Preface- Acknowledgments- Introduction- I Background- Preliminaries- Computability Theory- Kolmogorov Complexity of Finite Strings- Relating Plain and Prefix-Free Complexity- Effective Reals- II Randomness of Sets- Martin-Lof Randomness- Other Notions of Effective Randomness- Algorithmic Randomness and Turing Reducibility- III Relative Randomness- Measures of Relative Randomness- The Quantity of K- and Other Degrees- Randomness-Theoretic Weakness- Lowness for Other Randomness Notions- Effective Hausdorff Dimension- IV Further Topics- Omega as an Operator- Complexity of CE Sets- References- Index

Algorithmic Randomness and Complexity

We show that sets consisting of strings of high Kolmogorov complexity provide examples of sets that are complete for several complexity classes under probabilistic and non-uniform reductions. These sets are provably not complete under the usual many-one reductions. Let R/sub K/, R/sub Kt/, R/sub KS/, R/sub KT/ be the sets of strings x having complexity at least |x|/2, according to the usual Kolmogorov complexity measure K, Levin's time-bounded Kolmogorov complexity Kt [27], a space-bounded Kolmogorov measure KS, and the time-bounded Kolmogorov complexity measure KT that was introduced in [4], respectively. Our main results are: 1. R/sub KS/ and R/sub Kt/ are complete for PSPACE and EXP, respectively, under P/poly-truth-table reductions. 2. EXP = NP/sup R(Kt)/. 3. PSPACE = ZPP/sup R(KS)/ /spl sube/ P/sup R(K)/. 4. The Discrete Log, Factoring, and several lattice problems are solvable in BPP/sup R(KT)/.

Power from Random Strings

In this paper we provide a framework for computable analysis of measure, probability and integration theories. We work on computable metric spaces with computable Borel probability measures. We introduce and study the framework of layerwise computability which lies on Martin-Lof randomness and the existence of a universal randomness test. We then prove characterizations of effective notions of measurability and integrability in terms of layerwise computability. On the one hand it gives a simple way of handling effective measure theory, on the other hand it provides powerful tools to study Martin-Lof randomness, as illustrated in a sequel paper.

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An Application of Martin-Löf Randomness to Effective Probability Theory

Given a string, what is its complexity? We survey what is known about the computational complexity of this problem, and describe several open questions.

The Complexity of Complexity

We pursue the study of the framework of layerwise computability introduced in a preceding paper and give three applications (i) We prove a general version of Birkhoff's ergodic theorem for random points, where the transformation and the observable are supposed to be effectively measurable instead of computable  This result significantly improves V'yugin and Nandakumar's ones (ii) We provide a general framework for deriving sharper theorems for random points, sensitive to the speed of convergence This offers a systematic approach to obtain results in the spirit of Davie's ones (iii) Proving an effective version of Prokhorov theorem, we positively answer a question recently raised by Fouche: can random Brownian paths reach any random number? All this shows that layerwise computability is a powerful framework to study Martin-Lof randomness, with a wide range of applications

/pdf/applications-of-effective-probability-theory-to-martin-lof-2maavnhbon.pdf

Applications of Effective Probability Theory to Martin-Löf Randomness

We formulate effective versions of the Borel–Cantelli lemmas using a coefficient from Kolmogorov complexity. We then use these effective versions to lift the effective content of the law of large numbers and the law of the iterated logarithm.

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The Borel–Cantelli Lemmas, Probability Laws and Kolmogorov Complexity

We introduce the notion of being Weihrauch-complete for layerwise computability and provide several natural examples related to complex oscillations, the law of the iterated logarithm and Birkhoff's theorem. We also consider hitting time operators, which share the Weihrauch degree of the former examples but fail to be layerwise computable.

Weihrauch-completeness for layerwise computability

We examine a construction due to Fouche in which a Brownian motion is constructed from an algorithmically random infinite binary sequence. We show that although the construction is provably not computable in the sense of computable analysis, a lower bound for the rate of convergence is computable in any upper bound for the compressibilty of the sequence, making the construction layerwise computable.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0960129513000157

On the computability of a construction of Brownian motion

Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorov-random strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of problems C defined in terms of polynomial-time truth- table reducibility to RK (the set of Kolmogorov-random strings) that lies between BPP and PSPACE (5, 4). The results in this paper were obtained, as part of an investigation of whether this upper bound can be improved, to show BPP C PSPACE\ P/poly: ( ) In fact, we conjecture that C = BPP = P, and we close this paper with a discussion of the possibility this might be an avenue for trying to prove the equality of BPP and P. In this paper, we present a collection of true statements in the language of arithmetic, (each provable in ZF) and show that if these statements can be proved in certain extensions

Kolmogorov Complexity, Circuits, and the Strength of Formal Theories of Arithmetic.

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George Davie

Papers

The Borel–Cantelli Lemmas, Probability Laws and Kolmogorov Complexity

Weihrauch-completeness for layerwise computability

On the computability of a construction of Brownian motion

Kolmogorov Complexity, Circuits, and the Strength of Formal Theories of Arithmetic.

Kolmogorov Complexity, Circuits, and the Strength of Formal Theories of Arithmetic