Showing papers on "Recursively enumerable language published in 2015"
14 Jan 2015
TL;DR: This work wants to prove that a static analysis of a given program is complete, namely, no imprecision arises when asking some query on the program behavior in the concrete or in the abstract, and introduces the completeness class of an abstraction as the set of all programs for which the abstraction is complete.
Abstract: We want to prove that a static analysis of a given program is complete, namely, no imprecision arises when asking some query on the program behavior in the concrete (ie, for its concrete semantics) or in the abstract (ie, for its abstract interpretation). Completeness proofs are therefore useful to assign confidence to alarms raised by static analyses. We introduce the completeness class of an abstraction as the set of all programs for which the abstraction is complete. Our first result shows that for any nontrivial abstraction, its completeness class is not recursively enumerable. We then introduce a stratified deductive system to prove the completeness of program analyses over an abstract domain A. We prove the soundness of the deductive system. We observe that the only sources of incompleteness are assignments and Boolean tests --- unlikely a common belief in static analysis, joins do not induce incompleteness. The first layer of this proof system is generic, abstraction-agnostic, and it deals with the standard constructs for program composition, that is, sequential composition, branching and guarded iteration. The second layer is instead abstraction-specific: the designer of an abstract domain A provides conditions for completeness in A of assignments and Boolean tests which have to be checked by a suitable static analysis or assumed in the completeness proof as hypotheses. We instantiate the second layer of this proof system first with a generic nonrelational abstraction in order to provide a sound rule for the completeness of assignments. Orthogonally, we instantiate it to the numerical abstract domains of Intervals and Octagons, providing necessary and sufficient conditions for the completeness of their Boolean tests and of assignments for Octagons.
36 citations
16 Jul 2015
TL;DR: A notion of stochastic rewriting over marked graphs – i.e. directed multigraphs with degree constraints – is developed, which gives a general procedure to derive the moment semantics of any such rewriting system, as a countable (and recursively enumerable) system of differential equations indexed by motif functions.
Abstract: We develop a notion of stochastic rewriting over marked graphs – i.e. directed multigraphs with degree constraints. The approach is based on double-pushout (DPO) graph rewriting. Marked graphs are expressive enough to internalize the ‘no-dangling-edge’ condition inherent in DPO rewriting. Our main result is that the linear span of marked graph occurrence-counting functions – or motif functions – form an algebra which is closed under the infinitesimal generator of (the Markov chain associated with) any such rewriting system. This gives a general procedure to derive the moment semantics of any such rewriting system, as a countable (and recursively enumerable) system of differential equations indexed by motif functions. The differential system describes the time evolution of moments (of any order) of these motif functions under the rewriting system. We illustrate the semantics using the example of preferential attachment networks; a well-studied complex system, which meshes well with our notion of marked graph rewriting. We show how in this case our procedure obtains a finite description of all moments of degree counts for a fixed degree.
11 citations
TL;DR: An undecidability result for products of matrices is deduced which can be viewed as a variant of Rice's theorem stating that all nontrivial properties of recursively enumerable sets are undecidable.
8 citations
TL;DR: An axiomatic theory of “generalized Routley-Meyer (GRM) logics” is developed which shows that all GRM logics are subclassical, have recursively enumerable consequence relations, satisfy the compactness theorem, and satisfy the standard structural rules and conjunction and disjunction introduction/elimination rules.
Abstract: We develop an axiomatic theory of “generalized Routley-Meyer (GRM) logics.” These are first-order logics which are can be characterized by model theories in a certain generalization of Routley-Meyer semantics. We show that all GRM logics are subclassical, have recursively enumerable consequence relations, satisfy the compactness theorem, and satisfy the standard structural rules and conjunction and disjunction introduction/elimination rules. We also show that the GRM logics include classical logic, intuitionistic logic, LP/K3/FDE, and the relevant logics.
8 citations
Proceedings Article•
25 Jul 2015TL;DR: It is shown that, conversely, every homomorphism-closed recursively enumerable query can be expressed as an existential rule query, thus arriving at a precise characterization of existential rules by model-theoretic and computational properties.
Abstract: Existential rules (also known as Datalog± or tuple-generating dependencies) have been intensively studied in recent years as a prominent formalism in knowledge representation and database systems. We consider them here as a querying formalism, extending classical Datalog, the language of deductive databases. It is well known that the classes of databases recognized by (Boolean) existential rule queries are closed under homomorphisms. Also, due to the existence of a semi-decision procedure (the chase), these database classes are recursively enumerable. We show that, conversely, every homomorphism-closed recursively enumerable query can be expressed as an existential rule query, thus arriving at a precise characterization of existential rules by model-theoretic and computational properties. Although the result is very intuitive, the proof turns out to be non-trivial. This result can be seen as a very expressive counterpart of the prominent Lyndon-Łos-Tarski-Theorem characterizing the homomorphism-closed fragment of first-order logic. Notably, our result does not presume the existence of any additional built-in structure on the queried data, such as a linear order on the domain, which is a typical requirement for other characterizations in the spirit of descriptive complexity.
