Michel Rigo

Numeration Systems on a Regular Language

Abstract. Generalizations of positional number systems in which N is recognizable by finite automata are obtained by describing an arbitrary infinite regular language according to the lexicographic ordering. For these systems of numeration, we show that ultimately periodic sets are recognizable. We also study translation and multiplication by constants as well as the order-dependence of the recognizability.

Extensions and restrictions of Wythoff's game preserving its P positions

We consider extensions and restrictions of Wythoffs game having exactly the same set of P positions as the original game. No strict subset of rules gives the same set of P positions. On the other hand, we characterize all moves that can be adjoined while preserving the original set of P positions. Testing if a move belongs to such an extended set of rules is shown to be doable in polynomial time. Many arguments rely on the infinite Fibonacci word, automatic sequences and the corresponding numeration system. With these tools, we provide new two-dimensional morphisms generating an infinite picture encoding respectively P positions of Wythoffs game and moves that can be adjoined.

Generalization of automatic sequences for numeration systems on a regular language

Let L be an infinite regular language on a totally ordered alphabet (Σ,<). Feeding a finite deterministic automaton (with output) with the words of L, enumerated lexicographically with respect to <, leads to an infinite sequence over the output alphabet of the automaton. This process generalizes the concept of k-automatic sequence for abstract numeration systems on a regular language (instead of systems in base k). Here, we study the first properties of these sequences and their relations with numeration systems.

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

Abstract Generalizations of numeration systems in which N is recognizable by a finite automaton are obtained by describing a lexicographically ordered infinite regular language L⊂Σ ∗ . For these systems, we obtain a characterization of recognizable sets of integers in terms of N -rational formal series. After a study of the polynomial regular languages, we show that, if the complexity of L is Θ (n l ) (resp. if L is the complement of a polynomial language), then multiplication by λ∈ N preserves recognizability only if λ=β l+1 (resp. if λ≠(#Σ) β ) for some β∈ N . Finally, we obtain sufficient conditions for the notions of recognizability for abstract systems and some positional number systems to be equivalent.

On the Representation of Real Numbers Using Regular Languages

Abstract. Using a lexicographically ordered regular language, we show how to represent an interval of \R . We determine exactly the possible representations of any element in this interval and study the function which maps a representation onto its numerical value. We make explicit the relationship between the convergence of finite words to an infinite word and the convergence of the corresponding approximations to a real number.

A DECISION PROBLEM FOR ULTIMATELY PERIODIC SETS IN NONSTANDARD NUMERATION SYSTEMS

Consider a nonstandard numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over {0,1} without two consecutive 1. Given a set X of integers such that the language of their greedy representations in this system is accepted by a finite automaton, we consider the problem of deciding whether or not X is a finite union of arithmetic progressions. We obtain a decision procedure for this problem, under some hypothesis about the considered numeration system. In a second part, we obtain an analogous decision result for a particular class of abstract numeration systems built on an infinite regular language.

A morphic approach to combinatorial games: the Tribonacci case

We propose a variation of Wythoffs game on three piles of tokens, in the sense that the losing positions can be derived from the Tribonacci word instead of the Fibonacci word for the two piles game. Thanks to the corresponding exotic numeration system built on the Tribonacci sequence, deciding whether a game position is losing or not can be computed in polynomial time.

Real numbers having ultimately periodic representations in abstract numeration systems

Using a genealogically ordered infinite regular language, we know how to represent an interval of R. Numbers having an ultimately periodic representation play a special role in classical numeration systems. The aim of this paper is to characterize the numbers having an ultimately periodic representation in generalized systems built on a regular language. The syntactical properties of these words are also investigated. Finally, we show the equivalence of the classical θ-expansions with our generalized representations in some special case related to a Pisot number θ.

Syndeticity and independent substitutions

We associate in a canonical way a substitution to any abstract numeration system built on a regular language. In relationship with the growth order of the letters, we define the notion of two independent substitutions. Our main result is the following. If a sequence

Construction of regular languages and recognizability of polynomials

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