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lecture3 - CSci 5403 COMPLEXITY THEORY are just like...

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1 COMPLEXITY THEORY CSci 5403 LECTURE III: NOW WITH 66% MORE NP-COMPLETENESS!! NON-DETERMINISTIC PROGRAMS …are just like standard programs, except: 1. There is a special instruction, guess(), that can return 0 or 1. 2. The program accepts an input if there exists a set of guesses that make it accept. 3. The running time of the program is the maximum number of steps that can be caused by calls to guess(). NP = NTIME(n c ) c N { L | L is decided by a O(t(n))-time non-deterministic Turing machine } Definition: NTIME(t(n)) = A language is in NP if and only if there exist polynomial-length certificates for membership to the language Definition: A language B is NP-complete if: 1. B NP 2. Every A in NP is poly-time reducible to B (i.e. B is NP-hard) HARDEST PROBLEMS IN NP Theorem. C DTM is NP-Complete. Let C DTM = { M,x,t , | 9 y {0,1} . M(x,y) accepts in · t steps}

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2 Theorem (Cook-Levin): 3SAT is NP-complete Proof: (1) 3SAT NP (2) Every language A in NP is polynomial time reducible to 3SAT Our proof of (2) will have two steps. (a) C DTM · P CNF-SAT (b) CNF-SAT P 3SAT WARMUP Given (M,x,t, ), there is a function C : {0,1} ! {0,1} such that C(y) = 1 iff M(x,y) accepts in t steps.
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lecture3 - CSci 5403 COMPLEXITY THEORY are just like...

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