Small Spans in Scaled Dimension

Abstract

Juedes and Lutz [SIAM J. Comput., 24 (1995), pp. 279--295] 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 proved for increasingly stronger polynomial-time reductions, and a small span theorem for polynomial-time Turing reductions would imply $\BPP \not= \EXP$. In contrast to the progress in resource-bounded measure, Ambos-Spies et al. [{Proceedings of the 16th IEEE Conference on Computational Complexity, Philadelphia, PA, IEEE Computer Society, Los Alamitos, CA, 2001, pp. 210--217] showed that there is no small span theorem for the resource-bounded dimension of Lutz [SIAM J. Comput.}, 32 (2003), pp. 1236--1259], even for polynomial-time many-one reductions. Resource-bounded scaled dimension, recently introduced by Hitchcock, Lutz, and Mayordomo [J. Comput. System Sci., 69 (2004), pp. 97--122], 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 $\mathrm{P} = \BPP$ or $\mathrm{P} \not= \PSPACE$. 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 $\E$.