Showing papers on "Recursively enumerable language published in 2017"
TL;DR: Here Woodin’s theorem is generalized to all recursively enumerable extensions T of the fragment${I}\rm{\Sigma }}_1} of PA, and removed the countability restriction on ${\cal M}$ when T extends PA.
Abstract: Let be a canonical enumeration of recursively enumerable sets, and suppose T is a recursively enumerable extension of PA (Peano Arithmetic) in the same language. Woodin (2011) showed that there exists an index (that depends on T) with the property that if is a countable model of T and for some -finite set s, satisfies , then has an end extension that satisfies T + W e = s. Here we generalize Woodin’s theorem to all recursively enumerable extensions T of the fragment of PA, and remove the countability restriction on when T extends PA. We also derive model-theoretic consequences of a classic fixed-point construction of Kripke (1962) and compare them with Woodin’s theorem.
14 citations
05 Jun 2017
TL;DR: It is shown that pure MIS of size (3) (i.e., having ternary matrices inserting one symbol in two symbol context) can characterize all recursively enumerable languages.
Abstract: We study matrix insertion grammars (MIS) towards representation of recursively enumerable languages with small size. We show that pure MIS of size (3; 1, 2, 2) (i.e., having ternary matrices inserting one symbol in two symbol context) can characterize all recursively enumerable languages. This is achieved by either applying an inverse morphism and a weak coding, or a left (right) quotient with a regular language or an intersection with a regular language followed by a weak coding. The obtained results complete known results on insertion-deletion systems from DNA computing area.
13 citations
03 Jul 2017
TL;DR: It is shown that whenever GCID systems describe \(\mathrm {LIN}\) with t components, this can be extended toGCID systems with just one more component to describe, for instance, 2-\(\Mathrm { LIN}\) and with further addition of one more components, the rational closure of \(\mathRM {LIN}\).
Abstract: A regulated extension of an insertion-deletion system known as graph-controlled insertion-deletion (GCID) system has several components and each component contains some insertion-deletion rules. A rule is applied to a string in a component and the resultant string is moved to the target component specified in the rule. When resources are so limited (especially, when deletion is context-free) then GCID systems are not known to describe the class of recursively enumerable languages. Hence, it becomes interesting to find the descriptional complexity of such GCID systems of small sizes with respect to language classes below \(\mathrm {RE}\). To this end, we consider closure classes of linear languages. We show that whenever GCID systems describe \(\mathrm {LIN}\) with t components, we can extend this to GCID systems with just one more component to describe, for instance, 2-\(\mathrm {LIN}\) and with further addition of one more component, we can extend to GCID systems that describe the rational closure of \(\mathrm {LIN}\).
11 citations
Dissertation•
28 Jun 2017
TL;DR: In this paper, a dual approach is proposed to explore shift spaces with a dual objective: on the one hand, the authors are interested in their dynamical properties and on the other hand, they study these abjects as computational models.
Abstract: Shift spaces are sets of colorings of a group which avoid a set of forbidden patterns and are endowed with a shift action. These spaces appear naturally as discrete versions of dynamical systems: they are obtained by partitioning the phase space and mapping each element into the sequence of partitions visited by its orbit.Severa! breakthroughs in this domain have pointed out the intricate relationship between dynamics of shift spaces and their computability properties. One remarkable example is the classification of the entropies of multidimensional subshifts of finite type as the set of right recursively enumerable numbers. This work explores shift spaces with a dual approach: on the one hand we are interested in their dynamical properties and on the ether hand we studythese abjects as computational models.Four salient results have been obtained as a result of this approach: (1) a combinatorial condition ensuring non-emptiness of subshifts on arbitrary countable groups; (2) a simulation theorem which realizes effective actions of finitely generated groups as factors of a subaction of a subshift of finite type; (3) a characterization of effectiveness with oracles using generalized Turing machines and (4) the undecidability of the torsion problem for two group invariants of shift spaces.As byproducts of these results we obtain a simple proof of the existence of strongly aperiodic subshifts in countable groups. Furthermore, we realize them as subshifts of finite type in the case of a semidirect product of a d-dimensional integer lattice with a finitely generated group with decida ble word problem whenever d> 1.
9 citations
01 Dec 2017
TL;DR: Results on the language family generated by the labelled splicing system in comparison with the language families of the Chomsky hierarchy, including recursively enumerable languages, are obtained by involving only either one or two membranes in the P systems considered.
Abstract: Labelled splicing P systems are distributed parallel computing models, where sets of strings that evolve by splicing rules are labelled. In this work, we consider labelled splicing systems with the following modifications: (i) The strings in the membranes are present in arbitrary number of copies; (ii) the rules in the regions are finite in number. Results on the language family generated by the labelled splicing system in comparison with the language families of the Chomsky hierarchy, including recursively enumerable languages, are obtained, by involving only either one or two membranes in the P systems considered.
