Showing papers on "Recursively enumerable language published in 2018"

On describing the regular closure of the linear languages with graph-controlled insertion-deletion systems

Abstract: A graph-controlled insertion-deletion (GCID) system has several components and each component contains some insertion-deletion rules A transition is performed by any applicable rule in the current component on a string and the resultant string is then moved to the target component specified in the rule The language of the system is the set of all terminal strings collected in the final component When resources are very limited (especially, when deletion is demanded to be context-free and insertion to be one-sided only), then GCID systems are not known to describe the class of recursively enumerable languages Hence, it becomes interesting to explore the descriptional complexity of such GCID systems of small sizes with respect to language classes below RE and even below CF To this end, we consider so-called closure classes of linear languages defined over the operations concatenation, Kleene star and union We show that whenever GCID systems (with certain syntactical restrictions) describe all linear languages (LIN) with t components, we can extend this to GCID systems with just one more component to describe, for instance, the concatenation of two languages from the language family that can be described as the Kleene closure of linear languages With further addition of one more component, we can extend the construction to GCID systems that describe the regular closure of LIN

Axiomatizing provable $n$-provability

Abstract: A formula $\phi$ is called \emph{$n$-provable} in a formal arithmetical theory $S$ if $\phi$ is provable in $S$ together with all true arithmetical $\Pi_{n}$-sentences taken as additional axioms. While in general the set of all $n$-provable formulas, for a fixed $n>0$, is not recursively enumerable, the set of formulas $\phi$ whose $n$-provability is provable in a given r.e.\ metatheory $T$ is r.e. This set is deductively closed and will be, in general, an extension of $S$. We prove that these theories can be naturally axiomatized in terms of progressions of iterated local reflection principles. In particular, the set of provably 1-provable sentences of Peano arithmetic PA can be axiomatized by $\varepsilon_0$ times iterated local reflection schema over PA. Our characterizations yield additional information on the proof-theoretic strength of these theories (w.r.t. various measures of it) and on their axiomatizability. We also study the question of speed-up of proofs and show that in some cases a proof of $n$-provability of a sentence can be much shorter than its proof from iterated reflection principles.

Gödel’s second incompleteness theorem for Σn-definable theories

Abstract: Godel's second incompleteness theorem is generalized by showing that if the set of axioms of a theory T ⊇ PA isn+1-definable and T isn-sound, then T dose not prove the sentencen-Sound(T) that expresses then-soundness of T. The optimal- ity of the generalization is shown by presenting an+1-definable (indeed a complete �n+1-definable) andn−1-sound theory T such that PA ⊆ T andn−1-Sound(T) is provable in T. It is also proved that no recursively enumerable and �1-sound theory of arithmetic, even very weak theories which do not contain Robinson's Arithmetic, can prove its own �1-soundness.

A Recursively Enumerable Kripke Complete First-Order Logic Not Complete with Respect to a First-Order Definable Class of Frames.

Abstract: It is well-known that every quantified modal logic complete with respect to a firstorder definable class of Kripke frames is recursively enumerable. Numerous examples are also known of “natural” quantified modal logics complete with respect to a class of frames defined by an essentially second-order condition which are not recursively enumerable. It is not, however, known if these examples are instances of a pattern, i.e., whether every recursively enumerable, Kripke complete quantified modal logic can be characterized by a first-order definable class of frames. While the question remains open for normal logics, we show that, in the context of quasi-normal logics, this is not so, by exhibiting an example of a recursively enumerable, Kripke complete quasi-normal logic that is not complete with respect to any first-order definable class of (pointed) frames.

Three Analog Neurons Are Turing Universal

Abstract: The languages accepted online by binary-state neural networks with rational weights have been shown to be context-sensitive when an extra analog neuron is added (1ANNs). In this paper, we provide an upper bound on the number of additional analog units to achieve Turing universality. We prove that any Turing machine can be simulated by a binary-state neural network extended with three analog neurons (3ANNs) having rational weights, with a linear-time overhead. Thus, the languages accepted offline by 3ANNs with rational weights are recursively enumerable, which refines the classification of neural networks within the Chomsky hierarchy.

