Journal ArticleDOI
Arithmetical representations of enumerable sets with a small number of quantifiers
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This article is published in Journal of Mathematical Sciences.The article was published on 1976-10-01. It has received 8 citations till now. The article focuses on the topics: Arithmetical set & Recursively enumerable language.read more
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Universal diophantine equation
TL;DR: Matijasevic's theorem implies the existence of a diophantine equation U such that for all x and v, x ∈ W v is also recursively enumerable, and the nonexistence of such an algorithm follows immediately from theexistence of r.e. nonrecursive sets.
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Extensions of Hilbert's tenth problem
TL;DR: In this paper, the authors present an attempt to bridge the gap between the researchers that work in the areas adjacent to Hilbert's Tenth Problem (for short, HTP), mainly, number theory and mathematical logic.
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What can and cannot be done with Diophantine problems
TL;DR: In this paper, the authors present various theorems (obtained mainly by specialists in mathematical logic and computability theory) stating the impossibility of algorithms for solving certain Diophantine problems.
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Three universal representations of recursively enumerable sets
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.
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Journal ArticleDOI
Arithmetical problems and recursively enumerable predicates
TL;DR: In this article, it was shown that every recursively enumerable predicate is of the form (1) and conversely, every predicate of (1, √ √ n) is recurvably enumerable.
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Diophantine representation of enumerable predicates
TL;DR: In this article, it was shown that every enumerable predicate is diophantine, which implies that Hilbert's tenth problem is algorithmically unsolvable, and that every predicate has exponential growth.
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Diophantine representation of enumerable predicates
TL;DR: In this paper, the author's abstract of dissertation in partial fulfillment of the requirements for the degree of Doctor of Physicomathematical Sciences was defended at a session of the faculty council of the V. A. Steklov Mathematics Institute, Academy of Sciences of the SSSR.