8 citations
Posted Content•
TL;DR: In this paper, the Lambek calculus is extended using a unary connective, positive Kleene iteration, which corresponds to a variant of action logic that corresponds to general residuated Kleene algebras, not necessarily *-continuous.
Abstract: Formulae of the Lambek calculus are constructed using three binary connectives, multiplication and two divisions. We extend it using a unary connective, positive Kleene iteration. For this new operation, following its natural interpretation, we present two lines of calculi. The first one is a fragment of infinitary action logic and includes an omega-rule for introducing iteration to the antecedent. We also consider a version with infinite (but finitely branching) derivations and prove equivalence of these two versions. In Kleene algebras, this line of calculi corresponds to the *-continuous case. For the second line, we restrict our infinite derivations to cyclic (regular) ones. We show that this system is equivalent to a variant of action logic that corresponds to general residuated Kleene algebras, not necessarily *-continuous. Finally, we show that, in contrast with the case without division operations (considered by Kozen), the first system is strictly stronger than the second one. To prove this, we use a complexity argument. Namely, we show, using methods of Buszkowski and Palka, that the first system is $\Pi_1^0$-hard, and therefore is not recursively enumerable and cannot be described by a calculus with finite derivations.
7 citations
TL;DR: Arbitrary Arrow Update Logic with Common Knowledge (AAULC) as mentioned in this paper is a dynamic epistemic logic with an arrow update operator, which represents a particular type of information change and quantifies over arrow updates.
Abstract: Arbitrary Arrow Update Logic with Common Knowledge (AAULC) is a dynamic epistemic logic with (i) an arrow update operator, which represents a particular type of information change and (ii) an arbitrary arrow update operator, which quantifies over arrow updates.
By encoding the execution of a Turing machine in AAULC, we show that neither the valid formulas nor the satisfiable formulas of AAULC are recursively enumerable. In particular, it follows that AAULC does not have a recursive axiomatization.
7 citations
18 Jul 2017
TL;DR: It is shown that, in contrast with the case without division operations (considered by Kozen), the first system is strictly stronger than the second one and is not recursively enumerable and cannot be described by a calculus with finite derivations.
Abstract: Formulae of the Lambek calculus are constructed using three binary connectives, multiplication and two divisions. We extend it using a unary connective, positive Kleene iteration. For this new operation, following its natural interpretation, we present two lines of calculi. The first one is a fragment of infinitary action logic and includes an omega-rule for introducing iteration to the antecedent. We also consider a version with infinite (but finitely branching) derivations and prove equivalence of these two versions. In Kleene algebras, this line of calculi corresponds to the *-continuous case. For the second line, we restrict our infinite derivations to cyclic (regular) ones. We show that this system is equivalent to a variant of action logic that corresponds to general residuated Kleene algebras, not necessarily *-continuous. Finally, we show that, in contrast with the case without division operations (considered by Kozen), the first system is strictly stronger than the second one. To prove this, we use a complexity argument. Namely, we show, using methods of Buszkowski and Palka, that the first system is \(\varPi _1^0\)-hard, and therefore is not recursively enumerable and cannot be described by a calculus with finite derivations.
7 citations
TL;DR: This work extends to spiking neural P systems a notion investigated in the “standard” membrane systems: the language of the traces of a distinguished object, in which a spike is distinguished by “marking” it and its path through the neurons of the system is followed, thus obtaining a language.
6 citations
Proceedings Article•
25 Jul 2017TL;DR: By encoding the execution of a Turing machine in AAULC, it is shown that neither the valid formulas nor the satisfiable formulas of AAU LC are recursively enumerable.
Abstract: Arbitrary Arrow Update Logic with Common Knowledge (AAULC) is a dynamic epistemic logic with (i) an arrow update operator, which represents a particular type of information change and (ii) an arbitrary arrow update operator, which quantifies over arrow updates.
By encoding the execution of a Turing machine in AAULC, we show that neither the valid formulas nor the satisfiable formulas of AAULC are recursively enumerable. In particular, it follows that AAULC does not have a recursive axiomatization.
5 citations
TL;DR: The existence of a recursively enumerable lambda theory where the number is always one or infinite is shown and it is shown that there are λ-theories such that some terms have only two fixed points.
