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Kolmogorov complexity and the Recursion Theorem. Manuscript, submitted for publication
, 2005
"... Abstract. Several classes of diagonally nonrecursive (DNR) functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers A can wttcompute a DNR function iff there is a nontrivial recursive lower bound on the Kolmogorov complexity of the initial segments of ..."
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Abstract. Several classes of diagonally nonrecursive (DNR) functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers A can wttcompute a DNR function iff there is a nontrivial recursive lower bound on the Kolmogorov complexity of the initial segments of A. Furthermore, A can Turing compute a DNR function iff there is a nontrivial Arecursive lower bound on the Kolmogorov complexity of the initial segements of A. A is PAcomplete, that is, A can compute a {0, 1}valued DNR function, iff A can compute a function F such that F (n) is a string of length n and maximal Ccomplexity among the strings of length n. A ≥T K iff A can compute a function F such that F (n) is a string of length n and maximal Hcomplexity among the strings of length n. Further characterizations for these classes are given. The existence of a DNR function in a Turing degree is equivalent to the failure of the Recursion Theorem for this degree; thus the provided results characterize those Turing degrees in terms of Kolmogorov complexity which do no longer permit the usage of the Recursion Theorem. 1.
Computability and analysis: the legacy of Alan Turing
, 2012
"... For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems, ” which assert that a construction can be carried out to meet a given specification, and “theorems, ” which assert that some property holds of a par ..."
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For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems, ” which assert that a construction can be carried out to meet a given specification, and “theorems, ” which assert that some property holds of a particular geometric
Effectiveness for embedded spheres and balls Abstract
"... We consider arbitrary dimensional spheres and closed balls embedded in R n as Π 0 1 classes. Such a strong restriction on the topology of a Π0 1 class has computability theoretic repercussions. Algebraic topology plays a crucial role in our exploration of these consequences; the use of homology chai ..."
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We consider arbitrary dimensional spheres and closed balls embedded in R n as Π 0 1 classes. Such a strong restriction on the topology of a Π0 1 class has computability theoretic repercussions. Algebraic topology plays a crucial role in our exploration of these consequences; the use of homology chains as computational objects allows us to take algorithmic advantage of the topological structure of our Π0 1 classes. We show that a sphere embedded as a Π0 1 class is necessarily located, i.e., the distance to the class is a computable function, or equivalently, the class contains a computably enumerable dense set of computable points. Similarly, a ball embedded as a Π0 1 class has a dense set of computable points, though not necessarily c.e. To prove location for balls, it is sufficient to assume that both it and its boundary sphere are Π0 1. However, the converse fails, even for arcs; using a priority argument, we prove that there is a located arc in R2 without computable endpoints. Finally, the requirement that the embedding map itself be computable is shown to be stronger than the other effectiveness criteria considered. A characterization in terms of computable local contractibility is stated; the proof will be the subject of a sequel. 1
mlq header will be provided by the publisher Singular Coverings and NonUniform Notions of Closed Set Computability
, 2007
"... Key words cor.e. closed sets, nonuniform computability, connected component ..."
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Key words cor.e. closed sets, nonuniform computability, connected component