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Closure properties of hyper-minimized automata

Andrzej Szepietowski (2011)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Two deterministic finite automata are almost equivalent if they disagree in acceptance only for finitely many inputs. An automaton A is hyper-minimized if no automaton with fewer states is almost equivalent to A. A regular language L is canonical if the minimal automaton accepting L is hyper-minimized. The asymptotic state complexity s∗(L) of a regular language L is the number of states of a hyper-minimized automaton for a language finitely different from L. In this paper we show that: (1) the class...

Closure properties of hyper-minimized automata

Andrzej Szepietowski (2012)

RAIRO - Theoretical Informatics and Applications

Two deterministic finite automata are almost equivalent if they disagree in acceptance only for finitely many inputs. An automaton A is hyper-minimized if no automaton with fewer states is almost equivalent to A. A regular language L is canonical if the minimal automaton accepting L is hyper-minimized. The asymptotic state complexity s∗(L) of a regular language L is the number of states of a hyper-minimized automaton for a language ...

Closure under union and composition of iterated rational transductions

D. Simplot, A. Terlutte (2010)

RAIRO - Theoretical Informatics and Applications

We proceed our work on iterated transductions by studying the closure under union and composition of some classes of iterated functions. We analyze this closure for the classes of length-preserving rational functions, length-preserving subsequential functions and length-preserving sequential functions with terminal states. All the classes we obtain are equal. We also study the connection with deterministic context-sensitive languages.

Commutative images of rational languages and the abelian kernel of a monoid

Manuel Delgado (2001)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Natural algorithms to compute rational expressions for recognizable languages, even those which work well in practice, may produce very long expressions. So, aiming towards the computation of the commutative image of a recognizable language, one should avoid passing through an expression produced this way. We modify here one of those algorithms in order to compute directly a semilinear expression for the commutative image of a recognizable language. We also give a second modification of the algorithm...

Commutative images of rational languages and the Abelian kernel of a monoid

Manuel Delgado (2010)

RAIRO - Theoretical Informatics and Applications

Natural algorithms to compute rational expressions for recognizable languages, even those which work well in practice, may produce very long expressions. So, aiming towards the computation of the commutative image of a recognizable language, one should avoid passing through an expression produced this way. We modify here one of those algorithms in order to compute directly a semilinear expression for the commutative image of a recognizable language. We also give a second modification of the algorithm...

Comparing Complexity Functions of a Language and Its Extendable Part

Arseny M. Shur (2008)

RAIRO - Theoretical Informatics and Applications

Right (left, two-sided) extendable part of a language consists of all words having infinitely many right (resp. left, two-sided) extensions within the language. We prove that for an arbitrary factorial language each of these parts has the same growth rate of complexity as the language itself. On the other hand, we exhibit a factorial language which grows superpolynomially, while its two-sided extendable part grows only linearly.

Construction of tree automata from regular expressions

Dietrich Kuske, Ingmar Meinecke (2011)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Since recognizable tree languages are closed under the rational operations, every regular tree expression denotes a recognizable tree language. We provide an alternative proof to this fact that results in smaller tree automata. To this aim, we transfer Antimirov's partial derivatives from regular word expressions to regular tree expressions. For an analysis of the size of the resulting automaton as well as for algorithmic improvements, we also transfer the methods of Champarnaud and Ziadi from words...

Construction of tree automata from regular expressions

Dietrich Kuske, Ingmar Meinecke (2011)

RAIRO - Theoretical Informatics and Applications

Since recognizable tree languages are closed under the rational operations, every regular tree expression denotes a recognizable tree language. We provide an alternative proof to this fact that results in smaller tree automata. To this aim, we transfer Antimirov's partial derivatives from regular word expressions to regular tree expressions. For an analysis of the size of the resulting automaton as well as for algorithmic improvements, we also transfer the methods of Champarnaud and Ziadi...

Corrigendum to our paper: How Expressions Can Code for Automata

Sylvain Lombardy, Jacques Sakarovitch (2010)

RAIRO - Theoretical Informatics and Applications

In a previous paper, we have described the construction of an automaton from a rational expression which has the property that the automaton built from an expression which is itself computed from a co-deterministic automaton by the state elimination method is co-deterministic. It turned out that the definition on which the construction is based was inappropriate, and thus the proof of the property was flawed. We give here the correct definition of the broken derived terms of an...

Deciding whether a relation defined in Presburger logic can be defined in weaker logics

Christian Choffrut (2008)

RAIRO - Theoretical Informatics and Applications

We consider logics on ℤ and ℕ which are weaker than Presburger arithmetic and we settle the following decision problem: given a k-ary relation on ℤ and ℕ which are first order definable in Presburger arithmetic, are they definable in these weaker logics? These logics, intuitively, are obtained by considering modulo and threshold counting predicates for differences of two variables.

Defect theorem in the plane

Włodzimierz Moczurad (2007)

RAIRO - Theoretical Informatics and Applications

We consider the defect theorem in the context of labelled polyominoes, i.e., two-dimensional figures. The classical version of this property states that if a set of n words is not a code then the words can be expressed as a product of at most n - 1 words, the smaller set being a code. We survey several two-dimensional extensions exhibiting the boundaries where the theorem fails. In particular, we establish the defect property in the case of three dominoes (n × 1 or 1 × n rectangles).

Distance desert automata and the star height problem

Daniel Kirsten (2005)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We introduce the notion of nested distance desert automata as a joint generalization of distance automata and desert automata. We show that limitedness of nested distance desert automata is PSPACE-complete. As an application, we show that it is decidable in 2 2 𝒪 ( n ) space whether the language accepted by an n -state non-deterministic automaton is of a star height less than a given integer h (concerning rational expressions with union, concatenation and iteration), which is the first ever complexity bound...

Distance desert automata and the star height problem

Daniel Kirsten (2010)

RAIRO - Theoretical Informatics and Applications

We introduce the notion of nested distance desert automata as a joint generalization of distance automata and desert automata. We show that limitedness of nested distance desert automata is PSPACE-complete. As an application, we show that it is decidable in 22O(n) space whether the language accepted by an n-state non-deterministic automaton is of a star height less than a given integer h (concerning rational expressions with union, concatenation and iteration), which is the first ever complexity...

Equational description of pseudovarieties of homomorphisms

Michal Kunc (2003)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

The notion of pseudovarieties of homomorphisms onto finite monoids was recently introduced by Straubing as an algebraic characterization for certain classes of regular languages. In this paper we provide a mechanism of equational description of these pseudovarieties based on an appropriate generalization of the notion of implicit operations. We show that the resulting metric monoids of implicit operations coincide with the standard ones, the only difference being the actual interpretation of pseudoidentities....

Equational description of pseudovarieties of homomorphisms

Michal Kunc (2010)

RAIRO - Theoretical Informatics and Applications

The notion of pseudovarieties of homomorphisms onto finite monoids was recently introduced by Straubing as an algebraic characterization for certain classes of regular languages. In this paper we provide a mechanism of equational description of these pseudovarieties based on an appropriate generalization of the notion of implicit operations. We show that the resulting metric monoids of implicit operations coincide with the standard ones, the only difference being the actual interpretation of pseudoidentities. As...

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