Showing papers on "Recursively enumerable language published in 2016"
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TL;DR: It is shown that for various types of structures represented, there are minimal and maximal elements.
24 citations
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TL;DR: The concept of non-deterministic fuzzy Turing machine - NTFM is generalized, replacing the t-norm operator for several aggregation functions and establishing the languages accepted by these machines, called fuzzy recursively enumerable languages or simply LFRE and which classes of LFRE are closed under unions and intersections.
Abstract: There are several variations of fuzzy Turing machines in the literature, many of them require a t-norm in order to establish their accepted language. This paper generalize the concept of non-deterministic fuzzy Turing machine - NTFM, replacing the t-norm operator for several aggregation functions. We establish the languages accepted by these machines, called fuzzy recursively enumerable languages or simply LFRE and show, among other results, which classes of LFRE are closed under unions and intersections.
17 citations
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TL;DR: This work introduces a new variant numerical P systems with migrating variables (MNP systems), and investigates the computational power of MNP systems both as number generators and as string generators, working in the one-parallel or the sequential modes.
17 citations
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TL;DR: It is shown that there is no constructive proof for the incompleteness theorem using the $n$-consistency assumption, for $n\!>\!2$, and Godel-Rosser's Incompleteness Theorem is optimal in a sense.
Abstract: Godel's First Incompleteness Theorem is generalized to definable theories, which are not necessarily recursively enumerable, by using a couple of syntactic-semantic notions, one is the consistency of a theory with the set of all true $\Pi_n$-sentences or equivalently the $\Sigma_n$-soundness of the theory, and the other is $n$-consistency the restriction of $\omega$-consistency to the $\Sigma_n$-formulas. It is also shown that Rosser's Incompleteness Theorem does not generally hold for definable non-recursively enumerable theories, whence Godel-Rosser's Incompleteness Theorem is optimal in a sense. Though the proof of the incompleteness theorem using the $\Sigma_n$-soundness assumption is constructive, it is shown that there is no constructive proof for the incompleteness theorem using the $n$-consistency assumption, for $n\!>\!2$.
16 citations
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TL;DR: It is proved that recursively enumerable languages can be characterized as projections of inverse-morphic images of languages generated by such sequential SN P systems that are used as language generators.
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 following restriction: at each step the active neuron with the maximum (or minimum) number of spikes among the neurons that can spike will fire [if there is a tie for the maximum (or minimum) number of spikes stored in the active neurons, only one of the neurons containing the maximum (or minimum) 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 such sequential SN P systems. The relationships of the languages generated by these sequential SN P systems with finite and regular languages are also investigated.
13 citations
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11 Jul 2016
TL;DR: All the decidable and semidecidable properties can be obtained as combinations of two classes of basic decidable properties: (i) the function takes some particular values on a finite set of inputs, and (ii) every finite part of the function can be compressed to some extent.
Abstract: What can be decided or semidecided about a primitive recursive function, given a definition of that function by primitive recursion? What about subrecursive classes other than primitive recursive functions? We provide a complete and explicit characterization of the decidable and semidecidable properties. This characterization uses a variant of Kolmogorov complexity where only programs in a subrecursive programming language are allowed. More precisely, we prove that all the decidable and semidecidable properties can be obtained as combinations of two classes of basic decidable properties: (i) the function takes some particular values on a finite set of inputs, and (ii) every finite part of the function can be compressed to some extent.
10 citations
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TL;DR: An existence principle stating that the perception of the physical existence of any Turing program can serve as a physical causation for the application ofAny Turing-computable function to this Turing program is introduced.
Abstract: A fundamental question is whether Turing machines can model all reasoning processes. We introduce an existence principle stating that the perception of the physical existence of any Turing program can serve as a physical causation for the application of any Turing-computable function to this Turing program. The existence principle overcomes the limitation of the outputs of Turing machines to lists, that is, recursively enumerable sets. The principle is illustrated by productive partial functions for productive sets such as the set of the Goedel numbers of the Turing-computable total functions. The existence principle and productive functions imply the existence of physical systems whose reasoning processes cannot be modeled by Turing machines. These systems are called creative. Creative systems can prove the undecidable formula in Goedel's theorem in another formal system which is constructed at a later point in time. A hypothesis about creative systems, which is based on computer experiments, is introduced.
5 citations
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TL;DR: It is shown that five nodes are sufficient to accept (AHNEPs) or generate (GHN EPs) any recursively enumerable language and the more general result that any partial recursive relation can be computed by an HNEP with (at most) five nodes with the underlying graph structure for the communication between the evolutionary processors being the complete or the linear graph with five nodes.
