INTRODUCTION

While the complexity of a program is sometimes gauged in terms of a single number such as the number of instructions or the maximum depth of nesting of loops, much attention focuses on amounts of computational resources which may vary for different inputs to a program, for instance execution time or the amount of memory which must be created by dynamic storage allocation. The abstract theory of computational complexity uses recursion theory to model some of the principal properties of such dynamic computational resources. Emphasis is on the machine-independent features which various dynamic resources share in common; for instance, the theory is not intended to allow detailed comparisons of execution times of Turing machines and random access machines. Only general properties of such resources are modeled; for instance, the resource requirements of sorting algorithms or matrix inversion algorithms are never singled out for special consideration.

The theory is based on the notion of M. Blum (1967a) of a measure of computational complexity (complexity measure), which is discussed in Section 3 below. Although Blum's axioms appear at first glance to be quite weak, the area has been the subject of numerous papers, and surprisingly strong consequences of the axioms have been obtained. The most celebrated result is Blum's speed-up theorem, which can be paraphrased as establishing the existence of a computable function which does not possess an optimal (in the dynamic sense of fastest on infinitely many inputs, not in the static sense of shortest) program.