1,104 research outputs found
Bounded Parikh Automata
The Parikh finite word automaton model (PA) was introduced and studied by
Klaedtke and Ruess in 2003. Here, by means of related models, it is shown that
the bounded languages recognized by PA are the same as those recognized by
deterministic PA. Moreover, this class of languages is the class of bounded
languages whose set of iterations is semilinear.Comment: In Proceedings WORDS 2011, arXiv:1108.341
Two-Way Parikh Automata
Parikh automata extend automata with counters whose values can only be tested at the end of the computation, with respect to membership into a semi-linear set. Parikh automata have found several applications, for instance in transducer theory, as they enjoy a decidable emptiness problem.
In this paper, we study two-way Parikh automata. We show that emptiness becomes undecidable in the non-deterministic case. However, it is PSpace-C when the number of visits to any input position is bounded and the semi-linear set is given as an existential Presburger formula. We also give tight complexity bounds for the inclusion, equivalence and universality problems. Finally, we characterise precisely the complexity of those problems when the semi-linear constraint is given by an arbitrary Presburger formula
Weakly-Unambiguous Parikh Automata and Their Link to Holonomic Series
We investigate the connection between properties of formal languages and properties of their generating series, with a focus on the class of holonomic power series. We first prove a strong version of a conjecture by Castiglione and Massazza: weakly-unambiguous Parikh automata are equivalent to unambiguous two-way reversal bounded counter machines, and their multivariate generating series are holonomic. We then show that the converse is not true: we construct a language whose generating series is algebraic (thus holonomic), but which is inherently weakly-ambiguous as a Parikh automata language. Finally, we prove an effective decidability result for the inclusion problem for weakly-unambiguous Parikh automata, and provide an upper-bound on its complexity
Automates à contraintes semilinéaires = Automata with a semilinear constraint
Cette thèse présente une étude dans divers domaines de l'informatique
théorique de modèles de calculs combinant automates finis et contraintes
arithmétiques. Nous nous intéressons aux questions de décidabilité,
d'expressivité et de clôture, tout en ouvrant l'étude à la complexité, la
logique, l'algèbre et aux applications. Cette étude est présentée au travers
de quatre articles de recherche.
Le premier article, Affine Parikh Automata, poursuit l'étude de Klaedtke et Ruess
des automates de Parikh et en définit des généralisations et restrictions.
L'automate de Parikh est un point de départ de cette thèse; nous montrons que
ce modèle de calcul est équivalent à l'automate contraint que nous
définissons comme un automate qui n'accepte un mot que si le nombre de fois
que chaque transition est empruntée répond à une contrainte arithmétique.
Ce modèle est naturellement étendu à l'automate de Parikh affine qui
effectue une opération affine sur un ensemble de registres lors du
franchissement d'une transition. Nous étudions aussi l'automate de
Parikh sur lettres: un automate qui n'accepte un mot que si le nombre de
fois que chaque lettre y apparaît répond à une contrainte arithmétique.
Le deuxième article, Bounded Parikh Automata, étudie les langages
bornés des automates de Parikh. Un langage est borné s'il existe des
mots w_1, w_2, ..., w_k tels que chaque mot du langage peut s'écrire
w_1...w_1w_2...w_2...w_k...w_k. Ces langages sont
importants dans des domaines applicatifs et présentent usuellement de bonnes
propriétés théoriques. Nous montrons que dans le contexte des langages
bornés, le déterminisme n'influence pas l'expressivité des automates de
Parikh.
Le troisième article, Unambiguous Constrained Automata, introduit les
automates contraints non ambigus, c'est-à-dire pour lesquels il
n'existe qu'un chemin acceptant par mot reconnu par l'automate. Nous
montrons qu'il s'agit d'un modèle combinant une meilleure expressivité et de
meilleures propriétés de clôture que l'automate contraint déterministe. Le
problème de déterminer si le langage d'un automate contraint non ambigu est
régulier est montré décidable.
