A note on dense and nondense families of complexity classes

Abstract

In a model for a measure of computational complexity, Ф, for a partial recursive functiont, letR ^Ф _t denote all partial recursive functions having the same domain ast and computable within timet. Let Σ^Ф = {R ^Ф _t |t is recursive} and let Ω^Ф = {$R_{\Phi _i }$|Ф_i is actually the running time function of a computation}. Σ^Ф and Ω^Ф are partially ordered under set-theoretic inclusion. These partial orderings have been extensively investigated by Borodin, Constable and Hopcroft in [3]. In this paper we present a simple uniform proof of some of their results. For example, we give a procedure for easily calculating a model of computational complexity Ф for which Σ^Ф is not dense while Ω^Ф is dense. In our opinion, our technique is so transparent that it indicates that certain questions of density are not intrinsically interesting for general abstract measures of computational complexity, Ф. (This is not to say that similar questions are necessarily uninteresting for specific models.)