lecture Todays Lecture 13: Inference in FOL
CMPSCI 383: Artificial Intelligence Instructor: Shlomo Zilberstein
!! !!
Knowledge representation in FOL Matching rules and facts using unification Inference procedures: forward-chaining, backward-chaining, and resolution Limitations of logical reasoning
2
!!
!!
1
Shlomo Zilberstein University of Massachusetts
Inference rules for FOL
!! !! !! !! !!
Handling quantifiers
Conclusion q p, q p"q p#q p#r
!!
Modus Ponens: And-Elimination: And-Introduction: Or-Introduction: Resolution:
Premises p, p ! q p"q p, q p p#q, q#r
!!
Want to instantiate sentences like: x < x + 1, owns(x,y) ! belongs(y,x) A substitution, Subst($, p), is a set of pairs that binds one pattern to another in p.
!! !!
Subst({f(x)/A}, f(g(f(x)))) Subst({x/John,y/Mary}, likes(x,y))
!!
Given two atomic sentences p and q, $ unifies p and q if Subst($, p) = Subst($, q)
4
Shlomo Zilberstein University of Massachusetts
3
Shlomo Zilberstein University of Massachusetts
Unification example
P
Knows(John,x) Knows(John,x) Knows(John,x) F(x,y)
Unification algorithm
(defun sub-unify (pat-1 pat-2 theta) "Unify pat-1 with pat-2 given theta, the substitution so far" (cond ((equal pat-1 pat-2) theta) ((variable-p pat-1) (unify-var pat-1 pat-2 theta)) ((variable-p pat-2) (unify-var pat-2 pat-1 theta)) ((or (atom pat-1) (atom pat-2)) nil) ((eq (length pat-1) (length pat-2)) (let ((t* (sub-unify (first pat-1) (first pat-2) theta))) (if t* (sub-unify (rest pat-1) (rest pat-2) t*))))))
5 Shlomo Zilberstein University of Massachusetts 6
Q
Knows(y,Sarah) Knows(y,Mother(y)) Knows(x,Sarah) F(y,G(x))
!
{x/Sarah, y/John} {y/John, x/ Mother(John)} fail fail
Shlomo Zilberstein University of Massachusetts
Unification algorithm cont.
(defun unify-var (var pattern theta) "Unifying a single variable" (let ((value (cdr (assoc var theta)))) (if value (sub-unify value pattern theta) (if (member var (variables (sublis theta pattern))) nil (acons var pattern theta)))))
Universal elimination
For any sentence P, variable x, and ground term g: %x P &' Subst({x/g},P) Examples: %x Likes(x,IceCream) &' Likes(Ben,IceCream) %x At(x,UMass) ! Smart(x) &' At(Bill,UMass) ! Smart(Bill)
Shlomo Zilberstein University of Massachusetts
7
Shlomo Zilberstein University of Massachusetts
8
Existential elimination
For any sentence P, variable x, and constant k that does not appear elsewhere in the KB: (x P &' Subst({x/k},P} Examples: (x Kill(x,Victim) &' Kill(Murderer,Victim) (x Smart(x) &' Smart(SmartPerson)
Existential introduction
For any sentence P, variable x that does not occur in P, and ground term g: P &' (x Subst({g/x},P) Examples: Likes(Jerry,IceCream) &' (x Likes(x,IceCream) Smart(Einstein) &' (x Smart(x)
Shlomo Zilberstein University of Massachusetts
9
Shlomo Zilberstein University of Massachusetts
10
Finding proofs
!! !! !! !!
Horn sentences
A Horn sentence has the form:
P1 " P2 " ... " Pn ! Q where Pi and Q are positive atomic sentences Examples: Parent(g,p) " Parent(p,c) ! Grandparent(g,c) But? Sister(x,y) # Brother(x,y) ! Sibling(x,y) ) )Sibling(x,y) ! Sister(x,y) # Brother(x,y)
11 Shlomo Zilberstein University of Massachusetts 12
Finding a proof is a search process Operators/actions are the inference rules Search state is a set of sentences Using algorithms such as AO* for theorem proving Why is it an extremely hard search process?
Shlomo Zilberstein University of Massachusetts
Generalized Modus Ponens
P1 " P2 " ... " Pn ! Q* P1, P2, ... Pn $Q Can derive $Q if $ unifies Pi and Pi for all i Complete for Horn knowledge bases
Forward chaining
Shlomo Zilberstein University of Massachusetts
13
Shlomo Zilberstein University of Massachusetts
14
Properties of forward chaining
!!
Backward chaining
!! !!
!!
