Showing papers on "Recursively enumerable language published in 2014"

Universal computably enumerable equivalence relations

Abstract: We study computably enumerable equivalence relations (ceers), under the reducibility if there exists a computable function f such that if and only if , for every . We show that the degrees of ceers under the equivalence relation generated by form a bounded poset that is neither a lower semilattice, nor an upper semilattice, and its first-order theory is undecidable. We then study the universal ceers. We show that 1) the uniformly effectively inseparable ceers are universal, but there are effectively inseparable ceers that are not universal; 2) a ceer R is universal if and only if , where denotes the halting jump operator introduced by Gao and Gerdes (answering an open question of Gao and Gerdes); and 3) both the index set of the universal ceers and the index set of the uniformly effectively inseparable ceers are -complete (the former answering an open question of Gao and Gerdes).

On Controlled P Systems

Abstract: We introduce and briefly investigate P systems with controlled computations. First, P systems with label restricted transitions are considered in each step, all rules used have either the same label, or, possibly, the empty label, λ, then P systems with the computations controlled by languages as in context-free controlled grammars. The relationships between the families of sets of numbers computed by the various classes of controlled P systems are investigated, also comparing them with length sets of languages in Chomsky and Lindenmayer hierarchies characterizations of the length sets of ET0L and of recursively enumerable languages are obtained in this framework. A series of open problems and research topics are formulated.

P Colonies Processing Strings

Abstract: In this paper we introduce and study P colonies where the environment is given as a string. These constructs, called automaton-like P colonies or APCol systems, behave like automata: during functioning, the agents change their own states and process the symbols of the string. We show that the family of e-free languages accepted by jumping finite automata is properly included in the family of languages accepted by APCol systems with one agent and any e-free recursively enumerable language can be obtained as a projection of a language accepted by an automaton-like P colony with two agents.

Uniform Density in Lindenbaum Algebras

Abstract: In this paper we prove that the preordering ≲ of provable implication over any recursively enumerable theory T containing a modicum of arithmetic is uniformly dense. This means that we can find a recursive extensional density function F for ≲. A recursive function F is a density function if it computes, for A and B with A ≤≁ B, an element C such that A ≤≁ C ≤≁ B. The function is extensional if it preserves T -provable equivalence. Secondly, we prove a general result that implies that, for extensions of elementary arithmetic, the ordering ≲ restricted to σn-sentences is uniformly dense. In the last section we provide historical notes and background material.

Accepting Networks of Evolutionary Processors with Subregular Filters

Abstract: We propose a new variant of Accepting Networks of Evolutionary Processors, in which the operations are applied at arbitrary positions to the processed words (rather than at the ends of words only) and where the filters are languages from several special classes of regular sets. More precisely, we show that the use of filters from the class of non-counting, ordered, power-separating, suffix-closed regular, union-free, definite and combinational languages is as powerful as the use of arbitrary regular languages and yields networks that can accept all the recursively enumerable languages. On the other hand, by using filters that are only finite languages, monoids, nilpotent languages, commutative regular languages, or circular regular languages, one cannot generate all recursively enumerable languages. These results seem interesting as they provide both upper and lower bounds on the classes of languages that one can use as filters in an accepting network of evolutionary processors in order to obtain a complete computational model.

Solutions to four open problems concerning controlled pure grammar systems

Abstract: In this paper, we address several open problems concerning pure grammar systems (pGSs) and their controlled versions. More specifically, we prove the following four results. (I) Regular-controlled pGSs having a single component define the family of regular languages. (II) pGSs having two components controlled by infinite regular languages define the family of recursively enumerable languages. (III) Regular-controlled pGSs without any erasing rules define the family of regular languages not containing the empty string. (IV) pGSs define a proper subfamily of the family of regular languages.

Splicing Systems over Permutation Groups of Length Two

Abstract: The first theoretical model of DNA computing, called a splicing system, for the study of the generative power of deoxyribonucleic acid (DNA) in the presence of restriction enzymes and ligases was introduced by Head in 1987. Splicing systems model the recombinant behavior of double-stranded DNA (dsDNA) and the enzymes which perform operation of cutting and pasting on dsDNA. Splicing systems with finite sets of axioms and rules generate only regular languages when no additional control is assumed. With several restrictions to splicing rules, the generative power increase up to recursively enumerable languages. Algebraic structures can also be used in order to control the splicing systems. In the literature, splicing systems with additive and multiplicative valences have been investigated, and it has been shown that the family of languages generated by valence splicing systems is strictly included in the family of context-sensitive languages. This motivates the study of splicing systems over permutation groups. In this paper, we define splicing systems over permutation groups and investigate the generative power of the languages produced.

