Numeration systems on a regular language: Arithmetic operations, Recognizability and Formal power series
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 Urecognizability to be equivalent, where U is some positional numeration system related to a sequence of integers.
 Publication:

arXiv eprints
 Pub Date:
 November 1999
 arXiv:
 arXiv:cs/9911002
 Bibcode:
 1999cs.......11002R
 Keywords:

 Computer Science  Computational Complexity;
 F.1.1;
 F.4.3
 EPrint:
 34 pages