An introduction to the theory of computational complexity

Abstract

In this introductory paper, an attempt is made to present the key concepts of the theory of computational complexity and to review some of the most interesting results obtained in the last decade. The paper is organized in two parts: in the first 4 sections, the theoretical framework of computational complexity is introduced and a formal definition of problem is stated, together with that of dynamic measure of complexity. In sections 5 and 6, some recent applications of that theory to parallel computers and to artificial intelligence are presented. 1. WHAT DOES COMPUTING IMPLY The notion of computing is based on a few fundamental concepts. First of all, computing is done referring to a specific computation model. Next, computing consists of executing a sequence of steps on the computation model as specified by a suitable algorithm. Finally, computing refers to some input data upon which the algorithm will operate. As an example, a computer is a computation model, a program written in a programming language recognized by that computer is an algorithm, and a file of data to be read by the program are input data. Computational complexity, as an abstract discipline, does not refer to specific details about computers like instruction execution time or memory cycle. The main objective is to enucleate the intrinsic complexity of some problems, or the goodness of some algorithms, without being tied by existing architectural differences among computers. The underlying assumption, which has been confirmed in many cases, is that if a problem is really hard (or if an algorithm is realy good), this will show up on any type of computer. 2. A WELL KNOWN COMPUTATION MODEL As a consequence, a rather abstract computation model which embodies the key features of today computers is used. The model called Turing machine (TM)