6 citations
01 Jan 2015
TL;DR: It is shown that every recursively enumerable language can be accepted by an ANEP with an underlying graph in the form of a star with 13 nodes as well as by ANEPs having underlying graphs in the forms of a chain, a ring, or a wheel with 29 nodes each.
Abstract: In this paper, we approach the problem of accepting all recursively enumerable languages by accepting networks of evolutionary processors (ANEPs, for short) with a fixed architecture. More precisely, we show that every recursively enumerable language can be accepted by an ANEP with an underlying graph in the form of a star with 13 nodes or by an ANEP with an underlying grid with 13 × 4 = 52 nodes as well as by ANEPs having underlying graphs in the form of a chain, a ring, or a wheel with 29 nodes each. In all these cases, the size and form as well as the general working strategy of the constructed networks do not depend on the accepted language; only the rewriting rules and the filters associated to each node of the networks depend on this language. Noteworthy is also the fact that the filtering process is implemented using random context conditions only. Our results answer problems which were left open in a paper published by J. Dassow and F. Manea at the conference on Descriptional Complexity of Formal Systems (DCFS) 2010 and improve a result published by B. Truthe at the conference on Non-Classical Models of Automata and Applications (NCMA) 2013.
6 citations
25 Jun 2015
TL;DR: This paper brings a solution to the open problem of recursive enumerability of unsatisfiable formulae in the first-order Godel logic by modifying the hyperresolution calculus suitable for automated deduction with explicit partial truth.
Abstract: This paper brings a solution to the open problem of recursive enumerability of unsatisfiable formulae in the first-order Godel logic. The answer is affirmative even for a useful expansion by intermediate truth constants and the equality, Open image in new window , strict order, \(\prec \), projection \(\Delta \) operators. The affirmative result for unsatisfiable prenex formulae of \(G_\infty ^\Delta \) has been stated in [1]. In [7], we have generalised the well-known hyperresolution principle to the first-order Godel logic for the general case. We now propose a modification of the hyperresolution calculus suitable for automated deduction with explicit partial truth.
3 citations
Posted Content•
TL;DR: It is argued that Tarski's theorem on the Undefinability of Truth is Godel's First Incompleteness Theorem relativized to definable oracles; here a unification of these two theorems is given.
Abstract: We present a version of Godel's Second Incompleteness Theorem for recursively enumerable consistent extensions of a fixed axiomatizable theory, by incorporating some bi-theoretic version of the derivability conditions. We also argue that Tarski's theorem on the Undefinability of Truth is Godel's First Incompleteness Theorem relativized to definable oracles; a unification of these two theorems is given.
3 citations
TL;DR: A constrained version of UG that guarantees efficient processing, while allowing the expression of complex linguistic structures, is defined by proving that the constrained formalism is equivalent to Range Concatenation Grammar, a formalism that generates exactly the class of languages recognizable in deterministic polynomial time.
Abstract: Unification grammars (UG) are a grammatical formalism that underlies several contemporary linguistic theories, including Lexical-functional Grammar and Head-driven Phrase-structure Grammar. UG is an especially attractive formalism because of its expressivity, which facilitates the expression of complex linguistic structures and relations. Formally, UG is Turing-complete, generating the entire class of recursively enumerable languages. This expressivity, however, comes at a price: the universal recognition problem is undecidable for arbitrary unification grammars. We define a constrained version of UG that guarantees efficient processing, while allowing the expression of complex linguistic structures. We do so by proving that the constrained formalism is equivalent to Range Concatenation Grammar, a formalism that generates exactly the class of languages recognizable in deterministic polynomial time. We thus obtain a grammatical formalism that is on one hand highly expressive, and on the other efficient to compute with.
2 citations
25 Sep 2015
TL;DR: The universality of non-restricted virus machines is proved by showing that they can compute all diophantine sets, which the MRDP theorem proves that coincide with the recursively enumerable sets.
