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Showing papers on "Recursively enumerable language published in 2001"


Journal ArticleDOI
TL;DR: It is shown that the converse implication is true: any Ω-like real in the unit interval is the halting probability of a universal self-delimiting Turing machine.

82 citations


Journal ArticleDOI
Gheorghe Paun1
TL;DR: This paper introduces to the reader the main ideas of computing with membranes, a recent branch of (theoretical) molecular computing, in a cell-like system, where multisets of objects evolve according to given rules in the compartments defined by a membrane structure and compute natural numbers as the result of halting sequences of transitions.
Abstract: The aim of this paper is to introduce to the reader the main ideas of computing with membranes, a recent branch of (theoretical) molecular computing. In short, in a cell-like system, multisets of objects evolve according to given rules in the compartments defined by a membrane structure and compute natural numbers as the result of halting sequences of transitions. The model is parallel, nondeterministic. Many variants have already been considered and many problems about them were investigated. We present here some of these variants, focusing on two central classes of results: (1) characterizations of the recursively enumerable sets of numbers and (2) possibilities to solve NP-complete problems in polynomial — even linear — time (of course, by making use of an exponential space). The results are given without proofs. An almost complete bibliography of the domain, at the middle of October 2000, is also provided.

76 citations


Book ChapterDOI
23 May 2001
TL;DR: It is shown that the number of non-terminal symbols used in the appearance checking mode can be restricted to two, and in the case of graph controlled (and programmed grammars) with appearance checking this number can be reduced to three.
Abstract: We improve the results elaborated in [6] on the number of non-terminal symbols needed in matrix grammars, programmed grammars, and graph-controlled grammars with appearance checking for generating arbitrary recursively enumerable languages. Of special interest is the result that the number of non-terminal symbols used in the appearance checking mode can be restricted to two. In the case of graph controlled (and programmed grammars) with appearance checking also the number of non-terminal symbols can be reduced to three (and four, respectively); in the case of matrix grammars with appearance checking we either need four non-terminal symbols with three of them being used in the appearance checking mode or else again we only need two nonterminal symbols being used in the appearance checking mode, but in that case we cannot bound the total number of non-terminal symbols.

75 citations


Journal ArticleDOI
TL;DR: A general theorem is proved showing that in many cases two‐dimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders are undecidable, and a sufficient condition for such products to be not recursively enumerable is proved.
Abstract: We study two‐dimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4.3, S4.3, GL.3, Grz.3, or the logic determined by the Cartesian square of any infinite linear order. This theorem solves a number of open problems posed by Gabbay and Shehtman. We also prove a sufficient condition for such products to be not recursively enumerable and give a simple axiomatization for the square K4.3 × K4.3 of the minimal liner logic using non‐structural Gabbay‐type inference rules.

69 citations


Journal ArticleDOI
TL;DR: In this paper, a new approach to the construction of implicit operations is presented, where the projection of idempotents back to one-dimensional spaces produces implicit operations with interesting properties, which can be used to deduce from results of Ribes and Zalesskiiˇ, Margolis, Sapir and Weil, and Steinberg that p-groups are tame.
Abstract: This work gives a new approach to the construction of implicit operations. By considering higher-dimensional spaces of implicit operations and implicit operators between them, the projection of idempotents back to one-dimensional spaces produces implicit operations with interesting properties. Besides providing a wealth of examples of implicit operations which can be obtained by these means, it is shown how they can be used to deduce from results of Ribes and Zalesskiiˇ, Margolis, Sapir and Weil, and Steinberg that the pseudovariety of p-groups is tame. More generally, for a recursively enumerable extension closed pseudovariety of groups V, if it can be decided whether a finitely generated subgroup of the free group with the pro-V topology is dense, then V is tame.

44 citations


Journal ArticleDOI
23 May 2001
TL;DR: It is shown that, in the case of context-free programmed grammars with appearance checking working under free derivations, three nonterminals are enough to generate every recursively enumerable language.
Abstract: We show that, in the case of context-free programmed grammars with appearance checking working under free derivations, three nonterminals are enough to generate every recursively enumerable language. This improves the previously published bound of eight for the nonterminal complexity of these grammars. This also yields an improved nonterminal complexity bound of four for context-free matrix grammars with appearance checking. Moreover, we establish nonterminal complexity bounds for context-free programmed and matrix grammars working under leftmost derivations.

