SIGACT news complexity theory column 48
ACM SIGACT News
Dimension, entropy rates, and compression
Journal of Computer and System Sciences
Two open problems on effective dimension
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
Hardness hypotheses, derandomization, and circuit complexity
FSTTCS'04 Proceedings of the 24th international conference on Foundations of Software Technology and Theoretical Computer Science
Hi-index | 0.00 |
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 $\E$.