Separating the Words of a Language by Counting Factors

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.

Fuzzy regular languages over finite and infinite words

2006 ◽  
Vol 157 (11) ◽  
pp. 1532-1549 ◽  
Author(s):  
Werner Kuich ◽  
George Rahonis
Keyword(s):  
Regular Languages ◽  
Infinite Words

FINITE STATE PROCESSES, Z-TEMPORAL LOGIC AND THE MONADIC THEORY OF THE INTEGERS

1992 ◽  
Vol 03 (03) ◽  
pp. 233-244 ◽  
Author(s):  
A. SAOUDI ◽  
D.E. MULLER ◽  
P.E. SCHUPP
Keyword(s):  
Temporal Logic ◽  
Linear Temporal Logic ◽  
Second Order ◽  
Regular Languages ◽  
Infinite Words ◽  
First Order ◽  
Monadic Theory ◽  
Finite State ◽  
Temporal Formula

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.

AN ALGEBRAIC THEORY FOR REGULAR LANGUAGES OF FINITE AND INFINITE WORDS

1993 ◽  
Vol 03 (04) ◽  
pp. 447-489 ◽  
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.

scholarly journals REPRESENTATIONS AND CHARACTERIZATIONS OF LANGUAGES IN CHOMSKY HIERARCHY BY MEANS OF INSERTION-DELETION SYSTEMS

2008 ◽  
Vol 19 (04) ◽  
pp. 859-871 ◽  
Author(s):  
GHEORGHE PĂUN ◽  
MARIO J. PÉREZ-JIMÉNEZ ◽  
TAKASHI YOKOMORI
Keyword(s):  
Dna Computing ◽  
Research Direction ◽  
Regular Languages ◽  
Turing Computability ◽  
Chomsky Hierarchy ◽  
Recursively Enumerable ◽  
Enumerable Language ◽  
Language L ◽  
Context Free

Insertion-deletion operations are much investigated in linguistics and in DNA computing and several characterizations of Turing computability and characterizations or representations of languages in Chomsky hierarchy were obtained in this framework. In this note we contribute to this research direction with a new characterization of this type, as well as with representations of regular and context-free languages, mainly starting from context-free insertion systems of as small as possible complexity. For instance, each recursively enumerable language L can be represented in a way similar to the celebrated Chomsky-Schützenberger representation of context-free languages, i.e., in the form L = h(L(γ) ∩ D), where γ is an insertion system of weight (3, 0) (at most three symbols are inserted in a context of length zero), h is a projection, and D is a Dyck language. A similar representation can be obtained for regular languages, involving insertion systems of weight (2,0) and star languages, as well as for context-free languages – this time using insertion systems of weight (3, 0) and star languages.

The complexity of weakly recognizing morphisms

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.

Roots and Powers in Regular Languages: Recognizing Nonregular Properties by Finite Automata

2020 ◽  
Vol 175 (1-4) ◽  
pp. 173-185
Author(s):  
Fabian Frei ◽  
Juraj Hromkovič ◽  
Juhani Karhumäki
Keyword(s):  
Lower Bound ◽  
Regular Language ◽  
Finite Automata ◽  
Regular Languages ◽  
Fractional Powers ◽  
Concrete Construction ◽  
Square Roots ◽  
Language L ◽  
Interesting Behavior ◽  
Fractional Length

It is well known that the set of powers of any given order, for example squares, in a regular language need not be regular. Nevertheless, finite automata can identify them via their roots. More precisely, we recall that, given a regular language L, the set of square roots of L is regular. The same holds true for the nth roots for any n and for the set of all nontrivial roots; we give a concrete construction for all of them. Using the above result, we obtain decision algorithms for many natural problems on powers. For example, it is decidable, given two regular languages, whether they contain the same number of squares at each length. Finally, we give an exponential lower bound on the size of automata identifying powers in regular languages. Moreover, we highlight interesting behavior differences between taking fractional powers of regular languages and taking prefixes of a fractional length. Indeed, fractional roots in a regular language can typically not be identified by finite automata.

