On p/q-recognisable sets

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

Theoretical Informatics and Applications Will be set by the publisher Informatique Théorique et Applications MINIMAL DIGIT SETS FOR PARALLEL ADDITION IN NON-STANDARD NUMERATION SYSTEMS

Abstract. We study parallel algorithms for addition of numbers having finite representation in a positional numeration system defined by a base β in C and a finite digit set A of contiguous integers containing 0. For a fixed base β, we focus on the question of the size of the alphabet allowing to perform addition in constant time independently of the length of representation of the summands. We produce lower bounds on the size of such alphabet A. For several types of well studied bases (negative integer, complex numbers −1 + ı, 2ı, and ı √ 2, quadratic Pisot unit, and the non-integer rational base), we give explicit parallel algorithms performing addition in constant time. Moreover we show that digit sets used by these algorithms are the smallest possible

Rational base number systems for

This paper deals with rational base number systems for p-adic numbers. We mainly focus on the system proposed by Akiyama et al. in 2008, but we also show that this system is in some sense isomorphic to some other rational base number systems by means of finite transducers. We identify the numbers with finite and eventually periodic representations and we also determine the number of representations of a given p-adic number