37 citations


Journal ArticleDOI
TL;DR: It is proved that the recursively enumerable languages can be generated by systems with arbitrarily many membranes and bounded energy; when bounding the number of membranes and leaving free the quantity of energy associated with each rule, this feature is rather powerful.
Abstract: We consider P systems where each evolution rule “Produces” or “Consumes” some quantity of energy, in amounts which are expressed as integer numbers. In each moment and in each membrane the total energy involved in an evolution step should be positive, but if “Soo much” energy is present in a membrane, then the membrane will be destroyed (dissolved). We show that this feature is rather powerful. In the case of multisets of symbol-objects we find that systems with two membranes and arbitrary energy associated with rules, or with arbitrarily many membranes and a bounded energy associated with rules characterize the recursively enumerable sets of vectors of natural numbers (catalysts and priorities are used). In the case of string-objects we have only proved that the recursively enumerable languages can be generated by systems with arbitrarily many membranes and bounded energy; when bounding the number of membranes and leaving free the quantity of energy associated with each rule we have only generated all ma...

33 citations



Journal Article
TL;DR: It is proved that this regulation has no effect on the power of pushdown automata if the control languages are regular, however, the push down automata regulated by linear control languages characterize the family of recursively enumerable languages.
Abstract: The present paper suggests a new investigation area of the formal language theory—regulated automata Specifically, it investigates pushdown automata that regulate the use of their rules by control languages It proves that this regulation has no effect on the power of pushdown automata if the control languages are regular However, the pushdown automata regulated by linear control languages characterize the family of recursively enumerable languages All these results are established in terms of (A) acceptance by final state, (B) acceptance by empty pushdown, and (C) acceptance by final state and empty pushdown In its conclusion, this paper formulates several open problems

24 citations


Book ChapterDOI
01 Feb 2001
TL;DR: This paper shows, for any recursively enumerable language, how to construct a time-varying distributed H-system of degree 2 that generates that language exactly, and indicates that such a construction is impossible for time-Varying distributing H- system of degree 1.
Abstract: A time-varying distributed H system is a splicing system which has the following feature: at different moments one uses different sets of splicing rules. The number of these sets is called the degree of the system. The passing from one set of rules to another is specified in a cycle. It is known that any formal language can be generated by a time-varying distributed H-system of degree at least 4. We already proved that there are universal time-varying distributed H-systems of degree 2. In this paper we strengthen that result by showing, for any recursively enumerable language, how to construct a time-varying distributed H-system of degree 2 that generates that language exactly. We also indicate that such a construction is impossible for time-varying distributed H-systems of degree 1.

24 citations


Book ChapterDOI
04 Oct 2001
TL;DR: This paper introduces a new kind of communication between membranes, based upon the natural budding of vesicles in a cell, and proves that P systems of this type can generate all recursively enumerable languages and the Hamiltonian Path Problem can be solved in a quadratic time.
Abstract: P systems are computational models inspired by some biological features of the structure and the functioning of real cells. In this paper we introduce a new kind of communication between membranes, based upon the natural budding of vesicles in a cell. We define the operations of gemmation and fusion of mobile membranes, and we use membrane structures and rules over strings of biological inspiration only. We prove that P systems of this type can generate all recursively enumerable languages and, moreover, the Hamiltonian Path Problem can be solved in a quadratic time. Some open problems are also formulated.

Book ChapterDOI
23 May 2001
TL;DR: This is a survey of universality results in the area of Membrane Computing (P systems), at the level of December 2000, that considers both P systems with symbol-objects and with string-objects, and the techniques used in the proofs of such results.
Abstract: This is a survey of universality results in the area of Membrane Computing (P systems), at the level of December 2000. We consider both P systems with symbol-objects and with string-objects; in the latter case, we consider systems based on rewriting, splicing, as well as rewriting together with other operations (replication, crossingover), with sets or with multisets of strings. Besides recalling characterizations of recursively enumerable languages and of recursively enumerable sets of vectors of natural numbers, we also briefly discuss the techniques used in the proofs of such results. Several open problems are also formulated.

