Specific "scientific" data structures, and their processing

Programming physicists use, as all programmers, arrays, lists, tuples, records, etc., and this requires some change in their thought patterns while converting their formulae into some code, since the "data structures" operated upon, while elaborating some theory and its consequences, are rather: power series and Pad\'e approximants, differential forms and other instances of differential algebras, functionals (for the variational calculus), trajectories (solutions of differential equations), Young diagrams and Feynman graphs, etc. Such data is often used in a [semi-]numerical setting, not necessarily "symbolic", appropriate for the computer algebra packages. Modules adapted to such data may be "just libraries", but often they become specific, embedded sub-languages, typically mapped into object-oriented frameworks, with overloaded mathematical operations. Here we present a functional approach to this philosophy. We show how the usage of Haskell datatypes and - fundamental for our tutorial - the application of lazy evaluation makes it possible to operate upon such data (in particular: the "infinite" sequences) in a natural and comfortable manner.Comment: In Proceedings DSL 2011, arXiv:1109.032

Some Combinatorial Operators in Language Theory

Multitildes are regular operators that were introduced by Caron et al. in order to increase the number of Glushkov automata. In this paper, we study the family of the multitilde operators from an algebraic point of view using the notion of operad. This leads to a combinatorial description of already known results as well as new results on compositions, actions and enumerations.Comment: 21 page

P automata revisited

We continue here the investigation of P automata, in their non-extended case, a class of devices which characterize non-universal family of languages. First, a recent conjecture is confirmed: any recursively enumerable language is obtained from a language recognized by a P automaton, to which an initial (arbitrarily large) string is added. Then, we discuss possibilities of extending P automata, following suggestions from string finite automata. For instance, automata with a memory (corresponding to push-down automata) are considered and their power is briefly investigated, as well as some closure properties of the family of languages recognized by P automata. In the context, a brief survey of results about P and dP automata (a distributed version of P automata) is provided, and several further research topics are formulated.Junta de Andalucía P08-TIC-0420

Reordering Derivatives of Trace Closures of Regular Languages

We provide syntactic derivative-like operations, defined by recursion on regular expressions, in the styles of both Brzozowski and Antimirov, for trace closures of regular languages. Just as the Brzozowski and Antimirov derivative operations for regular languages, these syntactic reordering derivative operations yield deterministic and nondeterministic automata respectively. But trace closures of regular languages are in general not regular, hence these automata cannot generally be finite. Still, as we show, for star-connected expressions, the Antimirov and Brzozowski automata, suitably quotiented, are finite. We also define a refined version of the Antimirov reordering derivative operation where parts-of-derivatives (states of the automaton) are nonempty lists of regular expressions rather than single regular expressions. We define the uniform scattering rank of a language and show that, for a regexp whose language has finite uniform scattering rank, the truncation of the (generally infinite) refined Antimirov automaton, obtained by removing long states, is finite without any quotienting, but still accepts the trace closure. We also show that star-connected languages have finite uniform scattering rank

