Abstract
We study expansions in non-integer negative base − β introduced by Ito and Sadahiro [7]. Using countable automata associated with ( − β)-expansions, we characterize the case where the ( − β)-shift is a system of finite type. We prove that, if β is a Pisot number, then the ( − β)-shift is a sofic system. In that case, addition (and more generally normalization on any alphabet) is realizable by a finite transducer.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bertrand, A.: Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sci. Paris Sér A-B 285, A419–A421 (1977)
Eilenberg, S.: Automata, Languages and Machines, vol. A. Academic Press, London (1974)
Fotiades, N., Boudourides, M.: Topological conjugacies of piecewise monotone interval maps. Int. J. Math. Math. Sci. 25, 119–127 (2001)
Frougny, C.: Representations of numbers and finite automata. Math. Systems Theory 25, 37–60 (1992)
Frougny, C.: On-line finite automata for addition in some numeration systems. Theoretical Informatics and Applications 33, 79–101 (1999)
Grünwald, V.: Intorno all’aritmetica dei sistemi numerici a base negativa con particolare riguardo al sistema numerico a base negativo-decimale per lo studio delle sue analogie coll’aritmetica ordinaria (decimale). Giornale di Matematiche di Battaglini 367, 203–221 (1885)
Ito, S., Sadahiro, T.: ( − β)-expansions of real numbers. In: INTEGERS (to appear)
Knuth, D.E.: The Art of Computer Programming, Seminumerical Algorithms, 2nd edn., vol. 2. Addison-Wesley, Reading (1988)
Lind, D., Marcus, B.: An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge (1995)
Lothaire, M.: Algebraic combinatorics on words. Encyclopedia of Mathematics and its Applications, vol. 90. Cambridge University Press, Cambridge (2002)
Parry, W.: On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11, 401–416 (1960)
Rényi, A.: Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8, 477–493 (1957)
Sakarovitch, J.: Eléments de théorie des automates, Vuibert (2003); English translation: Elements of Automata Theory, Cambridge University Press, Cambridge (to appear)
Schmidt, K.: On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12, 269–278 (1980)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Frougny, C., Lai, A.C. (2009). On Negative Bases. In: Diekert, V., Nowotka, D. (eds) Developments in Language Theory. DLT 2009. Lecture Notes in Computer Science, vol 5583. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02737-6_20
Download citation
DOI: https://doi.org/10.1007/978-3-642-02737-6_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02736-9
Online ISBN: 978-3-642-02737-6
eBook Packages: Computer ScienceComputer Science (R0)