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

A Quasi-Linear Time Algorithm Deciding Whether Weak B\"uchi Automata Reading Vectors of Reals Recognize Saturated Languages

This work considers weak deterministic B\"uchi automata reading encodings of non-negative $d$-vectors of reals in a fixed base. A saturated language is a language which contains all encoding of elements belonging to a set of $d$-vectors of reals. A Real Vector Automaton is an automaton which recognizes a saturated language. It is explained how to decide in quasi-linear time whether a minimal weak deterministic B\"uchi automaton is a Real Vector Automaton. The problem is solved both for the two standard encodings of vectors of numbers: the sequential encoding and the parallel encoding. This algorithm runs in linear time for minimal weak B\"uchi automata accepting set of reals. Finally, the same problem is also solved for parallel encoding of automata reading vectors of relative reals

Minimal weight expansions in Pisot bases

For applications to cryptography, it is important to represent numbers with a small number of non-zero digits (Hamming weight) or with small absolute sum of digits. The problem of finding representations with minimal weight has been solved for integer bases, e.g. by the non-adjacent form in base~2. In this paper, we consider numeration systems with respect to real bases $\beta$ which are Pisot numbers and prove that the expansions with minimal absolute sum of digits are recognizable by finite automata. When $\beta$ is the Golden Ratio, the Tribonacci number or the smallest Pisot number, we determine expansions with minimal number of digits $\pm1$ and give explicitely the finite automata recognizing all these expansions. The average weight is lower than for the non-adjacent form

Commutative Languages and their Composition by Consensual Methods

Commutative languages with the semilinear property (SLIP) can be naturally recognized by real-time NLOG-SPACE multi-counter machines. We show that unions and concatenations of such languages can be similarly recognized, relying on -- and further developing, our recent results on the family of consensually regular (CREG) languages. A CREG language is defined by a regular language on the alphabet that includes the terminal alphabet and its marked copy. New conditions, for ensuring that the union or concatenation of CREG languages is closed, are presented and applied to the commutative SLIP languages. The paper contributes to the knowledge of the CREG family, and introduces novel techniques for language composition, based on arithmetic congruences that act as language signatures. Open problems are listed.Comment: In Proceedings AFL 2014, arXiv:1405.527

Tree Regular Model Checking for Lattice-Based Automata

Tree Regular Model Checking (TRMC) is the name of a family of techniques for analyzing infinite-state systems in which states are represented by terms, and sets of states by Tree Automata (TA). The central problem in TRMC is to decide whether a set of bad states is reachable. The problem of computing a TA representing (an over- approximation of) the set of reachable states is undecidable, but efficient solutions based on completion or iteration of tree transducers exist. Unfortunately, the TRMC framework is unable to efficiently capture both the complex structure of a system and of some of its features. As an example, for JAVA programs, the structure of a term is mainly exploited to capture the structure of a state of the system. On the counter part, integers of the java programs have to be encoded with Peano numbers, which means that any algebraic operation is potentially represented by thousands of applications of rewriting rules. In this paper, we propose Lattice Tree Automata (LTAs), an extended version of tree automata whose leaves are equipped with lattices. LTAs allow us to represent possibly infinite sets of interpreted terms. Such terms are capable to represent complex domains and related operations in an efficient manner. We also extend classical Boolean operations to LTAs. Finally, as a major contribution, we introduce a new completion-based algorithm for computing the possibly infinite set of reachable interpreted terms in a finite amount of time.Comment: Technical repor

Automatic sets of rational numbers

The notion of a k-automatic set of integers is well-studied. We develop a new notion - the k-automatic set of rational numbers - and prove basic properties of these sets, including closure properties and decidability.Comment: Previous version appeared in Proc. LATA 2012 conferenc

08171 Abstracts Collection -- Beyond the Finite: New Challenges in Verification and Semistructured Data

From 20.04. to 25.04.2008, the Dagstuhl Seminar 08171 ``Beyond the Finite: New Challenges in Verification and Semistructured Data\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available