Some Contributions to the Algebraic Theory of Automata

En el present treball estudiarem els autòmats des d'una perspectiva tant algebraica com coalgebraica. Volem aprofitar la natura dual d'aquests objectes per a presentar un marc unificador que explique i estenga alguns resultats recents de la teoria d'autòmats. Per tant, la secció 2 conté nocions i definicions preliminars per a mantenir el treball tan contingut com siga possible. Així, presentarem les nocions d'àlgebra i coàlgebra per a un endofunctor. També introduirem alguns conceptes sobre monoides i llenguatges. En aquest capítol també exposarem les nocions d'autòmats deterministes i no deterministes, homomorfismes i bisimulacions d'autòmats i productes i coproductes d'aquestes estructures. Finalment, recordarem algunes nocions bàsiques de teoria de reticles. Des d'una perspectiva algebraica, els autòmats són àlgebres amb operacions unàries. En aquest context, una equació és simplement un parell de paraules. Direm que una equació és satisfeta per un autòmat si per a cada estat inicial possible els estats als quals s'arriba des de l'estat considerat sota l'acció de les dues paraules coincideix. Es pot provar que, per a un autòmat donat, podem construir el major conjunt d'equacions que aquest satisfà. Aquest conjunt d'equacions resulta ser una congruència en el monoide lliure associat a l'alfabet d'entrada i ens permet definir l'autòmat lliure, denotat per free. Pel que respecta a la perspectiva coalgebraica, un autòmat és un sistema de transicions amb estats finals. Així, una coequació és un conjunt de llenguatges. Direm que una coequació és satisfeta per un autòmat, si per a cada observació possible (coloracions sobre els estats indicant-ne la finalitat o no), el llenguatge acceptat per l'autòmat es troba dins la coequació considerada. Intuïtivament, les coequacions poden ser pensades com comportaments o especificacions en el disseny que se suposa que una coàlgebra deu tindre. Com hem fet abans, per a un autòmat donat, podem construir el menor conjunt de coequacions que aquest satisfà. Aquest conjunt de coequacions resulta ser un subconjunt amb característiques ben determinades del conjunt de tots els llenguatges associats a l'alfabet d'entrada i ens permet definir l'autòmat colliure, denotat per cofree. Provem, a més, que aquestes construccions basades en equacions i coequacions són functorials. Al capítol 3 hem establert un nou resultat que presenta la dualitat entre quocients de congruència del monoide lliure i el seu conjunt de coequacions, que són àlgebres booleanes completes i atòmiques tancades sota derivació i que hem anomenat preformacions de llenguatges. Aquesta dualitat no imposa cap restricció en la grandària dels objectes, per tant, també s'aplica a objectes infinits. El capítol 3 està basat en els següents articles: - J.J.M.M. Rutten, A. Ballester-Bolinches, and E. Cosme-Llópez. Varieties and covarieties of languages (preliminary version). In D. Kozen and M. Mislove, editors, Proceedings of MFPS XXIX, volume 298 of Electron. Notes Theor. Comput. Sci., pages 7–28, 2013. - A. Ballester-Bolinches, E. Cosme-Llópez, and J. Rutten. The dual equivalence of equations and coequations for automata. Information and Computation, 244:49 – 75, 2015. Aquesta dualitat és emprada en el capítol 4 per a presentar un nou apropament al teorema de varietats d'Eilenberg. En primer lloc presentem una descripció equivalent, basada en equacions i coequacions, de la noció original de varietat de llenguatges d'Eilenberg. Aquesta nova descripció és un dels millors exemples possibles del poder expressiu del functors free i cofree. Una adaptació adient d'aquestes construccions permet presentar un resultat de tipus Eilenberg per a formacions de monoides no necessàriament finits. En el nostre cas, primerament provem que les formacions de monoides estan en correspondència biunívoca amb les formacions de congruències. Un segon pas en la prova relaciona formacions de congruències amb formacions de llenguatges. Així, provem que tots tres conceptes són equivalents Formacions de monoides -- Formacions de congruències -- Formacions de llenguatges La primera correspondència pareix ser completament nova i relaciona formacions de monoides amb filtres de congruències per a cada monoide. L'última correspondència és un dels millors exemples on poder aplicar la dualitat presentada al capítol 3. A més, donem una aplicació d'aquestes equivalències per al cas dels llenguatges relativament disjuntius. Aquests teoremes poden ser adequadament modificats per a cobrir el cas de les varietats de monoides en el sentit de Birkhoff. Discutim aquest cas particular al final del capítol 4. Els resultats d'aquest capítol han estat enviats per a la seua possible publicació en una revista científica sota el títol - A. Ballester-Bolinches, E. Cosme-Llópez, R. Esteban-Romero, and J. Rutten. Formations of monoids, congruences, and formal languages. 2015. El capítol 5 està completament dedicat a l'estudi de l'objecte final associat als autòmats no deterministes. En general, les tècniques emprades en el capítol 5 difereixen de les presentades en els capítols 3 i 4. En conseqüència, al principi d'aquest capítol introduïm alguns conceptes preliminars sobre bisimulacions i objectes finals. E l nostre resultat principal és presentat en el Teorema 5.17, que descriu l'autòmat final no determinista amb l'ajuda d'estructures basades en llenguatges. A continuació, relacionem altres descripcions de l'autòmat final no determinista amb la nostra construcció. El capítol 5 està basat en el següent article: - A. Ballester-Bolinches, E. Cosme-Llópez, and R. Esteban-Romero. A description based on languages of the final non-deterministic automaton. Theor. Comput. Sci., 536(0):1 – 20, 2014. Certament, els diferents punts de vista emprats en aquesta dissertació ja han estat explorats en alguns altres treballs. Per això, al final de cada capítol presentem un estudi detallat dels treballs relacionats i discutim les aportacions o millores realitzades en els resultats existents. Finalment, el capítol 6 presenta les conclusions i indica els treballs que caldrà realitzar en el futur. També presentem alguns del articles de recerca que es deriven de la realització d'aquest projecte.In the present work we want to study automata both from an algebraic perspective and a coalgebraic one. We want to exploit the dual nature of these objects and present a unifying framework to explain and extend some recent results in automata theory. Accordingly, Section 2 contains background material and definitions to keep the work as self-contained as possible. Thus, the notions of algebra and coalgebra for endofunctors are presented. We also introduce some basic concepts on monoids and languages. In this Chapter we also introduce the notions of deterministic and non-deterministic automata, homomorphisms and bisimulations of automata and the product and coproduct of these structures. Finally, we recall some basic notions of lattice theory. From the algebraic perspective, automata are algebras with unary operations. In this context, an equation is just a pair of words, and it holds in an automaton if for every initial state, the states reached from that state by both words are the same. It can be shown that, for a given automaton, we can construct the largest set of equations it satifies, which turns out to be a congruence on the free monoid on the input alphabet. We use this construction to define the free automaton associated to a given automaton, denoted by free. Coalgebraically, an automaton is a transition system with final states. A coequation is then a set of languages and it is satisfied by an automaton if, for every possible observation (colouring the states as either final or not) the language accepted by the automaton is within the specified coequation. Intuitively, coequations can be thought of as behaviours, or pattern specifications that a coalgebra is supposed to have. As we did before, for a given automaton, we can construct the smallest set of coequations it satifies, which turns out to be a special subset on the set of all languages over the input alphabet. We use this construction to define the cofree automaton associated to a given automaton, denoted by cofree. These constructions based on equations and coequations are proved to be functorial. In Chapter 3 we have established a new duality result between congruence quotients of the free monoid and its set of coequations, what we called preformations of languages, which are complete atomic boolean algebras closed under derivatives. This duality result does not impose any restriction on the size of the objects, therefore infinite objects are allowed. Chapter 3 is based on the following papers: - J.J.M.M. Rutten, A. Ballester-Bolinches, and E. Cosme-Llópez. Varieties and covarieties of languages (preliminary version). In D. Kozen and M. Mislove, editors, Proceedings of MFPS XXIX, volume 298 of Electron. Notes Theor. Comput. Sci., pages 7–28, 2013. - A. Ballester-Bolinches, E. Cosme-Llópez, and J. Rutten. The dual equivalence of equations and coequations for automata. Information and Computation, 244:49 – 75, 2015. This duality result is used in Chapter 4 to present a renewed approach to Eilenberg's variety theorem. In the first place, we introduce an equivalent description based on equations and coequations of the original notion of variety of regular language, originally introduced by Eilenberg. This description is one of the best examples of the expressiveness power of the aforementioned functors free and cofree. A suitable adaptation of this construction allows us to present an Eilenberg-like result for formations of (non-necessarily finite) monoids. In our case, we first prove that formations of monoids are in one-to-one correspondence with formations of congruences. A second step in our proof relates formations of congruences and formations of languages. All in all, these three concepts are shown to be equivalent Formations of monoids -- Formations of congruences -- Formations of languages The first correspondence seems to be completely new and relates formations of monoids to filters of congruences on every possible free monoid. The last correspondence is one of the best possible examples of application of the duality theorem presented in Chapter 3. We also give an application of this equivalence to the case of relatively disjunctive languages. These theorems can be slightly adapted to cover the case of varieties of monoids in the sense of Birkhoff. We discuss this particular case at the end of the Chapter 4. The results of this Chapter have been submitted to a journal for its possible publication under the title - A. Ballester-Bolinches, E. Cosme-Llópez, R. Esteban-Romero, and J. Rutten. Formations of monoids, congruences, and formal languages. 2015. Chapter 5 is completely devoted to the study of the final object associated to non-deterministic automata. In general, the techniques applied in Chapter 5 differ from those presented in Chapters 3 and 4. Consequently, at the beginning of this chapter we introduce some basic background on bisimulations and final objects. Our main result is presented in Theorem 5.17 which describes the final non-deterministic automaton with the help of structures based on languages. Hereafter, we relate other descriptions of the final non-deterministic automaton with our construction. Chapter 5 is based on the following paper: - A. Ballester-Bolinches, E. Cosme-Llópez, and R. Esteban-Romero. A description based on languages of the final non-deterministic automaton. Theor. Comput. Sci., 536(0):1 – 20, 2014. Certainly, the point of view that we adopt throughout this work has been explored in some other references too. Therefore, at the end of each Chapter, we present a detailed study of the related work and how our work subsumes or improves the existing results. Finally, Chapter 6 sets out the conclusions and indicates future work. We also present some of the derived research papers we have made during the realisation of this project

