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 (2003), even for
polynomial-time many-one reductions.
Resource-bounded scaled dimension, recently introduced by Hitchcock, Lutz, and Mayordomo (2004), 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.