December 7

December 7 - CSCI 2670 Introduction to Theory of Computing...

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Unformatted text preview: CSCI 2670 Introduction to Theory of Computing December 7, 2005 December 7, 2005 Agenda Today Discussion of Cook-Levin Theorem One more NP-completeness proof Course evaluations Tomorrow Return tests Provide pre-final grades Final review December 7, 2005 NP-completeness A problem C is NP-complete if finding a polynomial-time solution for C would imply P=NP Definition: A language B is NP-complete 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 Cook-Levin theorem SAT = {<B>|B is a satisfiable Boolean expression} Theorem: SAT is NP-complete If SAT can be solved in polynomial time then any problem in NP can be solved in polynomial time December 7, 2005 Proof of Cook-Levin theorem First show that SAT is in NP Easy The show that SAT P implies P = NP Proof converts a non-deterministic 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 Cook-Levin 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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December 7 - CSCI 2670 Introduction to Theory of Computing...

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