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

K4-free graphs as a free algebra

Graphs of treewidth at most two are the ones excluding the clique with four vertices (K4) as a minor, or equivalently, the graphs whose biconnected components are series-parallel. We turn those graphs into a finitely presented free algebra,answering positively a question by Courcelle and Engelfriet, in the case of treewidth two. First we propose a syntax for denoting these graphs: in addition to parallel composition and series composition, it suffices to consider the neutral elements of those operations and a unary transpose operation. Then we give a finite equationa lpresentation and we prove it complete: two terms from the syntax are congruent if and only if they denote the same graph

A characterization of the n-ary many-sorted closure operators and a many-sorted Tarski irredundant basis theorem

A theorem of single-sorted algebra states that, for a closure space (A, J ) and a natural number n, the closure operator J on the set A is n-ary if and only if there exists a single-sorted signature Σ and a Σ-algebra A such that every operation of A is of an arity ≤ n and J = SgA, where SgA is the subalgebra generating operator on A determined by A. On the other hand, a theorem of Tarski asserts that if J is an n-ary closure operator on a set A with n ≥ 2, then, for every i, j ∈ IrB(A, J ), where IrB(A, J ) is the set of all natural numbers which have the property of being the cardinality of an irredundant basis (≡ minimal generating set) of A with respect to J , if i < j and {i + 1, . . . , j − 1} ∩ IrB(A, J ) = Ø, then j − i ≤ n − 1. In this article we state and prove the many-sorted counterparts of the above theorems. But, we remark, regarding the first one under an additional condition: the uniformity of the many-sorted closure operator

When are profinite many-sorted algebras retracts of ultraproducts of finite many-sorted algebras?

For a set of sorts S and an S-sorted signature Σ we prove that a profinite Σ-algebra, i.e. a projective limit of a projective system of finite Σ-algebras, is a retract of an ultraproduct of finite Σ-algebras if the family consisting of the finite Σ-algebras underlying the projective system is with constant support. In addition, we provide a categorial rendering of the above result. Specifically, after obtaining a category where the objects are the pairs formed by a nonempty upward directed preordered set and by an ultrafilter containing the filter of the final sections of it, we show that there exists a functor from the just mentioned category whose object mapping assigns to an object a natural transformation which is a retraction

Eilenberg Theorems for Many-sorted Formations

A theorem of Eilenberg establishes that there exists a bijectionbetween the set of all varieties of regular languages and the set of all vari-eties of finite monoids. In this article after defining, for a fixed set of sortsSand a fixedS-sorted signature Σ, the concepts of formation of congruenceswith respect to Σ and of formation of Σ-algebras, we prove that the alge-braic lattices of all Σ-congruence formations and of all Σ-algebra formationsare isomorphic, which is an Eilenberg's type theorem. Moreover, under asuitable condition on the free Σ-algebras and after defining the concepts offormation of congruences of finite index with respect to Σ, of formation offinite Σ-algebras, and of formation of regular languages with respect to Σ, weprove that the algebraic lattices of all Σ-finite index congruence formations,of all Σ-finite algebra formations, and of all Σ-regular language formationsare isomorphic, which is also an Eilenberg's type theorem

Functoriality of the Schmidt construction

After proving, in a purely categorial way, that the inclusion functor InAlg(Σ) from Alg(Σ)⁠, the category of many-sorted Σ-algebras, to PAlg(Σ)⁠, the category of many-sorted partial Σ-algebras, has a left adjoint FΣ⁠, the (absolutely) free completion functor, we recall, in connection with the functor FΣ⁠, the generalized recursion theorem of Schmidt, which we will also call the Schmidt construction. Next, we define a category Cmpl(Σ)⁠, of Σ-completions, and prove that FΣ⁠, labelled with its domain category and the unit of the adjunction of which it is a part, is a weakly initial object in it. Following this, we associate to an ordered pair (α,f)⁠, where α=(K,γ,α) is a morphism of Σ-completions from F=(C,F,η) to G=(D,G,ρ) and f a homomorphism of D from the partial Σ-algebra A to the partial Σ-algebra B⁠, a homomorphism Υ G,0α(f):Schα(f)⟶B⁠. We then prove that there exists an endofunctor, Υ G,0α, of Mortw(D)⁠, the twisted morphism category of D⁠, thus showing the naturalness of the previous construction. Afterwards, we prove that, for every Σ-completion G=(D,G,ρ)⁠, there exists a functor ΥG from the comma category (Cmpl(Σ)↓G) to End(Mortw(D))⁠, the category of endofunctors of Mortw(D)⁠, such that ΥG,0⁠, the object mapping of ΥG⁠, sends a morphism of Σ-completion of Cmpl(Σ) with codomain G⁠, to the endofunctor ΥG,0

Some Contributions to the Theory of Transformation Monoids

The aim of this paper is to present some contributions to the theory of finite transformation monoids. The dominating influence that permutation groups have on transformation monoids is used to describe and characterise transitive transformation monoids and primitive transitive transformation monoids. We develop a theory that not only includes the analogs of several important theorems of the classical theory of permutation groups but also contains substantial information about the algebraic structure of the transformation monoids. Open questions naturally arising from the substantial paper of Steinberg [A theory of transformation monoids: combinatorics and representation theory. Electron. J. Combin. 17 (2010), no. 1, Research Paper 164, 56 pp] have been answered. Our results can also be considered as a further development in the hunt for a solution of the Černý conjecture

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

A comic page for the first isomorphism theorem

Given a homomorphism between algebras, there exists an isomorphism between the quotient of the domain by its kernel and the subalgebra in the codomain given by its image. This theorem, commonly known as the first isomorphism theorem, is a fundamental algebraic result. Different problems have been identified in its instruction, mainly related to the abstraction inherent to its content and to the lack of conceptual models to improve its understanding. In response to this situation, in this paper, we present an illustration that explores the narrative and graphical resources of comics with the aim of describing the set-theoretic elements that are involved in the proof of this theorem