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# lecture4 - P VS NP CSci 5403 Recognition vs Generation...

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COMPLEXITY THEORY CSci 5403 LECTURE IV: BEYOND P VS NP P VS NP Recognition vs Generation Checking if a song sounds like Mozart Composing a song that sounds like Mozart Checking if a proof is correct Finding a correct proof Checking if a fusion generator is efficient Finding an efficient fusion generator Verifying that p £ q = N Finding prime factors of N Appreciating funny jokes Writing funny jokes If P=NP, especially by a fast algorithm, then the tasks on left and right are all equally difficult! P vs NP is a central, motivating question for complexity theory, but also a “representative” of many related questions… NP vs coNP coNP = { L | L ̄ 2 NP } = { y | y L } Theorem. L 2 coNP iff there exists a poly-time TM V and polynomial p so that L = { x | 8 y 2 {0,1} p(y) , V(x,y) accepts } Proof. L 2 coNP means L ̄ 2 NP. Let U be the Verifier for L ̄ , and let V(x,y) = not U(x,y). x

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lecture4 - P VS NP CSci 5403 Recognition vs Generation...

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