In Ramsey's Theorem and recursion theory, Theorem 4.2, Jockusch proved that for any computable k-coloring of pairs of integers, there is an infinite Pi(0)(2) homogeneous set. The proof used a countable collection ...
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In Ramsey's Theorem and recursion theory, Theorem 4.2, Jockusch proved that for any computable k-coloring of pairs of integers, there is an infinite Pi(0)(2) homogeneous set. The proof used a countable collection of Pi(0)(2) sets as potential infinite homogeneous sets. In a remark preceding the proof, Jockusch stated without proof that it can be shown that there is no computable way to prove this result with a finite number of Pi(0)(2) sets. We provide a proof of this claim, showing that there is no computable way to take an index for an arbitrary computable coloring and produce a finite number of indices of Pi(0)(2) sets with the property that one of those sets will be homogeneous for that coloring. While proving this result, we introduce n-trains as objects with useful combinatorial properties which can be used as approximations to infinite Pi(0)(2) sets.
The present work investigates inductive inference from the perspective of reverse mathematics. Reverse mathematics is a framework that allows gauging the proof strength of theorems and axioms in many areas of mathemat...
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The present work investigates inductive inference from the perspective of reverse mathematics. Reverse mathematics is a framework that allows gauging the proof strength of theorems and axioms in many areas of mathematics. The present work applies its methods to basic notions of algorithmic learning theory such as Angluin's tell-tale criterion and its variants for learning in the limit and for conservative learning, as well as to the more general scenario of partial learning. These notions are studied in the reverse mathematics context for uniformly and weakly represented families of languages. The results are stated in terms of axioms referring to induction strength and to domination of weakly represented families of functions. (C) 2016 Elsevier B.V. All rights reserved.
In this paper, the notions of F-alpha-categorical and G(alpha)-categorical structures are introduced by choosing the isomorphism Such that the function itself or its graph sits on the alpha-th level of the Ershov hier...
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In this paper, the notions of F-alpha-categorical and G(alpha)-categorical structures are introduced by choosing the isomorphism Such that the function itself or its graph sits on the alpha-th level of the Ershov hierarchy, respectively. Separations obtained by natural graphs which are the disjoint unions of countably many finite graphs. Furthermore, for size-bounded graphs, an easy criterion is given to say when it is computable-categorical and when it is only G(2)-categorical;in the latter case it is not F-alpha-categorical for any recursive ordinal U. (C) 2008 Elsevier B.V. All rights reserved.
Identification by algorithmic devices of programs for computable functions from their graphs is a well studied problem in learning theory. Freivalds and Chen consider identification of ''minimal'' and ...
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Identification by algorithmic devices of programs for computable functions from their graphs is a well studied problem in learning theory. Freivalds and Chen consider identification of ''minimal'' and ''nearly minimal'' programs for functions from their graphs. The present paper solves the following question left open by Chen: Is it the case that for any collection of computable functions, C, such that some machine can finitely learn a nearly minimal (n + 1)-error program for every function in C, there exists another machine that can learn in the limit an n-error program (which need not be nearly minimal) for every function in C? We answer this question negatively.
recursion theory on the reals, the analog counterpart of recursive function theory, is an approach to continuous-time computation inspired by the models of Classical Physics. In recursion theory on the reals, the disc...
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recursion theory on the reals, the analog counterpart of recursive function theory, is an approach to continuous-time computation inspired by the models of Classical Physics. In recursion theory on the reals, the discrete operations of standard recursion theory are replaced by operations on continuous functions such as composition and various forms of differential equations like indefinite integrals, linear differential equations and more general Cauchy problems. We define classes of real recursive functions in a manner similar to the standard recursion theory and we study their complexity. We prove both upper and lower bounds for several classes of real recursive functions, which lie inside the elementary functions, and can be characterized in terms of space complexity. In particular, we show that hierarchies of real recursive classes closed under restricted integration operations are related to the exponential space hierarchy. The results in this paper, combined with earlier results, suggest that there is a close connection between analog complexity classes and subrecursive classes, at least in the region between FLINSPACE and the primitive recursive functions. (C) 2003 Elsevier B.V. All rights reserved.
Identification of grammars (r.e. indices) for recursively enumerable languages from positive data by algorithmic devices is a well-studied problem in learning theory. The present paper considers identification of r.e....
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Identification of grammars (r.e. indices) for recursively enumerable languages from positive data by algorithmic devices is a well-studied problem in learning theory. The present paper considers identification of r.e. languages by machines that have access to membership oracles for noncomputable sets. It is shown that for any set A there exists another set B such that the collections of re. languages that can be identified by machines with access to a membership oracle for B is strictly larger than the collections of r.e. languages that can be identified by machines with access to a membership oracle for A. In other words, there is no maximal inference degree for language identification.
We show that there is a computable procedure which, given an for all there exists-sentence phi in the language of the partially ordered sets with a top element 1 and a bottom element 0, computes whether phi is true in...
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We show that there is a computable procedure which, given an for all there exists-sentence phi in the language of the partially ordered sets with a top element 1 and a bottom element 0, computes whether phi is true in the Medvedev degrees of Pi(0)(1) classes in Cantor space, sometimes denoted by P-s.
In this paper we continue our work of Kalantari and Welch (1998). There we introduced machinery to produce a point-free approach to points and functions on topological spaces and found conditions for both which lend t...
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In this paper we continue our work of Kalantari and Welch (1998). There we introduced machinery to produce a point-free approach to points and functions on topological spaces and found conditions for both which lend themselves to effectivization. While we studied recursive points in that paper, here, we present two useful classes of recursive functions on topological spaces, apply them to the reals, and find precise accounting for the nature of the properties of some examples that exist in the literature. We end with a construction of a recursive function on a small subset of the unit interval which is strongly nonextendible. (C) 1999 Elsevier Science B.V. All rights reserved.
We establish the decidability of the Sigma(2) theory of both the arithmetic and hyperarithmetic degrees in the language of uppersemilattices, i.e., the language with <=, 0, and sic. This is achieved by using Kumabe...
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We establish the decidability of the Sigma(2) theory of both the arithmetic and hyperarithmetic degrees in the language of uppersemilattices, i.e., the language with <=, 0, and sic. This is achieved by using Kumabe-Slaman forcing, along with other known results, to show given finite uppersemilattices M and N, where M is a subuppersemilattice of N, that every embedding of M into either degree structure extends to one of N iff N is an end-extension of M.
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