A Refinement of the McCreight-Meyer Union Theorem

TL;DR

This paper refines the McCreight-Meyer Union Theorem by establishing a single computable function that characterizes the time complexity of various complexity classes, unifying their definitions through Blum complexity measures.

Contribution

It introduces a total computable, non-decreasing function that precisely characterizes multiple complexity classes, extending the union theorem to a broader class of definable language classes.

Findings

Unified time bounds for multiple complexity classes.

Characterization of classes via a single computable function.

Extension of the union theorem to classes definable by complexity operators.

Abstract

Using properties of Blum complexity measures and certain complexity class operators, we exhibit a total computable and non-decreasing function $t_{\mathsf{poly}}$ such that for all $k$, $\Sigma_k\mathsf{P} = \Sigma_k\mathsf{TIME}(t_{\mathsf{poly}})$, $\mathsf{BPP} = \mathsf{BPTIME}(t_{\mathsf{poly}})$, $\mathsf{RP} = \mathsf{RTIME}(t_{\mathsf{poly}})$, $\mathsf{UP} = \mathsf{UTIME}(t_{\mathsf{poly}})$, $\mathsf{PP} = \mathsf{PTIME}(t_{\mathsf{poly}})$, $\mathsf{Mod}_k\mathsf{P} = \mathsf{Mod}_k\mathsf{TIME}(t_{\mathsf{poly}})$, $\mathsf{PSPACE} = \mathsf{DSPACE}(t_{\mathsf{poly}})$, and so forth. A similar statement holds for any collection of language classes, provided that each class is definable by applying a certain complexity class operator to some Blum complexity class.