Game Arguments in Computability Theory and Algorithmic Information Theory

Shen, Alexander

doi:10.1007/978-3-642-30870-3_66

Cited by 13 publications

(6 citation statements)

References 8 publications

(11 reference statements)

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“…When interpreting the Champions Lemma below, it is useful to keep in mind that a (⌊k/2⌋ + 1, k)-recursive set is already recursive [Tra55]. Epstein and Levin recently discovered a related property which suffices to guarantee that sets contain elements with low Kolmogorov complexity [EL,She12].…”

Section: The Champion Methodsmentioning

confidence: 99%

On Approximate Decidability of Minimal Programs

Teutsch

Zimand

2015

ACM Trans. Comput. Theory

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An index e in a numbering of partial-recursive functions is called minimal if every lesser index computes a different function from e. Since the 1960's it has been known that, in any reasonable programming language, no effective procedure determines whether or not a given index is minimal. We investigate whether the task of determining minimal indices can be solved in an approximate sense. Our first question, regarding the set of minimal indices, is whether there exists an algorithm which can correctly label 1 out of k indices as either minimal or non-minimal. Our second question, regarding the function which computes minimal indices, is whether one can compute a short list of candidate indices which includes a minimal index for a given program. We give some negative results and leave the possibility of positive results as open questions.

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Section: The Champion Methodsmentioning

confidence: 99%

On Approximate Decidability of Minimal Programs

Teutsch

Zimand

2015

ACM Trans. Comput. Theory

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“…We get the notion of total conditional complexity CT (y|x), which is the length of the shortest total program that maps x to y. Total conditional complexity can be much greater than plain one, see for example [6]. Intuitively, good sufficient statistics A for x must have not only negligible conditional complexity C(A|x) (which follows from definition of a sufficient statistic) but also negligible total conditional complexity CT (A|x).…”

Section: Total Conditional Complexity and Strong Modelsmentioning

confidence: 99%

Algorithmic Statistics: Normal Objects and Universal Models

Milovanov

2016

Computer Science – Theory and Applications

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Kolmogorov suggested to measure quality of a statistical hypothesis (a model) P for a data x by two parameters: Kolmogorov complexity C(P ) of the hypothesis and the probability P (x) of x with respect to P . The first parameter measures how simple the hypothesis P is and the second one how it fits. The paper [2] discovered a small class of models that are universal in the following sense. Each hypothesis Sij from that class is identified by two integer parameters i, j and for every data x and for each complexity level α there is a hypothesis Sij with j i l(x) of complexity at most α that has almost the best fit among all hypotheses of complexity at most α. The hypothesis Sij is identified by i and the leading i − j bits of the binary representation of the number of strings of complexity at most i. On the other hand, the initial data x might be completely irrelevant to the the number of strings of complexity at most i. Thus Sij seems to have some information irrelevant to the data, which undermines Kolmogorov's approach: the best hypotheses should not have irrelevant information.To restrict the class of hypotheses for a data x to those that have only relevant information, the paper [10] introduced a notion of a strong model for x: those are models for x whose total conditional complexity conditional to x is negligible. An object x is called normal if for each complexity level α at least one its best fitting model of that complexity is strong.In this paper we show that there are "many types" of normal strings (Theorem 10). Our second result states that there is a normal object x such that all its best fitting models Sij are not strong for x. Our last result states that every best fit strong model for a normal object is again a normal object.

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“…Authors are grateful to Leonid Levin, Peter Gács, Bruno Bauwens, the participants of Kolmogorov seminar (Moscow) and all their colleagues in LIRMM (Montpellier) and LIAFA (Paris); special thanks to Rupert Hölzl for explaining Friedberg's argument. Robert Soare informed us (at CiE2012, where the preliminary version of this paper [14] was presented) about A.H. Lachlan's paper [8] where Lachlan initiated the game approach to computability theory (in slightly different context related to enumerable sets); see also [7]. Lance Fortnow showed us a proof of Friedberg theorem due to Kummer [6].…”

Section: Acknowledgmentsmentioning

confidence: 99%

Game arguments in computability theory and algorithmic information theory

Muchnik¹,

Shen²,

Mikhail³

2012

Preprint

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We provide some examples showing how game-theoretic arguments (the approach that goes back to Lachlan and was developed by An. Muchnik) can be used in computability theory and algorithmic information theory. To illustrate this technique, we start with a proof of a classical result, the unique numbering theorem of Friedberg, translated to the game language. Then we provide game-theoretic proofs for three other results: (1) the gap between conditional complexity and total conditional complexity; (2) Epstein-Levin theorem relating a priori and prefix complexity for a stochastic set (for which we provide a new game-theoretic proof) and ( 3) some result about information distances in algorithmic information theory (obtained by two of the authors [A.M. and M.V.] several years ago but not yet published).

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Game Arguments in Computability Theory and Algorithmic Information Theory

Cited by 13 publications

References 8 publications

On Approximate Decidability of Minimal Programs

On Approximate Decidability of Minimal Programs

Algorithmic Statistics: Normal Objects and Universal Models

Game arguments in computability theory and algorithmic information theory

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