Revisiting the Equivalence Problem for Finite Multitape Automata

  • James Worrell
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7966)


The decidability of determining equivalence of deterministic multitape automata (or transducers) was a longstanding open problem until it was resolved by Harju and Karhumäki in the early 1990s. Their proof of decidability yields a co-NP upper bound, but apparently not much more is known about the complexity of the problem. In this paper we give an alternative proof of decidability, which follows the basic strategy of Harju and Karhumäki but replaces their use of group theory with results on matrix algebras. From our proof we obtain a simple randomised algorithm for deciding equivalence of deterministic multitape automata, as well as automata with transition weights in the field of rational numbers. The algorithm involves only matrix exponentiation and runs in polynomial time for each fixed number of tapes. If the two input automata are inequivalent then the algorithm outputs a word on which they differ.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Amitsur, S.A., Levitzki, J.: Minimal identities for algebras. Proceedings of the American Mathematical Society 1, 449–463 (1950)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Arvind, V., Mukhopadhyay, P.: Derandomizing the isolation lemma and lower bounds for circuit size. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds.) APPROX and RANDOM 2008. LNCS, vol. 5171, pp. 276–289. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Bogdanov, A., Wee, H.: More on noncommutative polynomial identity testing. In: IEEE Conference on Computational Complexity, pp. 92–99. IEEE Computer Society (2005)Google Scholar
  4. 4.
    Cohn, P.M.: Further Algebra and Applications. Springer (2003)Google Scholar
  5. 5.
    Eilenberg, S.: Automata, Languages, and Machines, vol. A. Academic Press (1974)Google Scholar
  6. 6.
    Elgot, C.C., Mezei, J.E.: Two-sided finite-state transductions (abbreviated version). In: SWCT (FOCS), pp. 17–22. IEEE Computer Society (1963)Google Scholar
  7. 7.
    Friedman, E.P., Greibach, S.A.: A polynomial time algorithm for deciding the equivalence problem for 2-tape deterministic finite state acceptors. SIAM J. Comput. 11(1), 166–183 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Griffiths, T.V.: The unsolvability of the equivalence problem for ε-free nondeterministic generalized machines. J. ACM 15(3), 409–413 (1968)zbMATHCrossRefGoogle Scholar
  9. 9.
    Harju, T., Karhumäki, J.: The equivalence problem of multitape finite automata. Theor. Comput. Sci. 78(2), 347–355 (1991)zbMATHCrossRefGoogle Scholar
  10. 10.
    Kiefer, S., Murawski, A., Ouaknine, J., Wachter, B., Worrell, J.: On the complexity of equivalence and minimisation for Q-weighted automata. Logical Methods in Computer Science 9 (2013)Google Scholar
  11. 11.
    Malcev, A.I.: On the embedding of group algebras in division algebras. Dokl. Akad. Nauk 60, 1409–1501 (1948)MathSciNetGoogle Scholar
  12. 12.
    Mulmuley, K., Vazirani, U.V., Vazirani, V.V.: Matching is as easy as matrix inversion. In: STOC, pp. 345–354 (1987)Google Scholar
  13. 13.
    Neumann, B.H.: On ordered groups. Amer. J. Math. 71, 1–18 (1949)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Neumann, B.H.: On ordered division rings. Trans. Amer. Math. Soc. 66, 202–252 (1949)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Rabin, M., Scott, D.: Finite automata and their decision problems. IBM Journal of Research and Development 3(2), 114–125 (1959)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Sakarovich, J.: Elements of Automata Theory. Cambridge University Press (2003)Google Scholar
  17. 17.
    Saltman, D.: Lectures on Division Algebras. American Math. Soc. (1999)Google Scholar
  18. 18.
    Schützenberger, M.-P.: On the definition of a family of automata. Inf. and Control 4, 245–270 (1961)zbMATHCrossRefGoogle Scholar
  19. 19.
    Tzeng, W.: A polynomial-time algorithm for the equivalence of probabilistic automata. SIAM Journal on Computing 21(2), 216–227 (1992)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • James Worrell
    • 1
  1. 1.Department of Computer ScienceUniversity of OxfordUK

Personalised recommendations