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59 Slides--The Cook-Levin Theorem

# 59 Slides--The Cook-Levin Theorem - CS103 HO#59 Slides-The...

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CS103 HO#59 Slides--The Cook-Levin Theorem May 28, 2010 1 The Language Class NP Class NP includes all and only those languages that are decidable by a nondeterministic Turing machine in polynomial time. NP = NTIME(n k ) k NP is the class of languages that have polynomial time verifiers . Polynomial :O ( n 3 ) O(n log n) Non-polynomial: O(n log n ) O(2 n ) Language A is polynomial time mapping reducible to language B, which is written A P B, if there is a polynomial time computable function f: *  * where for every w, w A f(w) B. The function f is called the polynomial time reduction of A to B. To test whether w A, we use the reduction f and test whether f(w) B. This is similar to our original notion of reducibility, but now we are concerned with the efficiency of the process. Polynomial Time Mapping Reductions TM for f leaves f(w) on the tape. AB f Polynomial Time Mapping Reductions polynomial time maps to decides A B A P B Decidable(B) Decidable(A) ¬ Decidable(A) ¬ Decidable(B) Map the problem to B Problem for A Decider for B Decider for A Problem for b Decision for A Polynomial Time Mapping Reductions A B Theorem 7.31 : If A P B and B P, then A P. Suppose B has a polynomial time solution. Then so does A. P P Decidable(B) Decidable(A) ¬ Decidable(A) ¬ Decidable(B) Map the problem to B Problem for A Decider for B Decider for A Problem for b Decision for A P NP B A?

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59 Slides--The Cook-Levin Theorem - CS103 HO#59 Slides-The...

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