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# Recursively enumerable language

About: Recursively enumerable language is a(n) research topic. Over the lifetime, 1508 publication(s) have been published within this topic receiving 32382 citation(s).

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Romanian Academy

^{1}TL;DR: It is proved that the P systems with the possibility of objects to cooperate characterize the recursively enumerable sets of natural numbers; moreover, systems with only two membranes suffice.

Abstract: We introduce a new computability model, of a distributed parallel type, based on the notion of a membrane structure. Such a structure consists of several cell-like membranes, recurrently placed inside a unique “skin” membrane. A plane representation is a Venn diagram without intersected sets and with a unique superset. In the regions delimited by the membranes there are placed objects. These objects are assumed to evolve: each object can be transformed in other objects, can pass through a membrane, or can dissolve the membrane in which it is placed. A priority relation between evolution rules can be considered. The evolution is done in parallel for all objects able to evolve. In this way, we obtain a computing device (we call it a P system): start with a certain number of objects in a certain membrane and let the system evolve; if it will halt (no object can further evolve), then the computation is finished, with the result given as the number of objects in a specified membrane. If the development of the system goes forever, then the computation fails to have an output. We prove that the P systems with the possibility of objects to cooperate characterize the recursively enumerable sets of natural numbers; moreover, systems with only two membranes suffice. In fact, we do not need cooperating rules, but we only use catalysts, specified objects which are present in the rules but are not modified by the rule application. One catalyst suffices. A variant is also considered, with the objects being strings over a given alphabet. The evolution rules are now based on string transformations. We investigate the case when either the rewriting operation from Chomsky grammars (with respect to context-free productions) or the splicing operation from H systems investigated in the DNA computing is used. In both cases, characterizations of recursively enumerable languages are obtained by very simple P systems: with three membranes in the rewriting case and four in the splicing case. Several open problems and directions for further research are formulated

2,182 citations

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TL;DR: In this paper, the relation of the structure of an R set to its degree is discussed, and the infinite injury priority method is proposed to solve the problem of scaling and splitting R sets.

Abstract: TABLE OF CONTENTS Introduction Chapter I. The relation of the structure of an r.e. set to its degree. 1. Post's program and simple sets. 2. Dominating functions and quotient lattices. 3. Maximal sets and high degrees. 4. Low degrees, atomless sets, and invariant degree classes. 5. Incompleteness and completeness for noninvariant properties. Chapter II. The structure, automorphisms, and elementary theory of the r.e. sets. 6. Basic facts and splitting theorems. 7. Hh-simple sets. 8. Major subsets and r-maximal sets. 9. Automorphisms of &. 10. The elementary theory of S. Chapter III. The structure of the r.e. degrees. 11. Basic facts. 12. The finite injury priority method. 13. The infinite injury priority method. 14. The minimal pair method and lattice embeddings in R. 15. Cupping and splitting r.e. degrees. 16. Automorphisms and decidability of R.

1,873 citations

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22 Apr 1987TL;DR: In this paper, the authors discuss related theories of recursively enumerable sets, degree of un-solvability and turing degrees in particular, and generalizations of recursion theory.

Abstract: Central concerns of the book are related theories of recursively enumerable sets, of degree of un-solvability and turing degrees in particular. A second group of topics has to do with generalizations of recursion theory. The third topics group mentioned is subrecursive computability and subrecursive hierarchies

1,778 citations

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01 Jan 1989

TL;DR: Theories of Recursive functions, Hierarchies of recursive functions, and Arithmetical sets: Recursively enumerable sets.

Abstract: Preface. Introduction. Theories of Recursive functions. Hierarchies of recursive functions. Recursively enumerable sets. Recursively enumerable degrees. Limit sets. Arithmetical sets. Arithmetical degrees. Enumeration degrees. Bibliography. Notation index. Subject index.

1,046 citations

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TL;DR: This paper considers classes whose elements are re-cursively enumerable sets of non-negative integers whose properties are complete recursive enumerability and complete recursiveness.

Abstract: 1. Introduction. In this paper we consider classes whose elements are re-cursively enumerable sets of non-negative integers. No discussion of recur-sively enumerable sets can avoid the use of such classes, so that it seems desirable to know some of their properties. We give our attention here to the properties of complete recursive enumerability and complete recursiveness (which may be intuitively interpreted as decidability). Perhaps our most interesting result (and the one which gives this paper its name) is the fact that no nontrivial class is completely recursive. We assume familiarity with a paper of Kleene [5](2), and with ideas which are well summarized in the first sections of a paper of Post Í7].

677 citations