. We investigate the distribution of nonuniform complexities in uniform complexity classes. We prove that almost every problem decidable in exponential space has essentially maximum circuit-size and space-bounded Kolmogorov complexity almost everywhere. (The circuit-size lower bound actually exceeds, and thereby strengthens, the Shannon 2 n n lower bound for almost every problem, with no computability constraint.) In exponential time complexity classes, we prove that the strongest relativizable lower bounds hold almost everywhere for almost all problems. Finally, we show that infinite pseudorandom sequences have high nonuniform complexity almost everywhere. The results are unified by a new, more powerful formulation of the underlying measure theory, based on uniform systems of density functions, and by the introduction of a new nonuniform complexity measure, the selective Kolmogorov complexity. This research was supported in part by NSF Grants CCR-8809238 and CCR-9157382 and in ...

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 two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical systems. Lutz (2000) has recently proven a simple characterization of Hausdorff dimension in terms of gales, which are betting strategies that generalize martingales. Imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of Hausdorff dimension, including constructive, computable, polynomial-space, polynomial-time, and finite-state dimensions. Work by several investigators has already used these effective dimensions to shed significant new light on a variety of topics in theoretical computer science. In this paper we show that packing dimension can also be characterized in terms of gales. Moreover, even though the usual definition of packing dimension is considerably more complex than that of Hausdorff dimension, our gale characterization of packing dimension is an exact dual

. A set A is P-bi-immune if neither A nor its complement has an infinite subset in P. We investigate here the abundance of P-bi-immune languages in linearexponential time (E). We prove that the class of P-bi-immune sets has measure 1 in E. This implies that `almost' every language in E is P-bi-immune, that is to say, almost every set recognizable in linear exponential time has no algorithm that recognizes it and works in polynomial time on an infinite number of instances. A bit further, we show that every p-random (pseudo-random) language is E-bi-immune. Regarding the existence of P-bi-immune sets in NP, we show that if NP does not have measure 0 in E, then NP contains a P-bi-immune set. Another consequence is that the class of p m -complete languages for E has measure 0 in E. In contrast, it is shown that in E, and even in REC, the class of P-bi-immune languages lacks the property of Baire (the Baire category analogue of Lebesgue measurability). * This work was supported by a Spani...

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

The main theorem of this paper is that, for every real number ff ! 1 (e.g., ff = 0:99), only a measure 0 subset of the languages decidable in exponential time are P n ff \Gammatt -reducible to languages that are not exponentially dense. Thus every P n ff \Gammatt -hard language for E is exponentially dense. This strengthens Watanabe's 1987 result, that every P O(log n)\Gammatt -hard language for E is exponentially dense. The combinatorial technique used here, the sequentially most frequent query selection, also gives a new, simpler proof of Watanabe's result. The main theorem also has implications for the structure of NP under strong hypotheses. Ogiwara and Watanabe (1991) have shown that the hypothesis P 6= NP implies that every P btt -hard language for NP is non-sparse (i.e., not polynomially sparse). Their technique does not appear to allow significant relaxation of either the query bound or the sparseness criterion. It is shown here that a stronger hypothesis--- na...

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...

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. We introduce balanced t(n)-genericity which is a refinement of the genericity concept of Ambos-Spies, Fleischhack and Huwig [2] and which in addition controls the frequency with which a condition is met. We show that this concept coincides with the resource-bounded version of Church's stochasticity [6]. By uniformly describing these concepts and weaker notions of stochasticity introduced by Wilber [19] and Ko [11] in terms of prediction functions, we clarify the relations among these resource-bounded stochasticity concepts. Moreover, we give descriptions of these concepts in the framework of Lutz's resource-bounded measure theory [13] based on martingales: We show that t(n)-stochasticity coincides with a weak notion of t(n)-randomness based on so-called simple martingales but that it is strictly weaker than t(n)-randomness in the sense of Lutz. 1 Introduction Over the last years resource-bounded versions of Baire category and Lebesgue measure have been introduced in complexity theor...

The instance complexity of a string x with respect to a set A and time bound t, ic t (x : A), is the length of the shortest program for A that runs in time t, decides x correctly, and makes no mistakes on other strings (where "don't know" answers are permitted). The Instance Complexity Conjecture of Ko, Orponen, Schoning, and Watanabe states that for every recursive set A not in P and every polynomial t there is a polynomial t 0 and a constant c such that for infinitely many x, ic t (x : A) C t 0 (x) \Gamma c, where C t 0 (x) is the t 0 -time bounded Kolmogorov complexity of x. In this paper the conjecture is proved for all recursive tally sets and for all recursive sets which are NP-hard under honest reductions, in particular it holds for all natural NP-hard problems. The method of proof also yields the polynomial-space bounded and the exponential-time bounded versions of the conjecture in full generality. On the other hand, the conjecture itself turns out to be oracl...