Team Semantics and Recursive Enumerability.

Abstract: It is well known that dependence logic captures the complexity class NP, and it has recently been shown that inclusion logic captures P on ordered models. These results demonstrate that team semantics offers interesting new possibilities for descriptive complexity theory. In order to properly understand the connection between team semantics and descriptive complexity, we introduce an extension D∗ of dependence logic that can define exactly all recursively enumerable classes of finite models. Thus D∗ provides an approach to computation alternative to Turing machines. The essential novel feature in D∗ is an operator that can extend the domain of the considered model by a finite number of fresh elements. Due to the close relationship between generalized quantifiers and oracles, we also investigate generalized quantifiers in team semantics. We show that monotone quantifiers of type (1) can be canonically eliminated from quantifier extensions of first-order logic by introducing corresponding generalized dependence atoms.

Sticker Systems Over Monoids

Abstract: Molecular computing has gained many interests among researchers since Head introduced the first theoretical model for DNA based computation using the splicing operation in 1987. Another model for DNA computing was proposed by using the sticker operation which Adlemanused in his successful experiment for the computation of Hamiltonian paths in a graph: a double stranded DNA sequence is composed by prolonging to the left and to the right a sequence of (single or double) symbols by using given single stranded strings or even more complex dominoes with sticky ends, gluing these ends together with the sticky ends of the current sequence according to a complementarity relation. According to this sticker operation, a language generative mechanism, called a sticker system, can be defined: a set of (incomplete) double-stranded sequences (axioms) and a set of pairs of single or double-stranded complementary sequences are given. The initial sequences are prolonged to the left and to the right by using sequences from the latter set, respectively. The iterations of these prolongations produce “computations” of possibly arbitrary length. These processes stop when a complete double stranded sequence is obtained. Sticker systems will generate only regular languages without restrictions. Additional restrictions can be imposed on the matching pairs of strands to obtain more powerful languages. Several types of sticker systems are shown to have the same power as regular grammars; one type is found to represent all linear languages whereas another one is proved to be able to represent any recursively enumerable language. The main aim of this research is to introduce and study sticker systems over monoids in which with each sticker operation, an element of a monoid is associated and a complete double stranded sequence is considered to be valid if the computation of the associated elements of the monoid produces the neutral element. Moreover, the sticker system over monoids is defined in this study.

Five Nodes Are Sufficient for Hybrid Networks of Evolutionary Processors to Be Computationally Complete

Abstract: A hybrid network of evolutionary processors (HNEP) is a graph where each node is associated with a special rewriting system called an evolutionary processor, an input filter, and an output filter. Each evolutionary processor is given a finite set of one type of point mutations (insertion, deletion or a substitution of a symbol) which can be applied to certain positions in a string. An HNEP rewrites the strings in the nodes and then re-distributes them according to a filter-based communication protocol; the filters are defined by certain variants of random-context conditions. HNEPs can be considered both as languages generating devices (GHNEPs) and language accepting devices (AHNEPs); most previous approaches treated the accepting and generating cases separately. For both cases, in this paper we improve previous results by showing that five nodes are sufficient to accept (AHNEPs) or generate (GHNEPs) any recursively enumerable language by showing the more general result that any partial recursive relation can be computed by an HNEP with (at most) five nodes.

Diophantine sets of polynomials over algebraic extensions of the rationals

Abstract: Let L be a recursive algebraic extension of Q. Assume that, given alpha is an element of L, we can compute the roots in L of its minimal polynomial over Q and we can determine which roots are Aut(L)-conjugate to alpha. We prove that there exists a pair of polynomials that characterizes the Aut(L)-conjugates of alpha, and that these polynomials can be effectively computed. Assume furthermore that L can be embedded in R, or in a finite extension of Q(p) (with p an odd prime). Then we show that subsets of L[X](k) that are recursively enumerable for every recursive presentation of L[X], are diophantine over L[X].

Universal Turing Machine: A Model for allComputational Problems

Abstract: Turing machines are the most powerful computational machines and are the theoretical basis for modern computers. Universal Turing machine works for all classes of languages including regular languages (Res), Context-free languages (CFLs), as well as recursively enumerable languages (RELs). In this paper, we discuss the concept of Universal Turing machine as a computing device that can be used for solving any problem that a computer or a human can solve. We show how the machine works in solving computational problems and design some algorithms showing to show the operational procedure of input symbols such as moving left, right, or stationary depending on the input symbol. Finally, we argue that the Universal Turing machine is a generalpurpose machine that can be used to compute any computable problem.

