Fragments of first-order logic over infinite words

We give topological and algebraic characterizations as well as language theoretic descriptions of the following subclasses of first-order logic FO[<] for omega-languages: Sigma_2, FO^2, the intersection of FO^2 and Sigma_2, and Delta_2 (and by duality Pi_2 and the intersection of FO^2 and Pi_2). These descriptions extend the respective results for finite words. In particular, we relate the above fragments to language classes of certain (unambiguous) polynomials. An immediate consequence is the decidability of the membership problem of these classes, but this was shown before by Wilke and Bojanczyk and is therefore not our main focus. The paper is about the interplay of algebraic, topological, and language theoretic properties.Comment: Conference version presented at 26th International Symposium on Theoretical Aspects of Computer Science, STACS 200

Numeration systems on a regular language: Arithmetic operations, Recognizability and Formal power series

Generalizations of numeration systems in which N is recognizable by a finite automaton are obtained by describing a lexicographically ordered infinite regular language L over a finite alphabet A. For these systems, we obtain a characterization of recognizable sets of integers in terms of rational formal series. We also show that, if the complexity of L is Theta (n^q) (resp. if L is the complement of a polynomial language), then multiplication by an integer k preserves recognizability only if k=t^{q+1} (resp. if k is not a power of the cardinality of A) for some integer t. Finally, we obtain sufficient conditions for the notions of recognizability and U-recognizability to be equivalent, where U is some positional numeration system related to a sequence of integers.Comment: 34 pages; corrected typos, two sections concerning exponential case and relation with positional systems adde

Characterizations of recognizable picture series

AbstractThe theory of two-dimensional languages as a generalization of formal string languages was motivated by problems arising from image processing and pattern recognition, and also concerns models of parallel computing. Here we investigate power series on pictures. These are functions that map pictures to elements of a semiring and provide an extension of two-dimensional languages to a quantitative setting. We assign weights to different devices, ranging from picture automata to tiling systems. We will prove that, for commutative semirings, the behaviours of weighted picture automata are precisely alphabetic projections of series defined in terms of rational operations, and also coincide with the families of series characterized by weighted tiling or weighted domino systems

A Theory of Computation Based on Quantum Logic (I)

The (meta)logic underlying classical theory of computation is Boolean (two-valued) logic. Quantum logic was proposed by Birkhoff and von Neumann as a logic of quantum mechanics more than sixty years ago. The major difference between Boolean logic and quantum logic is that the latter does not enjoy distributivity in general. The rapid development of quantum computation in recent years stimulates us to establish a theory of computation based on quantum logic. The present paper is the first step toward such a new theory and it focuses on the simplest models of computation, namely finite automata. It is found that the universal validity of many properties of automata depend heavily upon the distributivity of the underlying logic. This indicates that these properties does not universally hold in the realm of quantum logic. On the other hand, we show that a local validity of them can be recovered by imposing a certain commutativity to the (atomic) statements about the automata under consideration. This reveals an essential difference between the classical theory of computation and the computation theory based on quantum logic

Logic Meets Algebra: the Case of Regular Languages

The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Buchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point of view on automata is an essential complement of this classification: by providing alternative, algebraic characterizations for the classes, it often yields the only opportunity for the design of algorithms that decide expressibility in some logical fragment. We survey the existing results relating the expressibility of regular languages in logical fragments of MSO[S] with algebraic properties of their minimal automata. In particular, we show that many of the best known results in this area share the same underlying mechanics and rely on a very strong relation between logical substitutions and block-products of pseudovarieties of monoid. We also explain the impact of these connections on circuit complexity theory.Comment: 37 page

On Lindenmayerian algebraic sequences

AbstractWe define and study Lindenmayerian algebraic sequences. These sequences are a generalization of algebraic sequences, k-regular sequences and automatic sequences

Systèmes de numération abstraits : reconnaissabilité, décidabilité, mots S-automatiques multidimensionnels et nombres réels

In this dissertation we study and we solve several questions regarding abstract numeration systems. Each particular problem is the focus of a chapter. The first problem concerns the study of the preservation of recognizability under multiplication by a constant in abstract numeration systems built on polynomial regular languages. The second is a decidability problem, which has been already studied notably by J. Honkala and A. Muchnik and which is studied here for two new cases: the linear positional numeration systems and the abstract numeration systems. Next, we focus on the extension to the multidimensional setting of a result of A. Maes and M. Rigo regarding S-automatic infinite words. Finally, we propose a formalism to represent real numbers in the general framework of abstract numeration systems built on languages that are not necessarily regular. We end by a list of open questions in the continuation of the present work.Dans cette dissertation, nous étudions et résolvons plusieurs questions autour des systèmes de numération abstraits. Chaque problème étudié fait l'objet d'un chapitre. Le premier concerne l'étude de la conservation de la reconnaissabilité par la multiplication par une constante dans des systèmes de numération abstraits construits sur des langages réguliers polynomiaux. Le deuxième est un problème de décidabilité déjà étudié notamment par J. Honkala et A. Muchnik et ici décliné en deux nouvelles versions : les systèmes de numération de position linéaires et les systèmes de numération abstraits. Ensuite, nous nous penchons sur l'extension au cas multidimensionnel d'un résultat d'A. Maes et de M. Rigo à propos des mots infinis S-automatiques. Finalement, nous proposons un formalisme de la représentation des nombres réels dans le cadre général des systèmes de numération abstraits basés sur des langages qui ne sont pas nécessairement réguliers. Nous terminons par une liste de questions ouvertes dans la continuité de ce travail

Establishing a Connection Between Graph Structure, Logic, and Language Theory

The field of graph structure theory was given life by the Graph Minors Project of Robertson and Seymour, which developed many tools for understanding the way graphs relate to each other and culminated in the proof of the Graph Minors Theorem. One area of ongoing research in the field is attempting to strengthen the Graph Minors Theorem to sets of graphs, and sets of sets of graphs, and so on. At the same time, there is growing interest in the applications of logic and formal languages to graph theory, and a significant amount of work in this field has recently been consolidated in the publication of a book by Courcelle and Engelfriet. We investigate the potential applications of logic and formal languages to the field of graph structure theory, suggesting a new area of research which may provide fruitful

Sets of integers in different number systems and the Chomsky hierarchy

The classes of the Chomsky hierarchy are characterized in respect of converting between canonical number systems. We show that the relations of the bases of the original and converted number systems fall into four distinct categories, and we examine the four Chomsky classes in each of the four cases. We also prove that all of the Chomsky classes are closed under constant addition and multiplication. The classes RE and CS are closed under every examined operation. The regular languages axe closed under addition, but not under multiplication