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11
A lower cone in the wtt degrees of nonintegral effective dimension
 In Proceedings of IMS workshop on Computational Prospects of Infinity
, 2006
"... ABSTRACT. For any rational number r, we show that there exists a set A (weak truthtable reducible to the halting problem) such that any set B weak truthtable reducible to it has effective Hausdorff dimension at most r, where A itself has dimension at least r. This implies, for any rational r, the e ..."
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Cited by 24 (2 self)
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ABSTRACT. For any rational number r, we show that there exists a set A (weak truthtable reducible to the halting problem) such that any set B weak truthtable reducible to it has effective Hausdorff dimension at most r, where A itself has dimension at least r. This implies, for any rational r, the existence of a wttlower cone of effective dimension r. 1.
KolmogorovLoveland randomness and stochasticity
 Annals of Pure and Applied Logic
, 2005
"... An infinite binary sequence X is KolmogorovLoveland (or KL) random if there is no computable nonmonotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while betting successively on bits of X. A sequence X is KLstochastic if there is no computable nonm ..."
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Cited by 16 (8 self)
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An infinite binary sequence X is KolmogorovLoveland (or KL) random if there is no computable nonmonotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while betting successively on bits of X. A sequence X is KLstochastic if there is no computable nonmonotonic selection rule that selects from X an infinite, biased sequence. One of the major open problems in the field of effective randomness is whether MartinLöf randomness is the same as KLrandomness. Our first main result states that KLrandom sequences are close to MartinLöf random sequences in so far as every KLrandom sequence has arbitrarily dense subsequences that are MartinLöf random. A key lemma in the proof of this result is that for every effective split of a KLrandom sequence at least one of the halves is MartinLöf random. However, this splitting property does not characterize KLrandomness; we construct a sequence that is not even computably random such that every effective split yields two subsequences that are 2random. Furthermore, we show for any KLrandom sequence A that is computable in the halting problem that, first, for any effective split of A both halves are MartinLöf random and, second, for any computable, nondecreasing, and unbounded function g
Measures and their random reals
 IN PREPARATION
"... We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every nonrecursive real is nontrivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of ..."
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Cited by 11 (2 self)
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We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every nonrecursive real is nontrivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of atoms, i.e. considers only continuous measures, it turns out that every nonhyperarithmetical real is random for a continuous measure. On the other hand, examples of reals not random for a continuous measure can be found throughout the hyperarithmetical Turing degrees.
Effectively closed sets of measures and randomness
 Ann. Pure Appl. Logic
"... We show that if a real x ∈ 2ω is strongly Hausdorff Hhrandom, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure µ such that the µmeasure of the basic open cylinders shrinks according to h. The proof uses a new method to con ..."
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Cited by 7 (1 self)
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We show that if a real x ∈ 2ω is strongly Hausdorff Hhrandom, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure µ such that the µmeasure of the basic open cylinders shrinks according to h. The proof uses a new method to construct measures, based on effective (partial) continuous transformations and a basis theorem for Π0 1classes applied to closed sets of probability measures. We use the main result to give a new proof of Frostman’s Lemma, to derive a collapse of randomness notions for Hausdorff measures, and to provide a characterization of effective Hausdorff dimension similar to Frostman’s Theorem. 1
NONCOMPUTABLE CONDITIONAL DISTRIBUTIONS
"... Abstract. We study the computability of conditional probability, a fundamental notion in probability theory and Bayesian statistics. In the elementary discrete setting, a ratio of probabilities defines conditional probability. In more general settings, conditional probability is defined axiomaticall ..."
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Cited by 7 (3 self)
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Abstract. We study the computability of conditional probability, a fundamental notion in probability theory and Bayesian statistics. In the elementary discrete setting, a ratio of probabilities defines conditional probability. In more general settings, conditional probability is defined axiomatically, and the search for more constructive definitions is the subject of a rich literature in probability theory and statistics. However, we show that in general one cannot compute conditional probabilities. Specifically, we construct a pair of computable random variables (X, Y) in the unit interval whose conditional distribution P[YX] encodes the halting problem. Nevertheless, probabilistic inference has proven remarkably successful in practice, even in infinitedimensional continuous settings. We prove several results giving general conditions under which conditional distributions are computable. In the discrete or dominated setting, under suitable computability hypotheses, conditional distributions are computable. Likewise, conditioning is a computable operation in the presence of certain additional structure, such as independent absolutely continuous noise.
