Victor Marsault - On p/q-recognisable sets

lmcs:6834 - Logical Methods in Computer Science, July 28, 2021, Volume 17, Issue 3 -
On p/q-recognisable sets

Authors: Victor Marsault

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 Büchi-Bruyère Theorem for integer bases numeration systems. Second, we show that the natural order relation and the modulo-q operator are not p/q-recognisable.

Volume: Volume 17, Issue 3
Published on: July 28, 2021
Accepted on: July 2, 2021
Submitted on: October 9, 2020
Keywords: Computer Science - Logic in Computer Science,Computer Science - Discrete Mathematics,Computer Science - Formal Languages and Automata Theory


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