Decidability Of Problem Research Articles

Deterministic regular expressions with back-references

Most modern libraries for regular expression matching allow back-references (i.e., repetition operators) that substantially increase expressive power, but also lead to intractability. In order to find a better balance between expressiveness and tractability, we combine these with the notion of determinism for regular expressions used in XML DTDs and XML Schema. This includes the definition of a suitable automaton model, and a generalization of the Glushkov construction. We demonstrate that, compared to their non-deterministic superclass, these deterministic regular expressions with back-references have desirable algorithmic properties (i.e., efficiently solvable membership problem and some decidable problems in static analysis), while, at the same time, their expressive power exceeds that of deterministic regular expressions without back-references.

On axiomatizability and decidability of universal theories of hereditary classes of matroids

Two hereditary classes of matroids – the class of matroids of rank doesn’t exceeding a fixed positive integer k and the class of matroids of finite rank – are studied by means of the model theory. The problems of axiomatizability of these two classes of matroids as structures and the problems of algorithmic decidability of universal theories of these classes are considered. It is shown that the first class is finitely axiomatizable whereas the second one is nonaxiomatizable. Decidability of the universal theories of the both classes is proved.

On counting functions and slenderness of languages

We study counting-regular languages — these are languages L for which there is a regular language L′ such that the number of strings of length n in L and L′ are the same for all n. We show that the languages accepted by unambiguous nondeterministic Turing machines with a one-way read-only input tape and a reversal-bounded worktape are counting-regular. Many one-way acceptors are a special case of this model, such as reversal-bounded deterministic pushdown automata, reversal-bounded deterministic queue automata, and many others, and therefore all languages accepted by these models are counting-regular. This result is the best possible in the sense that the claim does not hold for either 2-ambiguous NPDA's, unambiguous NPDA's with no reversal-bound, and other models.We also study closure properties of counting-regular languages, and we study decidability problems in regards to counting-regularity. For example, it is shown that the counting-regularity of even some restricted subclasses of NPDA's is undecidable. Lastly, k-slender languages — where there are at most k words of any length — are also studied. Amongst other results, it is shown that it is decidable whether a language in any semilinear full trio is k-slender.

Decidability of knapsack problem for Baumslag-Solitar group

In this work we investigate a decidability problem of group version of the knapsack problem for Baumslag-Solitar group BS(p, q). We proved, that the knapsack problem is decidable for group BS(p, q) for coprime integers p > 1, q > 1. In the case when p = 1, q ∈ ℕ, we proved that knapsack problem is decidable for group BS(1, q) with some restriction on the input of the problem. However, the problem of the decidability of the knapsack problem for group BS(1, q) on the whole set of inputs remains open.

Regular and linear permutation languages

A permutation rule is a non-context-free rule whose both sides contain the same multiset of symbols with at least one non-terminal. This rule does not add or substitute any symbols in the sentential form, but can be used to change the order of neighbouring symbols. In this paper, we consider regular and linear grammars extended with permutation rules. It is established that the generative power of these grammars relies not only on the length of the permutation rules, but also whether we allow or forbid the usage of erasing rules. This is quite surprising, since there is only one non-terminal in sentential forms of derivations for regular or linear grammars. Some decidability problems and closure properties of the generated families of languages are investigated. We also show a link to a similar model which mixes the symbols: grammars with jumping derivation mode.

Definability and Decidability Problems in Number Theory

This workshop brought together experts working on variations of Hilbert’s Tenth Problem and more general decidability issues for structures other than the ring of integers arising naturally in number theory and algebraic geometry.

Экспертно-аналитический процесс выбора управляющих организационных воздействий с использованием корпоративного знания. Часть 1. Формальный анализ системы целей

Paper is devoted to the models and the methods of support for the special process of the alternative management action expert choice when the action has to be realized by organizational measures. Such the choice takes into account the influence of the action on the decidable problem situation and its probable negative aftereffects in organization activity field. In Part I the model of the organization goals is presented. This model includes such the types of goals as standard, planned, produced by stakeholders' interests and brought about the decisions made in organization ones. The goals included are both hard and soft ones. The properties of goals such as equivalence, opposition and accessibility are formalized by means the model proposed. The analysis of these properties that is realized under actual state of the decision field and goals field of organization is employed as important analytical support for the expert analytical choice process. It includes the functions of the context and recommendations forming for the action perspectivity expert evaluation that will be considered in the next part of this paper. Problems in programming 2017; 4: 079-088

