ABSTRACT Recent results on the internal, measure-theoretic structure of the exponential time complexity classes E and EXP are surveyed. The measure structure of these classes is seen to interact in informative ways with bi-immunity, complexity cores, polynomial-time reductions, completeness, circuit-size complexity, Kolmogorov complexity, natural proofs, pseudorandom generators, the density of hard languages, randomized complexity, and lowness. Possible implications for the structure of NP are also discussed. 1

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...

Under the hypothesis that NP does not have p-measure 0 (roughly, that NP contains more than a negligible subset of exponential time), it is show n that there is a language that is P T -complete ("Cook complete "), but not P m -complete ("Karp-Levin complete"), for NP. This conclusion, widely believed to be true, is not known to follow from P 6= NP or other traditional complexity-theoretic hypotheses. Evidence is presented that "NP does not have p-measure 0" is a reasonable hypothesis with many credible consequences. Additional such consequences proven here include the separation of many truthtable reducibilities in NP (e.g., k queries versus k+1 queries), the class separation E 6= NE, and the existence of NP search problems that are not reducible to the corresponding decision problems. This research was supported in part by National Science Foundation Grant CCR9157382, with matching funds from Rockwell International. 1 Introduction The NP-completeness of decision problems has...

We present a notion of resource-bounded measure for P and other subexponential-time classes. This gen-emlization is based on Lutz’s notion of measure, but overcomes the limitations that cause Lptz’s definitions to apply only to classes at least as large as E. We present many of the basic properties of this measure, and use it to ezplore the class of sets that are hard for BPP. Bennett and Gill showed that almost all sets are hard for BPP; Lutz improved this from Lebesgue mea-sure to measure on ESPACE. We use OUT measure to improve this still further, showing that for all E> 0, almost every set in E, is hard for BPP, where E, = Us<rDTIME(2”6), which is the best that can be achieved without showing that BPP is properly con-tained in E. A number of related results are also ob-tained in this way. 1

Measure-theoretic aspects of the P m -reducibility structure of the exponential time complexity classes E=DTIME(2 linear ) and E 2 = DTIME(2 polynomial ) are investigated. Particular attention is given to the complexity (measured by the size of complexity cores) and distribution (abundance in the sense of measure) of languages that are P m - hard for E and other complexity classes. Tight upper and lower bounds on the size of complexity cores of hard languages are derived. The upper bound says that the P m -hard languages for E are unusually simple, in the sense that they have smaller complexity cores than most languages in E. It follows that the P m -complete languages for E form a measure 0 subset of E (and similarly in E 2 ). This latter fact is seen to be a special case of a more general theorem, namely, that every P m -degree (e.g., the degree of all P m -complete languages for NP) has measure 0 in E and in E 2 .

We introduce and study resource bounded random sets based on Lutz's concept of resource bounded measure ([7, 8]). We concentrate on n c - randomness (c 2) which corresponds to the polynomial time bounded (p-) measure of Lutz, and which is adequate for studying the internal and quantitative structure of E = DTIME(2 lin ). However we will also comment on E2 = DTIME(2 pol ) and its corresponding (p2 -) measure. First we show that the class of n c -random sets has p-measure 1. This provides a new, simplified approach to p-measure 1-results. Next we compare randomness with genericity (in the sense of [2, 3]) and we show that n c+1 - random sets are n c -generic, whereas the converse fails. From the former we conclude that n c -random sets are not p-btt-complete for E. Our technical main results describe the distribution of the n c -random sets under p-m-reducibility. We show that every n c -random set in E has n k -random predecessors in E for any k 1, whereas the amou...

We prove that if strong pseudorandom number generators exist, then the class of languages that have polynomialsized circuits (P/poly) is not measurable within exponential time, in terms of the resource-bounded measure theory of Lutz. We prove our result by showing that if P/poly has measure zero in exponential time, then there is a natural proof against P/poly, in the terminology of Razborov and Rudich [25]. We also provide a partial converse of this result.

Under the hypothesis that NP has positive p-dimension, we prove that any approximation for MAX3SAT must satisfy at least one of the following: time.

Abstract not found

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