Abstract not found

Abstract not found

An infinite binary sequence has randomness rate σ if, for almost every n, the Kolmogorov complexity of its prefix of length n is at least σn. It is known that for every rational σ ∈ (0, 1), on one hand, there exists sequences with randomness rate σ that can not be effectively transformed into a sequence with randomness rate higher than σ and, on the other hand, any two independent sequences with randomness rate σ can be transformed into a sequence with randomness rate higher than σ. We show that the latter result holds even if the two input sequences have linear dependency (which, informally speaking, means that all prefixes of length n of the two sequences have in common a constant fraction of their information). The similar problem is studied for finite strings. It is shown that from any two strings with sufficiently large Kolmogorov complexity and sufficiently small dependence, one can effectively construct a string that is random even conditioned by any one of the input strings.

Abstract. We construct a ∆0 2 infinite binary sequence with effective Hausdorff dimension 1/2 that does not compute a sequence of higher dimension. Introduced by Lutz, effective Hausdorff dimension can be viewed as a measure of the information density of a sequence. In particular, the dimension of A ∈ 2ω is the lim inf of the ratio between the information content and length of initial segments of A. Thus the main result demonstrates that it is not always possible to extract information from a partially random source to produce a sequence that has higher information density. 1.

Abstract Resource-bounded dimension is a notion of computational information density of in-finite sequences based on computationally bounded gamblers. This paper develops the theory of pushdown dimension and explores its relationship with finite-state dimension.The pushdown dimension of any sequence is trivially bounded above by its finite-state dimension, since a pushdown gambler can simulate any finite-state gambler. We show thatfor every rational 0 < d < 1, there exists a sequence with finite-state dimension d whosepushdown dimension is at most d/2. This provides a stronger quantitative analogue of thewell-known fact that pushdown automata decide strictly more languages than finite-state

The concern of this paper is with effective packing dimension. This concept can be traced back to the work of Borel and Lebesgue who studied measure as a way of specifying the size of sets. Carathéodory later generalized Lebesgue measure to the n-dimensional Euclidean space, and this was taken further by Hausdorff [Hau19]

We clarify the role of Kolmogorov complexity in the area of randomness extraction. We show that a computable function is an almost randomness extractor if and only if it is a Kolmogorov complexity extractor, thus establishing a fundamental equivalence between two forms of extraction studied in the literature: Kolmogorov extraction and randomness extraction. We present a distribution M k based on Kolmogorov complexity that is complete for randomness extraction in the sense that a computable function is an almost randomness extractor if and only if it extracts randomness from M k. 1

In this thesis, we study small, yet important, circuit complexity classes within NC 1, such as ACC 0 and TC 0. We also investigate the power of a closely related problem called Iterated Matrix Multiplication and its implications in low levels of algebraic complexity theory. More concretely, • We show that extremely modest-sounding lower bounds for certain problems can lead to non-trivial derandomization results. – If the word problem over S5 requires constant-depth threshold circuits of size n1+ɛ for some ɛ> 0, then any language accepted by uniform polynomial-size probabilistic threshold circuits can be solved in subexponential time (and more strongly, can be accepted by a uniform family of deterministic constant-depth threshold circuits of subexponential size.) – If there are no constant-depth arithmetic circuits of size n1+ɛ for the problem of multiplying a sequence of n 3-by-3 matrices, then for every constant d, black-box identity testing for depth-d arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constant-depth AC circuits of subexponential size).