Abstract: Virus Machines are a computational paradigm inspired by the manner in which viruses replicate and transmit from one host cell to another. This paradigm provides non-deterministic sequential devices. Non-restricted virus machines are unbounded virus machines, in the sense that no restriction on the number of hosts, the number of instructions and the number of viruses contained in any host along any computation is placed on them. The computational completeness of these machines has been obtained by simulating register machines. In this paper, virus machines as set generating devices are considered. Then, the universality of non-restricted virus machines is proved by showing that they can compute all diophantine sets, which the MRDP theorem proves that coincide with the recursively enumerable sets.
2 citations
TL;DR: Here it is shown that λ-confluence is decidable in polynomial time for limited context restarting automata of type R 2 , but that this property is not even recursively enumerable for clearing restarted automata.
01 Jul 2015
TL;DR: This work investigates the computational power of SNPA systems as language generators and shows that the astrocytes are a powerful ingredient for spiking neural P systems aslanguage generators.
Abstract: Spiking neural P systems with astrocytes (SNPA systems, for short) are a class of distributed parallel computing devices inspired from the way spikes pass along the synapses between neurons. In this work, we investigate the computational power of SNPA systems as language generators. Specifically, representations of recursively enumerable languages and of regular languages are given by means of SNPA systems without forgetting rules. Furthermore, a simple finite language is produced which can be generated by SNPA systems, while it cannot be generated by usual spiking neural P systems. These results show that the astrocytes are a powerful ingredient for spiking neural P systems as language generators.
Posted Content•
TL;DR: In this paper, a relational word is defined as a finite totally ordered set of positions endowed with three binary relations that describe which positions are labeled by equal data, by unequal data and those having an undefined relation between their labels.
Abstract: We introduce a new notion of a relational word as a finite totally ordered set of positions endowed with three binary relations that describe which positions are labeled by equal data, by unequal data and those having an undefined relation between their labels. We define the operations of insertion and deletion on relational words generalizing corresponding operations on strings. We prove that the transitive and reflexive closure of these operations has a decidable membership problem for the case of short insertion-deletion rules (of size two/three and three/two). At the same time, we show that in the general case such systems can produce a coding of any recursively enumerable language leading to undecidabilty of reachability questions.
Book•
01 Jan 2015TL;DR: It is proved that every recursively enumerable class of partial recursive functions with infinite domains must have a recursive witness array, and it is shown that no finitely generated group of recursive permutations can contain all recursive involutions.
Abstract: We prove that every recursively enumerable class of partial recursive functions with infinite domains must have a recursive witness array. The result gives a powerful method for proving properties of recursively enumerable classes. We show for example that no finitely generated group of recursive permutations can contain all recursive involutions, and neither can it contain all cycle-free recursive permutations.
TL;DR: It is proved that a first-order spatio-temporal theory over this flow is recursively enumerable if and only if the dimension of spacetime does not exceed 2.
Abstract: Spatio-temporal logic is a variant of branching temporal logic where one of the so-called causal relations on spacetime plays the role of a time flow. Allowing only rational numbers as space and time co-ordinates, we prove that a first-order spatio-temporal theory over this flow is recursively enumerable if and only if the dimension of spacetime does not exceed 2. The situation is somewhat different compared to the case of real co-ordinates, because we establish that even dimension 2 does not permit recursive enumerability in this case. The proof of the result on rational spacetime involves a more deeper portion of spacetime geometry than the corresponding, more evident result for the real co-ordinates.
01 Jan 2015
TL;DR: This paper shows that certain variants of generalized communicating P constructs are able to accept any recursively enumerable language even with a small number of cells, but there are rescticted types which recognize only the class of regular languages.
Abstract: In this paper we introduce and study generalized communicating P automata, computing devices that combine properties of classical automata and generalized communicating P systems. We show that certain variants of these constructs are able to accept any recursively enumerable language even with a small number of cells which interact with each other using only one type of very simple communication rules, but there are rescticted types which recognize only the class of regular languages.
TL;DR: By applying certain classical and recent results on Diophantine equations, it is shown that L RE = F ?
01 Jan 2015
TL;DR: This degree project is about two fundamental results in finite model theory, which is an area of mathematical logic with applications in computer science, and Trakhtenbrot's result stating that validity over finite models is not recursively enumerable and Fagin's result putting an equality sign between NP and existential second-order logic.
Abstract: This degree project is about two fundamental results in finite model theory, which is an area of mathematical logic with applications in computer science. Usually the structures of interest for computer scientists may be regarded as finite models for some formal language. One of the first results, sometimes regarded as the birth of finite model theory, is Trakhtenbrot’s result from 1950 stating that validity over finite models is not recursively enumerable. This means that completeness fails over finite models. The technique of the proof, which is based on encoding Turing machine computations as finite structures, was reused by Fagin some 25 years later to prove his result putting an equality sign between the complexity class NP and existential second-order logic, hence providing a machineindependent characterization of an important complexity class. As an example we may look at SQL (Structured Query Language), which is a well known and one of the first language for the relational database model described in Codd’s 1970 paper. SQL is based on firstorder predicate logic, and has the same expressive power.
