On-line finite automata for addition in some numeration systems

Free access article

Issue		RAIRO-Theor. Inf. Appl. Volume 33, Number 1, January-February 1999


Page(s)		79 - 101
DOI		10.1051/ita:1999107

DOI: 10.1051/ita:1999107

ITA, Vol. 33, N $^{\rm o}$ 1, 1999, p. 79-101

On-line finite automata for addition in some numeration systems

Christiane Frougny
Université Paris VIII and L.I.A.F.A., Case 7014, 2 place Jussieu, 75251 Paris Cedex 05, France; (Christiane.Frougny@liafa.jussieu.fr)

Received May, 1998. Accepted November, 1998.

Abstract: 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 $\beta$ and addition in base $\beta^m$ is given. Results on addition in base $\beta=\sqrt[m]{b}$ , 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.

Communicated by: Ch. Choffrut.

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