Theory of Computing Systems

, Volume 61, Issue 4, pp 1254–1287 | Cite as

Semiautomatic Structures

  • Sanjay Jain
  • Bakhadyr Khoussainov
  • Frank Stephan
  • Dan Teng
  • Siyuan Zou
Part of the following topical collections:
  1. Special Issue on Computability, Complexity and Randomness (CCR 2015)


Semiautomatic structures generalise automatic structures in the sense that for some of the relations and functions in the structure one only requires the derived relations and functions are automatic when all but one input are filled with constants. One can also permit that this applies to equality in the structure so that only the sets of representatives equal to a given element of the structure are regular while equality itself is not an automatic relation on the domain of representatives. It is shown that one can find semiautomatic representations for the field of rationals and also for finite algebraic field extensions of it. Furthermore, one can show that infinite algebraic extensions of finite fields have semiautomatic representations in which the addition and equality are both automatic. Further prominent examples of semiautomatic structures are term algebras, any relational structure over a countable domain with a countable signature and any permutation algebra with a countable domain. Furthermore, examples of structures which fail to be semiautomatic are provided.


Automatic structures Automata theory Basic algebraic structures Groups Rings Semiautomatic structures 



The authors would like to thank Anil Nerode as well as the participants of the IMS Workshop on Automata Theory and Applications in 2011 who discussed the topic and initial results with the authors.


