Iterating Invertible Binary Transducers

We study iterated transductions defined by a class of invertible transducers over the binary alphabet. The transduction semigroups of these automata turn out to be free Abelian groups and the orbits of finite words can be described as affine subspaces in a suitable geometry defined by the generators of these groups. We show that iterated transductions are rational for a subclass of our automata.

[1]  Samuel Eilenberg.  Automata, Languages and Machines, Vol. B , 1976 .

[2]  S. Sidki,et al.  Groups: Automorphisms of the binary tree: state-closed subgroups and dynamics of 1/2-endomorphisms , 2004 .

[3]  Anil Nerode,et al.  Automata theory and its applications , 2001 .

[4]  V. Glushkov.  THE ABSTRACT THEORY OF AUTOMATA , 1961 .

[5]  David B. A. Epstein,et al.  Word processing in groups , 1992 .

[6]  Alexei G. Myasnikov,et al.  From automatic structures to automatic groups , 2011, ArXiv.

[7]  Jean Vuillemin,et al.  On Circuits and Numbers , 1994, IEEE Trans. Computers.

[8]  Said Sidki,et al.  Automorphisms of one-rooted trees: Growth, circuit structure, and acyclicity , 2000 .

[9]  Jacques Sakarovitch,et al.  Elements of Automata Theory , 2009 .

[10]  Anil Nerode,et al.  Automatic Presentations of Structures , 1994, LCC.

[11]  Volodymyr Nekrashevych,et al.  Self-Similar Groups , 2005, 2304.11232.

[12]  Janusz A. Brzozowski,et al.  Derivatives of Regular Expressions , 1964, JACM.

[13]  Sasha Rubin,et al.  Automatic Structures: Overview and Future Directions , 2003, J. Autom. Lang. Comb..

[14]  Daniel Leivant,et al.  Logic and Computational Complexity , 1995, Lecture Notes in Computer Science.

[15]  Jorge E. Mezei,et al.  On Relations Defined by Generalized Finite Automata , 1965, IBM J. Res. Dev..

[16]  M. Artin,et al.  Société Mathématique de France , 1994 .

[17]  Samuel Eilenberg,et al.  Automata, languages, and machines. A , 1974, Pure and applied mathematics.

[18]  Laurent Bartholdi,et al.  Groups defined by automata , 2010, Handbook of Automata Theory.

[19]  Lawrence S. Moss,et al.  Review of "Automata theory and its applications" by Bakhadyr Khoussainov and Anil Nerode. Birkhäuser Boston, Inc. 2001.: and "Automata, logics, and infinite games" by E.Grädel, W. Thomas, and T. Wilke. Springer-Verlag. , 2004, SIGA.

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

[21]  George N. Raney,et al.  Sequential Functions , 1958, JACM.

[22]  Fernando Q. Gouvêa,et al.  P-Adic Numbers: An Introduction , 1993 .