Sequential Automatic Algebras

  • Michael Brough
  • Bakhadyr Khoussainov
  • Peter Nelson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5028)


A sequential automatic algebra is a structure of the type (A; f 1,..., f n ), where A is recognised by a finite automaton, and functions f 1, ..., f n are total operations on A that are computed by input-output automata. Our input-output automata are variations of Mealy automata. We study some of the fundamental properties of these algebras and provide many examples. We give classification results for certain classes of groups, Boolean algebras, and linear orders. We also introduce different classes of sequential automatic algebras and give separating examples. We investigate linear orders considered as sequential automatic algebras. Finally, we outline some of the basic properties of sequential automatic unary algebras.


Boolean Algebra Linear Order Unary Algebra Regular Language Congruence Relation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aleshin, M.: Finite automata and Burnside’s problem for periodic groups. Mat. Notes 11, 199–203 (1972)zbMATHGoogle Scholar
  2. 2.
    Bárány, V.: A Hierarchy of Automatic Words having a Decidable MSO Theory. In: Caucal, D. (ed.) Online Proceedings of the 11th Journées Montoises, Rennes (2006)Google Scholar
  3. 3.
    Béal, M., Carton, O., Prieur, C., Sakarovitch, J.: Squaring transducers: an efficient procedure for deciding functionality and sequentiality. Theor. Comput. Sci. 292(1), 45–63 (2003)CrossRefzbMATHGoogle Scholar
  4. 4.
    Blumensath, A.: Automatic Structures. Diploma Thesis, RWTH Aachen (1999)Google Scholar
  5. 5.
    Blumensath, A., Grädel, E.: Automatic Structures. In: 15th Annual IEEE Symposium on Logic in Computer Science, Santa Barbara, pp. 51–62 (2000)Google Scholar
  6. 6.
    Blumensath, A., Grädel, E.: Finite presentations of infinite structures: automata and interpretations. Theory of Computing Systems 37(6), 641–674 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Cannon, J., Epstein, D., Holt, D., Levy, S., Paterson, M., Thurston, W.: Word processing in groups. Jones and Bartlett (1992)Google Scholar
  8. 8.
    Carton, O., Thomas, W.: The monadic theory of morphic infinite words and generalizations. Information and Computation 176, 51–76 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Grigorchuk, R., Nekrashevich, V., Sushanski, V.: Automata, Dynamical systems, and groups. Tr. Mat. Inst. Steklova 231(Din. Sist. Avtom i Beskon. Gruppy), 134–214 (2000)Google Scholar
  10. 10.
    Khoussainov, B., Nerode, A.: Automatic Presentations of Structures. LNCS, vol. 960, pp. 367–392. Springer, Heidelberg (1995)Google Scholar
  11. 11.
    Khoussainov, B., Rubin, S.: Automatic Structures: Overview and Future Directions. Journal of Automata, Languages and Combinatorics 8, 287–301 (2003)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Khoussainov, B., Nies, A., Rubin, S., Stephan, F.: Automatic Structures: Richness and Limitations. In: LICS, pp. 44–53. IEEE Computer Society, Los Alamitos (2004)Google Scholar
  13. 13.
    Kuske, D.: Is Cantor’s theorem automatic?. LNCS, vol. 2850, pp. 332–343. Springer, Heidelberg (2003)Google Scholar
  14. 14.
    Oliver, G., Thomas, R.: Automatic Presentations for Finitely Generated Groups. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 693–704. Springer, Heidelberg (2005)Google Scholar
  15. 15.
    Saloma, A., Soittola, M.: Automata-theoretic Aspects of Formal Power Series. Springer, Heidelberg (1978)Google Scholar
  16. 16.
    Rabin, M.: Decidability of second order theories and automata on infinite trees. Trans. AMS 141, 1–35 (1969)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Rubin, S.: Automatic Structures. PhD Thesis, University of Auckland (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Michael Brough
    • 1
  • Bakhadyr Khoussainov
    • 1
  • Peter Nelson
    • 1
  1. 1.Department of Computer ScienceUniversity of Auckland 

Personalised recommendations