x 03 1 zprime

d y 02 z 1

The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates

into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Ex Jim revisited

We will use the Lukasiewicz logic in the following examples (⊓ =⊓1113693 )

⟨jim ∶ 120236120250120261120254 | 09⟩ (1)

⟨jim ∶ 120229120254120262120250120261120254 | 02⟩ (2)

⟨120236120250120261120254 ⊓ 120229120254120262120250120261120254 ⊑perp | 1⟩ (3)

The interpretation domain isΔℐ1 = Δℐ2 = j jimℐ1 = jimℐ2 = j

120236120250120261120254ℐ1 = (j 09)120229120254120262120250120261120254ℐ1 = (j 0)

120236120250120261120254ℐ2 = (j 09)120229120254120262120250120261120254ℐ2 = (j 02)

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Ex Jim revisited (check your knowledge)

Letrsquos check the interpretation against the definitions

ℐ ⊧ 120591 120591(1113568) 120591(1113569) 120591(1113570)ℐ1 yes no yes

ℐ2 yse yes no

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Ex Jim revisited (in fuzzyDL)

Letrsquos change the weights and encode the example in fuzzyDL

(instance jim Male 04)(instance jim Female 02)

(l-implies (and Male Female) bottom 09)

(min-instance jim Male)(max-instance jim Male)(min-instance jim Female)(max-instance jim Female)

Let ⟨jim ∶ 120236120250120261120254 | 120572⟩ and 1113993jim ∶ 120229120254120262120250120261120254 | 1205731113980 what are the bounds on 120572and 120573 fuzzyDL shows that 04 le 120572 le 09 and 02 le 120573 le 07 Why

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Ex Smokers

Recall the motivational example from the first lecture

1113993symmetric(120255 120267120258120254120263120253)1113980 (4)

1113993(anna bill) ∶ 120255 120267120258120254120263120253 | 11113980 (5)

1113993(bill cloe) ∶ 120255 120267120258120254120263120253 | 11113980 (6)

1113993(cloe dirk) ∶ 120255 120267120258120254120263120253 | 11113980 (7)

⟨anna ∶ 120242120262120264120260120254120267 | 1⟩ (8)

⟨exist 120255120267120258120254120263120253 sdot 120242120262120264120260120254120267 ⊑ 120242120262120264120260120254120267 | 07⟩ (9)

What are the bounds on ⟨i ∶ 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Ex Smokers

What changes if we add

⟨dirk ∶ not 120242120262120264120260120254120267 | 07⟩ (10)

(11)

What are the bounds on ⟨i ∶ not 120242120262120264120260120254120267⟩ for i isin anna bill cloe dirk

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Ex Smokers (in fuzzyDL)

(implies (some friendOf Smoker) Smoker 07)

(symmetric friendOf)(related anna bill friendOf)(related bill cloe friendOf)(related cloe dirk friendOf)

(instance anna Smoker)(instance dirk (not Smoker) 07)

(min-instance anna Smoker)(min-instance bill Smoker)(min-instance cloe Smoker)(min-instance dirk Smoker)

(max-instance anna Smoker)(max-instance bill Smoker)(max-instance cloe Smoker)(max-instance dirk Smoker)

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Concrete data types

The domainΔℐ is an unordered set This is good for modellingcathegorical data eg colors people

General idea Extended interpretationBut we also need to include real numbers IR The fuzzy descriptionlogic with concrete datatypes RGHEmdashC(uses ldquoabstract objectsrdquo andldquoconcrete objectsrdquo

Δℐ = Δℐa cup IR

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Concrete data types

bull Concrete individuals are interpreted as objects from IRbull Concrete concepts are interpreted as subsets from IRbull Concrete roles are interpreted as subsets from (Δℐa times IR)

All non-concrete notions are called abstract

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Concrete data types New concepts

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Ex Age of parents

(related adam bob parent) (related adam eve parent)

