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lect21 - 6.841 Advanced Complexity Theory Lecture 21...

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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, π ) ∈ { 0 , 1 } , polynomial-time in x and π , where x L there exists π such that P ( x, π ) = 1 (a proof exists) x 6∈ L for all π, P ( x, π ) = 0 (there are no fake proofs) Note that there is no restriction on the length of π . Propositional Proof Complexity We’re 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 we’re familiar with. Commercial SAT-solvers are absurdly quick, and routinely solve formulas that have millions of variables. In fact, good SAT-solvers are now roughly Θ( 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 polynomially-bounded [propositional] proof system, or a pbpps for short. Observation by Cook and Reckhow (1979): NP = coNP if and only if there exists a polynomially-bounded propositional proof system for TAUT (easy exercise). Corollary: If there is no pbpps for TAUT, P 6 = NP, since P is closed under complementation.
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