Testing Whether a Binary and Prolongeable Regular Language L Is Geometrical or Not on the Minimal Deterministic Automaton of Pref(L)

2008 ◽  
pp. 68-77
Author(s):  
J. -M. Champarnaud ◽  
J. -Ph. Dubernard ◽  
H. Jeanne
Keyword(s):  
Regular Language ◽  
Deterministic Automaton ◽  
Language L

ORDINAL AUTOMATA AND CANTOR NORMAL FORM

2012 ◽  
Vol 23 (01) ◽  
pp. 87-98
Author(s):  
ZOLTÁN ÉSIK
Keyword(s):  
Normal Form ◽  
Polynomial Time ◽  
Regular Language ◽  
Finite Automata ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Order Type ◽  
Deterministic Finite Automaton ◽  
Lexicographic Ordering ◽  
Language L

It is known that an ordinal is the order type of the lexicographic ordering of a regular language if and only if it is less than ωω. We design a polynomial time algorithm that constructs, for each well-ordered regular language L with respect to the lexicographic ordering, given by a deterministic finite automaton, the Cantor Normal Form of its order type. It follows that there is a polynomial time algorithm to decide whether two deterministic finite automata accepting well-ordered regular languages accept isomorphic languages. We also give estimates on the state complexity of the smallest "ordinal automaton" representing an ordinal less than ωω, together with an algorithm that translates each such ordinal to an automaton.

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.

scholarly journals Decidability of regular language genus computation

2019 ◽  
Vol 29 (9) ◽  
pp. 1428-1443
Author(s):  
Guillaume Bonfante ◽  
Florian L. Deloup
Keyword(s):  
Regular Language ◽  
New Genus ◽  
Regular Languages ◽  
Minimal Size ◽  
Deterministic Automaton ◽  
Letter Alphabet ◽  
New Family ◽  
Generic Class ◽  
Deterministic Automata ◽  
The Difference

AbstractThis article continues the study of the genus of regular languages that the authors introduced in a 2013 paper (published in 2018). In order to understand further the genus g(L) of a regular language L, we introduce the genus size of |L|gen to be the minimal size of all finite deterministic automata of genus g(L) computing L.We show that the minimal finite deterministic automaton of a regular language can be arbitrarily far away from a finite deterministic automaton realizing the minimal genus and computing the same language, in terms of both the difference of genera and the difference in size. In particular, we show that the genus size |L|gen can grow at least exponentially in size |L|. We conjecture, however, the genus of every regular language to be computable. This conjecture implies in particular that the planarity of a regular language is decidable, a question asked in 1976 by R. V. Book and A. K. Chandra. We prove here the conjecture for a fairly generic class of regular languages having no short cycles. The methods developed for the proof are used to produce new genus-based hierarchies of regular languages and in particular, we show a new family of regular languages on a two-letter alphabet having arbitrary high genus.

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.

NON-UNIQUENESS AND RADIUS OF CYCLIC UNARY NFAs

2005 ◽  
Vol 16 (05) ◽  
pp. 883-896 ◽  
Author(s):  
MICHAEL DOMARATZKI ◽  
KEITH ELLUL ◽  
JEFFREY SHALLIT ◽  
MING-WEI WANG
Keyword(s):  
Diophantine Equation ◽  
Regular Language ◽  
Regular Languages ◽  
Language L ◽  
Minimal Radius ◽  
Unary Languages

In this paper we study some properties of cyclic unary regular languages. We find a connection between the uniqueness of the minimal NFA for certain cyclic unary regular languages and a Diophantine equation studied by Sylvester. We also obtain some results on the radius of unary languages. We show that the nondeterministic radius of a cyclic unary regular language L is not necessarily obtained by any of the minimal NFAs for L. We give a class of examples which demonstrates that the nondeterministic radius of a regular language cannot necessarily even be approximated by the minimal radius of its minimal NFAs.

scholarly journals Ambiguity Hierarchy of Regular Infinite Tree Languages

2021 ◽  
Vol Volume 17, Issue 3 ◽  
Author(s):  
Alexander Rabinovich ◽  
Doron Tiferet
Keyword(s):  
Regular Language ◽  
Deterministic Automaton ◽  
Infinite Trees ◽  
Infinite Tree ◽  
Tree Languages ◽  
Natural Way