(define-fuzzy-concept around23 triangular(0100 182326))(define-fuzzy-concept moreTh17 right-shoulder(0100 1321))(instance bob (some age around23) 09)(instance eve (some age moreTh17))

(define-fuzzy-concept young left-shoulder(0100 1725))(define-concept YoungPerson (some age young))

(min-instance eve YoungPerson) (max-instance eve YoungPerson)(min-instance bob YoungPerson) (max-instance bob YoungPerson)(min-instance adam (all parent YoungPerson))(max-instance adam (all parent YoungPerson))(min-instance adam (some parent YoungPerson))(max-instance adam (some parent YoungPerson))

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Ex Age of parents

1 What are the bounds on 120572 from ⟨eve ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120572⟩

Start by drawing the concept 120250120267120264120270120263120253120804120805 then construct aninterpretation Howmuch freedom do you have when constructingthe interpretation

2 Let fuzzyDL reasoner give you both bounds on1113993i ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 120573i1113980 for i isin eve bob

How do you infer the bounds on 1113993adam ∶ 120248120264120270120263120256120239120254120267120268120264120263 | 1205741113980

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Ex Car dealing

1 The buyer wants a passenger that costs less than euro26000

2 If there is an alarm system in the car then he is satisfied withpaying no more than euro22300 but he can go up to euro22750 with alesser degree of satisfaction

3 The driver insurance air conditioning and the black color areimportant factors

4 Preferably the price is no more than euro22000 but he can go toeuro24000 to a lesser degree of satisfaction

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Ex Car dealing

1 The seller wants to sell no less than euro22000

2 Preferably the buyer buys the insurance plus package

3 If the color is black then it is highly possible the car has anair-conditioning

This can be formalized in fuzzy description logicWe have the background knowledge

⟨120242120254120253120250120263⊑120239120250120268120268120254120263120256120254120267120226120250120267 | 1⟩⟨120232120263120268120270120267120250120263120252120254120239120261120270120268 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254 ⊓ 120243120257120254120255120269120232120263120268120270120267120250120263120252120254 | 1⟩

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Ex Car dealing

The buyerrsquos preferences

1 B = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot le 26000

2 B1 = 120224120261120250120267120262120242120274120268120269120254120262↦exist120265120267120258120252120254 sdot lsh(22300 22750)

3 B2 = 120227120267120258120271120254120267120232120263120268120270120267120250120263120252120254B3 = 120224120258120267120226120264120263120253120258120269120258120264120263B4 = exist 120252120264120261120264120267 sdot 1202251202611202501202521202604 B5 = exist120265120267120258120252120254 sdot lsh(22000 24000)

The buyerrsquos preferences

1 S = 120239120250120268120268120254120263120256120254120267120226120250120267 ⊓ exist 120265120267120258120252120254 sdot ge 22000

2 S1 = 1202321202631202681202701202671202501202631202521202541202391202611202701202683 S2 = (05 (exist 120252120264120261120264120267 sdot 120225120261120250120252120260)↦120224120258120267120226120264120263120253120258120269120258120264120263)

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Ex Car dealing

We know that S and B are hard constraints and B15 and S12 are softpreferences All the concepts can be ldquosummed uprdquo

120225120270120274 = B ⊓ (01B1 + 02B2 + 01B3 + 04B4 + 02B5)and

120242120254120261120261 = S ⊓ (06S1 + 04S2)

A good choice of⊓ can make B a hard constraint

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types

Ex Car dealing

Optimal match

bsb(K 120225120270120274 ⊓ 120242120254120261120261)Finds the optimal match between a seller and a buyer (Finds an idealimaginary car that maximizes satisfaction of both parties)

Particular car

bdb(K ⟨audiTT ∶ 120225120270120274 ⊓ 120242120254120261120261⟩)Finds the degree of satisfaction for a particuklar car audiTT

• Fuzzy DL examples
• Concrete data types