Sound and complete for first-order definite clauses May not terminate in general if + is not entailed This is unavoidable: entailment with definite clauses is semi-decidable Matching itself can be expensive (matching conjunctive premises against known facts is NP-hard)
15 Shlomo Zilberstein University of Massachusetts 16
Shlomo Zilberstein University of Massachusetts
Properties of backward chaining
Resolution is sound and complete
(p1 # ... # pm) (q1 # ... # qn) Subst($, p2 # ... # pm # q2 # ... # qn ) Provided that $ unifies p1 and q1 Applies to KB in Conjunctive Normal Form e.g. 1. {P(x), Q(x,y)} GIVEN 2. {P(A), R(B,z)} GIVEN 3. {Q(A,y), R(B,z)} RESOLVE 1+2 Resolution is complete, theorem proving semi-decidable
!! !! !!
BUT, BC & FC combined are incomplete: e.g., A ! B, A ! C, C ! Z, B ! Z. We should be able to prove Z, but we cant.
17
Shlomo Zilberstein University of Massachusetts
Shlomo Zilberstein University of Massachusetts
18
Conjunctive Normal Form
Goal: KB contains only conjunction of disjunctions of literals (also called form). clausal Can convert any FOL KB into CNF (1) Remove implications
Replace P ! Q by P#Q Replace P , Q by (P # Q) " (P # Q)
Shlomo Zilberstein University of Massachusetts 19
Conjunctive Normal Form Cont.
(2) )Move negation inwards
%x P becomes (x P ) ) (x P becomes %x P P becomes P (P " Q) is replaced by P # Q (P # Q) is replaced by P " Q
(3)
Standardize variables each quantifier gets unique variables e.g. (xP(x)"(xQ(x) becomes (xP(x)"(yQ(y)
20
Shlomo Zilberstein University of Massachusetts
Conjunctive Normal Form Cont.
(4) (5) Move quantifiers to the left %xP # (yQ becomes %x(y P#Q Eliminate ( by Skolemization. (xP(x) becomes P(A) %x%y(zP(x,y,z) becomes %x%yP(x,y,F(x,y)) %x(yPred(x,y) becomes %xPred(x,Succ(x)) Drop universal quantifiers
21
Conjunctive Normal Form Cont.
(7) Distribute And over Or (P " Q) # R becomes (P#R) " (Q#R) Remove operators (" #) (P # Q) " (P # R) becomes { {P,Q} {P,R} }
(8)
(6)
Shlomo Zilberstein University of Massachusetts
Shlomo Zilberstein University of Massachusetts
22
Resolution example
Given: A. Twiddling someone guarantees they will twiddle you. B. Twiddling is transitive. C. Everyone twiddles somebody. Prove: D. Everyone twiddles themselves.
Facts
A. B. C. D.
First-Order Logic
%x%y T(x,y) ! T(y,x) %x%y%z T(x,y) " T(y,z) ! T(x,z) %x(y T(x,y) %x T(x,x)
Shlomo Zilberstein University of Massachusetts
23
Shlomo Zilberstein University of Massachusetts
24
Sentences
1. 2. 3. 4.
Clausal Form
Derive a contradiction
1. 2. 3. 4. 5. 6. 7. 8. { T(x1,y1), T(y1,x1) } { T(x2,y2), T(y2,z2), T(x2,z2) } { T(x3,F(x3)) } { T(A,A) } { T(A,y2), T(y2,A) } 2+4 { T(F(A),A) } 3+5 { T(A,F(A)) } 1+6 {} 3+7
26
{ T(x1,y1), T(y1,x1) } { T(x2,y2), T(y2,z2), T(x2,z2) } { T(x3,F(x3)) } { T(A,A) } (Negation of D)
Shlomo Zilberstein University of Massachusetts
25
Shlomo Zilberstein University of Massachusetts
Another resolution example
!! !!
!!
!!
!!
Vincent has been murdered, and Arthur, Bertram, and Carleton are suspects. Arthur says he did not do it. He says that Bertram was the victim's friend but that Carleton hated the victim. Bertram says he was out of town the day of the murder, and besides he didn't even know the guy. Carleton says he is innocent and he saw Arthur and Bertram with the victim just before the murder. Assuming that everyone--except possibly for the murderer--is telling the truth, how ca we solve the crime?
27 Shlomo Zilberstein University of Massachusetts 28
Shlomo Zilberstein University of Massachusetts
Converting the KB to CNF
Shlomo Zilberstein University of Massachusetts
29
Shlomo Zilberstein University of Massachusetts
30
Resolving the crime
Resolution strategies
Deletion strategies: Clauses with certain properties are deleted
before they are ever used. Examples: pure-literal elimination; tautology elimination; subsumption elimination (eliminate the more specific clause).
Unit resolution...