On p/q-recognisable sets

TL;DR

This paper characterizes p/q-recognisable sets in rational base numeration systems using first-order logic and demonstrates that certain natural relations are not recognisable within this framework.

Contribution

It provides a logical characterization of p/q-recognisable sets and shows limitations of recognisability for order and modulo-q relations.

Findings

p/q-recognisable sets are definable in a first-order logic similar to the B"uchi-Bruyère Theorem

The natural order relation is not p/q-recognisable

The modulo-q operator is not p/q-recognisable

Abstract

Let p/q be a rational number. Numeration in base p/q is defined by a function that evaluates each finite word over A_p={0,1,...,p-1} to some rational number. We let N_p/q denote the image of this evaluation function. In particular, N_p/q contains all nonnegative integers and the literature on base p/q usually focuses on the set of words that are evaluated to nonnegative integers; it is a rather chaotic language which is not context-free. On the contrary, we study here the subsets of (N_p/q)^d that are p/q-recognisable, i.e. realised by finite automata over (A_p)^d. First, we give a characterisation of these sets as those definable in a first-order logic, similar to the one given by the B\"uchi-Bruy\`ere Theorem for integer bases numeration systems. Second, we show that the natural order relation and the modulo-q operator are not p/q-recognisable.