Small spans in scaled dimension

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 -3^rd-order scaled dimension, but fails to hold in the -2^nd-order scaled dimension. Our results also hold in exponential space. As an application, we show that determining the -2^nd- or -l^st-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 -3^rd-order scaled dimension 0 in ESPACE and -2^nd- and -1^st-order scaled dimension 1 in E.