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Unformatted text preview: 6.841 Advanced Complexity Theory Apr 27, 2009 Lecture 21 Lecturer: Jakob Nordstr o m Scribe: Rishi Gupta Today and Thursday we will give an (extremely) selective overview of Proof Complexity . We will cover some major concepts, give an overview of the area, prove a classic result, and develop some nice tools along the way. Introduction to Proof Systems Proof Complexity is about figuring out how hard it is to prove things. But what is a proof? I claim 25957 is a product of two primes. What would we accept as proof of this fact? Some possibilities: Left to the reader. In a sense, every statement is its own proof. Proof by exhaustion: compute 25957 mod 2 and every odd number less than 25957, and see that exactly 2 of them are 0. 25957 = 101 257, check that 101 and 257 are prime. Observation by Cook: A proof should be efficiently verifiable. In particular, a proof system P for a language L is a deterministic function P ( x, ) { , 1 } , polynomialtime in x and , where x L there exists such that P ( x, ) = 1 (a proof exists) x 6 L for all , P ( x, ) = 0 (there are no fake proofs) Note that there is no restriction on the length of . Propositional Proof Complexity Were interested in proving tautologies , namely (boolean) formulas that are true no matter how you assign values to the variables. For instance, ( x x ) TAUT. Why are they interesting? For a while, people thought this would help with the P vs NP question. It provides a framework for quantifying/understanding limits of different kinds of mathematical rea soning. Proof systems have varying strengths, like the complexity classes were familiar with. Commercial SATsolvers are absurdly quick, and routinely solve formulas that have millions of variables. In fact, good SATsolvers are now roughly ( n ). This subfield is called automated theorem proving. A proof system has complexity g if every x P has a proof of size g (  x  ). If g is polynomial we say P is a polynomiallybounded [propositional] proof system, or a pbpps for short. Observation by Cook and Reckhow (1979): NP = coNP if and only if there exists a polynomiallybounded propositional proof system for TAUT (easy exercise)....
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This note was uploaded on 04/02/2010 for the course CS 6.841 taught by Professor Madhusudan during the Spring '09 term at MIT.
 Spring '09
 MadhuSudan

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