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 show 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 with the underlying graph structure for the communication between the evolutionary processors being the complete or the linear graph with five nodes, whereas with a star-like communication graph we need six nodes. If the final results are defined by only taking the terminal strings out of the designated output node, then for these extended HNEPs we can prove that only four nodes are needed in all cases--for computing any partial recursive relation as well as for generating and accepting any recursively enumerable language--and the underlying communication structure can be a complete or a linear graph, but now even a star-like graph, too.
4 citations
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TL;DR: This paper constitutes a substantial extension to prior work on partial learning and studies several typical learning criteria in the model of partial learning of r.e. sets in the recursion-theoretic framework of inductive inference, leading to interesting consequences about the structural properties of the collection of classes learnable under these criteria.
3 citations
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TL;DR: It is proved that, under suitable conditions, a set defined through a Horn theory in a set \(\mathfrak {A}\) is recursively enumerable in models of a higher recursion theory, like primitive set recursion, \(\alpha \)-recursion, or \(\beta -recursion).
Abstract: We extend a classical result in ordinary recursion theory to higher recursion theory, namely that every recursively enumerable set can be represented in any model \(\mathfrak {A}\) by some Horn theory, where \(\mathfrak {A}\) can be any model of a higher recursion theory, like primitive set recursion, \(\alpha \)-recursion, or \(\beta \)-recursion. We also prove that, under suitable conditions, a set defined through a Horn theory in a set \(\mathfrak {A}\) is recursively enumerable in models of the above mentioned recursion theories.
2 citations
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TL;DR: In this article, it was shown that there exists a universal finitely presented torsion-free group, one which contains all finite presentations of the torsions of a group.
Abstract: For a set $X\subseteq \mathbb{N}$, we define the $X$-torsion of a group $G$ to be all elements $g\in G$ with $g^{n}=e$ for some $n\in X$. With $X$ recursively enumerable, we give two independent proofs (group-theoretic, and model-theoretic) that there exists a universal finitely presented $X$-torsion-free group; one which contains all finitely presented $X$-torsion-free groups. We also show that, if $X$ is recursively enumerable, then the set of finite presentations of $X$-torsion-free groups is $\Pi_{2}^{0}$-complete in Kleene's arithmetic hierarchy.
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19 Sep 2016TL;DR: It is proved that the transitive and reflexive closure of these operations has a decidable reachability problem for the case of short insertion-deletion rules (of size two/three and three/two).
Abstract: We introduce a new notion of a relational word as a finite totally ordered set of positions endowed with two 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 reachability 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 undecidability of reachability questions.
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TL;DR: In this paper, the Iterated Matching Pennies (IMP) game is introduced, which is a powerful framework for adversarial learnability, conventional (i.e., non-adversarial) learnability and approximability.
Abstract: We introduce a problem set-up we call the Iterated Matching Pennies (IMP) game and show that it is a powerful framework for the study of three problems: adversarial learnability, conventional (i.e., non-adversarial) learnability and approximability. Using it, we are able to derive the following theorems. (1) It is possible to learn by example all of $\Sigma^0_1 \cup \Pi^0_1$ as well as some supersets; (2) in adversarial learning (which we describe as a pursuit-evasion game), the pursuer has a winning strategy (in other words, $\Sigma^0_1$ can be learned adversarially, but $\Pi^0_1$ not); (3) some languages in $\Pi^0_1$ cannot be approximated by any language in $\Sigma^0_1$.
We show corresponding results also for $\Sigma^0_i$ and $\Pi^0_i$ for arbitrary $i$.
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TL;DR: In this paper, the strength of Ramsey's theorem for pairs restricted to recursive assignments of $k-many colors, with respect to intuitionistic Heyting arithmetic, was studied and shown to be equivalent to the Limited Lesser Principle of Omniscience for formulas over intuitionistic arithmetic.
Abstract: The purpose is to study the strength of Ramsey's Theorem for pairs restricted to recursive assignments of $k$-many colors, with respect to Intuitionistic Heyting Arithmetic. We prove that for every natural number $k \geq 2$, Ramsey's Theorem for pairs and recursive assignments of $k$ colors is equivalent to the Limited Lesser Principle of Omniscience for $\Sigma^0_3$ formulas over Heyting Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is equivalent to: for every recursively enumerable infinite $k$-ary tree there is some $i < k$ and some branch with infinitely many children of index $i$.
