Structural Presburger digit vector automata

International audienceThe least significant digit first decomposition of integer vectors into words of digit vectors provides a natural way for representing sets of integer vectors by automata. In this paper, the minimal automata representing Presburger sets are proved structurally Presburger: automata obtained by moving the initial state and replacing the accepting condition represent Presburger sets

Structural Presburger-definable Digit Vector Automata

Les automates finis permettent de représenter symboliquement des ensembles infinis de vecteurs d'entiers décomposés comme des mots de vecteurs de chiffres. On montre que l'automate minimal représentant un ensemble Presburger-définissable est structurellement Presburger-définissable: c'est à dire, que les automates obtenus en changeant l'état initial et les états finaux représentent des ensembles Presburger-définissables./ Digit Vector Automata (DVA) provide a natural symbolic representation for regular sets of integer vectors encoded as strings of digit vectors (least significant digit first). We prove that the minimal DVA that represents a Presburger-definable set is structurally Presburger-definable: that means, the DVA obtained by modifying the initial state and the set of final states represents a Presburger-definable set

On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases

This article studies the expressive power of finite automata recognizing sets of real numbers encoded in positional notation. We consider Muller automata as well as the restricted class of weak deterministic automata, used as symbolic set representations in actual applications. In previous work, it has been established that the sets of numbers that are recognizable by weak deterministic automata in two bases that do not share the same set of prime factors are exactly those that are definable in the first order additive theory of real and integer numbers. This result extends Cobham's theorem, which characterizes the sets of integer numbers that are recognizable by finite automata in multiple bases. In this article, we first generalize this result to multiplicatively independent bases, which brings it closer to the original statement of Cobham's theorem. Then, we study the sets of reals recognizable by Muller automata in two bases. We show with a counterexample that, in this setting, Cobham's theorem does not generalize to multiplicatively independent bases. Finally, we prove that the sets of reals that are recognizable by Muller automata in two bases that do not share the same set of prime factors are exactly those definable in the first order additive theory of real and integer numbers. These sets are thus also recognizable by weak deterministic automata. This result leads to a precise characterization of the sets of real numbers that are recognizable in multiple bases, and provides a theoretical justification to the use of weak automata as symbolic representations of sets.Comment: 17 page

Presburger Arithmetic: From Automata to Formulas

Presburger arithmetic is the first-order theory of the integers with addition and ordering, but without multiplication. This theory is decidable and the sets it definesadmit several different representations, including formulas, generators, and finiteautomata, the latter being the focus of this thesis. Finite-automata representations of Presburger sets work by encoding numbers as words and sets by automata-defined languages. With this representation, set operations are easily computableas automata operations, and minimized deterministic automata are a canonicalrepresentation of Presburger sets. However, automata-based representations are somewhat opaque and do not allow all operations to be performed efficiently. Anideal situation would be to be able to move easily between formula-based andautomata-based representations but, while building an automaton from a formulais a well understood process, moving the other way is a much more difcult problem that has only attracted attention fairly recently.The main results of this thesis are new algorithms for extracting informationabout Presburger-definable sets represented by finite automata. More precisely, we present algorithms that take as input a finite-automaton representing a Presburgerdefinable set S and compute in polynomial time the affine hull over Qor over Z of the set S, i.e., the smallest set defined by a conjunction of linearequations (and congruence relations in Z) which includes S. Also, we presentan algorithm that takes as input a deterministic finite-automaton representing theinteger elements of a polyhedron P and computes a quantifier-free formula correspondingto this set.The algorithms rely on a very detailed analysis of the scheme used for encodinginteger vectors and this analysis sheds light on some structural properties offinite-automata representing Presburger definable sets.The algorithms presented have been implemented and the results are encouraging: automata with more than 100000 states are handled in seconds

How to Tackle Integer Weighted Automata Positivity

International audienceThis paper is dedicated to candidate abstractions to capture relevant aspects of the integer weighted automata. The expected effect of applying these abstractions is studied to build the deterministic reachability graphs allowing us to semi-decide the positivity problem on these automata. Moreover, the papers reports on the implementations and experimental results, and discusses other encodings

The First-Order Theory of Binary Overlap-Free Words is Decidable

We show that the first-order logical theory of the binary overlap-free words (and, more generally, the ${\alpha}$-free words for rational ${\alpha}$, $2 < {\alpha} \leq 7/3$), is decidable. As a consequence, many results previously obtained about this class through tedious case- based proofs can now be proved "automatically", using a decision procedure