Journal ArticleDOI
TL;DR: In this article, an axiomatization of the class SCmV of complex algebras of a variety V is given, which is recursively enumerable.
Abstract: Given a variety V, we provide an axiomatization ( V) of the class SCmV of complex algebras of algebras in V. ( V) can be obtained eectively from the axiomatization of V; in fact, if this axiomatization is recursively enumerable, then ( V) is recursive.

Journal ArticleDOI
TL;DR: It is shown that recursively enumerable (r.e.) prime theories over a finite number of variables are decidable, and an example of an undecidable r.e. prime theory over countably many variables is exhibited.

Journal ArticleDOI
TL;DR: All closure properties of families in the Chomksy hierarchy under both non-iterated and iterated PA-matching and overlapping operations are settled.

Journal ArticleDOI
01 Apr 2001
TL;DR: A variant of sticker systems which uses molecules with complex structures is proposed which can obtain the characterization of recursively enumerable languages by using only sticking (hybridization) operations for complex molecules, while the usual sticker systems require more complicated operations.
Abstract: In this paper, we propose a variant of sticker systems which uses molecules with complex structures. Since the original sticker systems (Paun et al. (1998) [2, 8]) working on double strands of DNA have been studied as a formal model for self-assembly in DNA computing, we extend the sticker systems to working on more complex (higher-order) structures of DNA molecules. The advantage of sticker systems with complex structures is that augmented with weak codings we can obtain the characterization of recursively enumerable languages by using only sticking (hybridization) operations for complex molecules, while the usual sticker systems require more complicated operations such as the simultaneous use of couples of dominoes or coherent computations besides morphisms.

Book ChapterDOI
01 Feb 2001
TL;DR: Characterizations of recursively enumerable languages are given, by means of splicing P systems, having splicing rules of small size (that is, involving short context strings), and it is shown that with only two membranes the authors can generate all the recursive enumerable Languages.
Abstract: This paper is a direct continuation of [11]. Characterizations of recursively enumerable languages are given, by means of splicing P systems, having splicing rules of small size (that is, involving short context strings). Also it is shown that with only two membranes we can generate all the recursively enumerable languages; this improves a result from [11], where three membranes are used.

01 Jan 2001
TL;DR: In this article, it was shown that the equational theory of RPA! is non-recursively enumerable in the generalized sense. But this result does not imply that RPA!, as a class of representable polyadic algebras, is also non-computable.
Abstract: In [3] Daigneault and Monk proved that the class of (! dimensional) representable polyadic algebras (RPA! for short) is axiomatizable by finitely many equationschemas. However, this result does not imply that the equational theory of RPA! would be recursively enumerable; one simple reason is that the language of RPA! contains a continuum of operation symbols. Here we prove the following. Roughly, for any reasonable generalization of computability to uncountable languages, the equational theory of RPA! remains non-recursively enumerable, or non-computable, in the generalized sense. This result has some implications on the non-computational character of Keisler’s completeness theorem for his “infinitary logic” in Keisler [6] as well.

Journal ArticleDOI
TL;DR: This paper presents a schema for constructing one-point bases for recursively enumerable sets of lambda terms, which implies that a single procedure can define any given recursive set of procedures, constants and free variables in a given programming language.
Abstract: In this paper, we present a schema for constructing one-point bases for recursively enumerable sets of lambda terms. The novelty of the approach is that we make no assumptions about the terms for which the one-point basis is constructed: They need not be combinators and they may contain constants and free variables. The significance of the construction is twofold: In the context of the lambda calculus, it characterises one-point bases as ways of ``packaging'' sets of terms into a single term; And in the context of realistic programming languages, it implies that we can define a single procedure that generates any given recursively enumerable set of procedures, constants and free variables in a given programming language.

