Lecture 8: First-order Back to resolution in predicate logic - PowerPoint PPT Presentation

CS440/ECE448: Intro to Artificial Intelligence � Lecture 8: 
 First-order 
 Back to resolution in predicate logic � propositional logic � Prof. Julia Hockenmaier � juliahmr@illinois.edu � � http://cs.illinois.edu/fa11/cs440 � � � The resolution rule � Proof by contradiction � How do we prove that " ⊨ # ? Unit resolution: p 1 ∨… ∨ p i-1 ∨ p i ∨ p i+1 ∨… ∨ p n ¬ p i !!!!!!!!!!!!!!!!!!!!!!!!!!!! " entails # (‘ " ⊨ # ’) iff " ∧ ¬ # not satisfiable. p 1 ∨… ∨ p i-1 ∨ p i+1 ∨… ∨ p n � Full resolution: Proof: � " ∧ ¬ # not satisfiable iff ⊨ ¬ ( " ∧ ¬ # ) p 1 ∨… ∨…∨ p i ∨ …∨ …∨ p n q 1 ∨… ∨…∨ ¬ p i ∨ …∨ …∨ q m � ⊨ ¬ ( " ∧ ¬ # ). Assume � � !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ⊨ ¬ " ∨ # ) � p 1 ∨ … ∨ p n ∨ q 1 ∨… ∨…∨ q m �⊨ " ! # . � � � � � Thus, ¬ ( " ∧ ¬ # ) ≡ " ! # . Final step: factoring (remove any duplicate literals from the result A ∨ A ≡ A) � � �

A resolution algorithm � function PLresolution( " , # ) input: formula " , // knowledge base formula # // query Goal: prove " ⊨ # ’ by showing that " ∧ ¬ # is clauses := CNF( " ∧ ¬ # ) not satisfiable ( false ) � new := {} � while true: for each c1, c2 in clauses do Observation: 
 resolvents := resolve( c1, c2) Resolution derives a contradiction ( false ) 
 if ∅ in resolvents then return true; if it derives the empty clause: � new := new ∪ resolvents p i ¬ p i if new ⊆ clauses then return false; !!!!!!! clauses := clauses ∪ new � ∅� � CS440/ECE448: Intro AI � 5 � CS440/ECE448: Intro AI � 6 � Completeness of Resolution � If RC(S) doesn ʼ t contain ∅… ∅… � Resolution closure RC(S): � …S is satisfiable, because we can build a model for its variables p 1 ….p n : � The set of all clauses that can be derived by resolution from a set of clauses S. � � If S is finite, RC(S) is finite. � For i from 1….n: � � � if a clause in RC(S) contains ¬ p i 
 Ground resolution theorem: � � � and all its other literals are false , 
 � If RC(S) contains ∅ , S is not satisfiable. � � then assign false to p i If RC(S) does not contain ∅ , S is satisfiable. � otherwise assign true to p i � � CS440/ECE448: Intro AI � 7 � CS440/ECE448: Intro AI � 8 �

Meet your neighbors! � Task 1: Give a WFF that is � 1. Valid � 2. Satisfiable � A little exercise… � 3. Not valid � 4. Not satisfiable � 5. Satisfiable but not valid � 6. Valid but not satisfiable � Axiom set ! : Derivation of R Task 2: 1. P " Q 1. P " Q ! Is R true? 11. Q MP: 1,9 2. L " R 2. L " R ! Can we prove it 3. Q " R 3. Q " R 12. R MP: 3,11 ! using Modus 4. ¬ L " Z Ponens? 4. ¬ L " Z ! 5. S " L 5. S " L ! 6. P " G #"$ 6. P " G ! 7. ¬ L # 7. ¬ L ! 8. A 8. A ! $ 9. P 9. P ! 10. G 10. G !

Derivation of R Give a WFF that is � 1. P " Q ! 11. Q MP: 1,9 2. L " R ! 3. Q " R 12. R MP: 3,11 P % ¬ P R " R 1. Valid �� ! 2. Satisfiable � P 4. ¬ L " Z ! 3. Not valid � 5. S " L ! P 6. P " G ! 4. Not satisfiable � P & ¬ P 7. ¬ L ! 5. Satisfiable but not valid � P 8. A ! 6. Valid but not satisfiable � 9. P ! Not possible 10. G ! Suppose ! is Satisfiable and ! Entails ' � Validity and Satisfiability � ' Valid ! Satisfiable All possible worlds (models) � What are the possible worlds of ' ? � All WFFs What are the possible worlds of ¬ ' ? � ( not universe of possible worlds!) What are the possible worlds of ( ! & ¬ ' )? �

