CS 4700: Artificial Intelligence

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1 CS 4700: Foundations of Artificial Intelligence Fall 2017 Instructor: Prof. Haym Hirsh Lecture 14

2 Today Knowledge Representation and Reasoning (R&N Ch 7-9) Prelim, Statler Auditorium Tuesday, March 21 Thursday, March 23 Knowledge Representation and Reasoning (R&N Ch 8-9)

3 Resolution Example: Rules KB: P Q L M P B L M A P L A B L A B KB P Q?

4 Resolution Example: Rules KB: P Q L M P B L M A P L A B L A B KB P Q? Is the following CNF unsatisfiable? ( P Q) ( L M P) ( B L M) ( A P L) ( A B L) A B ( P Q)

5 Horn Clause Logic KB: P Q L M P B L M A P L A B L A B KB P Q? Is the following CNF unsatisfiable? ( P Q) ( L M P) ( B L M) ( A P L) ( A B L) A B ( P Q)

6 Horn Clause Logic Horn clauses: A sentence in CNF where each clause (disjunction) has at most one unnegated literal Typically arises when KB is a set of implications whose antecedents are conjunctions of unnegated literals Inference methods: Forward chaining: Start from unnegated literal facts, add new facts from rules whose antecedents are true, apply recursively

7 Horn Clause Logic Forward chaining example: A [6] B [7] KB: 1. P Q 2. L M P 3. B L M 4. A P L 5. A B L 6. A 7. B

8 Horn Clause Logic Forward chaining example: A [6] B [7] L [5] KB: 1. P Q 2. L M P 3. B L M 4. A P L 5. A B L 6. A 7. B

9 Horn Clause Logic Forward chaining example: A [6] B [7] L [5] M [3] KB: 1. P Q 2. L M P 3. B L M 4. A P L 5. A B L 6. A 7. B

10 Horn Clause Logic Forward chaining example: A [6] B [7] L [5] M [3] P [2] KB: 1. P Q 2. L M P 3. B L M 4. A P L 5. A B L 6. A 7. B

11 Horn Clause Logic Forward chaining example: A [6] B [7] L [5] M [3] P [2] Q [1] KB: 1. P Q 2. L M P 3. B L M 4. A P L 5. A B L 6. A 7. B

12 Horn Clause Logic Which facts to combine with which rules? This is a search problem: States: Sets of facts Operators: The set of rules If the given state contains all the facts necessary for a rule, add the rule s consequence to the set of facts to create a new state Initial state: Starting facts Goal: One or more facts that are being asked about

13 Horn Clause Logic Example: Initial state: {A,B} Operators: {A B L, A P L, } Result of applying A B L to initial state: {A,B,L} Goal state: A state containing P and Q

14 Horn Clause Logic Horn clauses: A sentence in CNF where each clause (disjunction) has at most one unnegated literal Typically arises when KB is a set of implications whose antecedents are conjunctions of unnegated literals Inference methods: Forward chaining: Start from unnegated literal facts, add new facts from rules whose antecedents are true, apply recursively

15 Horn Clause Logic Horn clauses: A sentence in CNF where each clause (disjunction) has at most one unnegated literal Typically arises when KB is a set of implications whose antecedents are conjunctions of unnegated literals Inference methods: Forward chaining: Start from unnegated literal facts, add new facts from rules whose antecedents are true, apply recursively Backward chaining: Start from goal, find rule that concludes it, apply recursively to antecedents (this is the core of what the Prolog programming language does)

16 Horn Clause Logic Horn clauses: A sentence in CNF where each clause (disjunction) has at most one unnegated literal Typically arises when KB is a set of implications whose antecedents are conjunctions of unnegated literals Inference methods: Forward chaining: Start from unnegated literal facts, add new facts from rules whose antecedents are true, apply recursively Backward chaining: Start from goal, find rule that concludes it, apply recursively to antecedents (this is the core of what the Prolog programming language does) There is a linear time algorithm for satisfiability of Horn clauses

17 Satisfiability At one extreme: Tautology: Sentence is true no matter what truth assignments are given to the propositional variables At the other extreme: Unsatisfiable: The sentence is never true no matter what truth assignments are given to the propositional variables Satisfiability: Is the sentence true for any truth assignment? What truth assignment makes the sentence true?

