. 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 ...

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...

This paper reviews and investigates Bennett's notions of strong and weak computational depth (also called logical depth) for in nite binary sequences. Roughly, an in nite binary sequence x is de ned to be weakly useful if every element of a non-negligible set of decidable sequences is reducible to x in recursively bounded time. It is shown that every weakly useful sequence is strongly deep. This result (which generalizes Bennett's observation that the halting problem is strongly deep) implies that every high Turing degree contains strongly deep sequences. It is also shown that, in the sense of Baire category, almost

Introduction Investigation of the measure-theoretic structure of complexity classes began with the development of resource-bounded measure in 1991 [56]. Since that time, a growing body of research by more than forty scientists around the world has shown resource-bounded measure to be a powerful tool that sheds new light on many aspects of computational complexity. Recent survey papers by Lutz [60], Ambos-Spies and Mayordomo [3], and Buhrman and Torenvliet [22] describe many of the achievements of this line of inquiry. In this column, we give a more recent snapshot of resource-bounded measure, focusing not so much on what has been achieved to date as on what we hope will be achieved in the near future. Section 2 below gives a brief, nontechnical overview of resource-bounded measure in terms of its motivation and principal ideas. Sections 3, 4, and 5 describe twelve specific open problems in the area. We have used the following three criteria in choosing these problems. 1. Their