For all axiom sets we have these relationships E: entailed WFFs D: derivable WFFs Entailment � Sound but Complete but � not not sound E An axiom set ! contains 3 axioms: 
 D E D complete � ! ( & ) i i=1,3 All WFFs All WFFs � ( ! ( ) 1 & ) 2 & ) 3 ) � ) 2 � Neither ) 1 Sound and sound nor E complete D Any WFF that E complete D ) 3 completely includes Possible Worlds this intersection is All WFFs logically entailed All WFFs Shortcomings of 
 Shortcomings of 
 propositional logic � propositional logic � � We want to make generic statements: � There is at least one goal state. � � � (but we don ʼ t know which one….) 
 All humans are mortal. � � � human and mortal are properties � A queen on square X attacks a piece on square Y � if… � x is smaller than x+1 � � x,1 and x+1 refer to entities. 
 � There may be an infinite number of them. 
 � + is a function � � � CS440/ECE448: Intro AI � 19 � CS440/ECE448: Intro AI � 20 �


 We need a richer model � We need a richer language � Domain : a set of entities {ann, peter, …,book1,…} Terms: refer to entities � – Variables, constants, functions, � Entities can have properties . 
 � A property defines a (sub)set of entities: 
 Predicates: refer to properties of entities or blue= {book5, mug3, ...} 
 relations between entities � � � There may be relations between entities. 
 Sentences (= propositions) 
 An n -ary relation defines a set of n -ary tuples of can be true or false � entities: belongsTo ={<book5, peter>, <mug3, ann>,…} � � 21 � 22 � So… � We need a more expressive language: � – We need to talk about entities, their properties, and their relations to other entities � First-order 
 predicate logic � We need richer models: � – We need to represent entities, properties, and their relations to other entities � CS440/ECE448: Intro AI � 23 �


 Predicate logic � Some examples � John is a student: student(john) Syntax: What is the language of 
 well-formed formulas of predicate logic? � All students take at least one class: � ∀ x student(x) ! ∃ y(class(y) ∧ takes(x,y)) Semantics: What is the interpretation of a well-formed formula in predicate logic? 
 There is a class that all students take: � ∃ y(class(y) ∧ ∀ x (student(x) ! takes(x,y)) Inference rules and algorithms: How can we reason with predicate logic? � � CS440/ECE448: Intro AI � 26 � Terms � Predicates � Terms refer to entities. 
 Unary predicates 
 � define properties of entities: � They can be constants, variables, or n -ary functions applied to n terms: � student(john’) � � � john’, x, fatherOf(john’), N -ary predicates 
 define relations between n entities: � fatherOf(john’, tom’) � CS440/ECE448: Intro AI � 27 � CS440/ECE448: Intro AI � 28 �

Formulas � Quantified formulas � Formulas have truth values. � Universal quantifier ( ʻ for all x ʼ ) � All x are P: ∀ x P(x) � All P are Q: ∀ x P(x) ! Q(x) Atomic formulas consist of an n-ary predicate applied to n terms: 
 � fatherOf(john’, tom’) Existential quantifier ( ʻ there exists an x ʼ ) � There is a P: ∃ x P(x) Formulas can be negated or joined with the same At least one P is a Q: ∃ x P(x) ∧ Q(x) connectives as in propositional logic: � ¬fatherOf(john’, max’); fatherOf(john’, tom’) ∧ fatherOf(ann’, tom’) CS440/ECE448: Intro AI � 29 � CS440/ECE448: Intro AI � 30 � � The syntax of FOL expressions � Why ʻ first order ʼ� Term ⇒ Constant | In first order predicate logic, variables refer Variable | only to entities. We can only quantify over Function(Term,...,Term) entities: ∀ x (P(x) ! Q(x)) Formula ⇒ Predicate(Term, ...Term) | ¬ Formula | In second order predicate logic, variables ∀ Variable Formula | can also refer to predicates. We can now ∃ Variable Formula | quantify over predicates: Formula ∧ Formula | Formula ∨ Formula | ∀ x ∀ Q (P(x) ! Q(x)) Formula $ Formula NB: zero-th order predicate logic = propositional logic � 3 CS440/ECE448: Intro AI � 32 � 1

Quantifier scope � Quantifier scope equivalences � ∀ x (P(x,y) ! Q(x)) * x P(x) & * y Q(y) ( * x * y [P(x) & Q(y)] scope of ∀ * x * y [P(x) & Q(y)] ( * y * x [P(x) & Q(y)] x is bound by ∀ * x P(x) % * y Q(y) ( * x * y [P(x) % Q(y)] y is free. * x * y [P(x) % Q(y)] ( * y * x [P(x) % Q(y)] Variables that are in the scope of a quantifier + x P(x) % + y Q(y) ( + x + y [P(x) % Q(y)] are bound. Otherwise, they are free . � � + x + y [P(x) % Q(y)] ( + y + x [P(x) % Q(y)] � Only formulas without free variables have truth � values (= ʻ sentences/propositions ʼ ) � � � � CS440/ECE448: Intro AI � 33 � 34 � Negation and Quantifiers � Quantifier scope ambiguities � Not all x are P " 
 ʻ for each x there is a y such that P(x,y) ʼ
 � There is at least one x that is not P � * x + y P(x,y) ¬ * x P(x) ( + x ¬ P(x) � is NOT EQUIVALENT to � There is no x that is P " 
 � � All x are not P ʻ there is a y such that for each x P(x,y) ʼ� ¬ + x P(x) ( * x ¬ P(x) + y * x P(x,y) � � � � 35 � 36 