Nontransitive temporal multiagent logic, information and knowledge, deciding algorithms

Multiagent and temporal logics are active domains in Information Sciences, CS, and AI. Attention has predominantly focused on the logics based on transitive relational models, with particular emphasis on transitive time. But this does not seem rather reliable assumption. Nontransitivity of passing information may be demonstrated with relative ease through persuasive examples. Therefore, we introduce and study multiagent temporal logics that are based on nontransitive linear time. Another innovative step is consideration of incomplete information: the information/knowledge with lacunas,—the linear time with forgettable intervals of time in the past. Technically, the most important problems are problems of satisfiability and decidability of suggested logics. The main results are the algorithms that compute satisfiability and solve decidability (and so provide solutions to these problems). The paper concludes by posing a series of open problems.

Decidability classes for mobile agents computing

We establish a classification of decision problems that are to be solved by mobile agents operating in unlabeled graphs, using a deterministic protocol. The classification is with respect to the ability of a team of agents to solve decision problems, possibly with the aid of additional information. In particular, our focus is on studying differences between the decidability of a decision problem by agents and its verifiability when a certificate for a positive answer is provided to the agents (the latter is to the former what NP is to P in the framework of sequential computing). We show that the class MAV of mobile agents verifiable problems is much wider than the class MAD of mobile agents decidable problems. Our main result shows that there exist natural MAV-complete problems: the most difficult problems in this class, to which all problems in MAV are reducible via a natural mobile computing reduction. Beyond the class MAV we show that, for a single agent, three natural oracles yield a strictly increasing chain of relative decidability classes.

Normal forms for semigroup amalgams

We define a class of inverse semigroup amalgams and derive normal forms for the amalgamated free products in the variety of semigroups. The class includes all amalgams of finite inverse semigroups, recently studied by Cherubini, Jajcayova, Meakin, Nuccio, Piochi and Rodaro (2005–2014), and lower bounded amalgams, that were introduced by the author (1997). We provide sufficient conditions for decidable word problem. We show that the word problem is decidable for an amalgamated free product of finite inverse semigroups. The normal forms can be used to study amalgams in subvarieties of inverse semigroups. In a forthcoming paper by the author, the results are used for varieties of semilattices of groups.

On Hardness of Multilinearization and VNP-Completeness in Characteristic 2

For a Boolean function f : {0, 1} n → {0, 1}, let fˆ be the unique multilinear polynomial such that f ( x ) = fˆ ( x ) holds for every x ˆ {0, 1} n . We show that, assuming VP ≠ VNP, there exists a polynomial-time computable f such that fˆ requires superpolynomial arithmetic circuits. In fact, this f can be taken as a monotone 2-CNF, or a product of affine functions. This holds over any field. To prove the results in characteristic 2, we design new VNP-complete families in this characteristic. This includes the polynomial EC n counting edge covers in a graph and the polynomial mclique n counting cliques in a graph with deleted perfect matching. They both correspond to polynomial-time decidable problems, a phenomenon previously encountered only in characteristic ≠ 2.

One-way reversible multi-head finite automata

One-way multi-head finite automata are considered towards their ability to perform reversible computations. It is shown that, for every number k≥1 of heads, there are problems which can be solved by one-way k-head finite automata, but not by any one-way reversible k-head finite automaton. Additionally, a proper head hierarchy is obtained for one-way reversible multi-head finite automata. Closure properties and decidability problems are investigated as well. It turns out that one-way reversible finite automata with two heads are still a powerful model, since almost all commonly studied problems are not even semidecidable. Finally, descriptional complexity aspects are studied and non-recursive trade-offs are shown.

On the lower bondary for time complexity of a decidability problem of a theory of integers with a successor function and the least fixed point operator

On the lower bondary for time complexity of a decidability problem of a theory of integers with a successor function and the least fixed point operator

On ramification indices of formal solutions of constructive linear ordinary differential systems

We consider full rank linear ordinary differential higher-order systems whose coefficients are computable power series. It is shown that the algorithmic problems connected with the ramification indices of irregular formal solutions of a given system are mostly undecidable even if we fix a conjectural value r of the ramification index. This enables us to obtain a strengthening of the theorem which has been proven earlier and states that we are not able to compute algorithmically the dimension of the space of all formal solutions although we can construct a basis for the subspace of regular solutions. In fact, it is impossible to compute algorithmically this dimension even if, in addition to the system, we know the list of all values of the ramification indices. However, there is nearby an algorithmically decidable problem: if a system S and integers r, d are such that for S the existence of d linearly independent formal solutions of ramification index r is guaranteed then one can compute such d solutions of S.

