1 FOL CS 531 Dr M M Awais Formal Logic The most widely used formal logic method is FIRST-ORDER...

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CS 531 Dr M M Awais 1

Formal LogicFormal Logic

• The most widely used formal logic method is

FIRST-ORDER PREDICATE LOGIC

Reference: Chapter Two The predicate Calculus

Luger’s Book

Examples included from Norvig and Russel.

First-order logic

• Whereas propositional logic assumes the world contains facts,

• first-order logic (like natural language) assumes the world contains– Objects: people, houses, numbers, colors, baseball

games, wars, …– Relations: red, round, prime, brother of, bigger than,

part of, comes between, …– Functions: father of, best friend, one more than, plus,

• PL only deals with facts• PL cannot have variables• Variables refer to things in the world• You can deliberate over them• How would represent the following fact in

PLWhen you sterilize a jar all the bacteria are

Syntax of FOL: Basic elements

• Constants john, 2, lums,...

• Predicates brother, >,...

• Functions sqrt, leftsideOf,...

• Variables X,Y,A,B...

• Connectives , , , , • Equality =

• Quantifiers ,

Syntax of FOL: Basic elements

• Constants John, Lums,...

• Predicates Brother, >,...

• Functions Sqrt, LeftsideOf,...

• Variables x,y,a,b...

• Connectives , , , , • Equality =

• Quantifiers ,

Alternate Convention also exists (Norvig uses the alternate)

Truth in first-order logic• Sentences are true with respect to a model and an

interpretation

• Model contains objects (domain elements) and relations among them

• Interpretation specifies referents forconstant symbols → objectspredicate symbols → relationsfunction symbols → functional relations

• An atomic sentence predicate(term1,...,termn) is trueiff the objects referred to by term,,..., term ,are in the relation referred to by predicate

Alphabets-IAlphabets-I

Predicates, variables, functions,constants, connectives, quantifiers, and delimiters

Constants: (first letter small)bLUE a colorsanTRO a carcrow a birdVariables: (first letter capital)Dog: an element that is a dog, but unspecifiedColor: an unspecified color

Alphabets-IIAlphabets-IIFunction (evaluates to a constant / variable):

Maps Sentences to Objects

Ali is father of Babar father_of(ali) = baber father(babar) = ali

Babar is son of Ali son_of(babar) = ali

• Interpretation has to be very clear.• If you write father(baber), the answer should be ali• For the above functions the arity is 1 (number of arguments to

the function)

Alphabets-IIAlphabets-IIFunctions:1) shahid likes zahid likes(shahid) =

2) atif likes abid likes(atif) = abid

3) Constants to Variables likes(X) = Y

{X,Y} have two possible BINDINGSBINDINGS

{X, Y} could be {shahid, zahid}

{X,Y} could be {atif, abid}

likes(X) =Y

Substitutions:

For 1 to be true:{shahid/X, zahid/Y}

For 2 to be true:{atif/X, abid/Y}

Alphabets-IIAlphabets-IIPredicate

Maps Sentences to Truth Values (True/False)

1) Shahid is student student(shahid)

2) Sana is a girl girl(sana)

3) Father of baber is elder than Hamza

elder(father(babar), hamza)

For 1 and 2 arity is 1 and for 3 the arity is 2

Alphabets-IIAlphabets-IIPredicate1) Shahid is a good studentstudent(shahid,good) or good_student(shahid) oris_good(shahid,student)

2) Sana is a friend of Saima, Sana and Saima both are girls

friend_of(sana,saima)^girl(sana)^girl(saima)3) Bill helps Fredhelps(bill,fred)

Atomic sentencesAtomic sentence = predicate (term1,...,termn)

or term1 = term2

term = function (term1,...,termn) or constant

or variable • brother(john,richard)• greater(length(leftsideOf(squareA)), length(leftsideOf(squareB)))• >(length(leftsideOf(squareA)), length(leftsideOf(squareB)))

Functions cannot be atomic sentences

Alphabets-IIIAlphabets-IIIConnectives:^ andv or~ not

ImplicationQuantificationAll persons can seeThere is a person who cannot see

Universal quantifiers (ALL)Existential quantifiers (There exists)