Book ChapterDOI
10 Jun 2001
TL;DR: Here it is proved that there are universal time-varying distributed H-systems of degree 2, which are equivalent to any formal language of degree at least 7.
Abstract: Time-varying distributed H systems (TVDH systems shortly) of degree n are a well known model of splicing computations which has the following special feature: at different moments one uses different sets of splicing rules (the number of these sets of splicing rules is called the degree of the TVDH system). It is known that there is a universal TVDH system of degree 2. Now we prove that there is a universal TVDH system of degree 1. It is a surprising result because we did not thought that these systems are so powerful.Recently both authors proved that TVDH systems of degree 1 can generate any recursively enumerable languages. We present here the short description of the main idea of the proof that result.

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.


Book ChapterDOI
01 Feb 2001
TL;DR: It is shown that there exists a natural number k such that every recursively enumerable language can be generated by a context-free returning parallel communicating grammar system where the number of nonterminals is less than or equal to this constant.
Abstract: We show that there exists a natural number k such that every recursively enumerable language can be generated by a context-free returning parallel communicating grammar system where the number of nonterminals is less than or equal to this constant. Moreover, the component grammars of the system have a limited number of productions. The result demonstrates that context-free returning parallel communicating grammar systems are economical tools for language generation.

Journal ArticleDOI
TL;DR: It is shown that, for some such set A, the property of being cofinite in ℒ*(A) is still Σ03-complete, which implies that for this A, there is no uniformly computably enumerable “tower” of sets exhausting exactly the coinfinite sets in ⅓(A).
Abstract: We study the filter ℒ*(A) of computably enumerable supersets (modulo finite sets) of an r-maximal set A and show that, for some such set A, the property of being cofinite in ℒ*(A) is still Σ03-complete. This implies that for this A, there is no uniformly computably enumerable “tower” of sets exhausting exactly the coinfinite sets in ℒ*(A).

Journal ArticleDOI
TL;DR: This paper proposes some evaluation methods based on L-grammars which are fuzzy grammars for evaluation of documents in SGML-format and to the evaluation of HTML-pages in the World Wide Web and outlines how the generalization of these methods of evaluation can be applied in different contexts and for different roles.
Abstract: The large amount of information available and the difficulty on processing it has made knowledge management a promising area of research. Several topics are related to it, for example distributed and intelligent information retrieval, information filtering and information evaluation, which became crucial. In this paper, we focus our attention on the knowledge evaluation problem. With the aim of evaluating information coded in the standard non-proprietary format SGML (as also in XML), we propose some evaluation methods based on L-grammars which are fuzzy grammars. In particular we apply these methods to the evaluation of documents in SGML-format and to the evaluation of HTML-pages in the World Wide Web. L-grammars generate recursively enumerable L-languages, as it has been proved in Gerla ((1991), Information Sciences 53), and so they can be used to generate fuzzy languages based on extensions of the document type definitions (DTD) involved by SGML. Given a DTD, we extend its associated language by adding a judgement label. By selecting a particular label and by taking the start symbol of the grammar associated to the DTD, we can generate any DTD-compliant document with a fuzzy degree of membership derived from the judgement label. In this way we fit the computational model underlying the recursively enumerable L-languages to the process of collecting different evaluations of the same document. Finally, we outline how the generalization of these methods of evaluation can be applied in different contexts and for different roles, as for example for information filtering.

Journal ArticleDOI
TL;DR: In this article, a characterization of /nite algebras which generate a variety having a cardinal bound on its sub-directly irreducible algesbras with non-Abelian monolith is presented.

