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

TL;DR

This paper explores generalized numeration systems based on regular languages, characterizing recognizable sets of integers via formal series and analyzing how arithmetic operations affect recognizability within these systems.

Contribution

It introduces a framework for numeration systems using regular languages, characterizes recognizability through rational formal series, and examines the effects of multiplication on recognizability.

Findings

Recognizable sets of integers are characterized by rational formal series.

Multiplication by an integer preserves recognizability only under specific conditions related to the language complexity.

Conditions are provided under which recognizability and U-recognizability are equivalent.

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.