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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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Journal ArticleDOI
TL;DR: This paper attempts to characterize the class of recursively enumerable languages with much smaller language classes than that of linear languages with the result that the follwing statement is obtained.

8 citations

Book ChapterDOI
10 Oct 1994
TL;DR: Using topological concepts in studies of the Gold paradigm of inductive inference are accumulation points, derived sets of order α (α — constructive ordinal) and compactness and Identifiability of a class U of total recursive functions with a bound α on the number of mindchanges implies U^{(\alpha + 1} = ot 0\).
Abstract: The paper deals with using topological concepts in studies of the Gold paradigm of inductive inference. They are — accumulation points, derived sets of order α (α — constructive ordinal) and compactness. Identifiability of a class U of total recursive functions with a bound α on the number of mindchanges implies \(U^{(\alpha + 1)} = ot 0\). This allows to construct counter-examples — recursively enumerable classes of functions showing the proper inclusion between identification types: EXα⊂EXα+1.

8 citations

Journal ArticleDOI
TL;DR: The goal of this paper is to solve Odifreddi's question by showing there exists a nonzero r.e. tt-degree, and to show that 0′ T is singular, and that the singular T-degrees are dense, but also to construct a nonsingular T-degree.
Abstract: In [1], Degtev constructed a nonzero r.e. tt-degree containing a single r.e. m -degree. It is not difficult to construct an r.e. tt-degree containing infinitely many r.e. m -degrees (Fischer [6]); indeed, in [3], the author constructed an r.e. tt-degree with no greatest r.e. m -degree. Odifreddi [12, Problem 10] asked if every r.e. tt-degree contains either one or infinitely many r.e. m -degrees. The goal of this paper is to solve Odifreddi's question by showing: Theorem. There exists a nonzero r.e. tt-degree containing exactly 3 r.e. m-degrees . This theorem can be extended to show that there exist r.e. tt-degrees with arbitrarily large finite numbers of r.e. m -degrees. We remark that save for the aforementioned results, very little is known about the structures that can be realized as the collection of r.e. m -degrees within an r.e. tt-degree. It seems conceivable that the methods of the present paper may be useful in, for example, embedding distributive (semi) lattices into such structures. In part II of this paper [4], we continue our analysis of r.e. m - and tt-degrees. We define an r.e. tt-degree to be singular if it contains a single r.e. m -degree, and an r.e. T-degree a to be singular if a contains a singular r.e. tt-degree. In [4] we study the distribution (in the r.e. T-degrees) of singular tt-degrees. We show that 0′ T is singular (solving a question of Odifreddi [11]), and that the singular T-degrees are dense, but also we construct a nonsingular T-degree. The techniques used for the first results extend those of §2 of the present paper.

8 citations

Book ChapterDOI
TL;DR: In this paper, an absolute version of Post's problem for 3 E is devised, and studied when certain recursively enumerable projecta are equal, with the aid of the notions of indexicality and ordinal recursiveness.
Abstract: The unresolved character of the power set operation stymies the solution of elementary problems arising in Kleene's theory of recursion in objects of finite type. E.g. Post's problem for 3 E has a positive solution if V=L (NORMANN, 1975), and a negative if AD holds. Let δ be the class of all sets R ⊆ 2ω such that R is recursive in 3 E, b for some real b . A forcing construction shows δ is not recursively enumerable in 3 E when there is a recursively regular well-ordering of 2″ recursive in 3 E. It follows that the concepts of Σ * , and weak Σ * , definability differ. With the aid of the notions of indexicality and ordinal recursiveness, an absolute version of Post's problem for 3 E is devised, and studied when certain recursively enumerable projecta are equal.

8 citations

Journal ArticleDOI
TL;DR: The variables are considered as ranging over nonnegative integers; but it is found more useful here to restrict the range to positive integers, no essential change being thereby introduced.
Abstract: Hilbert's tenth problem is to find an algorithm for determining whether or not a diophantine equation possesses solutions. A diophantine predicate (of positive integers) is defined to be one of the formwhere P is a polynomial with integral coefficients (positive, negative, or zero). Previous work has considered the variables as ranging over nonnegative integers; but we shall find it more useful here to restrict the range to positive integers, no essential change being thereby introduced. It is clear that the recursive unsolvability of Hilbert's tenth problem would follow if one could show that some non-recursive predicate were diophantine. In particular, it would suffice to show that every recursively enumerable predicate is diophantine. Actually, it would suffice to prove far less.

8 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20237
202220
202127
202022
201918
201823