ON SUBWORD SYMMETRY OF WORDS

2008 ◽  
Vol 19 (01) ◽  
pp. 243-250 ◽  
Author(s):  
ANTON ČERNÝ
Keyword(s):  
Initial Segment ◽  
Language L ◽  
Finite Language

We call a word L-symmetric with respect to a finite language L if it contains the same number of scattered subwords u as of uR for every word u from L. We show that increasing the size of the language L may lead to an unlimited refinement of the language of L-symmetric words. Further we prove that if a long enough initial segment of a D0L-sequence consists entirely of L-symmetric words, then all words in the sequence are L-symmetric.

scholarly journals COMPLEXITY OF ATOMS OF REGULAR LANGUAGES

2013 ◽  
Vol 24 (07) ◽  
pp. 1009-1027 ◽  
Author(s):  
JANUSZ BRZOZOWSKI ◽  
HELLIS TAMM
Keyword(s):  
Upper Bound ◽  
Regular Language ◽  
Regular Languages ◽  
State Complexity ◽  
Empty Intersection ◽  
Language L ◽  
The Right

The quotient complexity of a regular language L, which is the same as its state complexity, is the number of left quotients of L. An atom of a non-empty regular language L with n quotients is a non-empty intersection of the n quotients, which can be uncomplemented or complemented. An NFA is atomic if the right language of every state is a union of atoms. We characterize all reduced atomic NFAs of a given language, i.e., those NFAs that have no equivalent states. We prove that, for any language L with quotient complexity n, the quotient complexity of any atom of L with r complemented quotients has an upper bound of 2n − 1 if r = 0 or r = n; for 1 ≤ r ≤ n − 1 the bound is[Formula: see text] For each n ≥ 2, we exhibit a language with 2n atoms which meet these bounds.

Roots and Powers in Regular Languages: Recognizing Nonregular Properties by Finite Automata

2020 ◽  
Author(s):  
Fabian Frei ◽  
Juraj Hromkovič ◽  
Juhani Karhumäki
Keyword(s):  
Lower Bound ◽  
Regular Language ◽  
Finite Automata ◽  
Regular Languages ◽  
Fractional Powers ◽  
Concrete Construction ◽  
Square Roots ◽  
Language L ◽  
Interesting Behavior ◽  
Fractional Length

It is well known that the set of powers of any given order, for example squares, in a regular language need not be regular. Nevertheless, finite automata can identify them via their roots. More precisely, we recall that, given a regular language L, the set of square roots of L is regular. The same holds true for the nth roots for any n and for the set of all nontrivial roots; we give a concrete construction for all of them. Using the above result, we obtain decision algorithms for many natural problems on powers. For example, it is decidable, given two regular languages, whether they contain the same number of squares at each length. Finally, we give an exponential lower bound on the size of automata identifying powers in regular languages. Moreover, we highlight interesting behavior differences between taking fractional powers of regular languages and taking prefixes of a fractional length. Indeed, fractional roots in a regular language can typically not be identified by finite automata.

A TIME AND SPACE EFFICIENT ALGORITHM FOR MINIMIZING COVER AUTOMATA FOR FINITE LANGUAGES

2003 ◽  
Vol 14 (06) ◽  
pp. 1071-1086 ◽  
Author(s):  
HEIKO KÖRNER
Keyword(s):  
Data Structure ◽  
Programming Language ◽  
Finite Automaton ◽  
Efficient Algorithm ◽  
Previous Method ◽  
Deterministic Finite Automaton ◽  
Time And Space ◽  
Language L ◽  
Finite Language ◽  
The Common

A deterministic finite automaton (DFA) [Formula: see text] is called a cover automaton (DFCA) for a finite language L over some alphabet Σ if [Formula: see text], with l being the length of some longest word in L. Thus a word w ∈ Σ* is in L if and only if |w| ≤ l and [Formula: see text]. The DFCA [Formula: see text] is minimal if no DFCA for L has fewer states. In this paper, we present an algorithm which converts an n–state DFA for some finite language L into a corresponding minimal DFCA, using only O(n log n) time and O(n) space. The best previously known algorithm requires O(n2) time and space. Furthermore, the new algorithm can also be used to minimize any DFCA, where the best previous method takes O(n4) time and space. Since the required data structure is rather complex, an implementation in the common programming language C/C++ is also provided.

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