Computability and Complexity Theory by Steven Homer & Alan L. Selman

Author:Steven Homer & Alan L. Selman
Language: eng
Format: epub
Publisher: Springer US, Boston, MA

In this chapter we expand more broadly on the idea of using a subroutine for one problem in order to efficiently solve another problem. By doing so, we make precise the notion that the complexity of a problem B is related to the complexity of A – that there is an algorithm to efficiently accept Brelative to an algorithm to efficiently decide A. As in Sect. 3.​9, this should mean that an acceptor for B can be written as a program that contains subroutine calls of the form “x ∈ A,” which returns True if the Boolean test is true and returns False otherwise. Recall that the algorithm for accepting B is called a reduction procedure and the set A is called an oracle. The reduction procedure is polynomial time-bounded if the algorithm runs in polynomial time when we stipulate that only one unit of time is to be charged for the execution of each subroutine call. Placing faith in our modified Church’s thesis and in Cobham’s thesis, these ideas, once again, are made precise via the oracle Turing machine.

Let M be an oracle Turing machine, let A be an oracle, and let T be a time-complexity function. We define an oracle Turing machine M with oracle A to be T(n) time-bounded if, for every input of length n, M makes at most T(n) moves before halting. If M is a nondeterministic oracle Turing machine, then every computation of M with A on words of length n must make at most T(n) moves before halting. The language accepted by M with oracle A is denoted L(M, A).

Let us consider once again the reduction procedure given in Fig. 3.​2. For each input word x, the procedure makes a total of | x | queries to the oracle set B. This reduction procedure can be implemented on an oracle Turing machine that operates in time cn 2 for some constant c. Suppose the input word is 110. The string 110 is the first query to B. The second query to the oracle is either 1101 or 1100, depending on whether or not 110 belongs to B. There is a potential of 2 n different queries as B ranges over all possible oracles.

Definition 7.1.

A set A is Turing-reducible toBin polynomial-time (A  ≤ PT B) if there exists a deterministic polynomial-time-bounded oracle Turing machine M such that A = L(M, B).

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