What Percentage of Programs Halt?

Bienvenu, Laurent; Desfontaines, Damien; Shen, Alexander

doi:10.1007/978-3-662-47672-7_18

Cited by 4 publications

(5 citation statements)

References 13 publications

(7 reference statements)

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“…The Halting Problem is historically the first proved undecidable problem; it has many applications in mathematics, logic and theoretical as well as applied computer science, mathematics, physics, biology, etc. Due to its practical importance approximate solutions for this problem have been proposed for quite a long time, see [2,[5][6][7]9,14,16,19,21,26].…”

Section: Introductionmentioning

confidence: 99%

A probabilistic anytime algorithm for the halting problem

Calude

Dumitrescu

2018

COM

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The Halting Problem, the most (in)famous undecidable problem, has important applications in theoretical and applied computer science and beyond, hence the interest in its approximate solutions.Experimental results reported on various models of computation suggest that halting programs are not uniformly distributedrunning times play an important role. A reason is that a program which eventually stops but does not halt "quickly", stops at a time which is algorithmically compressible.In this paper we work with running times to define a class of computable probability distributions on the set of halting programs in order to construct an anytime algorithm for the Halting problem with a probabilistic evaluation of the error of the decision.

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Section: Introductionmentioning

confidence: 99%

A probabilistic anytime algorithm for the halting problem

Calude

Dumitrescu

2018

COM

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“…Indeed, a prefix-free description of a program that maps k to n uses O(C (n | k)) bits, and if we append the first k bits of H n to it, we can then reconstruct the prefix-free description, then k, then n, and finally t. 3 This implies, for example, the following corollary: Corollary 6.3. Making BB (n/2) steps of computation of the universal machine for each input of length at most n, we have 2 n/2±O (1) unfinished (terminating) computations.…”

Section: Busy Beavers and Fraction Of Long Computationsmentioning

confidence: 99%

“…Then we need to implement a choice between the second and third conditions. The third one defines an enumerable set of pairs m, x , and we can use the property (3) for this set and the enumeration τ ( m, x ) = µ(x) obtained by using (1).…”

Section: Busy Beavers and Fraction Of Long Computationsmentioning

confidence: 99%

“…Note that we used Remark 2.5 in this argument, since the transformation of W into W should be effective to get a contradiction with the fixed-point theorem. 1 For brevity we use the name 'enumerable' for recursively (computably) enumerable sets.…”

Section: Introductionmentioning

confidence: 99%

“…First, we construct the uniform size-bounded reduction of all enumerations. The property (1) guarantees that there exists an enumeration ν such that…”

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confidence: 99%

See 2 more Smart Citations

Generic algorithms for halting problem and optimal machines revisited

Bienvenu¹,

Desfontaines²,

Shen³

2016

Logical Methods in Computer Science

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Abstract. The halting problem is undecidable -but can it be solved for "most" inputs? This natural question was considered in a number of papers, in different settings. We revisit their results and show that most of them can be easily proven in a natural framework of optimal machines (considered in algorithmic information theory) using the notion of Kolmogorov complexity.We also consider some related questions about this framework and about asymptotic properties of the halting problem. In particular, we show that the fraction of terminating programs cannot have a limit, and all limit points are Martin-Löf random reals. We then consider mass problems of finding an approximate solution of halting problem and probabilistic algorithms for them, proving both positive and negative results.We consider the fraction of terminating programs that require a long time for termination, and describe this fraction using the busy beaver function. We also consider approximate versions of separation problems, and revisit Schnorr's results about optimal numberings showing how they can be generalized.

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Algorithmic Statistics: Forty Years Later

Vereshchagin

Shen

2016

Lecture Notes in Computer Science

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Algorithmic statistics has two different (and almost orthogonal) motivations. From the philosophical point of view, it tries to formalize how the statistics works and why some statistical models are better than others. After this notion of a "good model" is introduced, a natural question arises: it is possible that for some piece of data there is no good model? If yes, how often these bad (non-stochastic) data appear "in real life"?Another, more technical motivation comes from algorithmic information theory. In this theory a notion of complexity of a finite object (=amount of information in this object) is introduced; it assigns to every object some number, called its algorithmic complexity (or Kolmogorov complexity). Algorithmic statistic provides a more fine-grained classification: for each finite object some curve is defined that characterizes its behavior. It turns out that several different definitions give (approximately) the same curve. 1 In this survey we try to provide an exposition of the main results in the field (including full proofs for the most important ones), as well as some historical comments. We assume that the reader is familiar with the main notions of algorithmic information (Kolmogorov complexity) theory. An exposition can be found in [44, chapters 1, 3, 4] Montpellier, 161 rue Ada, 34095, France 1 Road-map: Section 2 considers the notion of (α, β)-stochasticity; Section 3 considers two-part descriptions and the so-called "minimal description length principle"; Section 4 gives one more approach: we consider the list of objects of bounded complexity and measure how far some object is from the end of the list, getting some natural class of "standard descriptions" as a by-product; finally, Section 5 establishes a connection between these notions and resource-bounded complexity. The rest of the paper deals with an attempts to make theory close to practice by considering restricted classes of description (Section 6) and strong models (Section 7).

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What Percentage of Programs Halt?

Cited by 4 publications

References 13 publications

A probabilistic anytime algorithm for the halting problem

A probabilistic anytime algorithm for the halting problem

Generic algorithms for halting problem and optimal machines revisited

Algorithmic Statistics: Forty Years Later

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