Book
01 Jan 2001
TL;DR: In this paper, the authors present a set theory based on recursive functions and sets, which is a generalization of the notion of recursive functions in the theory of enumerable sets.
Abstract: Introduction 5. Recursion theory 5.1 Primitive recursive functions and sets 5.2 Recursive functions 5.3 Turing machines 5.4 Recursively enumerable sets 5.5 Exercises for Chapter 5 6. Formalization of arithmetic, Godel's theorems 6.1 Peano's axioms 6.2 Representable functions 6.3 Arithmetization of syntax 6.4 Incompleteness and undecidability theorem 7. Set theory 7.1 The theories Z and ZF 7.2 Ordinal numbers and integers 7.3 Inductive proofs and definitions 7.4 Cardinality 7.5 The axiom of foundation and the reflections schemes 7.6 Exercises for Chapter 7 8. Some model theory 8.1 Elementary substructures and extensions 8.2 Construction of elementary extensions 8.3 The interpolation and definability theorems 8.4 Reduced products and ultraproducts 8.5 Preservations theorems 8.6 -categorical theories 8.7 Exercises for Chapter 8 Solutions to the exercises of Part II Chapter 5 Chapter 6 Chapter 7 Chapter 8 Bibliography Index

Journal ArticleDOI
20 Jun 2001
TL;DR: It is shown that the learning power of finite and limit identification from good text examples coincide and, if learning from good informant examples is considered, limit identification is superior to finite identification in the class preserving as well as in theclass-comprising case.
Abstract: The present paper investigates identification of indexed families L of recursively enumerable languages from good examples. We distinguish class-preserving learning from good examples (the good examples have to be generated with respect to a hypothesis space having the same range as L) and class-comprising learning from the good examples (the good examples have to be selected with respect to a hypothesis space comprising the range of L). A learner is required to learn a target language on every finite superset of the good examples for it. If the learner's first and only conjecture is correct then the underlying learning model is referred to as finite identification from good examples and if the learner makes a finite number of incorrect conjectures before always outputting a correct one, the model is referred to as limit identification from good examples. In the context of class-preserving learning, it is shown that the learning power of finite and limit identification from good text examples coincide. When class comprising learning from good text examples is concerned, limit identification is strictly more powerful than finite learning. Furthermore, if learning from good informant examples is considered, limit identification is superior to finite identification in the class preserving as well as in the class-comprising case. Finally, we relate the models of learning from good examples to one another as well as to the standard learning models in the context of Gold-style language learnin

Journal ArticleDOI
TL;DR: This paper introduces a computability model?called shape completion system?for the restricted, but important, case in which the visual representation of the concepts to be communicated is built as a puzzle, and can characterize the recursively enumerable languages.
Abstract: Visual languages represent a response to the communicational challenges posed by end-user computing, but lack established computability frameworks for evaluating their computational power. In this paper, we introduce a computability model?called shape completion system?for the restricted, but important, case in which the visual representation of the concepts to be communicated is built as a puzzle. Shape completion systems are based on adjoining polyominoes, shapes from a basic set. A description in the form of a string on some alphabet can be associated with each basic shape. A computation in a shape completion system is correct when: (1) it starts by using a specified polyomino; (2) it ends when a rectangle is obtained (without holes); (3) at any step the current picture is connected; and (4) a sequencing mapping is given, so that at every step (except the first one) we use a polyomino depending on the previously used polyomino, as specified by this mapping (such a condition is essential for interactive visual languages, as formalized in 1, 2). We also establish how symbols associated with the polyominoes are concatenated to form strings in a string language associated with the computation. Surprisingly enough, in these circumstances we can characterize the recursively enumerable languages (hence the power of Turing machines). If we preserve only conditions (1), (2) and (3) above, then we cannot generate all linear languages but we can generate all regular languages and strictly more: also some one-letter non-regular languages can be obtained. In particular, we can obtain as correct computations squares only, which is often a difficult task in picture languages (see, e.g. 3).

Journal ArticleDOI
TL;DR: In this paper, it was shown that the existence of minimal pairs is equivalent to ∑2 induction in the B∑2 model and that every recursively enumerable (r.e.) set is either prompt or recursive.
Abstract: We prove that in everyB∑2 model (one satisfies ∑2 collection axioms but not ∑2 induction), every recursively enumerable (r.e.) set is either prompt or recursive. Consequently, over the base theory ∑2 collection, the existence of r.e. minimal pairs is equivalent to ∑2 induction. We also refute Shoenfield’s Conjecture inB∑2 models.