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SHORT LISTS FOR SHORTEST DESCRIPTIONS IN SHORT TIME
, 2013
"... Abstract. Is it possible to find a shortest description for a binary string? The wellknown 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. ..."
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Abstract. Is it possible to find a shortest description for a binary string? The wellknown 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 polynomialsize 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.
On approximate decidability of minimal programs
, 2014
"... An index e in a numbering of partialrecursive 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 invest ..."
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An index e in a numbering of partialrecursive 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 nonminimal. 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.