Complex sentences

• Complex sentences are made from atomic sentences using connectives

S, S1 S2, S1 S2, S1 S2, S1 S2,

sibling(ali,hamza) sibling(hamza,ali) >(1,2) ≤ (1,2) (1 is greater than 2 or less than equal to 2)

>(1,2) >(1,2) (1 is greater than 2 and is not greater than equal to 2)

ExamplesExamples

My house is a blue, two-story, with red shutters, and is a corner houseblue(my-house)^two-story(my-house)^red-shutters(my-

house)^corner(my-house)

Ali bought a scooter or a carbought(ali , car) v bought(ali , scooter)

IF fuel, air and spark are present the fuel will combustpresent(spark)^present(fuel)^present(air)

combustion(fuel)

Universal quantification <variables> <sentence>Everyone at LUMS is smart:

X at(X, lums) smart(X)

X P is true in a model m iff P is true with X being each possible object in the model

• Roughly speaking, equivalent to the conjunction of instantiations of P

at(rabia,lums) smart(rabia) at(shahid,lums) smart(shahid) at(lums,lums) smart(lums) ...

FOL ExamplesExamplesAll people need air

X[person(X) need_AIR(X)]

The owner of the car also owns the boat

[owner(X , car) ^ car(X , boat)]

Formulate the following expression in the PC:“Ali is a computer science student but not a pilot or

a football player”

cs_STUDENT(ali) ( pilot(ali) ft_PLAYER(ali) )

ExamplesExamples

Restate the sentence in the following way:

1. Ali is a computer science (CS) student

2. Ali is not a pilot

3. Ali is not a football player

cs_student(ali)^

~pilot(ali)^

~football_player(ali)

ExamplesExamplesStudying fuzzy systems is exciting and applying logic is great

fun if you are not going to spend all of your time slaving over the terminal

X(~slave_terminal(X) [fs_eciting(X)^logic_fun(X)])

Every voter either favors the amendment or despises it

X[voter(X) [favor(X , amendment) v despise(X,amendment)]

^ ~[favor(X , amendment) v

despise(X , amendment)] (this part simply endorses the statement, may not be required)

Undecidable Predicate

• For which exhaustive testing is required• Example:

X likes(zahra, X)• This sentence is computationally impossible to

calculate• Scope of problem domain is to be limited to

remove this problem, – i.e., X is a variable representing final year female

student in the AI class, compared to an X representing all the people in the city of Lahore

Robotic Arm Example

•Represent the initial details of the system•Generate sentences of descriptive and or implicative nature•Modify the facts using new sentences

Example: Robotic Arm

• Represent the initial details of the systems

on(a,b)on( c,d)ontable(b)ontable(d)clear(a)clear(c)hand_empty

Goal:To pick a block and place it over another block

Define predicate: stack_on(X,Y)General sentence:

Conditions ConclusionsWhat could the conditions be?

hand_emptyclear (X)clear (Y)pick (X)put_on (X,Y)

hand_empty ^ clear (X) ^ clear (Y) ^ pick (X) ^ put_on (X,Y) stack_on (X,Y)

Goal:To pick a block and place it over another block

hand_empty ^ clear (X) ^ clear (Y) ^ pick (X) ^ put_on (X,Y) stack_on (X,Y)

hand_empty ^ clear (X) pick (X) clear(Y) ^ pick (X) put_on (X,Y)

put_on (X,Y) stack_on (X,Y)

Semantically more correct

hand_empty could be written as empty(hand), if hand_empty is in the knowledge base, then hand is empty otherwise false.

put_on (X,Y) stack_on (X,Y)

Example: Robotic Arm• Modify details of the systems

on(b,a)on( c,d)ontable(b)ontable(d)clear(a)clear(c)hand_empty

on(b,a)on( c,d)on(e,a)ontable(b)ontable(d)clear(c)clear(e)hand_empty

Models for FOL: Example

A common mistake to avoid

• Represent: Everyone at LUMS is smart

X at(X,lums) smart(X)X at(X, lums) smart(X)• Common mistake:

using as the main connective with : means“Everyone is at LUMS and everyone is smart”

“Everyone at LUMS is smart” • Typically, is the main connective with

Existential quantification <variables> <sentence>

• Someone at LUMS is smart: X at(X,lums) smart(X)

X P is true in a model m iff P is true with X being some possible object in the model

• Roughly speaking, equivalent to the disjunction of instantiations of P

at(sana,lums) smart(sana) at(bashir,lums) smart(bashir) at(lums,lums) smart(lums) ...

