Christiane Frougny

Synchronized rational relations of finite and infinite words

Abstract The purpose of this paper is a comprehensive study of a family of rational relations, both of finite and infinite words, namely those that are computable by automata where the reading heads move simultaneously on the n input tapes, and that we thus propose to call synchronized rational relations.

Representations of numbers and finite automata

Numeration systems, the basis of which is defined by a linear recurrence with integer coefficients, are considered. We give conditions on the recurrence under which the function of normalization which transforms any representation of an integer into the normal one—obtained by the usual algorithm—can be realized by a finite automaton. Addition is a particular case of normalization. The same questions are discussed for the representation of real numbers in basis θ, where θ is a real number > 1, in connection with symbolic dynamics. In particular it is shown that if θ is a Pisot number, then the normalization and the addition in basis θ are computable by a finite automaton.

Computability by finite automata and Pisot bases

We prove that the function of normalization in base θ, which maps any θ-representation of a real number onto its θ-development, obtained by a greedy algorithm, is a function computable by a finite automaton over any alphabet if and only if θ is a Pisot number.

Linear numeration systems of order two

Abstract A numeration system is a sequence of integers such that any integer can be represented by means of the sequence using integers of bounded size. We study numeration systems defined by linear recurrences of order two. We give a necessary and sufficient condition on the system such that every integer has a canonical representation. We show that this canonical representation can be computed from any representation by a rational function. This rational function is the composition of two subsequential functions that are simply obtained from the system. The addition of two integers represented in the system can be performed by a subsequential machine.

Complexity of infinite words associated with beta-expansions

We study the complexity of the infinite word uβ associated with the Renyi expansion of 1 in an irrational base β > 1 . When β is the golden ratio, this is the well known Fibonacci word, which is Sturmian, and of complexity C(n) = n + 1 . For β such that dβ (1) = t1 t2 ...tm is finite we provide a simple description of the structure of special factors of the word uβ . When tm =1 we show that C(n) = (m - 1)n + 1 . In the cases when t1 = t2 = ... tm-1 or t1 > max{t2 ,...,tm-1 } we show that the first difference of the complexity function C(n + 1) - C(n ) takes value in {m - 1,m} for every n , and consequently we determine the complexity of uβ . We show that uβ is an Arnoux-Rauzy sequence if and only if dβ (1) = tt...t1 . On the example of β = 1 + 2 cos(2π/7), solution of X3 = 2X2 + X - 1 , we illustrate that the structure of special factors is more complicated for dβ (1) infinite eventually periodic. The complexity for this word is equal to 2n+1 .

On-line finite automata for addition in some numeration systems

We consider numeration systems where the base is a negative integer, or a complex number which is a root of a negative integer. We give parallel algorithms for addition in these numeration systems, from which we derive on-line algorithms realized by finite automata. A general construction relating addition in base β and addition in base βm is given. Results on addition in base , where b is a relative integer, follow. We also show that addition in base the golden ratio is computable by an on-line finite automaton, but is not parallelizable.

Confluent linear numeration systems

Abstract Numeration systems where the basis is defined by a linear recurrence with integer coefficients are considered. A rewriting system is associated with the recurrence. In this paper we study the case when it is confluent. We prove that the function of normalization which transforms any representation of an integer into the normal one — obtained by the usual algorithm — can be realized by a finite automation which is the composition of a left subsequential transducer and of a right subsequential transducer associated with the rewriting system. Addition of integers in a confluent linear numeration system is also computable by a finite automaton. These results extend to the representation of real numbers in basis θ, where θ is the dominant root of the characteristic polynomial of the recurrence.

On univoque Pisot numbers

We study Pisot numbers

On the sequentiality of the successor function

\beta \in (1, 2)

Rational relations with bounded delay

which are univoque, i.e., such that there exists only one representation of