Procedures for searching local solutions of linear differential systems with infinite power series in the role of coefficients

Construction of Laurent, regular, and formal (exponential---logarithmic) solutions of full-rank linear ordinary differential systems is discussed. The systems may have an arbitrary order, and their coefficients are formal power series given algorithmically. It has been established earlier that the first two problems are algorithmically decidable and the third problem is not decidable. A restricted variant of the third problem was suggested for which the desired algorithm exists. In the paper, a brief survey of algorithms for the abovementioned decidable problems is given. Implementations of these algorithms in the form of Maple procedures with a uniform interface and data representation are suggested.

Automata column

In this column, Sylvain Schmitz summarizes two recent advances on the complexity of the reachability problem for vector addition systems with states. This is one of the most celebrated decidable problems in theoretical computer science, and until now, there was absolutely no upper bound on the complexity. It was only known that the algorithms run in finite time. The first advance described in this column, by Leroux and Schmitz, is a computable upper bound on the running time of the (original) algorithm for the problem. Admittedly, the upper bound puts the Ackermann function to shame -- read the great column to see what it is --- but it's an upper bound. The second advance, by Blondin, Finkel, Göller, Haase and McKenzie, is finding the exact complexity, namely PSPACE complete, of the two dimensional case, something that has also been open for several decades.

Returning Parallel Communicating Finite Automata with Communication Bounds: Hierarchies, Decidabilities, and Undecidabilities

Systems of deterministic finite automata communicating by sending their states upon request are investigated, when the amount of communication is restricted, that is, when the number of necessary communications during the computations of the system is bounded by a function depending on the length of the input. The computational power and decidability problems are studied for returning systems, where components are set back to their initial states after having answered communication requests. It is proved that an infinite, strict hierarchy of language families exists, induced by the number of messages sent by their most economical acceptors. It is shown that at least one gap in this hierarchy exists. Some levels in the hierarchy are investigated in more detail.

Constructive Geometrization of Thurston Maps and Decidability of Thurston Equivalence

The key result in the present paper is a direct analogue of the celebrated Thurston’s Theorem Douady and Hubbard (Acta Math 171:263–297, 1993) for marked Thurston maps with parabolic orbifolds. Combining this result with previously developed techniques, we prove that every Thurston map can be constructively geometrized in a canonical fashion. As a consequence, we give a partial resolution of the general problem of decidability of Thurston equivalence of two postcritically finite branched covers of $$S^2$$ (cf. Bonnot et al. Moscow Math J 12:747–763, 2012).

Pseudo-inversion: closure properties and decidability

We consider a pseudo-inversion operation inspired by biological events, such as DNA sequence transformations, where only parts of a string are reversed. We define the pseudo-inversion of a string $$w = uxv$$w=uxv to be the set of all strings $$v^Rxu^R$$vRxuR, where $$uv \ne \lambda $$uv?? and consider the operation from a formal language theoretic viewpoint. We show that regular languages are closed under the pseudo-inversion operation whereas context-free languages are not. Furthermore, we study the iterated pseudo-inversion operation and show that the iterated pseudo-inversion of a context-free language is recognized by a nondeterministic reversal-bounded multicounter machine. Finally, we introduce the notion of pseudo-inversion-freeness and examine closure properties and decidability problems for regular and context-free languages. We demonstrate that pseudo-inversion-freeness is decidable in polynomial time for regular languages and undecidable for context-free languages.

Non-transitive linear temporal logic and logical knowledge operations

© 2015 The Author, 2015. Published by Oxford University Press. All rights reserved.We study a linear temporal logic LTLNT with non-transitive time (with NEXT and UNTIL) and possible interpretations for logical knowledge operations in this approach. We assume time to be non-transitive, linear and discrete, it is a major innovative part of our article. Motivation for our approach that time might be non-transitive and comments on possible interpretations of logical knowledge operations are given. The main result of Section 5 is a solution of the decidability problem for LTLNT, we find and describe in details the decision algorithm. In Section 6 we introduce non-transitive linear temporal logic LTLNT(m) with uniform bound (m) for non-transitivity. We compare it with standard linear temporal logic LTL and the logic LTLNT - where non-transitivity has no upper bound - and show that LTLNT may be approximated by logics LTLNT(m). Concluding part of the article contains a list of open interesting problems.