The complexity types of computable sets

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Abstract

We analyze the fine structure of time complexity classes for RAMS, in particular the equivalence relation A = cB (“A and B have the same time complexity”) ⇔ (for all time constructible f: A ϵ DTIME(f) ⇔ B ϵ DTIME(f)). The = c-equivalence class of A is called its complexity type. Characteristic sequences of time bounds are introduced as a technical tool for the analysis of complexity types. We investigate the relationship between the complexity type of a set and its polynomial time degree, as well as the structure of complexity types inside P with regard to linear time degrees. Furthermore, it is shown that every complexity type contains a sparse set and that the complexity types with their natural partial order form a lattice.

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Written under partial support by NSF Grant CCR 8903398. Part of this research was carried out during a visit of the first author at the University of Chicago. The first author thanks the Department of Computer Science at the University of Chicago for its hospitality.

Written under partial support by Presidential Young Investigator Award DMS-8451748 and NSF Grant DMS-8601856.