07h - Discrete Mathematics - Predicates and Quantifiers 7-2...

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Introduction Predicates and Quantifiers Discrete Mathematics Andrei Bulatov Discrete Mathematics - Predicates and Quantifiers 7-2 Conjunctive Normal Form A literal is a primitive statement (propositional variable) or its negation p, ¬ p, q, ¬ q A clause is a disjunction of one or more literals p q, p ∨ ¬ q r, ¬ q, ¬ s s ∨ ¬ r ∨ ¬ q A statement is said to be a Conjunctive Normal Form (CNF) if it is a conjunction of clauses p (p ∨ ¬ q) ( ¬ r ∨ ¬ p) p q ( ¬ r ∨ ¬ p) ( ¬ r q) (p ∨ ¬ q ∨ ¬ s r) ( ¬ r ∨ ¬ p) ¬ r Discrete Mathematics - Predicates and Quantifiers 7-3 CNF Theorem Theorem Every statement is logically equivalent to a certain CNF. Proof (sketch) Step 1. Express all logic connectives in Φ through negation, conjunction, and disjunction. Let Ψ be the obtained statement. Let Φ be a (compound) statement. Step 2. Using DeMorgan’s laws move all the negations in Ψ to individual primitive statements. Let Θ denote the updated statement Step 3. Using distributive laws transform Θ into a CNF. Discrete Mathematics - Predicates and Quantifiers 7-4 Example Find a CNF logically equivalent to (p q) r Step 1. ¬ ( ¬ p q) r Step 2. (p ∧ ¬ q) r Step 3. (p r) ( ¬ q r) Discrete Mathematics - Predicates and Quantifiers 7-5 Rule of Resolution p q ¬ p r q r The corresponding tautology ((p q) ( ¬ p r)) (q r) ``Jasmine is skiing or it is not snowing. It is snowing or Bart is playing hockey.’’ p - `it is snowing’ q - `Jasmine is skiing’ r - `Bart is playing hockey’ ``Therefore, Jasmine is skiing or Bart is playing hockey’’ q r is called resolvent Discrete Mathematics - Predicates and Quantifiers 7-6 Computerized Logic Inference Convert the premises into CNF Convert the negation of the conclusion into CNF Consider the collection consisting of all the clauses that occur in the obtained CNFs Use the rule of resolution to obtain the empty clause ( ). If it is possible, then the argument is valid. Otherwise, it is not. Why empty clause?
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07h - Discrete Mathematics - Predicates and Quantifiers 7-2...

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