A Robust Class of Regular Languages

2010

Regular expressions can be represented by expression tree and by flow diagram (syntax graph). Based on equivalences a possible normal form can be obtained. It has interesting properties in both graphical representations. The union-complexity of languages is defined. The union-complexity is 1 for the union-free languages. These languages can be given by regular expressions without the operation union. Some properties of them are described. There is an analogy with the star-free languages and the star-height problem of regular languages. We show that each regular language has finite unioncomplexity based on its description by a finite union of unionfree languages (union normal-form). For some special language classes, for instance for finite languages the union complexity is easily computed. For union-complexity of any regular languages lower and upper bounds are presented. This measure gives an infinite hierarchy of regular languages. The union-height can be defined and proved to be at most 1 allowing-ary unions.

Discrete Applied Mathematics, 1982

FSTTCS 2009, 2009

Recently, Mikołaj Bojańczyk introduced a class of max-regular languages, an extension of regular languages of infinite words preserving many of its usual properties. This new class can be seen as a different way of generalising the notion of regularity from finite to infinite words. This paper compares regular and max-regular languages in terms of topological complexity. It is proved that up to Wadge equivalence the classes coincide. Moreover, when restricted to ∆ 0 2-languages, the classes contain virtually the same languages. On the other hand, separating examples of arbitrary complexity exceeding ∆ 0 2 are constructed.

2015

In the last years, various meaningful extensions of ω-regular languages have been proposed in the literature, including ωB-regular languages (ω-regular languages extended with boundedness), ωS-regular languages (ω-regular languages extended with strict unboundedness), and ωBS-regular languages, which are obtained from the combination of ωBand ωS-regular languages. However, while its components satisfy a generalized closure property, namely, the complement of an ωB-regular (resp., ωS-regular) language is an ωS-regular (resp., ωB-regular) one, the class of ωBS-regular languages is not closed under complementation. The existence of non-ωBS-regular languages that are the complements of some ωBS-regular ones and express fairly natural properties of reactive systems motivates the search for larger, well-behaved classes of extended ω-regular languages. In this paper, we introduce the class of ωT -regular languages, that captures meaningful languages not belonging to the class of ωBS-regula...

Theoretical Computer Science

There have been several attempts to extend the notion of conjugacy from groups to monoids. The aim of this paper is study the decidability and independence of conjugacy problems for three of these notions (which we will denote by ∼p, ∼o, and ∼c) in certain classes of finitely presented monoids. We will show that in the class of polycyclic monoids, p-conjugacy is "almost" transitive, ∼c is strictly included in ∼p, and the p-and c-conjugacy problems are decidable with linear compexity. For other classes of monoids, the situation is more complicated. We show that there exists a monoid M defined by a finite complete presentation such that the c-conjugacy problem for M is undecidable, and that for finitely presented monoids, the c-conjugacy problem and the word problem are independent, as are the c-conjugacy and p-conjugacy problems.

RAIRO - Theoretical Informatics and Applications, 2009

We discuss some known and introduce some new hierarchies and reducibilities on regular languages, with the emphasis on the quantifier-alternation and difference hierarchies of the quasi-aperiodic languages. The non-collapse of these hierarchies and decidability of some levels are established. Complete sets in the levels of the hierarchies under the polylogtime and some quantifier-free reducibilities are found. Some facts about the corresponding degree structures are established. As an application, we characterize the regular languages whose balanced leaf-language classes are contained in the polynomial hierarchy. For any discussed reducibility we try to give motivations and open questions, in a hope to convince the reader that the study of these reducibilities is interesting for automata theory and computational complexity.

Lecture Notes in Computer Science, 2012

The paper continues the study of union-free and deterministic union-free languages. In contrast with the fact that every regular language can be described as a finite union of union-free languages, we show that the finite unions of deterministic union-free languages define a proper subfamily of regular languages. Then we examine the properties of this subfamily.

Acta Cybernetica, 2005

An algebraic characterization of the families of tree languages definable by syntactic monoids is presented. This settles a question raised by several authors.

In a recent thread of papers, we have introduced FQL, a precise specification language for test coverage, and developed the test case generation engine Fshell for ANSI C. In essence, an FQL test specification amounts to a set of regular languages, each of which has to be matched by at least one test execution. To describe such sets of regular languages, the FQL semantics uses an automata-theoretic concept known as rational sets of regular languages (RSRLs). RSRLs are automata whose alphabet consists of regular expressions. Thus, the language accepted by the automaton is a set of regular expressions. In this paper, we study RSRLs from a theoretic point of view. More specifically, we analyze RSRL closure properties under common set theoretic operations, and the complexity of membership checking, i.e., whether a regular language is an element of a RSRL. For all questions we investigate both the general case and the case of finite sets of regular languages. Although a few properties are left as open problems, the paper provides a systematic semantic foundation for the test specification language FQL.

Information and Computation/information and Control, 2007

This paper investigates the word problem for inverse monoids generated by a set Γ subject to relations of the form e = f , where e and f are both idempotents in the free inverse monoid generated by Γ. It is shown that for every fixed monoid of this form the word problem can be solved both in linear time on a RAM as well as in deterministic logarithmic space, which solves an open problem of Margolis and Meakin. For the uniform word problem, where the presentation is part of the input, EXPTIME-completeness is shown. For the Cayley-graphs of these monoids, it is shown that the first-order theory with regular path predicates is decidable. Regular path predicates allow to state that there is a path from a node x to a node y that is labeled with a word from some regular language. As a corollary, the decidability of the generalized word problem is deduced.