On p/q-recognisable sets

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.

Magic Numbers in Periodic Sequences

“…We briefly mention earlier works concerning periodicity and automaticity. Eventual periodicity and automatic sets have been considered in [5,15,22]. For ℓ ≥ 1, Alexeev [1] and Sutner [27] studied the automata generating the sequence with period (1, 0, 0, .…”

Automatic sequences: from rational bases to trees

“…In [18], a subset X of N p q is said to be p q -recognizable if there exists a DFA over A p accepting a language L such that val p q (L) = X. Since L p q is not regular, the set N is not p q -recognizable.…”

“…They are also S-automatic for any abstract numeration system S based on a regular language [14]. In general, this is not the case for p q -automaticity: the characteristic sequence of multiples of q is not p qautomatic [18,Proposition 5.39]. Nevertheless when the period length of an ultimately periodic sequence is coprime with q, then the sequence is p q -automatic [18, Théorème 5.34].…”