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Benign cost functions and lowness properties
"... Abstract. We show that the class of strongly jumptraceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of wellbehaved cost function, called benign. This characterisation implies the containment of the class of strongly jumptraceable c.e. ..."
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Abstract. We show that the class of strongly jumptraceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of wellbehaved cost function, called benign. This characterisation implies the containment of the class of strongly jumptraceable c.e. Turing degrees in a number of lowness classes, in particular the classes of the degrees which lie below incomplete random degrees, indeed all LRhard random degrees, and all ωc.e. random degrees. The last result implies recent results of Diamondstone’s and Ng’s regarding cupping with supwerlow c.e. degrees and thus gives a use of algorithmic randomness in the study of the c.e. Turing degrees. 1.
Short lists with short programs in short time
"... Given a machine U, a cshort program for x is a string p such that U(p) = x and the length of p is bounded by c + (the length of a shortest program for x). We show that for any universal machine, it is possible to compute in polynomial time on input x a list of polynomial size guaranteed to contai ..."
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Cited by 6 (5 self)
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Given a machine U, a cshort program for x is a string p such that U(p) = x and the length of p is bounded by c + (the length of a shortest program for x). We show that for any universal machine, it is possible to compute in polynomial time on input x a list of polynomial size guaranteed to contain a O(logx)short program for x. We also show that there exist computable functions that map every x to a list of size O(x  2) containing a O(1)short program for x and this is essentially optimal because we prove that such a list must have size Ω(x  2). Finally we show that for some machines, computable lists containing a shortest program must have length Ω(2 x ).
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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Cited by 2 (1 self)
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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.