Abstract: A natural question in the λ-calculus asks what is the possible number of fixed points of a combinator (closed term). A complete answer to this question is still missing (Problem 25 of TLCA Open Problems List) and we investigate the related question about the number of fixed points of a combinator in λ-theories. We show the existence of a recursively enumerable lambda theory where the number is always one or infinite. We also show that there are λ-theories such that some terms have only two fixed points. In a first example, this is obtained by means of a non-constructive (more precisely non-r.e.) λ-theory where the range property is violated. A second, more complex example of a non-r.e. λ-theory (with a higher unsolvability degree) shows that some terms can have only two fixed points while the range property holds for every term.
24 Jul 2017
TL;DR: This work investigates genPCol automata with input mappings that can be realized through the application of finite transducers to the string representations of multisets and shows that using unrestricted programs, these automata characterize the class of recursively enumerable languages.
Abstract: We investigate genPCol automata with input mappings that can be realized through the application of finite transducers to the string representations of multisets. We show that using unrestricted programs, these automata characterize the class of recursively enumerable languages. The same holds for systems with all-tape programs, having capacity at least two. In the case of systems with com-tape programs, we show that they characterize language classes which are closely related to those characterized by variants of P automata.
01 Jan 2017
TL;DR: It is shown that logics based on linear Kripke frames—with or without constant domains—that have a scattered end piece are not recursively enumerable.
Abstract: We show that logics based on linear Kripke frames--with or without constant domains--that have a scattered end piece are not recursively enumerable. This is done by reduction to validity in all finite classical models.
TL;DR: A natural sufficient condition on the family of basic neighbourhoods of computable elements that guarantees the existence of a principal computable numbering is presented and weakly-effective ω–continuous domains and the natural numbers with the discrete topology satisfy this condition.
Abstract: This paper is a part of the ongoing program of analysing the complexity of various problems in computable analysis in terms of the complexity of the associated index sets. In the framework of effectively enumerable topological spaces, we investigate the following question: given an effectively enumerable topological space whether there exists a computable numbering of all its computable elements. We present a natural sufficient condition on the family of basic neighbourhoods of computable elements that guarantees the existence of a principal computable numbering. We show that weakly-effective ω–continuous domains and the natural numbers with the discrete topology satisfy this condition. We prove weak and strong analogues of Rice's theorem for computable elements. Then, we construct principal computable numberings of partial majorant-computable real-valued functions and co-effectively closed sets and calculate the complexity of index sets for important problems such as root verification and function equality. For example, we show that, for partial majorant-computable real functions, the equality problem is Π1 1-complete.
Posted Content•
TL;DR: For a recursively enumerable theory U and a finite expansion of the signature of U that contains at least one predicate symbol of arity ≥ 2, this article showed that, for any finite extension of U in the expanded language that is conservative over U, there is a conservative extension in the language, such that the symbols of U are preserved.
Abstract: In this paper we provide a (negative) solution to a problem posed by Stanislaw Krajewski. Consider a recursively enumerable theory U and a finite expansion of the signature of U that contains at least one predicate symbol of arity $\ge$ 2. We show that, for any finite extension $\alpha$ of U in the expanded language that is conservative over U, there is a conservative extension $\beta$ of U in the expanded language, such that $\alpha\vdash\beta$ and $\beta
vdash\alpha$. The result is preserved when we consider either extensions or model-conservative extensions of U in stead of conservative extensions. Moreover, the result is preserved when we replace $\vdash$ as ordering on the finitely axiomatized extensions in the expanded language by a special kind of interpretability, to wit interpretability that identically translates the symbols of the U-language.
We show that the result fails when we consider an expansion with only unary predicate symbols for conservative extensions of U ordered by interpretability that preserves the symbols of U.
TL;DR: In this article, it was shown that for every natural number k ≥ 2, the strength of Ramsey's theorem for pairs restricted to recursive assignments of k-many colors is equivalent to the Limited Lesser Principle of Omniscience for Σ 0 3 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 ≥ 2, Ramsey's Theorem for pairs and recursive assignments of k colors is equivalent to the Limited Lesser Principle of Omniscience for Σ 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 .
TL;DR: The model of as mentioned in this paper is based on Turing machines with advice with a restricted scenario for the use of their advice during computations, and it has been shown that it is possible to construct a Turing machine with one-sided advice that can accept co-RE languages, even though the advice can be arbitrary.
Abstract: We resolve an old problem, namely to design a ‘natural’ machine model for accepting the complements of recursively enumerable languages. The model we present is based on Turing machines with ‘onesided’ advice, which are Turing machines with advice with a restricted scenario for the use of their advice during computations. We show that Turing machines with one-sided advice accept precisely the co-RE languages, even though the advices can be arbitrary. We prove that Turing machines with one-sided advice are not less powerful than their unrestricted version: for all languages L ∈ co-RE, L is accepted by an ordinary Turing machine with advice f if and only L is accepted by a Turing machine with one-sided advice f . We show that co-RE filters into an infinite proper hierarchy, based on the complexity (‘length’) of the advice f(n) a one-sided Turing machine needs for accepting a language. The model and the results dualise to the ordinary RE-languages.