Q-reducibility and m-reducibility on computably enumerable sets

Abstract: We study the distinctions between Q-reducibility and m-reducibility on computably enumerable sets. We construct a noncomputable m-incomplete computably enumerable set B such that all computably enumerable sets A ≤QB satisfy A ≤mB. We prove that for every noncomputable computably enumerable set A there exists a computably enumerable set B such that A ≤QB but A ≰mB. We prove that for every simple set B there exists a computably enumerable set A such that A ≤QB but A ≰mB. The last result implies in particular that the Q-degree of every simple set contains infinitely many computably enumerable m-degrees.

Monodic Fragments of Probabilistic First-Order Logic

Abstract: By classical results of Abadi and Halpern, validity for probabilistic first-order logic of type 2 (ProbFO) is \(\Pi^2_1\)-complete and thus not recursively enumerable, and even small fragments of ProbFO are undecidable. In temporal first-order logic, which has similar computational properties, these problems have been addressed by imposing monodicity, that is, by allowing temporal operators to be applied only to formulas with at most one free variable. In this paper, we identify a monodic fragment of ProbFO and show that it enjoys favorable computational properties. Specifically, the valid sentences of monodic ProbFO are recursively enumerable and a slight variation of Halpern’s axiom system for type-2 ProbFO on bounded domains is sound and complete for monodic ProbFO. Moreover, decidability can be obtained by restricting the FO part of monodic ProbFO to any decidable FO fragment. In some cases, which notably include the guarded fragment, our general constructions result in tight complexity bounds.

On an Algebraic Classification of Multidimensional Recursively Enumerable Sets Expressible in Formal Arithmetical Systems 1

Abstract: Algebraic representations of multidimensional recursively enumerable sets which are expressible in formal arithmetical systems based on the 1, are introduced and x , S), where S(x) , S), (0,,), (0,, S,signatures(0, investigated. The equivalence is established between the algebraic and logical representations of multidimensional recursively enumerable sets expressible in the mentioned systems.

On String Languages Generated by Spiking Neural P Systems with Astrocytes.

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.

Chapter 14 Controlled Pure Grammar Systems

Abstract: This chapter introduces pure grammar systems, which have only terminals. They generate their languages in the leftmost way, and in addition, this generative process is regulated by control languages over rule labels. The chapter concentrates its attention on investigating the generative power of these systems. It establishes three major results. First, without any control languages, these systems do not even generate some context-free languages. Second, with regular control languages, these systems characterize the family of recursively enumerable languages, and this result holds even if these systems have no more than two components. Finally, this chapter considers control languages as languages that are themselves generated by regular-controlled context-free grammars; surprisingly enough, with control languages of this kind, these systems over unary alphabets define nothing but regular languages. The chapter consists of two sections. First, Sect. 14.1 define controlled pure grammar systems and illustrate them by an example. Then, Sect. 14.2 rigorously establishes the results mentioned above and points out several open problems.

Turing machines with one-sided advice and the acceptance of the co-RE languages

Abstract: We resolve an old problem, namely to design a ‘natural’ machine model for accepting the complements of recursively enumerable languages.

Analyzing Expected Outcomes and Almost-Sure Termination of Probabilistic Programs is Hard.

Abstract: This paper considers the computational hardness of computing expected outcomes and deciding almost-sure termination of probabilistic programs. We show that deciding almost-sure termination and deciding whether the expected outcome of a program equals a given rational value is $\Pi^0_2$-complete. Computing lower and upper bounds on the expected outcome is shown to be recursively enumerable and $\Sigma^0_2$-complete, respectively.

On String Languages Generated by Sequential Spiking Neural P Systems Based on Maximum Spike Number

Abstract: Spiking neural P systems (SN P systems, for short) are a class of distributed parallel computing devices inspired from the way neurons communicate by means of spikes. In this work, we consider SN P systems with the restriction: at each step the neuron with the maximum number of spikes among the neurons that can spike will fire (if there is a tie for the maximum number of spikes stored in the active neurons, only one of the neurons containing the maximum is chosen non-deterministically). We investigate the computational power of such sequential SN P systems that are used as language generators. We prove that recursively enumerable languages can be characterized as projections of inverse-morphic images of languages generated by that sequential SN P systems. The relationships of the languages generated by these sequential SN P systems with finite and regular languages are also investigated.

