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 –F ina l rev iew
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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
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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
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December 7, 2005 Boolean expression
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This note was uploaded on 02/07/2011 for the course CS 501 taught by Professor Sm during the Spring '11 term at Indiana.

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

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