Negative bases and automata

We study expansions in non-integer negative base -{\beta} introduced by Ito and Sadahiro. Using countable automata associated with (-{\beta})-expansions, we characterize the case where the (-{\beta})-shift is a system of finite type. We prove that, if {\beta} is a Pisot number, then the (-{\beta})-shift is a sofic system. In that case, addition (and more generally normalization on any alphabet) is realizable by a finite transducer. We then give an on-line algorithm for the conversion from positive base {\beta} to negative base -{\beta}. When {\beta} is a Pisot number, the conversion can be realized by a finite on-line transducer

Finite beta-expansions with negative bases

The finiteness property is an important arithmetical property of beta-expansions. We exhibit classes of Pisot numbers $\beta$ having the negative finiteness property, that is the set of finite $(-\beta)$-expansions is equal to $\mathbb{Z}[\beta^{-1}]$. For a class of numbers including the Tribonacci number, we compute the maximal length of the fractional parts arising in the addition and subtraction of $(-\beta)$-integers. We also give conditions excluding the negative finiteness property

Ito-Sadahiro numbers vs. Parry numbers

We consider a positional numeration system with a negative base, as introduced by Ito and Sadahiro. In particular, we focus on the algebraic properties of negative bases −β for which the corresponding dynamical system is sofic, which happens, according to Ito and Sadahiro, if and only if the (−β)-expansion of −β/(β + 1) is eventually periodic. We call such numbers β Ito-Sadahiro numbers, and we compare their properties with those of Parry numbers, which occur in the same context for the Rényi positive base numeration system

Digital expansions with negative real bases

Similarly to Parry's characterization of $\beta$-expansions of real numbers in real bases $\beta > 1$, Ito and Sadahiro characterized digital expansions in negative bases, by the expansions of the endpoints of the fundamental interval. Parry also described the possible expansions of 1 in base $\beta > 1$. In the same vein, we characterize the sequences that occur as $(-\beta)$-expansion of $\frac{-\beta}{\beta+1}$ for some $\beta > 1$. These sequences also describe the itineraries of 1 by linear mod one transformations with negative slope

The Computational Power of Neural Networks and Representations of Numbers in Non-Integer Bases

We briefly survey the basic concepts and results concerning the computational power of neural net-orks which basically depends on the information content of eight parameters. In particular, recurrent neural networks with integer, rational, and arbitrary real weights are classi ed within the Chomsky and finer complexity hierarchies. Then we re ne the analysis between integer and rational weights by investigating an intermediate model of integer-weight neural networks with an extra analog rational-weight neuron (1ANN). We show a representation theorem which characterizes the classification problems solvable by 1ANNs, by using so-called cut languages. Our analysis reveals an interesting link to an active research field on non-standard positional numeral systems with non-integer bases. Within this framework, we introduce a new concept of quasi-periodic numbers which is used to classify the computational power of 1ANNs within the Chomsky hierarchy

Negative bases and automata

Automata, Logic and Semantic