Linear numeration systems, θ-developments and finite automata
In numeration systems defined by a linear recurrence relation, as well as in the set of developments of numbers in a non integer basis ϑ, we define the notion of normal representation of a number. We show that, taking for ϑ the greatest root of the characteristic polynomial of the linear recurrence, and under certain conditions of confluence, the normal representation can be obtained from any representation by a finite automaton which is the composition of two sequential transducers derived from the linear recurrence. The addition of two numbers can be performed by a left sequential transducer.
Unable to display preview. Download preview PDF.
- D. Beauquier and D. Perrin, Codeterministic automata on infinite words. I.P.L. 20 (1985), 95–98.Google Scholar
- J. Berstel, Transductions and Context-Free Languages. Teubner, 1979.Google Scholar
- J. Berstel, Fibonacci words — A survey, in The Book of L, Springer, 1986, 13–27.Google Scholar
- A. Bertrand-Mathis, Comment écrire les nombres entiers dans une base qui ne l'est pas. To appear.Google Scholar
- F. Blanchard et G. Hansel, Systèmes codés. T.C.S. 44 (1986), 17–49.Google Scholar
- S. Eilenberg, Automata, Languages, and Machines, vol. A. Academic Press, 1974.Google Scholar
- Ch. Frougny, Linear numeration systems of order two. Information and Computation 77 (1988), 233–259.Google Scholar
- Ch. Frougny, Confluent linear numeration systems. In preparation.Google Scholar
- G. Huet, Confluent reductions: Abstract properties and applications to term rewriting systems. J. Assoc. Comput. Mach. 27 (1980), 797–821.Google Scholar
- W. Parry, On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.Google Scholar
- A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493.Google Scholar