Classes of computable functions defined by bounds on computation

Abstract

The structure of the functions computable in time or space bounded by t is investigated for recursive functions t. The t-computable classes are shown to be closed under increasing recursively enumerable unions; as a corollary the primitive recursive functions are shown to equal the t-computable functions for a certain recursive t. Any countable partial order can be isomorphically embedded in the family of t-computable classes partially ordered by set inclusion. For any recursive t, there is a recursive t' which is (approximately) equal to an actual running time such that the t-computable functions equal the t'-computable functions.