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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: It is shown that every finite lattices, and in fact every recursively presentable lattice, can be embedded in the r.e. tt-degrees by a map preserving least and greatest elements.
Abstract: It is shown that every finite lattice, and in fact every recursively presentable lattice, can be embedded in the r.e. tt-degrees by a map preserving least and greatest elements. The decidability of the I-quantifier theory of the Le. ttdegrees in the language with ~, v, /\\, 0, and 1 is obtained as a corollary. Introduction. A set A ~ w is truth table (tt) reducible to B ~ w (A ~ tt B) if answers to questions of the form \"n E A?\" are given by a finite Boolean combination, effectively determined from n, of answers to questions of the form \"k E B?\". Sets A and B are of the same tt-degree if A ~ It Band B ~ It A. We consider the structure consisting of the tt-degrees of recursively enumerable sets of natural numbers. Odifreddi [4] and Rogers [6] contain background information on the tt-degrees. In particular, Fejer and Shore [1] contains information about the r.e. tt-degrees and about questions relating to the decidability of the theory of the r.e. tt-degrees. They show there that every recursively presentable lattice can be embedded in the r.e. tt-degrees preserving least element. Using this, they show that the 3 theory of the r.e. tt-degrees in the language with ~ , V, 1\\,0 is decidable, and ask whether the 3 theory is still decidable when 1 is added to the language. This decidability question can be answered by determining which finite lattices can be embedded in the r.e. tt-degrees preserving least and greatest elements. Jockusch and Mohrherr [3] have shown that the diamond lattice, the pentagon lattice, and the I-n-llattices can be embedded preserving least and greatest elements, but leave open the general question, and even such special cases as the three generator Boolean algebra. The embedding used in their proof requires that the lattice in question have the property that no element which is the inf of a pair of incomparable elements of the lattice can be joined up to the 1 of the lattice (except by 1 itself). We show here that all finite lattices, and in fact all recursively presentable lattices, can be embedded in the r.e. tt-degrees preserving least and greatest elements (provided the lattice has distinct least and greatest elements). Our proof for the general lattices combines a generalization of the coding method used by Jockusch and Mohrherr with the strategy for preserving nonzero infs used by Fejer and Shore. We prove first, in §l, that every lattice with a finite representation (and distinct least and greatest elements) can be embedded, and then in §2 outline the modifications needed to embed a recursively presented (possibly infinite) lattice. As a Received by the editors December 26, 1985 and. in revised form. May 15. 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 03D30; Secondary 03D25. 515 ©1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

3 citations

Journal ArticleDOI
Aaron Lye1
TL;DR: In this paper, it was shown that context-dependent fusion grammars can be used to generate recursively enumerable string languages, where the application of fusion rules is restricted by positive and negative context conditions.
Abstract: Context-dependent fusion grammars were recently introduced as devices for the generation of hypergraph languages. In this paper, we show that this new type of hypergraph grammars, where the application of fusion rules is restricted by positive and negative context conditions, is a universal computation model. Our main result is that Turing machines can be transformed into these grammars such that the recognized language of the Turing machine and the generated language of the corresponding context-dependent fusion grammar coincide up to representation of strings as graphs. As a corollary we get that context-dependent fusion grammars can generate all recursively enumerable string languages.

3 citations

Journal Article
TL;DR: This work addresses self–organizing systems that compile to the scale–free small world model, and extends the conditions of dynamic team constitution in eco–grammar systems to capture the behaviour of agents participating in network cluster formation.
Abstract: The concept and the reality of self–organizing networks have come to pervade modern society. But what exactly is a self–organizing network? Scientists from a range of disciplines have been pursuing questions on the particularities of self–organizing networks. Our work addresses self–organizing systems that compile to the scale–free small world model. We model self–organizing networks at syntactical level as well as reveal some semantical and experimental aspects related to them. At syntactical level, we use devices from grammar systems theory: in grammar systems theory the agents are represented by grammars and the generated strings describe the behaviour of the system. At experimental level, we utilize the methods of selective learning and value estimation under evolutionary pressure. The selection is influenced by the ever changing external world and by the competing individuals. First, we model peer–to–peer networks with the aid of networks of parallel multiset string processors. We establish the connection between the growth of the number of strings being present during the computation at the components of these networks of parallel multiset string processors and the growth function of a developmental system. We formalize security rules that conform to self–organizing dynamic systems and allow intra– and intercommunity collaborations. Our approach guarantees quick and efficient local analysis of the security requirements, thus reducing the need for global verification. Secondly, we illustrate the great diversity of employing regulated rewriting devices in eco– grammar systems to describe the search strategy of Internet crawlers. We prove that if we ignore the aging of the web pages in the model, then systems with rather simple component grammars suffice to identify any recursively enumerable language. Whereas if the web pages may become obsolete, then the efficiency of the cooperation of the agents decreases considerably. We also examine the extent to which communication makes a goal– oriented community efficient in different graph topologies through simulations. Finally, we extend the conditions of dynamic team constitution in eco–grammar systems to capture the behaviour of agents participating in network cluster formation. From the language classes that these systems are capable of generating, we deduce the difficulty of the problem they can solve. Depending on the team constitution mode, different classes of languages can be obtained. For all self–organizing networks to be presented in this dissertation, we also propose some further research directions.

3 citations

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