Representations of numbers and finite automata

Numeration systems, the basis of which is defined by a linear recurrence with integer coefficients, are considered. We give conditions on the recurrence under which the function of normalization which transforms any representation of an integer into the normal one—obtained by the usual algorithm—can be realized by a finite automaton. Addition is a particular case of normalization. The same questions are discussed for the representation of real numbers in basis θ, where θ is a real number > 1, in connection with symbolic dynamics. In particular it is shown that if θ is a Pisot number, then the normalization and the addition in basis θ are computable by a finite automaton.

[1]  Jacques Sakarovitch,et al.  Rational Ralations with Bounded Delay , 1991, STACS.

[2]  A. Rényi.  Representations for real numbers and their ergodic properties , 1957 .

[3]  Lahoraloire de Probahi,et al.  P-EXPANSIONS AND SYMBOLIC DYNAMICS , 2001 .

[4]  Christiane Frougny,et al.  Linear Numeration Systems of Order Two , 1988, Inf. Comput..

[5]  Christiane Frougny.  Linear Numeration Systems, Theta-Developments and Finite Automata , 1989, STACS.

[6]  A. Brauer,et al.  On algebraic equations with all but one root in the interior of the unit circle. To my teacher and former colleague Erhard Schmidt on his 75th birthday , 1950 .

[7]  François Blanchard.  beta-Expansions and Symbolic Dynamics , 1989, Theor. Comput. Sci..

[8]  Robert F. Tichy,et al.  On digit expansions with respect to linear recurrences , 1989 .

[9]  D. Lind.  The entropies of topological Markov shifts and a related class of algebraic integers , 1984, Ergodic Theory and Dynamical Systems.

[10]  W. Parry.  On theβ-expansions of real numbers , 1960 .

[11]  A. Avizeinis,et al.  Signed Digit Number Representations for Fast Parallel Arithmetic , 1961 .

[12]  Christiane Frougny.  Systemes de numeration lineaires et automates finis , 1989 .

[13]  Jean Berstel,et al.  Transductions and context-free languages , 1979, Teubner Studienbücher : Informatik.

[14]  J. Wrench.  Table errata: The art of computer programming, Vol. 2: Seminumerical algorithms (Addison-Wesley, Reading, Mass., 1969) by Donald E. Knuth , 1970 .

[15]  Aviezri S. Fraenkel,et al.  Systems of numeration , 1983, 1983 IEEE 6th Symposium on Computer Arithmetic (ARITH).

[16]  Aviezri S. Fraenkel,et al.  Systems of numeration , 1983, IEEE Symposium on Computer Arithmetic.

[17]  Algirdas Avizienis,et al.  Signed-Digit Numbe Representations for Fast Parallel Arithmetic , 1961, IRE Trans. Electron. Comput..