Computational Complexity Based on Boolean Circuits

Abstract

Computation performed by a Turing machine with time complexity t(n) can be thought of as being performed by a t(n)×t(n) table consisting of t(n)×t(n) cells. The ith row of the table transforms a configuration of the Turing machine at the ith step into the configuration at the next step. Each cell of the table can be regarded as a generalized gate that can be implemented by a certain number of Boolean gates under a suitable encoding. This circuit model which works as a counterpart to a Turing machine illustrates more directly how each configuration is transformed into the next configuration. By introducing this alternative circuit model, we can better understand the notion of nondeterminism discussed in Chap. 6 as well as the notion of NP-completeness which will be discussed in Chap. 10.