Another common mistake to avoid

• Typically, is the main connective with

• Common mistake: using as the main connective with :

X at(X,lums) smart(X)

is true even if there is anyone who is not at LUMS!• Even if the antecedent is false the sentence can still

be true (see the truth table of implication).

Properties of quantifiers X Y is the same as Y X X Y is the same as Y X

X Y is not the same as Y X X Y loves(X,Y)

“There is a person who loves everyone in the world” Y X loves(X,Y)

“Everyone in the world is loved by at least one person”

• Quantifier duality: each can be expressed using the other X likes(X,car) X likes(X,car) X likes(X,bread) X likes(X,bread)

Equality

• term1 = term2 is true under a given interpretation if and only if term1 and term2 refer to the same object

• Sibling in terms of Parent:X,Y sibling(X,Y)

[(X = Y) M,F (M = F) parent(M,X) parent(F,X) parent(M,Y) parent(F,Y)]

Using FOL

The kinship domain:• Brothers are siblings

X,Y brother(X,Y) sibling(X,Y)• One's mother is one's female parent

M,C mother(C) = M (female(M) parent(M,C))

• “Sibling” is symmetric

X,Y sibling(X,Y) sibling(Y,X)

Rules: Wumpus worldNorvig and Russel book for details(will provide you with handouts)

• Actions– Turn(Right), Turn(Left), Forward, Shoot, Grab,

Release, Climb

• Reflex T glitter(T) bestAction(grab,T)

• Perception T,S,B percept([S,B,glitter],T) glitter(T)

Deducing Squares/PropertiesWhat are Adjacent Squares

X,Y,A,B adjacent([X,Y],[A,B]) [A,B] {[X+1,Y], [X-1,Y],[X,Y+1],[X,Y-1]}

Properties of squares:S,T at(agent,S,T) breeze(T) breezy(S)

Squares are breezy near a pit:Diagnostic rule---infer cause from effect

S breezy(S) adjacent(R,S) pit(R)Causal rule---infer effect from cause

R pit(R) [S adjacent(R,S) breezy(S)]

Deducing Squares/Properties

Diagnostic rule---infer cause from effect

S breezy(S) adjacent(R,S) pit(R)

Causal rule---infer effect from cause

R pit(R) [S adjacent(R,S) breezy(S)]

Knowledge engineering in FOL

1. Identify the task2. Assemble the relevant knowledge3. Decide on a vocabulary of predicates, functions,

and constants4. Encode general knowledge about the domain5. Encode a description of the specific problem

instance6. Pose queries to the inference procedure and get

answers7. Debug the knowledge base

8.9.10.11.12.13.

Inferencing in FOL

• Interpretation of the objects/variables etc…

• Denotation of terms

• Find whether an interpretation holds for a particular sentence

Interpretation I/Denotation

• ‘U’ set of Objects ( U universe of discourse)• Map constants symbols to objects in U

I(zahid), U={zahid, shahid, raza, sana, …} ‘T’

I(X) what happens now?• Map predicates symbols to the relations between the

objects on U• Map function symbols to functions on U

I(predicate(t1, t2)) I(predicate)(I(t1), I(t2))

Holds (meta rule)

holds (predicate(t1,t2), I)

<I(t1),I(t2)> I(predicate)

Example: Robotic ArmFind if the following sentence holds or not

on(a,b)on( c,d)ontable(b)ontable(d)clear(a)clear(c)hand_empty

stack_on (X,Y)

I(stack_on)

holds (stack_on(X,Y),I) iff <I(X),I(Y)> I(stack_on))

stack_on is not in ‘U’

‘on’ Holds but under the following interpretations

{a/X, b/Y} or {c/X, d/Y}

clear(a) holds but clear(b) does not hold

Quantifiers

FOL: Example What is U?Define the following:Predicates: above 2 circle 1oval 1square 1triangle 1 Function;hat 2

FOL: Example

What is U?