Team Semantics and Recursive Enumerability

Abstract: It is well known that dependence logic captures the complexity class NP, and it has recently been shown that inclusion logic captures P on ordered models. These results demonstrate that team semantics offers interesting new possibilities for descriptive complexity theory. In order to properly understand the connection between team semantics and descriptive complexity, we introduce an extension D* of dependence logic that can define exactly all recursively enumerable classes of finite models. Thus D* provides an approach to computation alternative to Turing machines. The essential novel feature in D* is an operator that can extend the domain of the considered model by a finite number of fresh elements. Due to the close relationship between generalized quantifiers and oracles, we also investigate generalized quantifiers in team semantics. We show that monotone quantifiers of type (1) can be canonically eliminated from quantifier extensions of first-order logic by introducing corresponding generalized dependence atoms.

Recursively enumerable sets and well-ordering of their enumerations

Abstract: A b s t r a c t. We will introduce the special kind of the order relations into recursively enumerable sets and prove that they can be used to distinguish (albeit in a non-constructive way) between recursive and non-recursive sets.

Chapter 15 Self-Regulating Automata

Abstract: This chapter defines and investigates self-regulating automata. They regulate the selection of a rule according to which the current move is made by a rule according to which a previous move was made. Both finite and pushdown versions of these automata are investigated. The chapter is divided into two sections. Section 15.1 discusses self-regulating finite automata. It establishes two infinite hierarchies of language families resulting from them. Both hierarchies lie between the family of regular languages and the family of context-sensitive languages. Section 15.2 studies self-regulating pushdown automata. Based upon them, this section characterizes the families of context-free and recursively enumerable languages. However, as opposed to the results about self-regulating finite automata, many questions concerning their pushdown versions remain open; indeed, Sect. 15.2 formulates several specific open problem areas, including questions concerning infinite language-family hierarchies resulting from them.

A Simple Character String Proof of the "True but Unprovable" Version of G\"odel's First Incompleteness Theorem

Abstract: A rather easy yet rigorous proof of a version of Godel's first incompleteness theorem is presented. The version is "each recursively enumerable theory of natural numbers with 0, 1, +, *, =, logical and, logical not, and the universal quantifier either proves a false sentence or fails to prove a true sentence". The proof proceeds by first showing a similar result on theories of finite character strings, and then transporting it to natural numbers, by using them to model strings and their concatenation. Proof systems are expressed via Turing machines that halt if and only if their input string is a theorem. This approach makes it possible to present all but one parts of the proof rather briefly with simple and straightforward constructions. The details require some care, but do not require significant background knowledge. The missing part is the widely known fact that Turing machines can perform complicated computational tasks.

Separating the classes of recursively enumerable languages based on machine size

Abstract: In the late nineteen sixties it was observed that the recursively enumerable languages form an infinite proper hierarchy bassed 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. Several results are shown that how the hierarchy can be separated by finite languages of moderate complexity.

Chapter 8 Extension of Languages Resulting from Regulated Grammars

Abstract: The present chapter, consisting of two sections, studies modified regulated grammars so they generate their languages extended by some extra symbols that represent useful information related to the generated languages. Section 8.1 describes a transformation of any regular-controlled grammar with appearance checking G to a propagating regular-controlled with appearance checking H whose language L(H) has every sentence of the form wρ, where w is a string of terminals in G and ρ is a sequence of rules in H, so that (1) wρ ∈ L(H) if and only if w ∈ L(G) and (2) ρ is a parse of w in H. Consequently, for every recursively enumerable language K, there exists a propagating regular-controlled grammar with appearance checking H with L(H) of the above-mentioned form so K results from L(H) by erasing all rules in L(H). Analogical results are established (a) for regular-controlled grammars without appearance checking and (b) for these grammars that make only leftmost derivations. Section 8.2 studies a language operation referred to as coincidental extension, which extend strings by inserting some symbols into the languages generated by propagating scattered context grammars.

J ason Megill

Abstract: This paper concerns "human symbolic output," or strings of characters produced by humans in our various symbolic systems; eg, sen- tences in a natural language, mathematical propositions, and so on One can form a set that consists of all of the strings of characters that have been produced by at least one human up to any given moment in human history We argue that at any particular moment in human history, even at moments in the distant future, this set is finite But then, given fundamental results in recursion theory, the set will also be recursive, recursively enumerable, axiomatizable, and could be the output of a Turing machine We then argue that it is impossible to produce a string of symbols that humans could possi- bly produce but no Turing machine could Moreover, we show that any given string of symbols that we could produce could also be the output of a Turing machine Our arguments have implications for Hilbert's sixth problem and the possibility of axiomatizing particular sciences, they undermine at least two distinct arguments against the possibility of Artificial Intelligence, and they entail that expert systems that are the equals of human experts are possible, and so at least one of the goals of Artificial Intelligence can be realized, at least in principle