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We study expansions in non-integer negative base −β introduced by Ito and Sadahiro [7]. Using countable automata associated with (−β)-expansions, we characterize the case where the (−β)-shift is a system of finite type. We prove that, if β is a Pisot number, then the (−β)-shift is a sofic system. In that case, addition (and more generally normalization on… (More)

Let U be a strictly increasing sequence of integers. By a greedy algorithm, every nonnegative integer has a greedy U-representation. The successor function maps the greedy U-representation of N onto the greedy U-representation of N +1. We characterize the sequences U such that the successor function associated to U is a left, resp. a right sequential… (More)

It is shown that a function is computable by an on-they algorithm processing data in the most signiicant digit rst fashion with a nite number of registers if and only if it is computable by a right subsequential nite state machine processing deterministically data in the least signiicant digit rst fashion. Some applications to complex radix number systems… (More)

| To each number > 1 correspond abelian groups in R d , of the form = P d i=1 Z e i , which obey. The set Z of beta-integers is a countable set of numbers : it is precisely the set of real numbers which are polynomial in when they are written in \basis ", and Z = Zwhen 2 N. We prove here a list of arithmetic properties of Z : addition, multiplication,… (More)