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Recursively enumerable language
About: Recursively enumerable language is a research topic. Over the lifetime, 1508 publications have been published within this topic receiving 32382 citations.
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TL;DR: In an earlier paper, the authors investigated Friedberg splittings of maximal sets and showed that they formed an orbit with very interesting degree-theoretical properties.
10 citations
TL;DR: The structure of recursively enumerable degrees is not a Σ1-elementary substructure of , where (n > 1) is the structure of n-r in the Ershov hierarchy.
Abstract: We show that the structure M of recursively enumerable degrees is not a Ei-elementary substructure of 3f?, where 9f? (n > 1) is the structure of n-r.e. degrees in the Ershov hierarchy. ?
10 citations
TL;DR: In this paper, the authors discuss decision problems of arbitrarily preassigned recursively enumerable degree of unsolvability and decision problems concerned with recognizing that a system of a certain kind enjoys a specified property or with two systems of a different kind are in a specified relation.
Abstract: Publisher Summary This chapter discusses decision problems of arbitrarily preassigned recursively enumerable degree of unsolvability and decision problems concerned with recognizing that a system of a certain kind enjoys a specified property or with recognizing that two systems of a certain kind are in a specified relation The latter decision problems are also known as “global decision problems” The chapter also discusses several meta-decision problems about groups that are formed by compounding the kind of “first order” decision problems A recursive class of finite presentations of groups is always given by means of some generic presentation depending on a single parameter ranging over some recursive set of words Finally, the chapter presents several unsolvability results regarding the homomorphic images of a fixed finitely presented group
10 citations
TL;DR: It is proved that a relation over Fq[Z] is recursively enumerable if and only if it is Diophantine over F q[W,Z], where n is represented by Zn.
10 citations
TL;DR: This article constructs an explicit undecidable arithmetical formula, F(x, n) , in prenex normal form, which is explicit in the sense that it is written out in its entirety with no abbreviations and can be focused into Godel's Incompleteness Theorem.
Abstract: In his celebrated paper of 1931 [7], Kurt Godel proved the existence of sentences undecidable in the axiomatized theory of numbers Godel's proof is constructive and such a sentence may actually be written out Of course, if we follow Godel's original procedure the formula will be of enormous lengthForty-five years have passed since the appearance of Godel's pioneering work During this time enormous progress has been made in mathematical logic and recursive function theory Many different mathematical problems have been proved recursively unsolvable Theoretically each such result is capable of producing an explicit undecidable number theoretic predicate We have only to carry out a suitable arithmetization Until recently, however, techniques were not available for carrying out these arithmetizations with sufficient efficiencyIn this article we construct an explicit undecidable arithmetical formula, F(x, n), in prenex normal form The formula is explicit in the sense that it is written out in its entirety with no abbreviations The formula is undecidable in the recursive sense that there exists no algorithm to decide, for given values of n, whether or not F(n, n) is true or false Moreover F(n, n) is undecidable in the formal (axiomatic) sense of Godel [7] Given any of the usual axiomatic theories to which Godel's Incompleteness Theorem applies, there exists a value of n such that F(n, n) is unprovable and irrefutable Thus Godel's Incompleteness Theorem can be “focused” into the formula F(n, n) Thus some substitution instance of F(n, n) is undecidable in Peano arithmetic, ZF set theory, etc
10 citations