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l19-handout

# l19-handout - Lecture 19 Cooks theorem Outline Cooks...

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Lecture 19: Cook’s theorem David Dill Department of Computer Science 1 Outline Cook’s theorem 2 How hard is SAT? Thousands of problems are known to be NP-complete In NP There is a polynomial-time reduction from SAT (in many cases, this may be a very indirect reduction). If we could solve any of these problems in PTIME, we could solve all problems that have been reduced to it in PTIME. What if we could solve SAT in PTIME? Don’t know yet. 3 Cook’s Theorem Theorem If SAT can be solved in PTIME, every problem in NP can be solved in PTIME. Reminder: PTIME means: For every problem instance w , the question of whether w L can be answered in time p ( | w | ) on a DTM, where p is a polynomial function. Proof method: Give a generic recipe for reducing any problem in NP to a SAT problem in PTIME. 4

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Key Ideas in proof of Cook’s Theorem Recall that an NTM accepts w and a bound on computation length T ( n ) if there exists a computation: σ = α 0 α 1 . . . α T ( n ) where α T ( n ) is an accepting ID (the NTM is in a final state). Idea 1: Given an NTM for solving a problem and an input w , a propositional logical formula φ can be written saying “ σ is an accepting computation of M on w φ is satisfiable iff there φ is satisfiable (a satisfying assignment represents the accepting computation, σ ). Idea 2: The time to construct φ is polynomial in T ( n ) .
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