A Refinement of the McCreight-Meyer Union Theorem

TL;DR

This work refines the McCreight-Meyer Union Theorem by constructing a single non-decreasing bound $t_{\mathsf{poly}}$ that unifies a broad spectrum of complexity classes, showing that for all $k$, $\Sigma_k\P = \Sigma_k\TIME(t_{\mathsf{poly}})$, $\BPP = \BPTIME(t_{\mathsf{poly}})$, $\RP = \RTIME(t_{\mathsf{poly}})$, $\UP = \UTIME(t_{\mathsf{poly}})$, $\PP = \PTIME(t_{\mathsf{poly}})$, $\Mod_k\P = \Mod_k\TIME(t_{\mathsf{poly}})$, and $\PSPACE = \DSPACE(t_{\mathsf{poly}})$. The approach proceeds in two steps: (i) show that for any finite family of Blum measures, there exists a total computable, non-decreasing $t_{\mathsf{f}}$ with $\mathcal{C}_{\Phi}(t_{\mathsf{f}}) = \bigcup_{f\in\mathsf{f}} \mathcal{C}_{\Phi}(f)$; (ii) lift this to infinite families by introducing complexity-class operators $\Sigma_k^{\mathsf{f}}$ and $\mathsf{W}_{\omega}^{\mathsf{f}}$, proving that under a suitable $t_{\mathsf{f}}$ these operators satisfy $\mathcal{L}_{\Phi_i}(t_{\mathsf{f}}) = \bigcup_{f\in\mathsf{f}} \mathcal{L}_{\Phi_i}(f)$ and analogous equalities. This yields a robust framework toward Fortnow’s conjecture by tying multiple levels of the polynomial hierarchy to a common, model-agnostic runtime bound, with implications for foundational questions about separations and relativization.

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 $k$, $Σ_k\mathsf{P} = Σ_k\mathsf{TIME}(t_{\mathsf{poly}})$, $\mathsf{BPP} = \mathsf{BPTIME}(t_{\mathsf{poly}})$, $\mathsf{RP} = \mathsf{RTIME}(t_{\mathsf{poly}})$, $\mathsf{UP} = \mathsf{UTIME}(t_{\mathsf{poly}})$, $\mathsf{PP} = \mathsf{PTIME}(t_{\mathsf{poly}})$, $\mathsf{Mod}_k\mathsf{P} = \mathsf{Mod}_k\mathsf{TIME}(t_{\mathsf{poly}})$, $\mathsf{PSPACE} = \mathsf{DSPACE}(t_{\mathsf{poly}})$, 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.

Theorems & Definitions (27)

• Conjecture A: Fortnow fortnow3
• Definition 2.1
• Definition 2.2
• Theorem 2.1: McCreight-Meyer Union Theorem mccreightmeyer
• Corollary 2.2
• Proposition 2.3
• proof
• Corollary 2.4
• Conjecture A: Fortnow fortnow3
• Definition 3.1