An automaton is unambiguous if for every input it has at most one accepting computation. An automaton is k-ambiguous (for k > 0) if for every input it has at most k accepting computations. An automaton is boundedly ambiguous if it is k-ambiguous for some $k \in \mathbb{N}$. An automaton is finitely (respectively, countably) ambiguous if for every input it has at most finitely (respectively, countably) many accepting computations. The degree of ambiguity of a regular language is defined in a natural way. A language is k-ambiguous (respectively, boundedly, finitely, countably ambiguous) if it is accepted by a k-ambiguous (respectively, boundedly, finitely, countably ambiguous) automaton. Over finite words every regular language is accepted by a deterministic automaton. Over finite trees every regular language is accepted by an unambiguous automaton. Over $\omega$-words every regular language is accepted by an unambiguous B\"uchi automaton and by a deterministic parity automaton. Over infinite trees Carayol et al. showed that there are ambiguous languages. We show that over infinite trees there is a hierarchy of degrees of ambiguity: For every k > 1 there are k-ambiguous languages that are not k - 1 ambiguous; and there are finitely (respectively countably, uncountably) ambiguous languages that are not boundedly (respectively finitely, countably) ambiguous.

POWERS OF REGULAR LANGUAGES

2011 ◽  
Vol 22 (02) ◽  
pp. 323-330 ◽  
Author(s):  
SZILÁRD ZSOLT FAZEKAS
Keyword(s):  
Regular Language ◽  
Regular Languages ◽  
Language L ◽  
Unary Languages ◽  
Primitive Roots ◽  
Exponent Sets ◽  
Kleene Closure

In this paper we prove that it is decidable whether the set pow (L), which we get by taking all the powers of all the words in some regular language L, is regular or not. The problem was originally posed by Calbrix and Nivat in 1995. Partial solutions have been given by Cachat for unary languages and by Horváth et al. for various kinds of exponent sets for the powers and regular languages which have primitive roots satisfying certain properties. We show that the regular languages which have a regular power are the ones which are 'almost' equal to their Kleene-closure.

AN EFFICIENT ALGORITHM TO TEST WHETHER A BINARY AND PROLONGEABLE REGULAR LANGUAGE IS GEOMETRICAL

2009 ◽  
Vol 20 (04) ◽  
pp. 763-774 ◽  
Author(s):  
JEAN-MARC CHAMPARNAUD ◽  
JEAN PHILIPPE DUBERNARD ◽  
HADRIEN JEANNE
Keyword(s):  
Discrete Geometry ◽  
Efficient Algorithm ◽  
Regular Language ◽  
Practical Interest ◽  
Regular Languages ◽  
Regular Case ◽  
Deterministic Automaton ◽  
Model Based ◽  
Temporal Validation ◽  
The Family

Our aim is to present an efficient algorithm that checks whether a binary regular language is geometrical or not, based on specific properties of its minimal deterministic automaton. Geometrical languages have been introduced in the framework of off-line temporal validation of real-time softwares. Actually, validation can be achieved through both a model based on regular languages and a model based on discrete geometry. Geometrical languages are intended to develop a link between these two models. The regular case is of practical interest regarding to implementation features, which motivates the design of an efficient geometricity test addressing the family of regular languages.

EFFICIENT AUTOMATA CONSTRUCTIONS AND APPROXIMATE AUTOMATA

2008 ◽  
Vol 19 (01) ◽  
pp. 185-193 ◽  
Author(s):  
BRUCE W. WATSON ◽  
DERRICK G. KOURIE ◽  
TINUS STRAUSS ◽  
ERNEST KETCHA ◽  
LOEK CLEOPHAS
Keyword(s):  
Network Security ◽  
Pattern Matching ◽  
Data Structures ◽  
Regular Language ◽  
Practical Applications ◽  
Construction Algorithm ◽  
Language L ◽  
Security Pattern ◽  
Data Structures And Algorithms ◽  
A Performance

In this paper, we present data structures and algorithms for efficiently constructing approximate automata. An approximate automaton for a regular language L is one which accepts at leastL. Such automata can be used in a variety of practical applications, including network security pattern matching, in which false-matches are only a performance nuisance. The construction algorithm is particularly efficient, and is tunable to yield more or less exact automata.

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