Deconstructing Moore’s Law with Pask

Abstract: The implications of signed archetypes have been far-reaching and pervasive. In this position paper, we validate the simulation of the World Wide Web, demonstrates the natural importance of steganography. In this paper we motivate new real-time symmetries (Pask), disproving that the foremost unstable algorithm for the exploration of information retrieval systems by Zhao and Martinez [11] is recursively enumerable [7].

Axiomatizing Provable n -Provability

Abstract: The set of all formulas whose n-provability in a given arithmetical theory S is provable in another arithmetical theory T is a recursively enumerable extension of S. We prove that such extensions can be naturally axiomatized in terms of transfinite progressions of iterated local reflection schemata over S. Specifically, the set of all provably 1-provable sentences in Peano arithmetic PA can be axiomatized by an e0-times iterated local reflection schema over PA. The resulting characterizations provide additional information on the proof-theoretic strength of these theories and on the complexity of their axiomatization.

Development of IPv6

Abstract: Recent advances in collaborative theory and interactive archetypes cooperate in or- der to realize the lookaside buffer. Given the current status of cacheable epistemologies, researchers shockingly desire the understanding of redundancy. We introduce a novel application for the deployment of access points (SheldInditer), showing that the seminal psychoacoustic algorithm for the unproven unification of 64 bit architectures and symmetric encryption is recursively enumerable. Of course, this is not always the case.

Generic Amplification of Recursively Enumerable Sets

Abstract: Generic amplification is a method that allows algebraically undecidable problems to generate problems undecidable for almost all inputs. It is proved that every simple negligible set is undecidable for almost all inputs, but it cannot be obtained via amplification from any undecidable set. On the other hand, it is shown that every recursively enumerable set with nonzero asymptotic density can be obtained via amplification from a set of natural numbers.

Approximating Probabilistic Automata by Regular Languages

Abstract: A probabilistic finite automaton (PFA) A is said to be regular-approximable with respect to (x,y), if there is a regular language that contains all words accepted by A with probability at least x+y, but does not contain any word accepted with probability at most x. We show that the problem of determining if a PFA A is regular-approximable with respect to (x,y) is not recursively enumerable. We then show that many tractable sub-classes of PFAs identified in the literature - hierarchical PFAs, polynomially ambiguous PFAs, and eventually weakly ergodic PFAs - are regular-approximable with respect to all (x,y). Establishing the regular-approximability of a PFA has the nice consequence that its value can be effectively approximated, and the emptiness problem can be decided under the assumption of isolation.

Parameter-free Network Sparsification and Data Reduction by Minimal Algorithmic Information Loss.

Abstract: The study of large and complex datasets, or big data, organized as networks has emerged as one of the central challenges in most areas of science and technology. Cellular and molecular networks in biology is one of the prime examples. Henceforth, a number of techniques for data dimensionality reduction, especially in the context of networks, have been developed. Yet, current techniques require a predefined metric upon which to minimize the data size. Here we introduce a family of parameter-free algorithms based on (algorithmic) information theory that are designed to minimize the loss of any (enumerable computable) property contributing to the object's algorithmic content and thus important to preserve in a process of data dimension reduction when forcing the algorithm to delete first the least important features. Being independent of any particular criterion, they are universal in a fundamental mathematical sense. Using suboptimal approximations of efficient (polynomial) estimations we demonstrate how to preserve network properties outperforming other (leading) algorithms for network dimension reduction. Our method preserves all graph-theoretic indices measured, ranging from degree distribution, clustering-coefficient, edge betweenness, and degree and eigenvector centralities. We conclude and demonstrate numerically that our parameter-free, Minimal Information Loss Sparsification (MILS) method is robust, has the potential to maximize the preservation of all recursively enumerable features in data and networks, and achieves equal to significantly better results than other data reduction and network sparsification methods.

