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Switch to: References
Citations of:
Theory of Recursive Functions and Effective Computability
Journal of Symbolic Logic 36 (1):141-146 (1971)
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Dominical categories are categories in which the notions of partial morphisms and their domains become explicit, with the latter being endomorphisms rather than subobjects of their sources. These categories form the basis for a novel abstract formulation of recursion theory, to which the present paper is devoted. The abstractness has of course its usual concomitant advantage of generality: it is interesting to see that many of the fundamental results of recursion theory remain valid in contexts far removed from their classic (...) |
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It is shown that many different problems have the same degree of unsolvability. Among these problems are: THE INDUCTIVE INFERENCE PROBLEM. Infer in the limit an index for a recursive function f presented as f(0), f(1), f(2),.... THE RECURSIVE INDEX PROBLEM. Decide in the limit if i is the index of a total recursive function. THE ZERO NONVARIANT PROBLEM. Decide in the limit if a recursive function f presented as f(0), f(1), f(2),... has value unequal to zero for infinitely many (...) |
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Subcreative sets, introduced by Blum, are known to coincide with the effectively speedable sets. Subcreative sets are shown to be the complete sets with respect to S-reducibility, a special case of Turing reducibility. Thus a set is effectively speedable exactly when it contains the solution to the halting problem in an easily decodable form. Several characterizations of subcreative sets are given, including the solution of an open problem of Blum, and are used to locate the subcreative sets with respect to (...) |
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Osvald Demuth studied constructive analysis from the viewpoint of the Russian school of constructive mathematics. In the course of his work he introduced various notions of effective null set which, when phrased in classical language, yield a number of major algorithmic randomness notions. In addition, he proved several results connecting constructive analysis and randomness that were rediscovered only much later.In this paper, we trace the path that took Demuth from his constructivist roots to his deep and innovative work on the (...) |
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Many of our concepts are introduced to us via, and seem only to be constrained by, roughand-ready explanations and some sample paradigm positive and negative applications. This happens even in informal logic and mathematics. Yet in some cases, the concepts in question – although only informally and vaguely characterized – in fact have, or appear to have, entirely determinate extensions. Here’s one familiar example. When we start learning computability theory, we are introduced to the idea of an algorithmically computable function (...) |
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We discuss the interplay between the axiomatic and the semantic approach to truth. Often, semantic constructions have guided the development of axiomatic theories and certain axiomatic theories have been claimed to capture a semantic construction. We ask under which conditions an axiomatic theory captures a semantic construction. After discussing some potential criteria, we focus on the criterion of ℕ-categoricity and discuss its usefulness and limits. |
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We provide an introduction to methods and recent results on infinitely generated abelian groups with decidable word problem. |
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It is proved that if 1 $\langle \mathscr{D}_{2n}, \leq, P\rangle$ and $\langle \mathscr{D}_{2n}, \leq, P\rangle$ are not elementary equivalent where P is the predicate P(a) = "a is a Π 0 1 e-degree". |
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A set A is m-reducible to B if and only if there is a polynomial-time computable function f such that, for all x, x∈ A if and only if f ∈ B. Two sets are: 1-equivalent if and only if each is m-reducible to the other by one-one reductions; p-invertible equivalent if and only if each is m-reducible to the other by one-one, polynomial-time invertible reductions; and p-isomorphic if and only if there is an m-reduction from one set to the (...) |
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