Dimension and the Structure of Complexity Classes

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.