The Small-Is-Very-Small Principle

Abstract: The central result of this paper is the small-is-very-small principle for restricted sequential theories. The principle says roughly that whenever the given theory shows that a property has a small witness, i.e. a witness in every definable cut, then it shows that the property has a very small witness: i.e. a witness below a given standard number. We draw various consequences from the central result. For example (in rough formulations): (i) Every restricted, recursively enumerable sequential theory has a finitely axiomatized extension that is conservative w.r.t. formulas of complexity $\leq n$. (ii) Every sequential model has, for any $n$, an extension that is elementary for formulas of complexity $\leq n$, in which the intersection of all definable cuts is the natural numbers. (iii) We have reflection for $\Sigma^0_2$-sentences with sufficiently small witness in any consistent restricted theory $U$. (iv) Suppose $U$ is recursively enumerable and sequential. Suppose further that every recursively enumerable and sequential $V$ that locally inteprets $U$, globally interprets $U$. Then, $U$ is mutually globally interpretable with a finitely axiomatized sequential theory. The paper contains some careful groundwork developing partial satisfaction predicates in sequential theories for the complexity measure depth of quantifier alternations.

$\mathbb{F}_p((X))$ is decidable as a module over the ring of additive polynomials

Abstract: Let $p$ be a prime number, $K$ be the henselization of the rational functions over the finite field $\mathbb{F}_p$ and $R$ be the ring of additive polynomials over K. We show that the field of Laurent series over $\mathbb{F}_p$ is decidable seen as an R-module. Moreover, we provide a recursively enumerable axiom system (satisfied by $K$) in the language of $R$-modules together with a unary predicate for the valuation ring, modulo which every positive primitive formula is equivalent to a universal formula. Consequently the $R$-module theory of the field of Laurent series is model-complete in this language and admits $K$ as its prime model.

Minimizing Rules and Nonterminals in Semi-conditional Grammars: Non-trivial for the Simple Case

Abstract: A simple semi-conditional (SSC) grammar is a form of regulated rewriting system where the derivations are controlled either by a permitting string alone or by a forbidden string alone and is specified in the rule. The maximum length i (j, resp.) of the permitting (forbidden, resp.) strings serves as a measure of descriptional complexity known as the degree of such grammars. In addition to the degree, the numbers of nonterminals and of conditional rules are also counted into the descriptional complexity measures of these grammars. We improve on some previously obtained results on computational completeness of SSC grammars by minimizing the number of nonterminals and/or the number of conditional rules for a given degree (i, j). More specifically, we prove that every recursively enumerable language is generated by an SSC grammar of (i) degree (2, 1) with at most eight conditional rules and nine nonterminals, (ii) degree (3, 1) with at most seven conditional rules and eight nonterminals and (iii) degree (3, 1) with at most nine conditional rules and seven nonterminals.

On Derivation Languages of Flat Splicing Systems.

Abstract: In this work, we associate the idea of derivation languages with at splicing systems and compare the languages generated as derivation languages (Szilard and Control languages) with the family of languages in Chomsky hierarchy. We show that there exist regular languages which cannot be generated as a Szilard language by any labeled flat splicing system. But some context-sensitive languages can be generated as a Szilard language by alphabetic labeled flat splicing systems. Also, any regular, context-free and recursively enumerable language can be represented as a homomorphic image of the Szilard language generated by the labeled flat splicing systems of type (1, 2); (2, 3) and (4, 3) respectively. We also introduce the idea of Control languages for labeled finite flat splicing systems and show that any regular and context-free language can be generated as a Control language by these systems of type (1, 2) and (2,*) respectively. At the end we show that any recursively enumerable language can be generated as a Control language of labeled flat splicing systems of type (4, 3) when {\lambda}-labeled rules are allowed.

