lec18 - Completeness As explained earlier Generalized Modus...

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Completeness As explained earlier, Generalized Modus Ponens requires sentences to be in Horn form: atomic, or an implication with a conjunction of atomic sentences as the antecedent and an atom as the consequent. CS 460, Session 18 1 However, some sentences cannot be expressed in Horn form. e.g.: x ¬ bored_with_this_lecture (x) Cannot be expressed in Horn form due to presence of negation.
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Completeness A significant problem since Modus Ponens (with FC/BC) cannot automatically operate on such a sentence, and thus cannot use it in inference. Knowledge exists but cannot be used. CS 460, Session 18 2 Thus inference using Modus Ponens for ALL of first order logic is incomplete.
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Completeness However, Kurt Gödel in 1930-31 developed the completeness theorem , which shows that it is possible to find complete inference rules for First Order Logic. The theorem states: CS 460, Session 18 3 any sentence entailed by a set of sentences can be proven from that set. => Resolution Algorithm which is a complete inference method.
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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. CS 460, Session 18 4 But entailment in first-order logic is only semi- decidable , meaning that if a sentence is not entailed by another set of sentences, it cannot necessarily be proven that it is not entailed.
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Completeness in FOL CS 460, Session 18 5
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Historical note CS 460, Session 18 6
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Refutation Proof/Graph ¬parent(art,jon) ¬ father(X, Y) \/ parent(X, Y) \ / ¬ father (art, jon) father (art, jon) CS 460, Session 18 7 \ / []
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Resolution CS 460, Session 18 8
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Resolution inference rule CS 460, Session 18 9
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