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December 7

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

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CSCI 2670 Introduction to Theory of Computing December 7, 2005

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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

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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  φ M accepts string w iff  φ  is satisfiable <M,w> is converted into  φ  in polynomial time

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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 M’s 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 • Using the variables T i,j,k , H

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• Spring '11
• sm
• Computational complexity theory, NP-complete, Boolean satisfiability problem, polynomial time, Clique problem, Boolean  expression

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

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