Short lists with short programs in short time - a short proof

Bauwens, Mahklin, Vereshchagin and Zimand [ECCC TR13-007] and Teutsch [arxiv:1212.6104] 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

Short lists for shortest descriptions in short time

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)

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. \cite{bmvz:c:shortlist} 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

How incomputable is Kolmogorov complexity?

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 incomputabilty 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.Comment: 9 pages LaTe

On approximate decidability of minimal programs

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

List Approximation for Increasing Kolmogorov Complexity

It is impossible to effectively modify a string in order to increase its Kolmogorov complexity. But is it possible to construct a few strings, not longer than the input string, so that most of them have larger complexity? We show that the answer is yes. We present an algorithm that on input a string x of length n returns a list with O(n^2) many strings, all of length n, such that 99% of them are more complex than x, provided the complexity of x is less than n. We obtain similar results for other parameters, including a polynomial-time construction