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

Michel Rigo

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arXiv · cs/9911002

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

Michel Rigo

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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 over a finite alphabet A. For these systems, we obtain a characterization of recognizable sets of integers in terms of rational formal series. We also show that, if the complexity of L is Theta (n^q) (resp. if L is the complement of a polynomial language), then multiplication by an integer k preserves recognizability only if k=t^{q+1} (resp. if k is not a power of the cardinality of A) for some integer t. Finally, we obtain sufficient conditions for the notions of recognizability and U-recognizability to be equivalent, where U is some positional numeration system related to a sequence of integers.

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Michel Rigo. 2000-01-20. Numeration systems on a regular language: Arithmetic operations, Recognizability and Formal power series. https://arxiv.org/abs/cs/9911002

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