Inference in FOL Copyright, 1996 © Dale Carnegie & Associates, Inc. Chapter 9 Spring 2004

CS 471/598 by H. Liu2 Inference with quantifiers Previous Rules for propositional logic Modus Ponens (p 211) And-Elimination, plus Fig 7.11 (p 210) Resolution (p 213) Variables and quantifiers More rules with SUBST( ,  )  - binding list,  - sentence SUBST({x/Sam}, …)  Sibling(x,John)) = Sibling(Sam,John)

CS 471/598 by H. Liu3 Inference rules for quantifiers Universal Instantiation (UI) For any sentence , variable v, and ground term g, E.g., Existential Instantiation (EI) For any , v, and constant k new to KB, E.g.,

CS 471/598 by H. Liu4 Generalized Modus Ponens It raises Modus Ponens from Prop Logic to FOL – it’s called lifting It takes bigger steps in inference It’s focused - not randomly trying UIs

CS 471/598 by H. Liu5 Unification UNIFY (p,q)=  (unifier - the binding list) where SUBST( ,p) = SUBST( ,q) Some examples: Unifying Knows(Jo,x) with Knows(Jo,Ja)Knows(y,Bi)Knows(y,Mother(y)) Knows(x,Elizabeth) Standardizing apart: Rename one sentence to avoid name clashes Most general unifier Finding MGU algorithm in Fig 9.1  Occur Check if the variable itself occurs inside the complex term; omitting it can result in unsound inferences

CS 471/598 by H. Liu6 An example proof The law says that 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. (p 280) To prove (infer) that West is a criminal. FOL, using these predicates American, Weapon, Sells, Hostile, Criminal, Owns, Missile, Enemy  Datalog knowledge bases – FO definite clauses with no function symbols How do you prove it?

CS 471/598 by H. Liu7 Example (continued) The proof can be very LONG for such a simple problem if using UI (substituting with a ground term). The branching factor increases as KB grows. Universal Instantiation has an enormous branching factor. We need more principled ways for proofs Combining atomic sentences to conjunctions, instantiating universal rules to match, then applying Generalized Modus Ponens Two ways to proceed

CS 471/598 by H. Liu8 Forward chaining A first-order definite clause either is atomic or is an implication p1^p2^…^p3  p p’s are positive literals Datalog KBs – FO DC without function symbols Start with the KB and generate new conclusions using Modes Ponens FC is usually used when a new fact is added into the KB -> any new consequences Algorithm FOL-FC-ASK (Fig 9.3) An example of answering who is criminal - Criminal(x) (Fig 9.4)

CS 471/598 by H. Liu9 Backward chaining Start with something to be proved and find implication sentences to conclude it The list of goals is a stack waiting to be worked on When all goals are satisfied, the proof succeeds BC is used when there is a goal to be proved Algorithm FOL-BC-ASK (Fig 9.6) An example of answering who is criminal -Criminal(x) (Fig 9.7) BC is a depth-first search algorithm that suffers from problems with repeated states and incompleteness Which chaining method to use? Any suggestion?

CS 471/598 by H. Liu10 Completeness An incomplete proof procedure - there are sentences entailed by the KB, the procedure cannot find them (Fig 9.10). Significance of completeness (Math) All conjectures can be established mechanically We only need a set of fundamental axioms Significance to AI - a machine can solve any problem that can be stated in FOL Looking for a complete proof procedure

CS 471/598 by H. Liu11 Completeness theorem If KB  , then KB  R  - Godel’s completeness theorem What’s the procedure R ? “There exists one” does tell us which one Resolution algorithm is one such procedure Entailment in FOL is semidecidable: We can show sentences follow, if they do; but we can’t always show if they don’t.

CS 471/598 by H. Liu12 Resolution A refutation complete inference procedure Algorithm (Fig. 7.12): the same for both logics Conjunctive normal form for FOL Resolution inference rule where UNIFY(l i, ¬m j ) =  First-order literals are complementary if one unifies with the negation of the other

CS 471/598 by H. Liu13 Canonical forms for resolution Conjunctive normal form (CNF) CNF: Conjunction of disjunctions of literals The KB is one big, implicit conjunctions Implicative normal form (INF) CNF and INF are notational variants Skolemization Eliminate existstential quantifiers Skolem functions  The arguments of the Skolem function are all universally quantified variables

CS 471/598 by H. Liu14 Conversion to normal form 1. Eliminate implications 2. Move NOT inwards 3. Standardize variables 4. Skolemize - removing E-Quantifier introducing a function associated with the variable 5. Drop universal quantifiers 6. Distribute ^ over v An example  Everyone who loves all animals is loved by someone

CS 471/598 by H. Liu15 Proof revisit Criminal(West)

CS 471/598 by H. Liu16 Another example proof Everyone who loves all animals is loved by someone. Anyone who kills an animal is loved by no one. Jack loves all animals. Either Jack or Curiosity killed the cat, who is named Tuna. Did Curiosity kill the cat?

CS 471/598 by H. Liu17 Resolution strategies To guide the fast proof using refutation Unit preference  Prefer inferences that produce shorter sentences Set of support  Use the negated query as the set of support Input resolution  Combines one of the input sentences (KB or Q) Subsumption  Keeps KB small by eliminating all sentences that are subsumed by an existing sentence

CS 471/598 by H. Liu18 Completeness of resolution Resolution is refutation-complete If a set of sentences is unsatisfiable, then resolution will always be able to derive a contradiction.

CS 471/598 by H. Liu19 Summary Simple FOL proofs are complex and long Generalized MP is natural and powerful, used in forward or backward chaining Generalized resolution is more general than generalized MP, but not complete Refutation using resolution is complete There are strategies to guide search