Errorcorrecting codes and phase transitions
"... Abstract. The theory of errorcorrecting codes is concerned with constructing codes that optimize simultaneously transmission rate and relative minimum distance. These conflicting requirements determine an asymptotic bound, which is a continuous curve in the space of parameters. The main goal of thi ..."
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Cited by 4 (2 self)
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Abstract. The theory of errorcorrecting codes is concerned with constructing codes that optimize simultaneously transmission rate and relative minimum distance. These conflicting requirements determine an asymptotic bound, which is a continuous curve in the space of parameters. The main goal of this paper is to relate the asymptotic bound to phase diagrams of quantum statistical mechanical systems. We first identify the code parameters with Hausdorff and von Neumann dimensions, by considering fractals consisting of infinite sequences of code words. We then construct operator algebras associated to individual codes. These are Toeplitz algebras with a time evolution for which the KMS state at critical temperature gives the Hausdorff measure on the corresponding fractal. We extend this construction to algebras associated to limit points of codes, with nonuniform multifractal measures, and to tensor products over varying parameters. Contents. 0. Introduction: asymptotic bounds
ON THE COMPUTABILITY OF CONDITIONAL PROBABILITY
"... Abstract. We study the problem of computing conditional probabilities, a fundamental operation in statistics and machine learning. In the elementary discrete setting, conditional probability is defined axiomatically and the search for more constructive definitions is the subject of a rich literature ..."
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Cited by 3 (3 self)
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Abstract. We study the problem of computing conditional probabilities, a fundamental operation in statistics and machine learning. In the elementary discrete setting, conditional probability is defined axiomatically and the search for more constructive definitions is the subject of a rich literature in probability theory and statistics. In the discrete or dominated setting, under suitable computability hypotheses, conditional probabilities are computable. However, we show that in general one cannot compute conditional probabilities. We do this by constructing a pair of computable random variables in the unit interval whose conditional distribution encodes the halting problem at almost every point. We show that this result is tight, in the sense that given an oracle for the halting problem, one can compute this conditional distribution. On the other hand, we show that conditioning in abstract settings is computable in the presence of certain additional structure, such as independent absolutely continuous noise. 1.
Probability measures and effective randomness
, 2007
"... Abstract. We study the question, “For which reals x does there exist a measure µ such that x is random relative to µ? ” We show that for every nonrecursive x, there is a measure which makes x random without concentrating on x. We give several conditions on x equivalent to there being continuous meas ..."
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Cited by 2 (0 self)
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Abstract. We study the question, “For which reals x does there exist a measure µ such that x is random relative to µ? ” We show that for every nonrecursive x, there is a measure which makes x random without concentrating on x. We give several conditions on x equivalent to there being continuous measure which makes x random. We show that for all but countably many reals x these conditions apply, so there is a continuous measure which makes x random. There is a metamathematical aspect of this investigation. As one requires higher arithmetic levels in the degree of randomness, one must make use of more iterates of the power set of the continuum to show that for all but countably many x’s there is a continuous µ which makes x random to that degree. 1.
Complexity and Mixed Strategy Equilibria ∗
"... Unpredictable behavior is central for optimal play in many strategic situations because a predictable pattern leaves a player vulnerable to exploitation. A theory of unpredictable behavior is presented in the context of repeated twoperson zerosum games in which the stage games have no pure strateg ..."
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Unpredictable behavior is central for optimal play in many strategic situations because a predictable pattern leaves a player vulnerable to exploitation. A theory of unpredictable behavior is presented in the context of repeated twoperson zerosum games in which the stage games have no pure strategy equilibrium. Computational complexity considerations are introduced to restrict players ’ strategy sets. The use of Kolmogorov complexity allows us to obtain a sufficient condition for equilibrium existence. The resulting theory has implications for the empirical literature that tests the equilibrium hypothesis in a similar context. In particular, the failure of some tests for randomness does not justify rejection of equilibrium play.
UNIVERSALITY PROBABILITY OF A PREFIXFREE MACHINE
"... Abstract. We study the notion of universality probability of a universal prefixfree machine, as introduced by C.S. Wallace (see [Dow08, Section 0.2.2] and [Dow11, Section 2.5]). We show that it is random relative to the third iterate of the halting problem and determine its Turing degree and its pl ..."
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Abstract. We study the notion of universality probability of a universal prefixfree machine, as introduced by C.S. Wallace (see [Dow08, Section 0.2.2] and [Dow11, Section 2.5]). We show that it is random relative to the third iterate of the halting problem and determine its Turing degree and its place in the arithmetical hierarchy of complexity. Furthermore, we give a computational characterization of the real numbers which are universality probabilities of universal prefixfree machines. 1.