Fuzzy regular languages over finite and infinite words

We introduce four classes of Z-regular grammars for generating bi-infinite words (i.e. Z-words) and prove that they generate exactly Z-regular languages. We extend the second order monadic theory of one successor to the set of the integers (i.e. Z) and give some characterizations of this theory in terms of Z-regular grammars and Z-regular languages. We prove that this theory is decidable and equivalent to the weak theory. We also extend the linear temporal logic to Z-temporal logic and then prove that each Z-temporal formula is equivalent to a first order monadic formula. We prove that the correctness problem for finite state processes is decidable.

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AN ALGEBRAIC THEORY FOR REGULAR LANGUAGES OF FINITE AND INFINITE WORDS

International Journal of Algebra and Computation ◽

10.1142/s0218196793000287 ◽

1993 ◽

Vol 03 (04) ◽

pp. 447-489 ◽

Cited By ~ 39

Author(s):

THOMAS WILKE

Keyword(s):

Algebraic Theory ◽

Algebraic Approach ◽

Regular Languages ◽

Cantor Space ◽

Infinite Words ◽

Finite Semigroups ◽

Borel Hierarchy ◽

Algebraic Description ◽

One To One

An algebraic approach to the theory of regular languages of finite and infinite words (∞-languages) is presented. It extends the algebraic theory of regular languages of finite words, which is based on finite semigroups. Their role is taken over by a structure called right binoid. A variety theorem is proved: there is a one-to-one correspondence between varieties of ∞-languages and pseudovarieties of right binoids. The class of locally threshold testable languages and several natural subclasses (such as the class of locally testable languages) as well as classes of the Borel hierarchy over the Cantor space (restricted to regular languages) are investigated as examples for varieties of ∞-languages. The corresponding pseudovarieties of right binoids are characterized and in some cases defining equations are derived. The connections with the algebraic description and classification of regular languages of infinite words in terms of finite semigroups are pointed out.

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The complexity of weakly recognizing morphisms

RAIRO - Theoretical Informatics and Applications ◽

10.1051/ita/2018006 ◽

2018 ◽

Vol 53 (1-2) ◽

pp. 1-17

Author(s):

Lukas Fleischer ◽

Manfred Kufleitner

Keyword(s):

Computational Complexity ◽

Regular Languages ◽

Decision Problems ◽

Membership Problem ◽

Surjective Morphism ◽

Infinite Words ◽

Descriptional Complexity ◽

Emptiness Problem ◽

Free Semigroups ◽

Büchi Automaton

Weakly recognizing morphisms from free semigroups onto finite semigroups are a classical way for defining the class of ω-regular languages, i.e., a set of infinite words is weakly recognizable by such a morphism if and only if it is accepted by some Büchi automaton. We study the descriptional complexity of various constructions and the computational complexity of various decision problems for weakly recognizing morphisms. The constructions we consider are the conversion from and to Büchi automata, the conversion into strongly recognizing morphisms, as well as complementation. We also show that the fixed membership problem is NC1-complete, the general membership problem is in L and that the inclusion, equivalence and universality problems are NL-complete. The emptiness problem is shown to be NL-complete if the input is given as a non-surjective morphism.

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Separating the Words of a Language by Counting Factors

Fundamenta Informaticae ◽

10.3233/fi-2021-2047 ◽

2021 ◽

Vol 180 (4) ◽

pp. 375-393

Author(s):

Aleksi Saarela

Keyword(s):

Regular Languages ◽

Infinite Words ◽

Language L ◽

Finite Language

For a given language L, we study the languages X such that for all distinct words u, v ∈ L, there exists a word x ∈ X that appears a different number of times as a factor in u and in v. In particular, we are interested in the following question: For which languages L does there exist a finite language X satisfying the above condition? We answer this question for all regular languages and for all sets of factors of infinite words.

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Characterizing classes of regular languages using prefix codes of bounded synchronization delay

International Journal of Algebra and Computation ◽

10.1142/s021819671750028x ◽

2017 ◽

Vol 27 (06) ◽

pp. 561-589

Author(s):

Volker Diekert ◽

Tobias Walter

Keyword(s):

Finite Groups ◽

Abelian Groups ◽

Regular Languages ◽

Prefix Code ◽

Infinite Words ◽

Syntactic Monoid ◽

Prefix Codes ◽

Language Classes ◽

Restricted Type

This paper is motivated by the work of Schützenberger on codes with bounded synchronization delay. He was interested in characterizing those regular languages where groups in the syntactic monoid belong to a variety [Formula: see text]. On the language side he allowed the operations union, intersection, concatenation and modified Kleene-star involving a mapping of a prefix code of bounded synchronization delay to a group [Formula: see text], but no complementation. In our notation, this leads to the language classes [Formula: see text]. Our main result shows that [Formula: see text] coincides with the class of languages having syntactic monoids where all subgroups are in [Formula: see text]. We show that this statement holds for all varieties [Formula: see text] of finite groups, whereas Schützenberger proved this result for varieties [Formula: see text] containing Abelian groups, only. Our method shows the result for all [Formula: see text] simultaneously on finite and infinite words. Furthermore, we introduce the notion of local Rees extension which refers to a restricted type of the classical Rees extension. We give a decomposition of a monoid in terms of its groups and local Rees extensions. This gives a somewhat similar, but simpler decomposition than in the synthesis theorem of Rhodes and Allen. Moreover, we need a singly exponential number of operations, only. Finally, our decomposition yields an answer to a question in a recent paper of Almeida and Klíma about varieties that are closed under Rees extensions.

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Ambiguity, Weakness, and Regularity in Probabilistic Büchi Automata

Lecture Notes in Computer Science - Foundations of Software Science and Computation Structures ◽

10.1007/978-3-030-45231-5_27 ◽

2020 ◽

pp. 522-541

Author(s):

Christof Löding ◽

Anton Pirogov

Keyword(s):

Natural Generalization ◽

Regular Languages ◽

Natural Class ◽

Regularity Problem ◽

Infinite Words ◽

Büchi Automata ◽

Buchi Automata

AbstractProbabilistic Büchi automata are a natural generalization of PFA to infinite words, but have been studied in-depth only rather recently and many interesting questions are still open. PBA are known to accept, in general, a class of languages that goes beyond the regular languages. In this work we extend the known classes of restricted PBA which are still regular, strongly relying on notions concerning ambiguity in classical $$\omega $$ ω -automata. Furthermore, we investigate the expressivity of the not yet considered but natural class of weak PBA, and we also show that the regularity problem for weak PBA is undecidable.

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