Encoding Knowledge with First Order Predicate Logic Outline: Motivation Predicate calculus terminology Expressing knowledge in logic An example proof by resolution Formal interpretations and satisfiability CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic

Motivation: More Accurate and General Representations of Knowledge “If Willy is a bird, then Willie can fly.” Propositional calculus: P: Willy is a bird Q: Willy can fly P -> Q “Any bird can fly.” Predicate calculus Forall x (Bird(x) -> CanFly(x) ) CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic

CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic Another Example “John’s grandfather was a blacksmith, and his mother was a seamstress.” Propositional calculus: P1: John’s grandfather was a blacksmith. P2: John’s grandmother was a seamstress. P & Q Predicate calculus: Q1(x): x is a blacksmith. Q2(x): x is a seamstress. f(x): grandfather of x. Q1(f(x)) & Q2(m(x)) CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic

Terminology of the Predicate Calculus Predicate symbols: P, Q, R, P1, P2, ... , Q1, Q2, ..., R1, R2, ... If P takes n arguments, we call P an n-ary predicate P( term1, term2, . . . , termn) n >= 0 Function symbols: f, g, h, f1, f2, . . . , g1, g2, . . . , h1, h2, . . . A term is either: a constant: e.g., a, b, c, a1, a2, . . . , b1, b2, . . . , c1, c2, . . . a variable: e.g., x, y, z, x1, x2, . . . , y1, y2, . . . , z1, z2, . . . an n-ary function symbol, followed by n terms in parentheses. Examples of terms: x, a, f(x), g1(a, y), h(f(x), g(y, b)) CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic

CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic Well-Formed Formulas Atomic formula: an n-ary predicate symbol followed by n terms in parentheses: e.g., P(x, f(x, a, y)) If F is an atomic formula, then it is a well-formed formula (WFF). Compound formula: a statement formed from one or more WFFs using logical connectives. If F is a compound formula, then it is a WFF. e.g., P(x, f(x, a, y)) v (Q -> R(x)) Quantified formula: a statement formed from a WFFs by prefixing it with a quantifier and a variable. e.g., (Forall x) (P(x, f(x, a, y)) v (Q -> R(x))) (Exists y) Q(y) If F is a quantified formula, then it is a WFF. CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic

A Syllogism for Predicate Logic Coloured flowers are always scented. I dislike flowers that are not grown in the open air. No flowers grown in the open air are colourless. Therefore, I dislike all flowers that are not scented. Lewis Carroll CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic

Encoding the Syllogism Predicate symbols: C(x) = x is coloured. S(x) = x is scented. D(x) = I dislike x. A(x) = x is grown in the open air. CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic

Encoding the Syllogism Coloured flowers are always scented. (forall x) ( C(x) -> S(x) ) I dislike flowers that are not grown in the open air. (forall x) ( ~A(x) -> D(x) ) No flowers grown in the open air are colourless. (forall x) ( A(x) -> ~~C(x) ) Therefore, I dislike all flowers that are not scented. (forall x) (~S(x) -> D(x) ) CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic

Proof by Resolution: (1) Negate the conclusion Coloured flowers are always scented. (forall x) ( C(x) -> S(x) ) I dislike flowers that are not grown in the open air. (forall x) ( ~A(x) -> D(x) ) No flowers grown in the open air are colourless. (forall x) ( A(x) -> ~~C(x) ) It is not the case that I dislike all flowers that are not scented. ~(forall x) (~S(x) -> D(x) ) CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic

Proof by Resolution: (2) Obtain Clause Form Coloured flowers are always scented. (forall x) ( C(x) -> S(x) ) (forall x) (~C(x) v S(x) ) Rewrite -> using ~ and v. ~C(x) v S(x) Drop leading universal quantifier. The other clauses are: A(x) v D(x) ~A(x) v C(x) (exists x) ~(~S(x) -> D(x) ) Move negation inward (exists x) (~S(x) & ~D(x) ) DeMorgan’s law ~S(a) & ~D(a) Introduce Skolem constant ~S(a), ~D(a) Break into 2 clauses. CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic

Proof by Resolution: (3) Resolve P1: ~C(x) v S(x) P2: A(y) v D(y) variables are standardized apart. P3: ~A(z) v C(z) P4: ~S(a) P5: ~D(a) P6: A(a) P2, P5 { a/y } P7: C(a) P3, P6 { a/z } P8: S(a) P1, P7 { a/x } P9: [] P4, P8 { } CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic

Formal Interpretations An interpretation for a given formula W consists of the following: A domain D of elements that can be referred to by terms. E.g., D = (0, 1, 2, . . . } An assignment that maps each constant symbol of W to an element of D. For each n-ary function symbol of W a mapping from n-tuples of elements of D to single elements of D. For each n-ary predicate symbol of W a mapping from n-tuples of elements of D to {T, F}. CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic

Example of Interpretation Let W = (forall x) (P(a) & Q(x, a) ) An interpretation for W: D = { apple, peach } { a = apple } mapping for constants { } mappings for functions { P(apple) = T, P(peach) = T; mappings for predicates Q(apple,apple) = T, Q(apple, peach) = T, Q(peach,apple) = T, Q(peach, peach) = T } CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic

CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic Satisfiability A formula W is satisfiable iff there exists some interpretation of W that make W true. A formula W is unsatisfiable iff there does NOT exist any interpretation of W that make W true. Then W is inconsistent. W is a contradiction. If every interpretation of W satisfies W (makes W true), then W is a tautology. If I is an interpretation for W that satisfies W , then I is a model for W. CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic

Another Interpretation Let W = (forall x) (P(a) & Q(x, a) ) An interpretation for W: D = { 0, 1, 2, . . . } { a = 0 } mapping for constants { } mappings for functions { P(n) iff n = 0; mappings for predicates Q(m,n) iff n > m. } Note that ~Q(x, x). This interpretation fails to satisfy W. It’s not a model for W. CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic

The Basis of Proof by Resolution Resolution is used to prove that a given set of clauses is inconsistent. (i.e., it cannot have a model). Each time a resolvent is formed, it is added to the set of clauses. This does not change the consistency of the set. However, if the null clause is ever added to the set, then it becomes very obvious that the set is inconsistent. CSE 415 -- (c) S. Tanimoto, 2002 Predicate Logic