  1. 1.
    Case, J., Jain, S., Seah, S., Stephan, F.: Automatic functions, linear time and learning. In: How the World Computes - Turing Centenary Conference and Eighth Conference on Computability in Europe, CiE 2012, Cambridge, UK, June 18–23, 2012. Proceedings. Springer LNCS, vol. 7318, pp 96–106 (2012)Google Scholar
  2. 2.
    Cobham, A.: On the base-dependence of sets of numbers recognizable by finite automata. Mathematical Systems Theory 3, 186–192 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Delhommé, C.: Automaticité des ordinaux et des graphes homogènes. Comptes Rendus Mathematique 339(1), 5–10 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Epstein, D.B.A., Cannon, J.W., Holt, D.F., Levy, S.V.F., Paterson, M.S., Thurston, W.P.: Word Processing in Groups. Jones and Bartlett Publishers, Boston (1992)zbMATHGoogle Scholar
  5. 5.
    Fuchs, L.: Partially Ordered Algebraic Systems. Pergamon Press (1963)Google Scholar
  6. 6.
    Hodgson, B.R.: Théories décidables par automate fini. Ph.D.Thesis, Département de mathématiques et de statistique, Université de Montréal (1976)Google Scholar
  7. 7.
    Hodgson, B.R.: Décidabilité par automate fini. Annales des Sciences Mathé,matiques du Québec 7(1), 39–57 (1983)zbMATHGoogle Scholar
  8. 8.
    Hölzl, R., Jain, S., Stephan, F.: Learning pattern languages over groups. In: Algorithmic Learning Theory - Twentyseventh International Conference, ALT 2016, Bari, Italy, October 19–21, 2016, Proceedings. Springer LNCS, vol. 9925, pp 189–203 (2016)Google Scholar
  9. 9.
    Hopcroft, J. E., Motwani, R., Ullman, J. D.: Introduction to Automata Theory, Languages and Computation, 3rd edn. Addison Wesley (2007)Google Scholar
  10. 10.
    Jain, S., Khoussainov, B., Stephan, F., Teng, D., Zou, S.: Semiautomatic structures. In: Computer Science – Theory and Applications – Ninth International Computer Science Symposium in Russia, CSR 2014, Moscow, Russia, June 7–11, 2014. Proceedings. Springer LNCS, vol. 8476, pp 204–217 (2014)Google Scholar
  11. 11.
    Khoussainov, B., Jain, S., Stephan, F.: Finitely generated semiautomatic groups. In: Pursuit of the Universal, Twelfth Conference on Computability in Europe, CiE 2016, Paris, France, 27 June–1 July 2016, Proceedings. Springer LNCS, vol. 9709, pp 282–291 (2016)Google Scholar
  12. 12.
    Kharlampovich, O., Khoussainov, B., Miasnikov, A.: From automatic structures to automatic groups. arXiv:1107.3645 (2011)
  13. 13.
    Khoussainov, B., Nerode, A.: Automatic presentations of structures. In: Logic and Computational Complexity, International Workshop, LCC 1994, Indianapolis, Indiana, USA, October 13–16, 1994; Springer LNCS, vol. 960, pp 367–392 (1995)Google Scholar
  14. 14.
    Khoussainov, B., Rubin, S., Stephan, F.: Definability and regularity in automatic structures. In: Twentyfirst Annual Symposium on Theoretical Aspects of Computer Science, STACS 2004, Montpellier, France, March 25–27, 2004, Proceedings; Springer LNCS, vol. 2996, pp 440–451 (2004)Google Scholar
  15. 15.
    Kozen, D.: Complexity of Finitely Presented Algebras. PhD thesis, Computer Science Department Cornell University (May 1977)Google Scholar
  16. 16.
    Lagrange, J.-L.: Solution d’un probléme d’arithmétique. In: Serret, J.-A. (ed.) Oeuvres de Lagrange, vol. 1, pp 671–731 (1867). Google Scholar
  17. 17.
    Matiyasevich, Y.V.: Diofantovost’ perechislimykh mnozhestv. Doklady Akademii Nauk SSSR 191, 297–282 (1970). (Russian). English translation: Enumerable sets are Diophantine, Soviet Mathematics Doklady 11, 354–358 (1970)Google Scholar
  18. 18.
    Matiyasevich, Y.V.: Hilbert’s Tenth Problem. MIT Press, Cambridge, Massachusetts (1993)zbMATHGoogle Scholar
  19. 19.
    Miasnikov, A., Šunić, Z.: Cayley graph automatic groups are not necessarily Cayley graph biautomatic. In: Language and Automata Theory and Applications - Sixth International Conference, LATA 2012, A Corũna, Spain, March 5-9, 2012, Proceedings. Springer LNCS, vol. 401–407, p 7183 (2012)Google Scholar
  20. 20.
    Neumann, B.H.: On ordered groups. Am. J. Math. 71, 1–18 (1949)CrossRefzbMATHGoogle Scholar
  21. 21.
    Nies, A.: Describing groups. Bull. Symb. Log. 13(3), 305–339 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nies, A., Thomas, R.: FA-Presentable groups and rings. Journal of Algebra 320, 569–585 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Nies, A., Semukhin, P.: Finite automata presentable Abelian groups. Annals of Pure and Applied Logic 161, 458–467 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Semenov, A.L.: The Presburger nature of predicates that are regular in two number systems. Sib. Math. J. 18, 289–299 (1977)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Tan, W.Y.: Automatic Structures. Honours Year Thesis, Department of Mathematics National University of Singapore (2008)Google Scholar
  26. 26.
    Teng, D.: Automatic Structures. Honours Year Thesis, Department of Mathematics National University of Singapore (2012)Google Scholar
  27. 27.
    Zou, S.: Automatic Semigroups and Ordering. Honours Year Thesis, Department of Mathematics National University of Singapore (2013)Google Scholar
  28. 28.
    Tsankov, T.: The additive group of the rationals does not have an automatic presentation. J. Symb. Log. 76(4), 1341–1351 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Sanjay Jain
    • 1
  • Bakhadyr Khoussainov
    • 2
  • Frank Stephan
    • 1
    • 3
  • Dan Teng
    • 3
  • Siyuan Zou
    • 3
  1. 1.Department of Computer ScienceNational University of SingaporeSingaporeRepublic of Singapore
  2. 2.Department of Computer ScienceUniversity of AucklandAucklandNew Zealand
  3. 3.Department of MathematicsThe National University of SingaporeSingaporeRepublic of Singapore

Personalised recommendations