Short lists for shortest descriptions in short time

Abstract

Is it possible to find a shortest description for a binary string? The well-known answer is "no, Kolmogorov complexity is not computable." Faced with this barrier, one might instead seek a short list of candidates which includes a laconic description. Remarkably such approximations exist. This paper presents an efficient algorithm which generates a polynomial-size list containing an optimal description for a given input string. Along the way, we employ expander graphs and randomness dispersers to obtain an Explicit Online Matching Theorem for bipartite graphs and a refinement of Muchnik's Conditional Complexity Theorem. Our main result extends recent work by Bauwens, Mahklin, Vereschchagin, and Zimand.

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

“…In view of these strongly negative facts, the recent results of Bauwens, Mahklin, Vereshchagin and Zimand [BMVZ13] and Teutsch [Teu14] are surprising. They show that it is possible to effectively construct a short list guaranteed to contain a close-to-optimal program for x.…”

“…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).…”

“…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.…”

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

“…Given that the Kolmogorov complexity is not computable, it is natural to ask if given a string x it is posible to construct a short list containing a minimal (+ small overhead) description of x. Bauwens, Mahklin, Vereshchagin and Zimand [1] and Teutsch [5] show that, surprisingly, the answer is YES. Even more, in fact the short list can be computed in polynomial time.…”

“…

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.

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