18 Example: N-Queens Encode in propositional logic Qij: There is a queen in position (i,j) One queen in each row, column, diagonal: (Qi1 Qi2 QiN) for i {1,,N} (Q1j Q2j QNj) for i {1,,N} (Q11 Q22 QNN) etc. for each diagonal No two attacking queens: (Qab Qac) for all b c (Qab Qdb) for all a d (Qij Qkl) for all k=i m, l=j m, m 0 (Qij Qkl) for all i+j = k+l, i k

19 Example: N-Queens Unsatisfiable: Means there is no placement of the queens (N=2, 3) Satisfiable: Means there is a placement of the queens (N > 3) Don t just want to be told there is a placement, want an actual placement!

20 Searching for Truth Assignments Brute-force search: Assign a variable True or False Is any clause violated? If not, go on to another variable If yes, backtrack to most recent variable assignment and flip it If both options have been tried and failed, backtrack to the next most recent assignment Complete Davis-Putnam: Backtracking exploiting some additional observations that simplify the problem in certain cases Further efficiency: Grab bag of clever ideas

21 Searching for Truth Assignments Hill climbing: Surprisingly effective Pick random assignment If it doesn t satisfy the sentence, flip whichever one most increases the number of satisfied terms and repeat If there are none, restart

22 First Order Logic Sentence AtomicSentence ComplexSentence AtomicSentence Predicate Predicate(Arguments) Term = Term Arguments Term Term,Arguments Term Function(Arguments) Constant Variable ComplexSentence (Sentence) [Sentence] Sentence Sentence Sentence Sentence Sentence Sentence Sentence Quantifiers Sentence Quantifiers Quantifier Variables Quantifier Variables Quantifiers Quantifier Constant/Predicate/Function <strings starting with upper case letters> Variables Variable Variable,Variables Variable <strings comprised of lower case letters>

23 Examples Mother(Alice,Charlie) Father(Bob,Charlie)

24 Examples Mother(Alice,Charlie) Father(Bob,Charlie) x,y Mother(x,y) Parent(x,y) x,y Father(x,y) Parent(x,y)

25 Examples Mother(Alice,Charlie) Father(Bob,Charlie) x,y Mother(x,y) Parent(x,y) x,y Father(x,y) Parent(x,y) x,y Parent(x,y) Ancestor(x,y)

26 Examples Mother(Alice,Charlie) Father(Bob,Charlie) x,y Mother(x,y) Parent(x,y) x,y Father(x,y) Parent(x,y) x,y Parent(x,y) Ancestor(x,y) x,y,z Ancestor(x,y) Ancestor(y,z) Ancestor(x,z)

27 Examples Mother(Alice,Charlie) Father(Bob,Charlie) x,y Mother(x,y) Parent(x,y) x,y Father(x,y) Parent(x,y) x,y Parent(x,y) Ancestor(x,y) x,y,z Ancestor(x,y) Ancestor(y,z) Ancestor(x,z) x y Mother(y,x) x y Father(y,x)

28 Examples Mother(Alice,Charlie) Father(Bob,Charlie) x,y Mother(x,y) Parent(x,y) x,y Father(x,y) Parent(x,y) x,y Parent(x,y) Ancestor(x,y) x,y,z Ancestor(x,y) Ancestor(y,z) Ancestor(x,z) x y Mother(y,x) x y Father(y,x) x y Mother(x,y)

29 Inference in First Order Logic First-order Horn clauses are an analogous subset of first-order logic They have analogs of forward chaining and backward chaining Prolog is a programming language based on backward chaining on first-order Horn clauses There is a conjunctive normal form for First-Order Logic There is a resolution inference rule for First-Order Logic

30 CNF Replace α β with α β everywhere [no more ] Replace (α β) with α β [push inward until Replace (α β) with α β you can t any more] Replace α with α [ only before prop symbols] Distribute over : Rewrite α (β γ) as (α β) (α γ)

