Predicate Calculus Russell and Norvig: Chapter 8,9

Propositional Logic vs. Predicate Calculus Propositional Logic The world consists of propositions (sentences) which can be true or false. Predicate Calculus (First Order Logic) The world consists of objects, functions and relations between the objects.

Syntax Term: constant|variable|function(term, …, term)  War-and-Peace  author-of(War-and-Peace)  father-of(author-of(War-and-Peace)) Atomic Sentence predicate(term, …, term) Complex Sentence

Universal Quantifier Brothers are siblings Mother: a female parent of a plant or animal

Existential Quantifier A red object is on top of a green one An author is a person who writes documents A grandparent is the parent of one’s parent

Properties of Quantifiers

Examples All purple mushrooms are poisonous No purple mushroom is poisonous Every CS student knows a programming language. A programming language is known by every CS student

Properties of Quantifiers

Resolution Refutation in Predicate Calculus Add ¬σ to KB Conversion to CNF Apply Resolution Procedure Derive {}: σ is proved No more possible application of resolution rules: σ is not a consequence of KB

Resolution Rule in FOL Example: father(John, Kim),  x ¬father(x,y) V parent(x, y) parent(John, Kim)? Resolution with propositional logic: Find complementary literals Resolution with FOL Create complementary literals with substitution

Substitution Given sentence S and substitution σ, Sσ denotes result of plugging σ into S Example,  S = father(x, y)  σ = {x/John, y/Kim}  Subst(S, σ) = father(John, Kim) Also written as Sσ = father(John, Kim)

Skolemizing To convert arbitrary FOL formula to CNF, one needs to eliminate . Solution: just “name” it, using a new name different from any existing name …. to avoid conflict. Example  x rich(x) becomes rich(g 7 ) g 7 is called a Skolem constant.

Skolemization Theorem

Conversion to CNF

Applying Substitution

Composition of Substitutions

Unification To apply the resolution rule, we need to find a pair of complementary literals. Unification is a method for making two literals identical. If two literals can be made identical, they are unifiable.

Most General Unifier

Question MGU for f(W, g(Z), Z ) f(X, Y, h(X))

Notes on MGU If two terms are unifiable, then there exists a MGU There can be more than one MGU, but they differ only in variable names Not every unifier is MGU A MGU uses constants only as necessary

Inference Examples rich(me)?

Example #2 Jack owns a dog Every dog owner is an animal lover. No animal lover kills an animal. Either Jack or Curiosity killed the cat (named Tuna) Did Curiosity kill the cat?

Properties of Resolution Sound Refutation Complete If KB |= alpha, refutation will prove it Otherwise, refutation procedure may not terminate. Intractable Exponential in the size of KB for Propositional Logic

Summary Quantifiers Resolution Substitution Skolemization Unification  MGU