Logic characterisation of p/q-recognisable sets
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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 a rational number in some set N_p/q. In particular, N_p/q contains all integers and the literature on base p/q usually focuses on the set of words that are evaluated to 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. Second, we show that the order relation and the modulo-q operator are not p/q-recognisable.
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