Extensional Properties of Sets of Time Bounded Complexity (Extended Abstract)

We analyze the fine structure of time complexity classes for RAM's, in particular the equivalence relation A:c B (A and B have the same time complexity") ++ (for ali time constructible / : Ae DTIMEn,cM(Í) ë B e DTIMEaa¡a(/))' The -c-equivalence class of A is called its complexity type. 'We prove that every set X can be partitioned into two sets ,4. and B such that X :c A :c B, and that the partial order of sets in an arbitrary complexity type under C* (inclusion modulo finite sets) is dense. The proofs employ a nerv strategy for finite injury priority arguments. We consider the following set of time bounds: T :: {f : N --+ N I /(") ) n and / is time constructible on a RAM}, where / is called time constructible on a RAM if some RAM can compute the function 1' ¡+ 1/(') ltr OU@D steps. TVe do not allow arbitrary recursive functions as time bounds in our approach in order to avoid pathological phenomena (e.g. gap theorems [HU], [HH]). In this way we can focus on those aspects of complexity ciasses that are relevant for concrete complexity (note that all functions that are actually used as time bounds in the analysis of algorithms are time constructible). 'We use the random access machine (RAM) with uniform cost criterion as machine model (e.g. as defined in [CR]; see also [AHU], [MY]) because this is the most frequently considered model in algorithm design, and because a RAM allows more sophisticated diagonalization constructions than a Turing machine. One defines DTIMERAMU),: {A g {0,1}*l there is a RAM *Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL. 60680. Written under partial support by NSF-Grant CCR 8703889. **Department of Mathematics, University of Chicago, Chicago, IL. 60637' Written under partial support by Presidential Young Investigator Award DMS-8451748 and NSF-Grant DMS-8601856.