Improving Rasterization Using Distributed Archetypes

Abstract: Recent advances in cacheable models and psychoacoustic algorithms have paved the way for redundancy. After years of technical research into agents, we disconfirm the construction of consistent hashing. Our focus in this paper is not on whether the little-known introspec- tive algorithm for the visualization of redun- dancy by Martinez is recursively enumerable, but rather on proposing an analysis of Internet QoS (Tivoli).

Derivation languages, descriptional complexity measures and decision problems of a class of flat splicing systems

Abstract: In this paper, we associate the idea of derivation languages with flat splicing systems and compare the families of derivation languages (Szilard and control languages) of these systems with the family of languages in Chomsky hierarchy. We show that the family of Szilard languages of labeled flat finite splicing systems of type $( m, n)$ (i.e., $SZLS_{n, FIN}^{m}$ ) and $REG$, $CF$ and $CS$ are incomparable. Also, it is decidable whether or not $SZ_{n, FIN}^{m}(\mathscr{LS}) \subseteq R$ and $R \subseteq SZ_{n, FIN}^{m}(\mathscr{LS})$ for any regular language $R$ and labeled flat finite splicing system $\mathscr{LS}$. Also, any non-empty regular, non-empty context-free and recursively enumerable language can be obtained as homomorphic image of Szilard language of the labeled flat finite splicing systems of type $(1, 2), (2, 2)$ and $(4, 2)$ respectively. We also introduce the idea of control languages for labeled flat finite splicing systems and show that any non-empty regular and context-free language can be obtained as a control language of labeled flat finite splicing systems of type $(1,2)$ and $(2, 2)$ respectively. At the end, we show that any recursively enumerable language can be obtained as a control language of labeled flat finite splicing systems of type $(4,2)$ when $\lambda$-labeled rules are allowed.

Labelled graph rules in P systems

Abstract: P systems with graph productions were introduced by Freund in 2004. It was shown that such a variant of P system can generate any recursively enumerable language of weakly connected graphs. In this paper, we consider such P systems where the input is any connected graph and each graph production is labelled over an alphabet. We name the system as Process Guided P System where the computation is guided by a string over the set of labels. The set of all strings over the label alphabet which results in the halting computation of the system forms a guiding language of the system. We compare such guiding languages with Chomsky hierarchy. Two direct applications of this variant are also given.

On the Jumps of the Degrees Below a Recursively Enumerable Degree

Abstract: We consider the set of jumps below a Turing degree, given by JB(a) = {x(1) : x <= a}, with a focus on the problem: Which recursively enumerable (r.e.) degrees a are uniquely determined by JB(a)? Initially, this is motivated as a strategy to solve the rigidity problem for the partial order R of r.e. degrees. Namely, we show that if every high(2) r.e. degree a is determined by JB(a), then R cannot have a nontrivial automorphism. We then defeat the strategy-at least in the form presented-by constructing pairs a(0), a(1) of distinct r.e. degrees such that JB(a(0)) = JB(a(1)) within any possible jump class {x : x' = c}. We give some extensions of the construction and suggest ways to salvage the attack on rigidity.

Putnam’s Theorem on the Complexity of Models

Abstract: A streamlined proof of a theorem of Putnam’s: any satisfiable schema of predicate calculus has a model in which the predicates are interpreted as Boolean combinations of recursively enumerable relations. Related open problems are canvassed.

Decoupling RPCs From Byzantine Fault Tolerance in E-Business

Abstract: Developers agree that robust theory are an in- teresting new topic in the field of programming languages, and developers concur. Given the trends in signed modalities, physicists famously note the improvement of the producer-consumer problem, demonstrates the technical importance of programming languages. Here, we verify that although the famous knowledge-based al- gorithm for the understanding of compilers is recursively enumerable, the famous embedded algorithm for the synthesis of erasure coding is in Co-NP.