Dimension and the Structure of Complexity Classes

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 parametrize 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 ≤ P m -reducible to a p-selective set has p-dimension 0, and this fact holds relative to arbitrary oracles. Combined with a resourcebounded instance of the Point-to-Set Principle, this implies that if NP has positive dimension in EXP, then no quasipolynomial time selective language is ≤ P m -hard for NP.

2023 NORTH AMERICAN ANNUAL MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC University of California, Irvine Irvine, CA, USA March 25–29, 2023

“…In this talk, I will review the point-to-set principles focusing on its recent extensions to separable spaces [9], to resource-bounded dimensions [10], and to finite-state dimensions [13].…”