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 −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-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

Abstract We consider hypotheses about nondeterministic computation that have been studied in dif-ferent contexts and shown to have interesting consequences: * The measure hypothesis: NP does not have p-measure 0.* The pseudo-NP hypothesis: there is an NP language that can be distinguished from anyDTIME(2 nffl) language by an NP refuter. * The NP-machine hypothesis: there is an NP machine accepting 0 * for which no 2n ffl-time machine can find infinitely many accepting computations. We show that the NP-machine hypothesis is implied by each of the first two. Previously, norelationships were known among these three hypotheses. Moreover, we unify previous work by showing that several derandomizations and circuit-size lower bounds that are known to followfrom the first two hypotheses also follow from the NP-machine hypothesis. In particular, the NPmachine hypothesis becomes the weakest known uniform hardness hypothesis that derandomizesAM. We also consider UP versions of the above hypotheses as well as related immunity and scaled dimension hypotheses. 1 Introduction The following uniform hardness hypotheses are known to imply full derandomization of ArthurMerlin games (NP = AM): * The measure hypothesis: NP does not have p-measure 0 [24].

The Turing and many-one completeness notions for NP have been previously separated under measure, genericity, and bi-immunity hypotheses on NP. The proofs of all these results rely on the existence of a language in NP with almost everywhere hardness. In this

In this paper, we use resource-bounded dimension theory to investigate polynomial size circuits. We show that for every i 0, P/poly has ith-order scaled p 3 -strong dimension 0. We also show that P/poly i.o.

We establish a relationship between the online mistake-bound model of learning and resourcebounded dimension. This connection is combined with the Winnow algorithm to obtain new results about the density of hard sets under adaptive reductions. This improves previous work

We study constructive and resource-bounded scaled dimension as an information content measure and obtain several results that parallel previous work on unscaled dimension. Scaled dimension for finite strings is developed and shown to be closely related to Kolmogorov complexity. The scaled dimension of an infinite sequence is characterized by the scaled dimensions of its prefixes. We obtain an exact Kolmogorov complexity characterization of scaled dimension. Juedes and Lutz (1996) established a small span theorem for P/poly-Turing reductions which asserts that for any problem A in ESPACE, either the class of problems reducible to A (the lower span) or the class of problems to which A is reducible (the upper span) has measure 0 in ESPACE. We apply our Kolmogorov complexity characterization to improve this to (−3) rd-order scaled dimension 0 in ESPACE. As a consequence we obtain a new upper bound on the Kolmogorov complexity of Turing-hard sets for ESPACE. 1

Bennett and Gill (1981) proved that P A � = NP A relative to a random oracle A, or in other words, that the set O [P=NP] = {A | P A = NP A} has Lebesgue measure 0. In contrast, we show that O [P=NP] has Hausdorff dimension 1. This follows from a much more general theorem: if there is a relativizable and paddable oracle construction for a complexity-theoretic statement Φ, then the set of oracles relative to which Φ holds has Hausdorff dimension 1. We give several other applications including proofs that the polynomial-time hierarchy is infinite relative to a Hausdorff dimension 1 set of oracles and that P A � = NP A ∩ coNP A relative to a Hausdorff dimension 1 set of oracles. 1

Hausdorff dimension assigns a dimension value to each subset of an arbitrary metric space. In Euclidean space, this concept coincides with our intuition that

Abstract Resource-bounded dimension is a notion of computational information density of in-finite sequences based on computationally bounded gamblers. This paper develops the theory of pushdown dimension and explores its relationship with finite-state dimension.The pushdown dimension of any sequence is trivially bounded above by its finite-state dimension, since a pushdown gambler can simulate any finite-state gambler. We show thatfor every rational 0 < d < 1, there exists a sequence with finite-state dimension d whosepushdown dimension is at most d/2. This provides a stronger quantitative analogue of thewell-known fact that pushdown automata decide strictly more languages than finite-state

Abstract This paper develops new relationships between resource-bounded dimension, entropy rates,and compression. New tools for calculating dimensions are given and used to improve previous results about circuit-size complexity classes.Approximate counting of SpanP functions is used to prove that the NP-entropy rate is an upper bound for dimension in \Delta E3, the third level of the exponential-time hierarchy. This generalresult is applied to simultaneously improve the results of Mayordomo (1994) on the measure on P/poly in \Delta E3 and of Lutz (2000) on the dimension of exponential-size circuit complexity classesin ESPACE. Entropy rates of efficiently rankable sets, sets that are optimally compressible, are studied inconjunction with time-bounded dimension. It is shown that rankable entropy rates give upper bounds for time-bounded dimensions. We use this to improve results of Lutz (1992) aboutpolynomial-size circuit complexity classes from resource-bounded measure to dimension.