The resource-bounded measures of complexity classes are shown to be robust with respect to certain changes in the underlying probability measure. Specifically, for any real number ffi ? 0, any uniformly polynomial-time computable sequence ~ fi = (fi 0 ; fi 1 ; fi 2 ; : : : ) of real numbers (biases) fi i 2 [ffi; 1 \Gamma ffi], and any complexity class C (such as P, NP, BPP, P/Poly, PH, PSPACE, etc.) that is closed under positive, polynomial-time, truth-table reductions with queries of at most linear length, it is shown that the following two conditions are equivalent. (1) C has p-measure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the coin-toss probability measure given by the sequence ~ fi. (2) C has p-measure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the uniform probability measure. The proof introduces three techniques that may be useful in other contexts, namely, (i) the transformation of an efficient martingale for one probability measu...

We survey recent results on resource-bounded measure and randomness in structural complexity theory. In particular, we discuss applications of these concepts to the exponential time complexity classes E and E2 . Moreover, we treat time-bounded genericity and stochasticity concepts which are weaker than time-bounded randomness but which suffice for many of the applications in complexity theory. 1 Introduction The first attempt for defining the concept of a random sequence goes back to von Mises [vM19] in 1919. He proposed that an infinite 0-1-sequence S should be considered to be random if, in the limit, the number of the occurrences of the 0s and 1s in S is the same (i.e. the sequence S satisfies the law of large numbers) and if this stability property is inherited by every infinite subsequence of S obtained by an admissible selection rule. A fuzzyness in this concept, due to the lack of a formal definition of admissibility was later eliminated by Church [Ch40] in 1940, who proposed t...

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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

In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completeness of several problems for NP by Cook [Coo71] and Levin [Lev73] and for many other problems by Karp [Kar72], the interest in completeness notions in complexity classes has tremendously increased. Virtually every form of reduction known in computability theory has found its way to complexity theory. This is usually done by imposing time and/or space bounds on the computational power of the device representing the reduction. Early on, Ladner et al. [LLS75] categorized the then known types of reductions and made a comparison between these by constructing sets that are reducible to each other via one type of reduction and not reducible via the other. They however were interested just in the rela...

. We show that there is a set which is almost complete but not complete under polynomial-time many-one (p-m) reductions for the class E of sets computable in deterministic time 2 lin . Here a set A in a complexity class C is almost complete for C under some reducibility r if the class of the problems in C which do not r-reduce to A has measure 0 in C in the sense of Lutz's resource-bounded measure theory. We also show that the almost complete sets for E under polynomial-time bounded one-one length-increasing reductions and truth-table reductions of norm 1 coincide with the almost p-m-complete sets for E. Moreover, we obtain similar results for the class EXP of sets computable in deterministic time 2 poly . 1 Introduction Lutz [15] introduced measure concepts for the standard deterministic time and space complexity classes which contain the class E of sets computable in deterministic time 2 lin . These measure concepts have been used for investigating quantitative aspe...

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The development of Small Span Theorems for various complexity classes and reducibilities plays a basic role in (resource bounded) measure-theoretic investigations of e cient reductions. A Small Span Theorem for a complexity class C and reducibility r is the assertion that, for all sets A in C, at least one of the cones below or above A is a negligible small class with respect to C, where the cones below orabove A refer to the sets fB: B r Ag and fB: A r Bg, respectively. That is, a Small Span Theorem rules out one of the four possibilities of the size of upper and lower cones for a set in C. Here we use the recent formulation of resource-bounded measure of Allender and Strauss which allows meaningful notions of measure on polynomial-time complexity classes. We showtwo Small Span Theorems for polynomial-time complexity classes and sublinear-time reducibilities, namely a Small Span Theorem for P and Dlogtimeuniform NC0-computable reductions, and for PNP and Dlogtime-transformations. Furthermore, we showthat, for every xed k, the hard set for P under Dlogtimeuniform AC 0-reductions of depth k and size n k is a small class. In contrast, we show that every upper cone under P-uniform NC 0-reductions is not small. 1

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