31 CNF Replace α β with α β everywhere Replace (α β) with α β Replace (α β) with α β Replace α with α Distribute over : Rewrite α (β γ) as (α β) (α γ)

32 CNF Replace α β with α β everywhere Replace (α β) with α β Replace (α β) with α β Replace α with α Distribute over : Rewrite α (β γ) as (α β) (α γ)

33 CNF Replace α β with α β everywhere Replace (α β) with α β Replace (α β) with α β Replace α with α Replace x with x Replace x with x Distribute over : Rewrite α (β γ) as (α β) (α γ)

34 CNF Replace α β with α β everywhere Replace (α β) with α β Replace (α β) with α β Replace α with α Replace x with x Replace x with x Distribute over : Rewrite α (β γ) as (α β) (α γ)

35 CNF Replace α β with α β everywhere Replace (α β) with α β Replace (α β) with α β Replace α with α Replace x with x Replace x with x Standardize variables: If the same variable name is used in multiple places with multiple quantifiers, rename them so there is no duplication x P(x) x Q(x) becomes x P(x) z Q(z) [where z is new] Distribute over : Rewrite α (β γ) as (α β) (α γ)

36 CNF Replace α β with α β everywhere Replace (α β) with α β Replace (α β) with α β Replace α with α Replace x with x Replace x with x Standardize variables: If the same variable name is used in multiple places with multiple quantifiers, rename them so there is no duplication x P(x) x Q(x) becomes x P(x) z Q(z) [where z is new] Distribute over : Rewrite α (β γ) as (α β) (α γ)

37 CNF Replace α β with α β everywhere Replace (α β) with α β Replace (α β) with α β Replace α with α Replace x with x Replace x with x Standardize variables: If the same variable name is used in multiple places with multiple quantifiers, rename them so there is no duplication x P(x) x Q(x) becomes x P(x) z Q(z) [where z is new] Skolemization Distribute over : Rewrite α (β γ) as (α β) (α γ)

38 CNF Replace α β with α β everywhere Replace (α β) with α β Replace (α β) with α β Replace α with α Replace x with x Replace x with x Standardize variables: If the same variable name is used in multiple places with multiple quantifiers, rename them so there is no duplication x P(x) x Q(x) becomes x P(x) z Q(z) [where z is new] Skolemization Distribute over : Rewrite α (β γ) as (α β) (α γ)

39 CNF Replace α β with α β everywhere Replace (α β) with α β Replace (α β) with α β Replace α with α Replace x with x Replace x with x Standardize variables: If the same variable name is used in multiple places with multiple quantifiers, rename them so there is no duplication x P(x) x Q(x) becomes x P(x) z Q(z) [where z is new] Skolemization Drop universal quantifiers Distribute over : Rewrite α (β γ) as (α β) (α γ)

40 Skolemization x P(x) becomes P(A) where A is a constant that is new

41 Skolemization x P(x) becomes P(A) where A is a new constant x,y z P(z) becomes P(F(x,y)) where F is a new function

42 Skolemization x P(x) becomes P(A) where A is a new constant x,y z P(z) becomes P(F(x,y)) where F is a new function Replace existentially quantified variables with Skolem functions new functions whose arguments are all universally quantified variables of higher scope

43 Resolution in First Order Logic Recall: (l α 1 α k ) where each α i and β j are propositional ( l β 1 β n ) symbols or their negations and l is a (α 1 α k β 1 β n ) propositional symbol

44 Resolution in First Order Logic Here: (l α 1 α k ) [l and l can be made identical by ( l β 1 β n ) substituting variables with terms] Subst(ѳ, α 1 α k β 1 β n ) [the necessary substitution] where ѳ = Unify(l, l )

CS 4700: Artificial Intelligence

CS 4700: Artificial Intelligence CS 4700: Foundations of Artificial Intelligence Fall 2017 Instructor: Prof. Haym Hirsh Lecture 12 Prelim Tuesday, March 21 8:40-9:55am Statler Auditorium Homework 2 To be posted on Piazza 4701 Projects: