Regular Varieties of Automata and Coequations

In this paper we use a duality result between equations and coequations for automata, proved by Ballester-Bolinches, Cosme-Ll´opez, and Rutten to characterize nonempty classes of deterministic automata that are closed under products, subautomata, homomorphic images, and sums. One characterization is as classes of automata defined by regular equations and the second one is as classes of automata satisfying sets of coequations called varieties of languages. We show how our results are related to Birkhoff’s theorem for regular varieties

The dual equivalence of equations and coequations for automata

Because of the isomorphism (X x A) -> X = X -> (A -> X), the transition structure t: X -> (A -> X) of a deterministic automaton with state set X and with inputs from an alphabet A can be viewed both as an algebra and as a coalgebra. Here we will use this algebra-coalgebra duality of automata as a common perspective for the study of equations and coequations. Equations are sets of pairs of words (v,w) tha

Guarded Kleene Algebra with Tests: Coequations, Coinduction, and Completeness

Guarded Kleene Algebra with Tests (GKAT) is an efficient fragment of KAT, as it allows for almost linear decidability of equivalence. In this paper, we study the (co)algebraic properties of GKAT. Our initial focus is on the fragment that can distinguish between unsuccessful programs performing different actions, by omitting the so-called early termination axiom. We develop an operational (coalgebraic) and denotational (algebraic) semantics and show that they coincide. We then characterize the behaviors of GKAT expressions in this semantics, leading to a coequation that captures the covariety of automata corresponding to these behaviors. Finally, we prove that the axioms of the reduced fragment are sound and complete w.r.t. the semantics, and then build on this result to recover a semantics that is sound and complete w.r.t. the full set of axioms

Guarded Kleene Algebra with Tests: Coequations, Coinduction, and Completeness

Guarded Kleene Algebra with Tests (GKAT) is an efficient fragment of KAT, as it allows for almost linear decidability of equivalence. In this paper, we study the (co)algebraic properties of GKAT. Our initial focus is on the fragment that can distinguish between unsuccessful programs performing different actions, by omitting the so-called early termination axiom. We develop an operational (coalgebraic) and denotational (algebraic) semantics and show that they coincide. We then characterize the behaviors of GKAT expressions in this semantics, leading to a coequation that captures the covariety of automata corresponding to these behaviors. Finally, we prove that the axioms of the reduced fragment are sound and complete w.r.t. the semantics, and then build on this result to recover a semantics that is sound and complete w.r.t. the full set of axioms

How to write a coequation

There is a large amount of literature on the topic of covarieties, coequations and coequational specifications, dating back to the early seventies. Nevertheless, coequations have not (yet) emerged as an everyday practical specification formalism for computer scientists. In this review paper, we argue that this is partly due to the multitude of syntaxes for writing down coequations, which seems to have led to some confusion about what coequations are and what they are for. By surveying the literature, we identify four types of syntaxes: coequations-as-corelations, coequations-as-predicates, coequations-as-equations, and coequations-as-modal-formulas. We present each of these in a tutorial fashion, relate them to each other, and discuss their respective uses

Duality of equations and coequations via contravariant adjunctions

In this paper we show duality results between categories of equations and categories of coequations. These dualities are obtained as restrictions of dualities between categories of algebras and coalgebras, which arise by lifting contravariant adjunctions on the base categories. By extending this approach to (co)algebras for (co)monads, we retrieve th

Formations of monoids, congruences, and formal languages

The main goal in this paper is to use a dual equivalence in automata theory started in [25] and developed in [3] to prove a general version of the Eilenberg-type theorem presented in [4]. Our principal results confirm the existence of a bijective correspondence between three concepts; formations of monoids, formations of languages and formations of congruences. The result does not require finiteness on monoids, nor regularity on languages nor finite index conditions on congruences. We relate our work to other results in the field and we include applications to non-r-disjunctive languages, Reiterman's equational description of pseudovarieties and varieties of monoid

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