I(object) =

I(above) =

Find whether I(square) = ????

Reason it Out

• holds (sqaure(object),I) ?

• YES

• holds (above(object,hat(object)),I) ?

• I(hat(object))= sqaure

• holds(above(triangle,sqaure),I)?

• NO• Hence I(square) = FALSE

Find the truth of

• holds ( X, oval(X)), I) ?• I(sqaure)={<triangle>}

Note for the studentsShould have solved it when you come next time.

The electronic circuits domain

One-bit full adder

1. Identify the task– Does the circuit actually add properly?

(circuit verification)2. Assemble the relevant knowledge

– Composed of wires and gates; Types of gates (AND, OR, XOR, NOT)

– Irrelevant: size, shape, color, cost of gates3. Decide on a vocabulary

– Alternatives:type(x1) = xortype(x1, xor)xor(x1)

–•

–•–

The electronic circuits domain4. Encode general knowledge of the domain

T1,T2 connected(T1, T2) signal(T1) = signal(T2) T signal(T) = 1 signal(T) = 0– 1 ≠ 0 T1,T2 connected(T1, T2) connected(T2, T1) G type(G) = OR signal(out(1,G)) = 1 N

signal(in(N,G)) = 1 G type(G) = AND signal(out(1,G)) = 0 N

signal(in(N,G)) = 0 G type(G) = XOR signal(out(1,G)) = 1

signal(in(1,G)) ≠ signal(in(2,G)) G type(G) = NOT signal(out(1,G)) ≠

signal(in(1,G))–

5. Encode the specific problem instancetype(x1) = xor type(x2) = xor

type(a1) = and type(a2) = and

type(o1) = or

connected(out(1,x1),in(1,x2)) connected(in(1,c1),in(1,x1))

connected(out(1,x1),in(2,a2)) connected(in(1,c1),in(1,a1))

connected(out(1,o2),in(1,o1)) connected(in(2,c1),in(2,x1))

connected(out(1,a1),in(2,o1)) connected(in(2,c1),in(2,a1))

connected(out(1,x2),out(1,c1)) connected(in(3,c1),in(2,x2))

connected(out(1,o1),out(2,c1)) connected(in(3,c1),in(1,a2))

6. Pose queries to the inference procedureWhat are the possible sets of values of all the

terminals for the adder circuit? I1,I2,I3,O1,O2 signal(in(1,c_1)) = I1 signal(in(2,c1)) = I2 signal(in(3,c1)) = I3 signal(out(1,c1)) = O1 signal(out(2,o1)) = O2

7. Debug the knowledge baseMay have omitted assertions like 1 ≠ 0

–•

Summary

• First-order logic:– objects and relations are semantic primitives– syntax: constants, functions, predicates,

equality, quantifiers

• Increased expressive power: sufficient to define wumpus world

Operations/Processes• Entailment: • Unification: Algorithm for determining the

substitutions needed to make two predicate calculus expressions match

• Skolemization: A method of removing or replacing existential quantifiers using Skolem function

• Composition: If S and S` are two substitutions sets, then the composition of S and S` (SS`) is obtained by applying the elements of S’ to the elements of ` and adding the S’ to S

Examples (Alternate Convention)

Superscript represents arity of predicates/functions

Entailment

A knowledge base entails a sentence IFF the sentence holds in every interpretation in

which the knowledge base holds

In Propositional Logic:A knowledge base entails a sentence IFF the

sentence is TRUE in every interpretation that makes the knowledge base TRUE

Entailment (follows from)

KB entails S: for every Interpretation I,

if KB holds in I,

S holds in I

Entailment

•Computing entailment is impossible in general, because there are Infinitely many possible interpretations•Even computing holds is impossible for interpretations with infinite universes

Example

Alternate Convention

Example

1 FOL CS 531 Dr M M Awais Formal Logic The most widely used formal logic method is FIRST-ORDER...

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