Ch15 - Chapter 15 P, NP, and Cooks Theorem Computability...

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Chapter 15 P , NP , and Cook’s Theorem
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2 Computability Theory Establishes whether decision problems are theoretically decidable, i.e., decides whether each solvable problem has a practical solution that can be solved efficiently A theoretically solvable problem may not have a practical solution, i.e., there is no efficient algorithm to solve the problem in polynomial time – an intractable problem Solving intractable problems require extraordinary amount of time and memory. Efficiently solvable problems are polynomial ( P ) problems. Intractable problems are non-polynomial ( NP ) problems. Can any problem that is solvable in polynomial time by a non-deterministic algorithm also be solved deterministically in polynomial time, i.e., P = NP ?
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3 15.1 Time Complexity of NTMs A deterministic TM searches for a solution to a problem by sequentially examining a number of possibilities, e.g., to determine a perfect square number. A NTM employs a “guess-and check” strategy on any one of the possibilities. Defn. 15.1.1 The time complexity of a NTM M is the function tc M : N N such that tc M ( n ) is the maximum number of transitions in any computation for an input of length n . Time complexity measures the efficiency over all computations the non-deterministic analysis must consider all possible computations for an input string. the guess-and-check strategy is generally simpler than its deterministic counterparts.
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4 Example 15.1.1 Consider the following two-tape NTM M that accepts the palindromes over { a , b }. the time complexity of M is tc M ( n ) = The strategy employed in the transformation of a NTM to an equivalent DTM (given in Section 8.7) does not preserve polynomial time solvability . n + 2 if n is odd n + 3 if n is even 15.1 Time Complexity of NTMs (odd) (even)
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5 15.1 Time Complexity of NTMs Theorem 15.1.2 Let L be the language accepted by a one- tape NTM M with time complexity tc M ( n ) = f ( n ). Then L is accepted by a DTM M ’ with time complexity tc M’ ( n ) O ( f ( n ) c f ( n ) ), where c is the maximum number of transitions for any <state, symbol> pair of M . Proof. Let M be a one-tape NTM that halts for all inputs, and let c be the maximum number of distinct transitions for any <state, symbol> pair of M . A three-tape DTM M ’ was described in Section 8.7 (pages 275- 277) whose computation with input w iteratively simulated all possible computations of M with input w . The transformation from non-determinism to determinism is obtained by associating a unique computation of M with a sequence ( m 1 , …, m n ), where 1 m i c . The value m i indicates which of the c possible transitions of M should be executed on the i th step of the computation.
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6 Theorem 15.1.2 (Continued)
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7 15.1 Time Complexity of NTMs Theorem 15.1.2 (Cont.) Given a NTM M with tc M ( n ) = f ( n ), show a DTM M ’ with time complexity tc M’ ( n ) O ( f ( n ) c
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Ch15 - Chapter 15 P, NP, and Cooks Theorem Computability...

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