A note on dense and nondense families of complexity classes

In a model for a measure of computational complexity, , for a partial recursive functiont, letR _t denote all partial recursive functions having the same domain ast and computable within timet. Let = {R _t |t is recursive} and let = { |_i is actually the running time function of a computation}. and 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 for which is not dense while 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, . (This is not to say that similar questions are necessarily uninteresting for specific models.)Supported by NSF Research Grants GP6120 and GJ27127.