Small Spans in Scaled Dimension

TL;DR

This paper investigates the small span theorem within resource-bounded scaled dimension, demonstrating its validity at certain scales but not others, and explores implications for complexity class separations.

Contribution

It extends the understanding of small span theorems to scaled dimension, showing where they hold or fail, and connects these results to complexity class separations.

Findings

Small span theorem holds in -3rd-order scaled dimension.

Fails in -2nd-order scaled dimension.

Determining scaled dimension of complete languages could resolve P vs BPP or P vs PSPACE.

Abstract

Juedes and Lutz (1995) proved a small span theorem for polynomial-time many-one reductions in exponential time. This result says that for language A decidable in exponential time, either the class of languages reducible to A (the lower span) or the class of problems to which A can be reduced (the upper span) is small in the sense of resource-bounded measure and, in particular, that the degree of A is small. Small span theorems have been proven for increasingly stronger polynomial-time reductions, and a small span theorem for polynomial-time Turing reductions would imply BPP != EXP. In contrast to the progress in resource-bounded measure, Ambos-Spies, Merkle, Reimann, and Stephan (2001) showed that there is no small span theorem for the resource-bounded dimension of Lutz (2000), even for polynomial-time many-one reductions. Resource-bounded scaled dimension, recently introduced by Hitchcock, Lutz, and Mayordomo (2003), provides rescalings of resource-bounded dimension. We use scaled dimension to further understand the contrast between measure and dimension regarding polynomial-time spans and degrees. We strengthen prior results by showing that the small span theorem holds for polynomial-time many-one reductions in the -3rd-order scaled dimension, but fails to hold in the -2nd-order scaled dimension. Our results also hold in exponential space. As an application, we show that determining the -2nd- or -1st-order scaled dimension in ESPACE of the many-one complete languages for E would yield a proof of P = BPP or P != PSPACE. On the other hand, it is shown unconditionally that the complete languages for E have -3rd-order scaled dimension 0 in ESPACE and -2nd- and -1st-order scaled dimension 1 in E.