Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 -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.

21
The categories of this paper are: adap-org
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas
19
The categories of this paper are: eess
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
23
The categories of this paper are: solv-int
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
14
The categories of this paper are: math-ph
24
The categories of this paper are: dg-ga
26
The categories of this paper are: bayes-an
18
The categories of this paper are: econ