Odometers on Regular Languages

Odometers or “adding machines” are usually introduced in the context of positional numeration systems built on a strictly increasing sequence of integers. We generalize this notion to systems defined on an arbitrary infinite regular language. In this latter situation, if (A, <) is a totally ordered alphabet, then enumerating the words of a regular language L over A with respect to the induced genealogical ordering gives a one-to-one correspondence between N and L. In this general setting, the odometer is not defined on a set of sequences of digits but on a set of pairs of sequences where the first (resp. the second) component of the pair is an infinite word over A (resp. an infinite sequence of states of the minimal automaton of L). We study some properties of the odometer like continuity, injectivity, surjectivity, minimality,. . . We then study some particular cases: we show the equivalence of this new function with the classical odometer built upon a sequence of integer...

• Type: Journal
• Year: 2007
• Where: MST
• Authors: Valérie Berthé, Michel Rigo
• Keywords: MST 2007, Numeration Systems, Positional Numeration Systems, Regular Language