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

We define a class of recursive functions on the reals analogous to the classical recursive functions on the natural numbers, corresponding to a conceptual analog computer that operates in continuous time. This class turns out to be surprisingly large, and includes many functions which are uncomputable in the traditional sense.

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...

An explicit recursion-theoretic definition of a random sequence or random set of natural numbers was given by Martin-Löf in 1966. Other approaches leading to the notions of n-randomness and weak n-randomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of n-random and weakly n-random sequences with an emphasis on the structure of their Turing degrees. After an introduction and summary, in Chapter II we present several equivalent definitions of n-randomness and weak n-randomness including a new definition in terms of a forcing relation analogous to the characterization of n-generic sequences in terms of Cohen forcing. We also prove that, as conjectured by Kurtz, weak nrandomness is indeed strictly weaker than n-randomness. Chapter III is concerned with intrinsic properties of n-random sequences. The main results are that an (n + 1)-random sequence A satisfies the condition A (n) ≡T A⊕0 (n) (strengthening a result due originally to Sacks) and that n-random sequences satisfy a number of strong independence properties, e.g., if A ⊕ B is n-random then A is n-random relative to B. It follows that any countable distributive lattice can be embedded

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.

Abstract not found

. The Li--Yorke definition of chaos proved its value for interval maps. In this paper it is considered in the setting of general topological dynamics. We adopt two opposite points of view. On the one hand su#cient conditions for Li--Yorke chaos in a topological dynamical system are given. We solve a long--standing open question by proving that positive entropy implies Li--Yorke chaos. On the other hand properties of dynamical systems without Li--Yorke pairs are investigated; in addition to having entropy 0, they are minimal when transitive, and the property is stable under factor maps, arbitrary products and inverse limits. Finally it is proven that minimal systems without Li--Yorke pairs are disjoint from scattering systems. 0. Introduction The term `chaos' in connection with a map was introduced by Li and Yorke [LY], although without a formal definition. Today there are various definitions of what it means for a map to be chaotic, some of them working reasonably only in particular p...

Abstract not found

Object-oriented class libraries offer the potential for individual researchers to manage the large bodies of code generated in the experimental development of complex interactive systems. This article analyzes the structure of such a class library that supports the rapid prototyping of a wide range of systems including collaborative networking, shared documents, hypermedia, machine learning, knowledge acquisition and knowledge representation, and various combinations of these technologies. The overall systems architecture is presented in terms of a heterogeneous collection of systems providing a wide range of application functionalities. Examples are given of group writing, multimedia and knowledge-based systems which are based on combining these functionalities. The detailed design issues of the knowledge representation server component of the system are analyzed in terms of requirements, current state-of-the-art, and the underlying theoretical principles that lead to an effective obj...

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