CS 416 Artificial Intelligence

1. CS 416Artificial Intelligence Lecture 12 First-Order Logic Chapter 9

2. Midterm • October 25th • Up through chapter 9 (excluding chapter 5)

3. Review • Inference in First-Order Logic • Could convert first-order logic (FOL) to propositional logic (PL) and use PL inference • Must convert (reduce) Universal and Existential Quantifiers into PL • Potential problem with infinite number of substitutions • Could lift Modus Ponens to FOL • Unification required to make expressions look identical

4. How to perform inference in FOL • Use a knowledge base • Insert data with “STORE” • Retrieve “unified” sentences using “FETCH” • Search for substitutions that unite query with every sentence in KB • How can se make FETCH fast? • Only attempt unifications if they have a chance of succeeding

5. Predicate Indexing • Index the facts in the KB • Example: unify Knows (John, x) with KB: • Knows (John, Jane) • Knows (y, Bill) • Brother (Richard, John) • Knows (y, Mother(y)) • Knows (x, Elizabeth) No need to check

6. Predicate Indexing • Predicate indexing puts all the Knows facts in one bucket and all the Brother facts in another • Might not be a win if there are lots of clauses for a particular predicate symbol • Consider how many people Know one another • Instead index by predicate and first argument • Clauses may be stored in multiple buckets

7. Subsumption lattice • How to construct indices for all possible queries that unify with it • Example: Employs (AIMA.org, Richard)

8. Subsumption Lattice

9. Subsumption lattice • Each node reflects making one substitution • The “highest” common descendent of any two nodes is the result of applying the most general unifier • Predicate with n arguments will create a lattice with O(2n) nodes • Benefits of indexing may be outweighed by cost of storing and maintaining indices

10. Forward Chaining • Remember this from propositional logic? • Start with atomic sentences in KB • Apply Modus Ponens • add new sentences to KB • discontinue when no new sentences • Hopefully find the sentence you are looking for in the generated sentences

11. Lifting forward chaining • First-order definite clauses • disjunction of literals with exactly one positive or • implication whose antecedent is a conjunction of positive literals and whose consequent is a single positive literal • all sentences are defined this way to simplify processing Universal quantificationassumed

12. Example • The law says it is a crime for an American to sell weapons to hostile nations. The country Nono, an enemy of America, has some missiles, and all of its missiles were sold to it by Colonel West, who is American • We will prove West is a criminal

13. Example • It is a crime for an American to sell weapons to hostile nations • Nono… has some missles • Owns (Nono, M1) • Missile (M1) • All of its missiles were sold to it by Colonel West

14. Example • We also need to know that missiles are weapons • and we must know that an enemy of America counts as “hostile” • “West, who is American” • The country Nono, an enemy of America

15. Forward-chaining • Starting from the facts • find all rules with satisfied premises • add their conclusions to known facts • repeat until • query is answered • no new facts are added

16. First iteration of forward chaining • Look at the implication sentences first • must satisfy unknown premises • We can satisfy this rule • by substituting {x/M1} • and adding Sells(West, M1, Nono) to KB

17. First iteration of forward chaining • We can satisfy • with {x/M1} • and Weapon (M1) is added • We can satisfy • with {x/Nono} • and Hostile {Nono} is added

18. Second iteration of forward chaining • We can satisfy • with {x/West, y/M1, z/Nono} • and Criminal (West) is added

19. Analyze this algorithm • Sound? • Does it only derive sentences that are entailed? • Yes, because only Modus Ponens is used and it is sound • Complete? • Does it answer every query whose answers are entailed by the KB? • Yes, if the clauses are definite clause • Semi-decidable? • Algorithm returns “yes” for every entailed sentence, but does not return “no” for every nonentailed sentence NatNum(0) NatNum(n)  NatNum(S(n)) NatNum(S(0)), NatNum(S(S(0))),…

20. Proving completeness • Assume KB only has sentences with no function symbols • What’s the most number of iterations through algorithm? • Depends on the number of facts that can be added • Let k be the arity, the max number of arguments of any predicate and • Let p be the number of predicates • Let n be the number of constant symbols • At most pnk distinct ground facts • Fixed point is reached after this many iterations • A proof by contradiction shows that the final KB is complete

