Abstract Complexity Theory

Abstract

In previous chapters we developed an implementation-independent model of computation. The fundamental features of this model were examined in the last chapter. As always, we assume that φ₀, φ₁, ... is an acceptable programming system. Now we will develop a complexity theory that is independent of the machine model and of the resource being measured. Consequently, this theory will be insensitive to particular language features and input/output conventions, etc. Given a program i and some measure of complexity (like time or space), the complexity of φ_i is determined by a function that determines, for any input x, how much of the resource counted by the complexity measure is used in the computation of φ_i (x). In general, the complexity of a function is another function. This is because programs consume different amounts of some resources depending on what inputs they are given. In the case where we are measuring time, any divergent computation will consume infinitely many time steps without converging. However, it is possible for a computation to diverge and only use a small, finite amount of space. If a computation is undefined, then we would like its complexity to be undetermined as well. Consequently, we adopt the convention that if it turns out that the computation of φ_i(x) diverges, then we will say that program i on input x uses infinitely much of the resource counted by the complexity measure. To make the case of an undefined computation fit neatly into our results we will adopt the convention that in comparisons any natural number will be considered to be less than an undefined quantity.