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 firstorder 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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 Spring '08
 Narayanaswamy
 Logic, Artificial Intelligence, Quantification, Firstorder logic, Disjunctions

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