Regularity Preserving but not Reflecting Encodings

Encodings, that is, injective functions from words to words, have been studied extensively in several settings. In computability theory the notion of encoding is crucial for defining computability on arbitrary domains, as well as for comparing the power of models of computation. In language theory much attention has been devoted to regularity preserving functions. A natural question arising in these contexts is: Is there a bijective encoding such that its image function preserves regularity of languages, but its pre-image function does not? Our main result answers this question in the affirmative: For every countable class C of languages there exists a bijective encoding f such that for every language L in C its image f[L] is regular. Our construction of such encodings has several noteworthy consequences. Firstly, anomalies arise when models of computation are compared with respect to a known concept of implementation that is based on encodings which are not required to be computable: Every countable decision model can be implemented, in this sense, by finite-state automata, even via bijective encodings. Hence deterministic finite-state automata would be equally powerful as Turing machine deciders. A second consequence concerns the recognizability of sets of natural numbers via number representations and finite automata. A set of numbers is said to be recognizable with respect to a representation if an automaton accepts the language of representations. Our result entails that there is one number representation with respect to which every recursive set is recognizable

Une application de la representation matricielle des transductions

RésuméOn étudie le problème suivant, fréquemment rencontré en théorie des langages: soient n langages L1,…,Ln reconnus par les monoïdes M1,…,Mn respectivement. Etant donné une opération ϕ, on cherche à construire un monoïde M, fonction de M1,…,Mn, qui reconnaisse le langage (L1,…,Ln)ϕ. Nous montrons que la plupart des constructions proposées dans la littérature pour ce type de problème sont en fait des cas particuliers d'une méthode générale que nous exposons ici. Cette méthode s'applique également à certains problèmes moins classiques relatifs par exemple à la réduction du groupe libre ou aux opérations de contrôle sur les T0L-systèmes.AbstractWe study the following classical problem of formal language theory: let L1,…,Ln be n languages recognized by the monoids M1,…,Mn respectively. Given an operation ϕ, we want to build a monoid M, function of M1,…,Mn, which recognizes the language (L1,…,Ln)ϕ. We show that most of the constructions given in the literature for this kind of problem are particular cases of a general method. This method can also be applied to some less classical problems related for example to the Dyck-reduction of the free-group or to control operations on T0L-systems

Modal logics on rational Kripke structures

This dissertation is a contribution to the study of infinite graphs which can be presented in a finitary way. In particular, the class of rational graphs is studied. The vertices of a rational graph are labeled by a regular language in some finite alphabet and the set of edges of a rational graph is a rational relation on that language. While the first-order logics of these graphs are generally not decidable, the basic modal and tense logics are. A survey on the class of rational graphs is done, whereafter rational Kripke models are studied. These models have rational graphs as underlying frames and are equipped with rational valuations. A rational valuation assigns a regular language to each propositional variable. I investigate modal languages with decidable model checking on rational Kripke models. This leads me to consider regularity preserving relations to see if the class can be generalised even further. Then the concept of a graph being rationally presentable is examined - this is analogous to a graph being automatically presentable. Furthermore, some model theoretic properties of rational Kripke models are examined. In particular, bisimulation equivalences between rational Kripke models are studied. I study three subclasses of rational Kripke models. I give a summary of the results that have been obtained for these classes, look at examples (and non-examples in the case of automatic Kripke frames) and of particular interest is finding extensions of the basic tense logic with decidable model checking on these subclasses. An extension of rational Kripke models is considered next: omega-rational Kripke models. Some of their properties are examined, and again I am particularly interested in finding modal languages with decidable model checking on these classes. Finally I discuss some applications, for example bounded model checking on rational Kripke models, and mention possible directions for further research

Newton’s Forward Difference Equation for Functions from Words to Words

International audienceNewton's forward difference equation gives an expression of a function from ℕ to ℤ in terms of the initial value of the function and the powers of the forward difference operator. An extension of this formula to functions from A* to ℤ was given in 2008 by P. Silva and the author. In this paper, the formula is further extended to functions from A* into the free group over B.L'équation aux différences avant de Newton donne une expression d'une fonction de ℕ dans ℤ en termes de la valeur initiale de la fonction et des puissances de l'opérateur de difference avant. Une extension de cette formule aux fonctions de A* dans ℤ a été donnée en 2008 par P. Silva et l'auteur. Dans cet article, cette formule est à nouveau généralisée, cette fois aux fonctions de A* dans le groupe libre sur B

A topological approach to transductions

AbstractThis paper is a contribution to the mathematical foundations of the theory of automata. We give a topological characterization of the transductions τ from a monoid M into a monoid N, such that if R is a recognizable subset of N,τ-1(R) is a recognizable subset of M. We impose two conditions on the monoids, which are fullfilled in all cases of practical interest: the monoids must be residually finite and, for every positive integer n, must have only finitely many congruences of index n. Our solution proceeds in two steps. First we show that such a monoid, equipped with the so-called Hall distance, is a metric space whose completion is compact. Next we prove that τ can be lifted to a map τ^ from M into the set of compact subsets of the completion of N. This latter set, equipped with the Hausdorff metric, is again a compact monoid. Finally, our main result states that τ-1 preserves recognizable sets if and only if τ^ is continuous

