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Unformatted text preview: CSCI 2670 Introduction to Theory of Computing December 7, 2005 December 7, 2005 Agenda Today Discussion of CookLevin Theorem One more NPcompleteness proof Course evaluations Tomorrow Return tests Provide prefinal grades Final review December 7, 2005 NPcompleteness A problem C is NPcomplete if finding a polynomialtime solution for C would imply P=NP Definition: A language B is NPcomplete if it satisfies two conditions: 1. B is in NP, and 2. Every A in NP is polynomial time reducible to B December 7, 2005 CookLevin theorem SAT = {<B>B is a satisfiable Boolean expression} Theorem: SAT is NPcomplete If SAT can be solved in polynomial time then any problem in NP can be solved in polynomial time December 7, 2005 Proof of CookLevin theorem First show that SAT is in NP Easy The show that SAT P implies P = NP Proof converts a nondeterministic Turing machine M with input w into a Boolean expression M accepts string w iff is satisfiable <M,w> is converted into in polynomial time December 7, 2005 Variables in proof of CookLevin Thm T i,j,s is true if tape position i contains symbol s at step j of computation O(p(n) 2 ) variables H i,k is true if Ms tape head is at position i at step k of computation O(p(n) 2 ) variables Q q,k is true if M is in state q at step k of the computation O(p(n)) variables December 7, 2005 Boolean expression...
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