Groups Whose Word Problem is a Petri Net Language

  • Gabriela Aslı Rino Nesin
  • Richard M. Thomas
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9118)


There has been considerable interest in exploring the connections between the word problem of a finitely generated group as a formal language and the algebraic structure of the group. However, there are few complete characterizations that tell us precisely which groups have their word problem in a specified class of languages. We investigate which finitely generated groups have their word problem equal to a language accepted by a Petri net and give a complete classification, showing that a group has such a word problem if and only if it is virtually abelian.


Finitely generated group Word problem Petri net language 



Some of the research for this paper was done whilst the authors were visiting the Nesin Mathematics Village in Turkey; the authors would like to thank the Village both for the financial support that enabled them to work there and for the wonderful research environment it provided that stimulated the results presented here. The authors would like to thank the referees for their helpful and constructive comments. The second author also would like to thank Hilary Craig for all her help and encouragement.


  1. 1.
    Anisimov, A.V.: Group languages. Cybernet. Syst. Anal. 7, 594–601 (1971)CrossRefGoogle Scholar
  2. 2.
    Autebert, J.M., Boasson, L., Sénizergues, G.: Groups and NTS languages. J. Comput. Syst. Sci. 35, 243–267 (1987)zbMATHCrossRefGoogle Scholar
  3. 3.
    Dunwoody, M.J.: The accessibility of finitely presented groups. Invent. Math. 81, 449–457 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Elder, M., Kambites, M., Ostheimer, G.: On groups and counter automata. Internat. J. Algebra Comput. 18, 1345–1364 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Greibach, S.A.: Remarks on blind and partially blind one-way multicounter machines. Theor. Comput. Sci. 7, 311–324 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Gromov, M.: Groups of polynomial growth and expanding maps. Publications Mathematiques de l’Institut des Hautes Etudes Scientifiques, vol. 53, pp. 53–78 (1981)Google Scholar
  7. 7.
    Hack, M.: Petri net languages. Computation Structures Group Memo 124, Project MAC, M.I.T. (1975)Google Scholar
  8. 8.
    Herbst, T.: On a subclass of context-free groups. RAIRO Theor. Infor. Appl. 25, 255–272 (1991)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Herbst, T., Thomas, R.M.: Group presentations, formal languages and characterizations of one-counter groups. Theor. Comput. Sci. 112, 187–213 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Holt, D.F., Owens, M.D., Thomas, R.M.: Groups and semigroups with a one-counter word problem. J. Aust. Math. Soc. 85, 197–209 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Holt, D.F., Rees, S., Röver, C.E., Thomas, R.M.: Groups with context-free co-word problem. J. Lond. Math. Soc. 71, 643–657 (2005)zbMATHCrossRefGoogle Scholar
  12. 12.
    Jantzen, M.: On the hierarchy of Petri net languages. RAIRO Theor. Inf. Appl. 13, 19–30 (1979)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Jantzen, M.: Language theory of Petri nets. In: Brauer, W., Reisig, W., Rozenberg, G. (eds.) Petri Nets: Central Models and Their Properties. LNCS, vol. 254, pp. 397–412. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  14. 14.
    Johnson, D.L.: Presentations of groups. In: London mathematical society student texts, CUP, 2nd edn. Cambridge University Press, Cambridge (1997).
  15. 15.
    Lambert, J.: A structure to decide reachability in Petri nets. Theoret. Comput. Sci. 99, 79–104 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Lehnert, J., Schweitzer, P.: The co-word problem for the Higman-Thompson group is context-free. Bull. Lond. Math. Soc. 39, 235–241 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Muller, D., Schupp, P.: Groups, the theory of ends, and context-free languages. J. Comput. Syst. Sci. 26, 295–310 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Parikh, R.J.: On context-free languages. J. ACM 13, 570–581 (1966)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Peterson, J.L.: Computation sequence sets. J. Comput. Syst. Sci. 13, 1–24 (1976)zbMATHCrossRefGoogle Scholar
  20. 20.
    Robinson, D.: A Course in the Theory of Groups, 2nd edn. Springer, New York (1995)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Gabriela Aslı Rino Nesin
    • 1
  • Richard M. Thomas
    • 1
  1. 1.Department of Computer ScienceUniversity of LeicesterLeicesterUK

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