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Lectures 13-14First-Order LogicLectures 13-14

First-Order Logic

CSE 573

Artificial Intelligence IHenry Kautz

Fall 2001

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Axiom SchemasAxiom Schemas

Useful to allow schemas that stand for sets of sentences.

• Blowup: (index range)nesting

for i, j in {1, 2, 3, 4} such that adjacent(i,j):

for c in {R, B, G}:

( Xc,i Xc,j)

12 4

3

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Representing Information about Different Objects

Representing Information about Different Objects

Suppose we want to talk about many different objects…

Where is truck #87? What about airplane #32?

Which vehicle carries package #806?

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Complex PropositionsComplex Propositions

At( TRUCK87, UW )

At( PLANE32, SEATAC )

In( PACKAGE806, PLANE32 )

Ingredients:

Example of a sentence:

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Complex TermsComplex Terms

plus(1,3)

4 = 1+3

teacher_of( CSE573, AU2001 )

Overworked(teacher_of(CSE573, AU2001 )) Underpaid(teacher_of(CSE573, AU2001 )))

teacher_of(CSE573, AU2001 ) = HENRY

What is the obvious conclusion?

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Models of Ground SentencesModels of Ground Sentences

What does entailment mean?

{ Overworked(teacher_of(CSE573, AU2001)),HENRY = teacher_of(CSE573, AU2001)) }

Overworked(HENRY)

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First Order InterpretationFirst Order Interpretation

1. Domain of objects

2. : Constants 3. : n-ary Function symbols (n )4. : Propositional symbols {true, false}

5. : n-ary Predicate symbols Subsets of n

6. ( “=”) = { (d, d) | d }

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Do Obvious Recursive Thing!Do Obvious Recursive Thing!

(P(a)) = true iff (a) (P) (Q(a,b)) = true iff ((a), (b)) (Q) (S T) = true iff (S) and (T) (S T) = true iff (S) or (T) (S) = true iff (S) = false

(f(a)) = (f)[(a) ] (f(a,b)) = (f)[(a), (b) ]

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ExampleExample

Overworked(teacher_of(CSE573, AU2001)) HENRY = teacher_of(CSE573, AU2001))

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Not Done Yet…Not Done Yet…

So far we’ve provided a way to break propositions down into meaningful bits

• Brings logic closer to the way we think about the world – objects, relationships…

• Doesn’t fundamentally change the expressive or computational properties of propositional logic

• “Naming convention” for propositions

• Inference: same algorithms: Davis-Putnam, GSAT, Resolution

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Universal StatementsUniversal Statements

Consider:• All men are mortal.

• A parent’s parent is a grandparent.

• No adjacent countries are colored the same color.

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Universal Statements vs Propositional Schemas

Universal Statements vs Propositional Schemas

1. Do not need to fully instantiate universal statements – more compact.

2. Can use universals in queries, because are real sentences.

3. Can use universals even if you do not have constants to uniquely identify all the objects in the domain.

4. Can use universals to talk about infinite sets

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SyntaxSyntax

x . = x . x . = x .

x . ( Man(x) Mortal(x) )

x . (Man(x) Mortal(x) )

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Some Syntactic SugarSome Syntactic Sugar

Vehicle = { TRUCK99, TRUCK33, PLANE66 }shorthand for

Vehicle(TRUCK99) Vehicle(TRUCK33) Vehicle(PLANE66 ) x . (Vehicle(x)

(x= TRUCK99 x= TRUCK33 x= PLANE66 ))

Predicate closure axiomsDomain closure axiom

x . (x= TRUCK99 x= TRUCK33 x= PLANE66 )

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SemanticsSemantics

A model also maps variables to domain objects: : Variables

x/d = the model that maps “x” to d, but is otherwise just like

( x . ) = true iff

for all d it is the case that x/d () is true

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Inference in First-Order LogicInference in First-Order Logic

Proof theory:

