Generic density and small span theorem
Information and Computation
Generic density and small span theorem
FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
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Juedes and Lutz (1995) proved a small span theoremfor polynomial-time many-one reductions in exponentialtime. This result says that for language A decidablein exponential time, either the class of languagesreducible to A (the lower span) or the class of problemsto which A can be reduced (the upper span) is smallin the sense of resource-bounded measure and, in particular,that the degree of A is small. Small span theorems havebeen proven for increasingly stronger polynomial-time reductions,and a small span theorem for polynomial-timeTuring reductions would imply BPP 驴 EXP. In contrast tothe progress in resource-bounded measure, Ambos-Spies,Merkle, Reimann, and Stephan (2001) showed that thereis no small span theorem for the resource-bounded dimensionof Lutz (2000), even for polynomial-time many-onereductions.Resource-bounded scaled dimension, recently introducedby Hitchcock, Lutz, and Mayordomo (2003), providesrescalings of resource-bounded dimension. We usescaled dimension to further understand the contrast betweenmeasure and dimension regarding polynomial-timespans and degrees. We strengthen prior results by showingthat the small span theorem holds for polynomial-timemany-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-1^st-order scaled dimension in ESPACE of the many-onecomplete languages for E would yield a proof of P =BPP or P 驴 PSPACE. On the other hand, it is shown unconditionallythat the complete languages for E have -3^rd-orderscaled dimension 0 in ESPACE and-2^nd- and-1^st-orderscaled dimension 1 in E.