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TL;DR: It is demonstrated that greatest solutions of such equations represent exactly the Σ 1 1 -sets in the analytical hierarchy, and all those sets can already be represented by systems in the resolved form X i = ? i ( X 1, ?, X n ) .
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TL;DR: In this article, the concept of a generic relation for algorithmic problems is introduced, which preserves the property of being decidable for a problem for almost all inputs and possesses the transitive property.
Abstract: We introduce the concept of a generic relation for algorithmic problems, which preserves the property of being decidable for a problem for almost all inputs and possesses the transitive property. As distinct from the classical m-reducibility relation, the generic relation under consideration does not possess the reflexive property: we construct an example of a recursively enumerable set that is generically incomparable with itself. We also give an example of a set that is complete with respect to the generic relation in the class of recursively enumerable sets.
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TL;DR: Production systems are introduced, viewed as counterparts of the combinatorial ones, that generate all recursively enumerable predicates in this way using as tools only elementary operations and functions from classical Analysis.
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TL;DR: In this paper, a hierarchy of partial combinatory algebras (pcas) is constructed by forcing them to represent certain functions (e.g., complement functions) relative to the complement function.
Abstract: We use a way to extend partial combinatory algebras (pcas) by forcing them to represent certain functions. In the case of Scott's Graph model, equality is computable relative to the complement function. However, the converse is not true. This creates a hierarchy of pcas which relates to similar structures of extensions on other pcas. We study one such structure on Kleene's second model and one on a pca equivalent but not isomorphic to it. For the recursively enumerable sub pca of the Graph model, results differ as we can compute the (partial) complement function using the equality.
01 Jan 2016
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01 Jan 2016TL;DR: In this paper, a solution to the open problem of recursive enumerability of unsatisfiable prenex formulae in the first-order Godel logic is presented, and the answer is affirmative even for a useful expansion by intermediate truth constants and the projection operator Open image in new window.
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 projection operator Open image in new window . The affirmative result for unsatisfiable prenex formulae of \(G_\infty ^\varDelta \) has been stated in [1]. In [2], 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.
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TL;DR: This paper demonstrates that these systems characterize the family of recursively enumerable languages, and is demostrated in both deterministic and nondeterministic versions of these systems.
Abstract: In this paper, we introduce a new kind of automata systems, called state-synchronized automata systems of degree n. In general, they consists of n pushdown automata, referred to as their components. These systems can perform a computation step provided that the concatenation of the current states of all their components belongs to a prescribed control language. As its main result, the paper demonstrates that these systems characterize the family of recursively enumerable languages. In fact, this characterization is demostrated in both deterministic and nondeterministic versions of these systems. Restricting their components, these systems provides less computational power.
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24 May 2016
TL;DR: The main contribution is the fact that a group for which all irreducible word problems are recursively enumerable must necessarily have solvable word problem.
Abstract: Descriptions of Groups using Formal Language Theory Gabriela Asli Rino Nesin This work treats word problems of finitely generated groups and variations thereof, such as word problems of pairs of groups and irreducible word problems of groups. These problems can be seen as formal languages on the generators of the group and as such they can be members of certain well-known language classes, such as the class of regular, one-counter, context-free, recursively enumerable or recursive languages, or less well known ones such as the class of terminal Petri net languages. We investigate what effect the class of these various problems has on the algebraic structure of the relevant group or groups. We first generalize some results on pairs of groups, which were previously proven for context-free pairs of groups only. We then proceed to look at irreducible word problems, where our main contribution is the fact that a group for which all irreducible word problems are recursively enumerable must necessarily have solvable word problem. We then investigate groups for which membership of the irreducible word problem in the class of recursively enumerable languages is not independent of generating set. Lastly, we prove that groups whose word problem is a terminal Petri net language are exactly the virtually abelian groups.
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25 Jul 2016
TL;DR: It is shown that no regular language has a MPFS which is recursively enumerable but not recursive, and for any regular language the authors can decide whether or not it has a context-free non-regular MPFS.
Abstract: We investigate non-regular maximal prefix-free subsets MPFS of regular languages. We give a method to decide whether or not a regular language has any non-regular MPFS.
Next, we prove that if a regular language has any non-regular MPFS, then it also has a MPFS which is context-sensitive but not context-free, it has a MPFS which is recursive but not context-sensitive, and it has a MPFS which is not recursively enumerable.
We show that no regular language has a MPFS which is recursively enumerable but not recursive. Finally, for any regular language we can decide whether or not it has a context-free non-regular MPFS.