21. Complexity of this algorithm • Three sources of complexity • inner loop requires finding all unifiers such that premise of rule unifies with facts of database • this “pattern matching” is expensive • must check every rule on every iteration to check if its premises are satisfied • many facts are generated that are irrelevant to goal

22. Pattern matching • Conjunct ordering • Missile (x) ^ Owns (Nono, x) => Sells (West, x, Nono) • Look at all items owned by Nono, call them X • for each element x in X, check if it is a missile • Look for all missiles, call them X • for each element x in X, check if it is owned by Nono • Optimal ordering is NP-hard, similar to matrix mult • Good heuristic: group first according to whichever group is smaller

23. Incremental forward chaining • Pointless (redundant) repetition • Some rules generate new information • this information may permit unification of existing rules • some rules generate preexisting information • we need not revisit the unification of the existing rules • Every new fact inferred on iteration t must be derived from at least one new fact inferred on iteration t-1

24. Irrelevant facts • Some facts are irrelevant and occupy computation of forward-chaining algorithm • What if Nono example included lots of facts about food preferences? • Not related to conclusions drawn about sale of weapons • How can we eliminate them?

25. Magic Set • Rewriting the rule set • We have learned earlier that backward chaining limits the search space • We’re still using forward chaining, but we will add elements to premises of sentences that further restrict the forward inferences we will draw • added elements are based on goal (thus the backward influence)

26. Magic Set • Rewriting the rule set • Let goal = Criminal (West) • American(x) ^ Weapon(y) ^ Sells(x,y,z) ^ Hostile(z)  Criminal(x) • We know we are concerned with criminals and West • Let’s distinguish West as important by placing it in a Magic set: Magic(West) • Let’s rewrite the rule to constrain our forward chaining to only include sentences related to West • Magic(x) ^ American(x) ^ Weapon(y) ^ Sells(x, y, z) ^ Hostile(z) => Criminal (x)

27. Backward Chaining • Start with the premises of the goal • Each premise must be supported by KB • Start with first premise and look for support from KB • looking for clauses with a head that matches premise • the head’s body (premise) must then be supported by KB • Place premises on a stack with the goal • A recursive, depth-first, algorithm • Suffers from repetition and incompleteness

28. Resolution • We saw earlier that resolution is a complete algorithm for refuting statements • Must put first-order sentences into conjunctive normal form • conjunction of clauses, each is a disjunction of literals • literals can contain variables (which are assumed to be universally quantified)

29. Resolution - aside • Mathematical theorem provers • Resolution has been used to prove mathematical theorems • Kurt Gödel proved incompleteness of logic systems containing induction (a building block of discrete math) • There are sentences entailed by a knowledge base, but no finite proof exists • That is, generating all possible sentences, when induction is one possible operator, will not derive all sentences that are entailed

30. First-order CNF • For all x, American(x) ^ Weapon(y) ^ Sells(x, y, z) ^ Hostile (z) => Criminal(x) • ~American(x) V ~Weapon(y) V ~Sells(x, y, z) V ~Hostile(z) V Criminal(x) • Every sentence of first-order logic can be converted into an inferentially equivalent CNF sentence (they are both unsatisfiable in same conditions)

31. Example • Everyone who loves all animals is loved by someone

32. Example

33. Example • F and G are Skolem Functions • arguments of function are universally quantified variables in whose scope the existential quantifier appears

34. Example • Two clauses • F(x) refers to the animal potentially unloved by x • G(x) refers to someone who might love x

35. Resolution inference rule li and m are complementaryliterals • A lifted version of propositional resolution rule • two clauses must be standardized apart • no variables are shared • can be resolved if their literals are complementary • one is the negation of the other • if one unifies with the negation of the other

36. Resolution li and mj are complementaryliterals

37. Inference in first-order logic • Our goal is to prove that KB entails a fact, a • We use logical inference • Forward chaining • Backward chaining • Resolution • All three logical inference systems rely on search to find a sequence of actions that derive the empty clause