Le quatrième article, Algebra and Complexity Meet Contrained Automata,
présente une étude des représentations algébriques qu'admettent les automates
contraints et les automates de Parikh affines. Nous déduisons de ces
caractérisations des résultats d'expressivité et de complexité. Nous
montrons aussi que certaines hypothèses classiques en complexité
computationelle sont reliées à des résultats de séparation et de non clôture
dans les automates de Parikh affines.
La thèse est conclue par une ouverture à un possible approfondissement, au
travers d'un certain nombre de problèmes ouverts.This thesis presents a study from the theoretical computer science
perspective of computing models combining finite automata and arithmetic
constraints. We focus on decidability questions, expressiveness, and closure
properties, while opening the study to complexity, logic, algebra, and
applications. This thesis is presented through four research articles.
The first article, Affine Parikh Automata, continues the study of Klaedtke
and Ruess on Parikh automata and defines generalizations and restrictions of
this model. The Parikh automaton is one of the starting points of this
thesis. We show that this model of computation is equivalent to the
constrained automaton that we define as an automaton which accepts a word
only if the number of times each transition is taken satisfies a given
arithmetic constraint. This model is naturally extended to affine Parikh
automata, in which an affine transformation is applied to a set of registers
on taking a transition. We also study the Parikh automaton on letters, that
is, an automaton which accepts a word only if the number of times each letter
appears in the word verifies an arithmetic constraint.
The second article, Bounded Parikh Automata, focuses on the
bounded languages of Parikh automata. A language is bounded if there
are words w_1, w_2, ..., w_k such that every word in the language can be
written as w_1...w_1w_2...w_2 ... w_k...w_k. These languages
are important in applications and usually display good theoretical
properties. We show that, over the bounded languages, determinism does not
influence the expressiveness of Parikh automata.
The third article, Unambiguous Constrained Automata, introduces the
concept of unambiguity in constrained automata. An automaton is
unambiguous if there is only one accepting path per word of its language. We
show that the unambiguous constrained automaton is an appealing model of
computation which combines a better expressiveness and better closure
properties than the deterministic constrained automaton. We show that it is
decidable whether the language of an unambiguous constrained automaton is
regular.
The fourth article, Algebra and Complexity Meet Constrained Automata,
presents a study of algebraic representations of constrained automata and
affine Parikh automata. We deduce expressiveness and complexity results from
these characterizations. We also study how classical computational
complexity hypotheses help in showing separations and nonclosure properties
in affine Parikh automata.
The thesis is concluded by a presentation of possible future avenues of
research, through several open problems
On the Path-Width of Integer Linear Programming
We consider the feasibility problem of integer linear programming (ILP). We
show that solutions of any ILP instance can be naturally represented by an
FO-definable class of graphs. For each solution there may be many graphs
representing it. However, one of these graphs is of path-width at most 2n,
where n is the number of variables in the instance. Since FO is decidable on
graphs of bounded path- width, we obtain an alternative decidability result for
ILP. The technique we use underlines a common principle to prove decidability
which has previously been employed for automata with auxiliary storage. We also
show how this new result links to automata theory and program verification.Comment: In Proceedings GandALF 2014, arXiv:1408.556
An approach to computing downward closures
The downward closure of a word language is the set of all (not necessarily
contiguous) subwords of its members. It is well-known that the downward closure
of any language is regular. While the downward closure appears to be a powerful
abstraction, algorithms for computing a finite automaton for the downward
closure of a given language have been established only for few language
classes.
This work presents a simple general method for computing downward closures.
For language classes that are closed under rational transductions, it is shown
that the computation of downward closures can be reduced to checking a certain
unboundedness property.
This result is used to prove that downward closures are computable for (i)
every language class with effectively semilinear Parikh images that are closed
under rational transductions, (ii) matrix languages, and (iii) indexed
languages (equivalently, languages accepted by higher-order pushdown automata
of order 2).Comment: Full version of contribution to ICALP 2015. Comments welcom
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