TL;DR: This model allows us to gain more insight into the reasons for unsolvability of algorithmic decision problems from different perspectives and investigates some classes of a hierarchy that is defined semantically by the authors' deterministic oracle machines and that can be syntactically characterized by formulas whose quantifiers range only over an enumerable set.
Abstract: We consider a uniform model of computation over algebraic structures resulting from a generalization of the Turing machine and the BSS model of computation. This model allows us to gain more insight into the reasons for unsolvability of algorithmic decision problems from different perspectives. For example, classes of undecidable problems can be introduced in several ways by analogy with the classical arithmetical hierarchy and, for many structures, the different definitions lead to different hierarchies of undecidable problems. Here, we will investigate some classes of a hierarchy that is defined semantically by our deterministic oracle machines and that can be syntactically characterized by formulas whose quantifiers range only over an enumerable set. Starting from machines over algebraic structures endowed with some relations and containing an infinite recursively enumerable sequence of individuals, we will also consider this hierarchy for BSS RAM's over the reals and some undecidable problems defined by algebraic properties of the real numbers.
TL;DR: It is proved that a new type of transducer is actually a network of polarized evolutionary processors which receives a word as input and collects in the output node, when the computation halts, the translation of the input word.
24 Jul 2017
TL;DR: The result on bounded ComW seems to support such conjecture while the conjecture is not true for bounded ComN and ComR, and the class of recursively enumerable sets can be computed using ECPe systems with two membranes.
Abstract: We explore the computing power of Evolution-Communication P systems with energy (ECPe systems) considering dynamical communication measures, ComN, ComR and ComW. These measures consider the number of communication steps, communication rules and total energy used per communication step, respectively. In this paper, we address a previous conjecture that states that only semilinear sets can be generated with bounded ComX, \(X \in \{N,R,W\}\). Our result on bounded ComW seems to support such conjecture while the conjecture is not true for bounded ComN and ComR. We also show that the class of recursively enumerable sets can be computed using ECPe systems with two membranes. This improves a previous result that makes use of four membranes to show computational completeness.
TL;DR: Using the universal Diophantine representation of recursively enumerable sets of positive integers due to Matiyasevich, a Z-rational series r over a binary alphabet X is constructed which has a maximal image complexity and various undecidability results are obtained.
Abstract: By using the universal Diophantine representation of recursively enumerable sets of positive integers due to Matiyasevich we construct a Z-rational series r over a binary alphabet X which has a maximal image complexity in the sense that all recursively enumerable sets of positive integers are obtained as the sets of positive coefficients of the series w -1 r where w ∈ X ∗ . As a consequence we obtain various undecidability results for Z-rational series.
TL;DR: A constructive semantics for the language of set theory with atoms based on interpreting set variables by enumerable species is defined and the soundness of the axioms of the Zermelo–Fraenkel set theory is studied.
Abstract: A constructive semantics for the language of set theory with atoms based on interpreting set variables by enumerable species is defined. The soundness of the axioms of the Zermelo–Fraenkel set theory with this semantics is completely studied.
01 Sep 2017
TL;DR: The existence of the tt-mitotic hypersimpleasure set is proved in this article, which is not btt-mitotically monotone, but it is known that the set A is ttmitotic if there is an r.i.d. splitting (B, C) of A such that the sets B and C both belong to the same tt -degree of unsolvability as A.
Abstract: Let us adduce some definitions: If a recursively enumerable (r.e.) set A is a disjoint union of two sets B and C, then we say that B, C is an r.e. splitting of A The r.e. set A is tt-mitotic (btt-mitotic) if there is an r.e. splitting (B, C) of A such that the sets B and C both belong to the same tt — (btt -) degree of unsolvability, as the set A. In this paper the existence of the tt-mitotic hypersimple set, which is not btt-mitotic is proved.
25 Jul 2017
TL;DR: Arbitrary Arrow Update Logic with Common Knowledge (AAULC) as discussed by the authors is a dynamic epistemic logic with an arrow update operator, which represents a particular type of information change and quantifies over arrow updates.
Abstract: Arbitrary Arrow Update Logic with Common Knowledge (AAULC) is a dynamic epistemic logic with (i) an arrow update operator, which represents a particular type of information change and (ii) an arbitrary arrow update operator, which quantifies over arrow updates.
By encoding the execution of a Turing machine in AAULC, we show that neither the valid formulas nor the satisfiable formulas of AAULC are recursively enumerable. In particular, it follows that AAULC does not have a recursive axiomatization.