• Makes the leap from truth and modelsto symbol pushing

Consider a special case:

1. No function symbols

2. Closed domain, or quantify only over closed predicates

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Grounding Out a First-Order Theory (Special Case)

Grounding Out a First-Order Theory (Special Case)

Vehicle = { TRUCK99, TRUCK33, PLANE66 }

City = { SEATTLE, BOSTON }

x . ( Vehicle(x) ( y . City(y) Based(x, y) ) )

(Based(TRUCK99, SEATTLE) Based(TRUCK99, BOSTON )) (Based(TRUCK33, SEATTLE) Based(TRUCK33, BOSTON )) (Based(PLANE66 , SEATTLE) Based(PLANE66 , BOSTON ))

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Grounding RulesGrounding Rules

Foo = { F1, F2, … }

x . ( Foo(x) Bar(x))

( Bar(F1) Bar(F2) … )

x . ( Foo(x) Bar(x) )

( Bar(F1) Bar(F2) … )

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When Grounding is a Bad IdeaWhen Grounding is a Bad Idea

Everyone has friend, all of whose friends drink heavily.

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Hardness of Full First-Order Logic

Hardness of Full First-Order Logic

Can we always in principle propositionalize a theory?

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Lifted ResolutionLifted Resolution

• First-order clausal form

• Begin with universal quantifiers

• Rest is a clause

• No existentials, but may include function symbols

( Man(x)Mortal(x)) (Mortal(y)Fallible(y))

(Man(z)Fallible(z))

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UnificationUnification

Match two literals if:

• Same predicate, one positive, one negative

• Match variable(s) to other vars, constants, or complex terms (function symbols)

(Mortal(HENRY)) (Mortal(y)Fallible(y))

(Fallible(HENRY))

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Unification with Multiple Variables

Unification with Multiple Variables

You always hurt the ones you love.

Politicians love themselves.

Therefore, politicians hurt themselves.

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Unification with Function Symbols

Unification with Function Symbols

Say s(x) means the successor of x

s(1) = 2, s(2)=3, etc.

• A number is less than it’s successor.

• “Less than” is transitive.

• Therefore, a number is less than it’s successor’s successor.

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Unification with Function Symbols

Unification with Function Symbols

(Less(a,s(a))) (Less(b,c) Less(c,d) Less(b,d))

(Less(s(a),d) Less(a,d))

rename variables:

(Less(s(e),f) Less(e,f))

Less(e,s(s(e)))

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Converting to Clausal Form: Skolem Functions

Converting to Clausal Form: Skolem Functions

Everyone loves someone.x . y . ( Loves(x,y) )

x . ( Loves(x, f33(y)) )

There is somebody whom everyone loves.

y . x . ( Loves(x,y) )

x . ( Loves(x, F99) )

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Everyone Drinks?Everyone Drinks?

Everyone has friend, all of whose friends drink heavily.

x . y . x . (Friend(x,y) ( Friend(y,z) Drinks(z)))

( Friend(x,f(x) )

( Friend(f(w),z) Drinks(w) )

Conclusion: Everyone drinks heavily.x . ( Drinks(x) )

x . ( Drinks(x) )x . ( Drinks(x) )

Drinks(G)

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The Case of the Missing AxiomThe Case of the Missing Axiom

( Drinks(G)) ( Friend(f(w),z) Drinks(z) )

Friend(f(w),G)Friend(x,f(x))

Cannot unifiy f(x) and G !

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Friend is ReflexiveFriend is Reflexive

x ,y . ( Friend(x,y) Friend(y,x) )

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FOL Refutation ProofFOL Refutation Proof

Drinks(G)

( Friend(f(w),z) Drinks(z) )

Friend(f(w),G)

Friend(x,f(x))

( Friend(a,b) Friend(b,a) )

Friend(f(x),x)

()

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NextNext

Applications of Logic

• Ontologies

• Reasoning about Change

• Planning