A General Framework for the Derivation of Regular Expressions

The aim of this paper is to design a theoretical framework that allows us to perform the computation of regular expression derivatives through a space of generic structures. Thanks to this formalism, the main properties of regular expression derivation, such as the finiteness of the set of derivatives, need only be stated and proved one time, at the top level. Moreover, it is shown how to construct an alternating automaton associated with the derivation of a regular expression in this general framework. Finally, Brzozowski's derivation and Antimirov's derivation turn out to be a particular case of this general scheme and it is shown how to construct a DFA, a NFA and an AFA for both of these derivations.Comment: 22 page

Bottom Up Quotients and Residuals for Tree Languages

In this paper, we extend the notion of tree language quotients to bottom-up quotients. Instead of computing the residual of a tree language from top to bottom and producing a list of tree languages, we show how to compute a set of k-ary trees, where k is an arbitrary integer. We define the quotient formula for different combinations of tree languages: union, symbol products, compositions, iterated symbol products and iterated composition. These computations lead to the definition of the bottom-up quotient tree automaton, that turns out to be the minimal deterministic tree automaton associated with a regular tree language in the case of the 0-ary trees

Biens culturels, transmission de culture et croissance

Dans ce modèle à générations imbriquées à la Diamond, les ménages produisent domestiquement une culture (inobservable et transmissible) en combinant biens culturels et culture transmise. La transmission involontaire de culture aux enfants produit une externalité positive non prise en compte par les parents et constitue donc une source d’inefficacité. Le gouvernement peut atteindre l’optimum de premier rang s’il peut discriminer les consommateurs selon leur âge et subventionner uniquement les biens culturels achetés par les jeunes. Dans le cas Cobb-Douglas considéré ici, ce taux est égal à l’élasticité de l’externalité sur la production de culture des jeunes.We construct a two-periods OLG model in which households produce domestically a non-observable and transmissible commodity: "household culture" using cultural goods and transmitted culture as inputs. Parents fail to internalise the effect of unvoluntarily transmitted culture on their children's welfare. The social planner can implement the first best optimum if it is possible to subsidize cultural goods purchased by the young generation alone. When considering the Cobb-Douglas case, we find that the marginal subsidy rate is equal to the elasticity of cultural externality upon the cultural production of the young

Biens culturels, transmission de culture et croissance

We construct a two-periods OLG model in which households produce domestically a non-observable and transmissible commodity: "household culture" using cultural goods and transmitted culture as inputs. Parents fail to internalise the effect of unvoluntarily transmitted culture on their children's welfare. The social planner can implement the first best optimum if it is possible to subsidize cultural goods purchased by the young generation alone. When considering the Cobb-Douglas case, we find that the marginal subsidy rate is equal to the elasticity of cultural externality upon the cultural production of the young. Dans ce modèle à générations imbriquées à la Diamond, les ménages produisent domestiquement une culture (inobservable et transmissible) en combinant biens culturels et culture transmise. La transmission involontaire de culture aux enfants produit une externalité positive non prise en compte par les parents et constitue donc une source d’inefficacité. Le gouvernement peut atteindre l’optimum de premier rang s’il peut discriminer les consommateurs selon leur âge et subventionner uniquement les biens culturels achetés par les jeunes. Dans le cas Cobb-Douglas considéré ici, ce taux est égal à l’élasticité de l’externalité sur la production de culture des jeunes.

An optimal parallel algorithm to convert a regular expression into its Glushkov automaton

AbstractThe aim of this paper is to describe a CREW-PRAM optimal algorithm which converts a regular expression of size s into its Glushkov automaton in O(log s) time using O(s2log s) processors. This algorithm makes use of the star-normal form of an expression as defined by Brüggemann-Klein (1993) and is based on the sequential algorithm due to Ziadi et al. (1997) which computes an original representation of Glushkov automaton in O(s) time

From regular expressions to smaller NFAs

Several methods have been developed to construct -free automata that represent a regular expression. Among the most widely known are the position automaton (Glushkov), the partial derivatives automaton (Antimirov) and the follow automaton (Ilie and Yu). All these automata can be obtained with quadratic time complexity, thus, the comparison criterion is usually the size of the resulting automaton. The methods that obtain the smallest automata (although, for general expressions, they are not comparable), are the follow and the partial derivatives methods. In this paper, we propose another method to obtain a -free automaton from a regular expression. The number of states of the automata we obtain is bounded above by the size of both the partial derivatives automaton and of the follow automaton. Our algorithm also runs with the same time complexity of these methods. © 2011 Elsevier B.V. All rights reserved.This work was partially supported by the Spanish Ministerio de Educacion y Ciencia under project TIN2007-60769.García Gómez, P.; López Rodríguez, D.; Ruiz Ochando, J.; Álvarez Vargas, GI. (2011). From regular expressions to smaller NFAs. Theoretical Computer Science. 412(41):5802-5807. https://doi.org/10.1016/j.tcs.2011.05.058S580258074124

A maxmin problem on finite automata

AbstractWe solve the following problem proposed by Straubing. Given a two-letter alphabet A, what is the maximal number of states f(n) of the minimal automaton of a subset of An, the set of all words of length n. We give an explicit formula to compute f(n) and we show that 1= lim infn→∞nƒ(n)/2n≤lim supn→∞nƒ(n)/2n=2

Geometrical regular languages and linear Diophantine equations: The strongly connected case

AbstractGiven an arbitrarily large alphabet Σ, we consider the family of regular languages over Σ for which the deterministic minimal automaton has a strongly connected state diagram. We present a new method for checking whether such a language is semi-geometrical or not and whether it is geometrical or not. This method makes use of the enumeration of the simple cycles of the state diagram. It is based on the construction of systems of linear Diophantine equations, where the coefficients are deduced from the set of simple cycles

Two-Sided Derivatives for Regular Expressions and for Hairpin Expressions

The aim of this paper is to design the polynomial construction of a finite recognizer for hairpin completions of regular languages. This is achieved by considering completions as new expression operators and by applying derivation techniques to the associated extended expressions called hairpin expressions. More precisely, we extend partial derivation of regular expressions to two-sided partial derivation of hairpin expressions and we show how to deduce a recognizer for a hairpin expression from its two-sided derived term automaton, providing an alternative proof of the fact that hairpin completions of regular languages are linear context-free.Comment: 28 page