Operations preserving recognizable languages

Given a strictly increasing sequence s of non-negative integers, filtering a word a_0a_1 ... a_n by s consists in deleting the letters ai such that i is not in the set {s_0, s_1, ...}. By a natural generalization, denote by L[s], where L is a language, the set of all words of L filtered by s. The filtering problem is to characterize the filters s such that, for every regular language L, L[s] is regular. In this paper, the filtering problem is solved, and a unified approach is provided to solve similar questions, including the removal problem considered by Seiferas and McNaughton. Our approach relies on a detailed study of various residual notions, notably residually ultimately periodic sequences and residually rational transductions

Synthesis of Computable Regular Functions of Infinite Words

Regular functions from infinite words to infinite words can be equivalently specified by MSO-transducers, streaming ?-string transducers as well as deterministic two-way transducers with look-ahead. In their one-way restriction, the latter transducers define the class of rational functions. Even though regular functions are robustly characterised by several finite-state devices, even the subclass of rational functions may contain functions which are not computable (by a Turing machine with infinite input). This paper proposes a decision procedure for the following synthesis problem: given a regular function f (equivalently specified by one of the aforementioned transducer model), is f computable and if it is, synthesize a Turing machine computing it. For regular functions, we show that computability is equivalent to continuity, and therefore the problem boils down to deciding continuity. We establish a generic characterisation of continuity for functions preserving regular languages under inverse image (such as regular functions). We exploit this characterisation to show the decidability of continuity (and hence computability) of rational and regular functions. For rational functions, we show that this can be done in NLogSpace (it was already known to be in PTime by Prieur). In a similar fashion, we also effectively characterise uniform continuity of regular functions, and relate it to the notion of uniform computability, which offers stronger efficiency guarantees

Synthesis of Computable Regular Functions of Infinite Words

Regular functions from infinite words to infinite words can be equivalently specified by MSO-transducers, streaming $\omega$-string transducers as well as deterministic two-way transducers with look-ahead. In their one-way restriction, the latter transducers define the class of rational functions. Even though regular functions are robustly characterised by several finite-state devices, even the subclass of rational functions may contain functions which are not computable (by a Turing machine with infinite input). This paper proposes a decision procedure for the following synthesis problem: given a regular function $f$ (equivalently specified by one of the aforementioned transducer model), is $f$ computable and if it is, synthesize a Turing machine computing it. For regular functions, we show that computability is equivalent to continuity, and therefore the problem boils down to deciding continuity. We establish a generic characterisation of continuity for functions preserving regular languages under inverse image (such as regular functions). We exploit this characterisation to show the decidability of continuity (and hence computability) of rational and regular functions. For rational functions, we show that this can be done in $\mathsf{NLogSpace}$ (it was already known to be in $\mathsf{PTime}$ by Prieur). In a similar fashion, we also effectively characterise uniform continuity of regular functions, and relate it to the notion of uniform computability, which offers stronger efficiency guarantees

Tool for Abstract Regular Model Checking

Metody formální verifikace mohou poskytnout automatizované ověření korektnosti softwaru (stavěné na matematických základech), což je velmi důležité. Jednou z těchto metod je abstraktní regulární model checking, jenž používá konečné automaty a převodníky pro reprezentaci množiny dosažitelných konfigurací, respektive jednokrokového přechodu mezi těmito konfiguracemi. Přestože tato metoda řeší obecně nerozhodnutelné problémy, umožňuje terminaci v mnoha praktických případech a navíc výrazně zmírňuje problém stavové exploze. Tohoto dosahuje urychlením výpočtu dosažitelných stavů pomocí inkrementálního zjemňování abstrakcí, k odstranění neplatných protipříkladů vzniklých nadaproximací pak slouží technika zjemňování abstrakce založená na protipříkladech. Cílem této práce je vytvořit dobře navržený nástroj pro abstraktní regulární model checking, jenž byl dosud implementován pouze v prototypech. Nový nástroj bude systémy modelovat pomocí symbolických automatů a převodníků namísto jejich (méně stručných) klasických alternativ.Formal verification methods offer a large potential to provide automated software correctness checking (based on sound mathematical roots), which is of vital importance. One such technique is abstract regular model checking, which encodes sets of reachable configurations and one-step transitions between them using finite automata and transducers, respectively. Though this method addresses problems that are undecidable in general, it facilitates termination in many practical cases, while also significantly reducing the state space explosion problem. This is achieved by accelerating the computation of reachability sets using incrementally refinable abstractions, while eliminating spurious counterexamples caused by overapproximation using a counterexample-guided abstraction refinement technique. The aim of this thesis is to create a well designed tool for abstract regular model checking, which has so far only been implemented in prototypes. The new tool will model systems using symbolic automata and transducers instead of their (less concise) classic alternatives.