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The theorem states any sentence entailed by a set of

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Unformatted text preview: rules for First Order Logic. • The theorem states: • any sentence entailed by a set of sentences can be proven from that set. => Resolution Algorithm which is a complete inference method. CS 460, Session 18 3 Completeness • The completeness theorem says that a sentence can be proved if it is entailed by another set of sentences. • This is a big deal, since arbitrarily deeply nested functions combined with universal quantification make a potentially infinite search space. • But entailment in first-order logic is only semidecidable, meaning that if a sentence is not entailed by another set of sentences, it cannot necessarily be proven that it is not entailed. CS 460, Session 18 4 Completeness in FOL CS 460, Session 18 5 Historical note CS 460, Session 18 6 Refutation Proof/Graph ¬parent(art,jon) ¬ father(X, Y) \/ parent(X, Y) \ / ¬ father (art, jon) father (art, jon) \ / CS 460, Session 18 7 Resolution CS 460, Session 18 8 Resolution inference rule CS 460, Session 18 9 Remember: normal forms “product of sums of simple variables or negated simple variables” “sum of products of simple variables or negated simple variables” CS 460, Sessi...
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