01 Jan 2015
TL;DR: Chih et al. as discussed by the authors investigated information theoretic properties of disjoint sets and properties of their enumerations, and whether these properties are preserved under splittings, and showed that a speedable set cannot be split into two speedable sets.
Abstract: Author(s): Chih, Ellen S. | Advisor(s): Harrington, Leo A; Slaman, Theodore A | Abstract: A split of an r.e. set A is a pair of disjoint r.e. sets whose union is A. We investigate information theoretic properties of r.e. sets and properties of their enumerations, and whether these properties are preserved under splittings. The first part is involved with dynamic notions. An r.e. set is speedable if for every computable function, there exists a finite algorithm enumerating membership faster, by the desired computable factor, on infinitely many integers. Remmel (1986) asked whether every speedable set could be split into two speedable sets. Jahn published a proof, answering their question positively. However, his proof was seen to be incorrect and we construct a speedable set that cannot be split into speedable sets, answering their question negatively. We also prove additional splitting results related to speedability.The second part is involved with effectively closed sets. We introduce the notion of being recursively avoiding and prove several results.
12 Mar 2015
TL;DR: It is proved that the seminal decentralized algorithm for the improvement of fiber-optic cables by Kobayashi et al. is recursively enumerable.
Abstract: In recent years, much research has been devoted to the exploration of semaphores; unfortunately, few have investigated the deployment of randomized algorithms. In fact, few mathematicians would disagree with the improvement of Scheme, which embodies the private principles of artificial intelligence. We prove that the seminal decentralized algorithm for the improvement of fiber-optic cables by Kobayashi et al. is recursively enumerable.
TL;DR: In this paper, the fundamental position of the finite languages and their complements in the hierarchy is examined and several auxiliary results for Turing machines with advice are given. But the main result is that the hierarchy can be separated by increasing chains of finite languages.
Abstract: In the late nineteen sixties it was observed that the r.e. languages form an infinite proper hierarchy RE1⊂RE2⊂⋯ based on the size of the Turing machines that accept them. We examine the fundamental position of the finite languages and their complements in the hierarchy. We show that for every finite language L one has that L, L¯∈REn for some n≤p⋅(m−⌊ log2p ⌋+1)+1 where m is the length of the longest word in L, c is the cardinality of L, and p=min{c,2m−1}. If L∈REn, then L¯∈REs for some s=O(n+m). We also prove that for every n, there is a finite language Ln with m=O(nlog2n) such that Ln∉REn but Ln, L¯n∈REs for some s=O(nlog2n). Several further results are shown that how the hierarchy can be separated by increasing chains of finite languages. The proofs make use of several auxiliary results for Turing machines with advice.
Posted Content•
TL;DR: In this paper, a new model of Watson-Crick automata which is reversible in nature is introduced and the computational power of this model is explored, even though the model is reversible and one way it accepts all regular languages.
Abstract: Watson-Crick automata are finite automata working on double strands. Extensive research work has already been done on non-deterministic Watson-Crick automata and on deterministic Watson-Crick automata. In this paper, we introduce a new model of Watson-Crick automata which is reversible in nature named reversible Watson-Crick automata and explore its computational power. We show even though the model is reversible and one way it accepts all regular languages and also analyze the state complexity of the above stated model with respect to non-deterministic block automata and non-deterministic finite automata and establish its superiority. We further explore the relation of the reversible model with twin-shuffle language and recursively enumerable languages.
Journal Article•
TL;DR: This paper demonstrates how to solve in the limit Turing Machine Halting Problem, to approximate the universal search algorithm, to decide diagonalization language, nontrivialproperties of recursively enumerable languages, and how to solves Post Correspondence Problem and Busy BeaverProblem.
Abstract: The $-calculus process algebra for problem solving applies the cost performance measures to converge in finitetime or in the limit to optimal solutions with minimal problem solving costs. The $-calculus belongs to superTuringmodels of computation. Its main goal is to provide the support to solve hard computational problems. It allows alsoto solve in the limit some undecidable problems. In the paper we demonstrate how to solve in the limit Turing MachineHalting Problem, to approximate the universal search algorithm, to decide diagonalization language, nontrivialproperties of recursively enumerable languages, and how to solve Post Correspondence Problem and Busy BeaverProblem.