The Prefix PO- and Aspect in Russian and Polish: A Cognitive Grammar Account

This study examines the meanings of the prefix po- and verbal aspect in Russian and Polish in a Cognitive Grammar framework. The principles of Cognitive Grammar adopted in this study are based on Langacker (1991). This study follows Dickey's (2000) East-West division of aspect, within which the prototypical meanings of the Russian perfective and imperfective aspects are temporal definiteness and qualitative temporal indefiniteness, respectively, and the prototypical meanings of the Polish perfective and imperfective aspects are temporal definiteness/totality and quantitative temporal indefiniteness. According to Cognitive Grammar, the prototype is the most salient node in a network; this study is based around an analysis of the meaning and grammatical function of the Russian delimitative in po- and the Polish distributive in po- as prototypes in their respective semantic networks for po-. Regarding the methodological approach of this dissertation, in addition to relying on the views presented in the traditional literature, quantitative data is also presented, consisting of dictionary counts and hit counts and relative frequencies drawn from online corpora in support of the view that the delimitative is the prototype verb in po- in Russian and the distributive is the prototype verb in po- in Polish. The results of the corpus-based research show that the productivity and level of use are higher for the prototype Russian delimitative in po- relative to the Polish delimitative in po-and for the prototype Polish distributive in po- relative to the Russian distributive in po-. The main conclusion arrived upon in this study is that the meanings of the prototype verb in po- and the prototypical perfective meaning in each language overlap, which is manifested as the ability of the prototype verb in po- to function as a perfective partner in the grammar