Numeration systems on a regular language: Arithmetic operations, Recognizability and Formal power series
摘要
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.
引用
@article{arxiv.cs/9911002,
title = {Numeration systems on a regular language: Arithmetic operations, Recognizability and Formal power series},
author = {Michel Rigo},
journal= {arXiv preprint arXiv:cs/9911002},
year = {2007}
}
备注
34 pages; corrected typos, two sections concerning exponential case and relation with positional systems added