Short Lists with Short Programs in Short Time – A Short Proof

Zimand, Marius

doi:10.1007/978-3-319-08019-2_42

Cited by 13 publications

(8 citation statements)

References 9 publications

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“…More precisely, [BMVZ13] showed that one can compute lists of quadratic size guaranteed to contain a program of x whose length is C(x) + O(1) and that one can compute in polynomialtime a list guaranteed to contain a program whose length is additively within C(x) + O(log n). [Teu14] improved the latter result by reducing the O(log n) term to O(1) (see also [Zim14] for a simpler proof).…”

Section: Introductionmentioning

confidence: 94%

“…The size of the list in [BMVZ13] is quadratic in n and in fact in the same paper it is shown that this is optimal because any effectively computed list that contains a program that is additively c close to optimal length must have size Ω(n 2 /(c + 1) 2 ) (for any c). The size of the list in the polynomial-time construction from [Teu14] is n 7+ǫ and [Zim14] improves it to O(n 6+ǫ ). We show here that the size of the list can be linear, thus beating the above quadratic lower bound, if we allow probabilistic computation, in fact even polynomial-time probabilistic computation.…”

Section: Introductionmentioning

confidence: 99%

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Linear List-Approximation for Short Programs (or the Power of a Few Random Bits)

Bauwens

Zimand

2014

2014 IEEE 29th Conference on Computational Complexity (CCC)

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A c-short program for a string x is a description of x of length at most C(x) + c, where C(x) is the Kolmogorov complexity of x. We show that there exists a randomized algorithm that constructs a list of n elements that contains a O(log n)-short program for x. We also show a polynomial-time randomized construction that achieves the same list size for O(log 2 n)-short programs. These results beat the lower bounds shown by Bauwens et al.[BMVZ13] for deterministic constructions of such lists. We also prove tight lower bounds for the main parameters of our result. The constructions use only O(log n) (O(log 2 n) for the polynomial-time result) random bits. Thus using only few random bits it is possible to do tasks that cannot be done by any deterministic algorithm regardless of its running time. * A preliminary version has been presented at the 29-th IEEE Conference on Computational Complexity, June 11-13, 2014, Vancouver, British Columbia, Canada.Bruno Bauwens is grateful to Mathieu Hoyrup and Université de Lorraine for financial support.

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Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Linear List-Approximation for Short Programs (or the Power of a Few Random Bits)

Bauwens

Zimand

2014

2014 IEEE 29th Conference on Computational Complexity (CCC)

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“…Quoting from [25]: "Given that the Kolmogorov complexity is not computable, it is natural to ask if given a string x it is possible to construct a short list containing a minimal (plus possibly a small overhead) description of x. Bauwens, Mahklin, Vereshchagin and Zimand [1] and Teutsch [21] show that, surprisingly, the answer is YES. Even more, in fact the short list can be computed in polynomial time.…”

Section: Short Listsmentioning

confidence: 99%

How Incomputable Is Kolmogorov Complexity?

Vitányi

2020

Entropy

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Kolmogorov complexity is the length of the ultimately compressed version of a file (that is, anything which can be put in a computer). Formally, it is the length of a shortest program from which the file can be reconstructed. We discuss the incomputability of Kolmogorov complexity, which formal loopholes this leaves us, recent approaches to compute or approximate Kolmogorov complexity, which approaches are problematic and which approaches are viable.

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“…Moreover, similar lists with slightly weaker parameters can actually be constructed in polynomial time. Teutsch [Teu], Zimand [Zim14] and Bauwens and Zimand [BZ14] have obtained polynomial-time constructions with improved parameters.…”

Section: Occam's Razor For Algorithmsmentioning

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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Short Lists with Short Programs in Short Time – A Short Proof

Abstract: Bauwens, Mahklin, Vereshchagin and Zimand [1] and Teutsch [5] have shown that given a string x it is possible to construct in polynomial time a list containing a short description of it. We simplify their technique and present a shorter proof of this result.

Cited by 13 publications

References 9 publications

Linear List-Approximation for Short Programs (or the Power of a Few Random Bits)

Linear List-Approximation for Short Programs (or the Power of a Few Random Bits)

How Incomputable Is Kolmogorov Complexity?

On Approximate Decidability of Minimal Programs

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