Topic
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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01 Jan 2002
TL;DR: Two aspects of the jump are discussed, which are implicitly characterized by general properties of relative definability, and the Shore and Slaman theorem that the function x 7→ x′ is first order definable in the Turing degrees is presented.
Abstract: X ′ is the canonical example of a set which is definable from X but not recursive in X. The Turing degree of X ′ depends only on the Turing degree of X, so the jump induces an increasing function on the Turing degrees D. In this paper, we will discuss two aspects of the jump and its iterations. First, we will show that they are implicitly characterized by general properties of relative definability. Second, we will present the Shore and Slaman [1999] theorem that the function x 7→ x′ is first order definable in the Turing degrees. Finally, we will pose analogous questions about the relation y is recursively enumerable in x and discuss what is known about them. Our discussion will rest on two technical facts, which are generalizations of the following two theorems.
11 citations
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TL;DR: This paper shows that the algebra Pℛ of primitive recursive functions over the natural numbers has a recursive equational specification under second order initial algebra semantics, and it follows that higher orderinitial algebra specifications are strictly more powerful than first order initialgebra specifications.
Abstract: Theoretical results on the scope and limits of first order algebraic specifications can be used to show that certain natural algebras have no recursively enumerable equational specification under first order initial algebra semantics. A well known example is the algebraP? of primitive recursive functions over the natural numbers. In this paper we show thatP? has a recursive equational specification under second order initial algebra semantics. It follows that higher order initial algebra specifications are strictly more powerful than first order initial algebra specifications.
11 citations
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TL;DR: It is shown that using generators of the length at most two, context sensitive and recursively enumerable languages can be characterized in a natural manner.
Abstract: The notion of a (direct) derivation is introduced on word monoids generated by finite languages over total vocabularies of context free grammars. It is shown that using generators of the length at most two, context sensitive and recursively enumerable languages can be characterized in a natural manner.
11 citations
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03 Jul 2017TL;DR: It is shown that whenever GCID systems describe \(\mathrm {LIN}\) with t components, this can be extended toGCID systems with just one more component to describe, for instance, 2-\(\Mathrm { LIN}\) and with further addition of one more components, the rational closure of \(\mathRM {LIN}\).
Abstract: A regulated extension of an insertion-deletion system known as graph-controlled insertion-deletion (GCID) system has several components and each component contains some insertion-deletion rules. A rule is applied to a string in a component and the resultant string is moved to the target component specified in the rule. When resources are so limited (especially, when deletion is context-free) then GCID systems are not known to describe the class of recursively enumerable languages. Hence, it becomes interesting to find the descriptional complexity of such GCID systems of small sizes with respect to language classes below \(\mathrm {RE}\). To this end, we consider closure classes of linear languages. We show that whenever GCID systems describe \(\mathrm {LIN}\) with t components, we can extend this to GCID systems with just one more component to describe, for instance, 2-\(\mathrm {LIN}\) and with further addition of one more component, we can extend to GCID systems that describe the rational closure of \(\mathrm {LIN}\).
11 citations
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TL;DR: The main result is that there exists a reducing operator δ with the following property: A special generalized grammar can be reduced to a special grammar by means of a reducing operators if and only if it can be reduction to aSpecial generalized grammar by Means of δ.
Abstract: Recursively enumerable languages can be constructed on the basis of grammatizable languages which are generated by the so-called special grammars. If certain finiteness conditions are omitted in the definition of a special grammar, a special generalized grammar is defined. Any language is generated by a special generalized grammar in a trivial way. Reducing operators are studied which assign a smaller special generalized grammar to each special generalized grammar in such a way that both generate the same language. The main result is that there exists a reducing operator δ with the following property: A special generalized grammar can be reduced to a special grammar by means of a reducing operator if and only if it can be reduced to a special grammar by means of δ .
11 citations