Dimension and the Structure of Complexity Classes

TL;DR

This paper explores the dimension structure of complexity classes, establishing new principles and results that relate resource-bounded dimensions of sets and individual languages, with implications for complexity class hardness and NP's dimension.

Contribution

It introduces resource-bounded instances of the Point-to-Set Principle and applies them to analyze the dimensions of complexity classes and reducibility, providing new insights into class hardness and NP's dimension.

Findings

Resource-bounded Point-to-Set Principle characterizes class dimensions.

Languages reducible to p-selective sets have p-dimension 0.

If disjoint NP pairs have dimension 1 in EXP, then NP has positive dimension in EXP.

Abstract

We prove three results on the dimension structure of complexity classes. 1. The Point-to-Set Principle, which has recently been used to prove several new theorems in fractal geometry, has resource-bounded instances. These instances characterize the resource-bounded dimension of a set $X$ of languages in terms of the relativized resource-bounded dimensions of the individual elements of $X$, provided that the former resource bound is large enough to parameterize the latter. Thus for example, the dimension of a class $X$ of languages in EXP is characterized in terms of the relativized p-dimensions of the individual elements of $X$. 2. Every language that is $\leq^P_m$-reducible to a p-selective set has p-dimension 0, and this fact holds relative to arbitrary oracles. Combined with a resource-bounded instance of the Point-to-Set Principle, this implies that if NP has positive dimension in EXP, then no quasipolynomial time selective language is $\leq^P_m$-hard for NP. 3. If the set of all disjoint pairs of NP languages has dimension 1 in the set of all disjoint pairs of EXP languages, then NP has positive dimension in EXP.