Unformatted text preview: Pf : BY CONTRADICTION. (See notes.) Def : A computational problem is a specification of two sets A and U, such that A is a subset of U. We say a computational problem is decidable if there exists a finitely describable computer program (M) or an algorithm such that for all x U, M(x) = {yes if ϵ x A, no if x is not in A) ϵ Example (See notes.) Theorem : Propositional logic is decidable. Def : Let S Sent ϵ PL . S = number of symbols in S. Question: How many steps does the truth table test for tautology take as a function of S. Answer: Take 2 ^ (1/3) S Conjecture: For every n N (natural numbers) there exist sentences S Sent ϵ ϵ PL such that the shortest proof of S (Say in F T ) is 2 ^ R S, R > 0. Claim: (See notes.)...
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 Spring '08
 Dean
 Logic, fitch, Completeness Theorem

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