Skip to main content
Springer Nature Link
Log in

Bounded Automata Groups are co-ET0L

  • Conference paper
  • First Online:
Language and Automata Theory and Applications (LATA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11417))

  • 553 Accesses

Abstract

Holt and Röver proved that finitely generated bounded automata groups have indexed co-word problem. Here we sharpen this result to show they are in fact co-ET0L.

Research supported by Australian Research Council grant DP160100486 and an Australian Government Research Training Program PhD Scholarship.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Anīsīmov, A.V.: The group languages. Kibernetika (Kiev) 7(4), 18–24 (1971)

    MathSciNet  Google Scholar 

  2. Asveld, P.R.J.: Controlled iteration grammars and full hyper-AFL’s. Inf. Control 34(3), 248–269 (1977)

    Article  MathSciNet  Google Scholar 

  3. Bleak, C., Matucci, F., Neunhöffer, M.: Embeddings into Thompson’s group \(V\) and \(co\cal{CF}\) groups. J. Lond. Math. Soc. (2) 94(2), 583–597 (2016). https://doi.org/10.1112/jlms/jdw04

    Article  MathSciNet  MATH  Google Scholar 

  4. Ciobanu, L., Diekert, V., Elder, M.: Solution sets for equations over free groups are EDT0L languages. Int. J. Algebra Comput. 26(5), 843–886 (2016). https://doi.org/10.1142/S0218196716500363

    Article  MathSciNet  MATH  Google Scholar 

  5. Ciobanu, L., Elder, M., Ferov, M.: Applications of L systems to group theory. Int. J. Algebra Comput. 28(2), 309–329 (2018). https://doi.org/10.1142/S0218196718500145

    Article  MathSciNet  MATH  Google Scholar 

  6. Culik II, K.: On some families of languages related to developmental systems. Int. J. Comput. Math. 4, 31–42 (1974). https://doi.org/10.1080/00207167408803079

    Article  MathSciNet  MATH  Google Scholar 

  7. Diekert, V., Elder, M.: Solutions of twisted word equations, EDT0L languages, and context-free groups. In: 44th International Colloquium on Automata, Languages, and Programming, LIPIcs. Leibniz International Proceedings in Informatics, vol. 80, Article No. 96, 14. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2017)

    Google Scholar 

  8. Diekert, V., Jeż, A., Kufleitner, M.: Solutions of word equations over partially commutative structures. In: 43rd International Colloquium on Automata, Languages, and Programming, LIPIcs. Leibniz International Proceedings in Informatics, vol. 55, Article No. 127, 14. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2016)

    Google Scholar 

  9. Elder, M., Kambites, M., Ostheimer, G.: On groups and counter automata. Int. J. Algebra Comput. 18(8), 1345–1364 (2008). https://doi.org/10.1142/S0218196708004901

    Article  MathSciNet  MATH  Google Scholar 

  10. 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)

    Book  Google Scholar 

  11. Gilman, R.H.: Formal languages and infinite groups. In: Geometric and Computational Perspectives on Infinite Groups, Minneapolis, MN and New Brunswick, NJ, 1994. DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, vol. 25, pp. 27–51. American Mathematical Society, Providence (1996)

    Chapter  Google Scholar 

  12. Grigorchuk, R.: On Burnside’s problem on periodic groups. Funktsional. Anal. i Prilozhen. 14(1), 53–54 (1980)

    Article  MathSciNet  Google Scholar 

  13. Gupta, N., Sidki, S.: On the Burnside problem for periodic groups. Mathematische Zeitschrift 182(3), 385–388 (1983). https://doi.org/10.1007/BF01179757

    Article  MathSciNet  MATH  Google Scholar 

  14. Holt, D.F., Rees, S., Röver, C.E., Thomas, R.M.: Groups with context-free co-word problem. J. Lond. Math. Soc. (2) 71(3), 643–657 (2005). https://doi.org/10.1112/S002461070500654X

    Article  MathSciNet  MATH  Google Scholar 

  15. Holt, D.F., Röver, C.E.: Groups with indexed co-word problem. Int. J. Algebra Comput. 16(5), 985–1014 (2006). https://doi.org/10.1142/S0218196706003359

    Article  MathSciNet  MATH  Google Scholar 

  16. König, D., Lohrey, M., Zetzsche, G.: Knapsack and subset sum problems in nilpotent, polycyclic, and co-context-free groups. In: Algebra and Computer Science, Contemporary Mathematics, vol. 677, pp. 129–144. American Mathematical Society, Providence (2016)

    Google Scholar 

  17. van Leeuwen, J.: Variations of a new machine model. In: 17th Annual Symposium on Foundations of Computer Science, Houston, Texas 1976, pp. 228–235. IEEE Computer Society, Long Beach, October 1976. https://doi.org/10.1109/SFCS.1976.35

  18. Lehnert, J., Schweitzer, P.: The co-word problem for the Higman-Thompson group is context-free. Bull. Lond. Math. Soc. 39(2), 235–241 (2007). https://doi.org/10.1112/blms/bdl043

    Article  MathSciNet  MATH  Google Scholar 

  19. Lindenmayer, A.: Mathematical models for cellular interactions in development I. Filaments with one-sided inputs. J. Theoret. Biol. 18(3), 280–99 (1968). https://doi.org/10.1016/0022-5193(68)90079-9

    Article  Google Scholar 

  20. Muller, D.E., Schupp, P.E.: The theory of ends, pushdown automata, and second-order logic. Theoret. Comput. Sci. 37(1), 51–75 (1985). https://doi.org/10.1016/0304-3975(85)90087-8

    Article  MathSciNet  MATH  Google Scholar 

  21. Nekrashevych, V.: Self-similar Groups, Mathematical Surveys and Monographs, vol. 117. American Mathematical Society, Providence (2005). https://doi.org/10.1090/surv/117

    Book  MATH  Google Scholar 

  22. Rozenberg, G., Salomaa, A. (eds.): Handbook of Formal Languages. Springer, Berlin (1997). https://doi.org/10.1007/978-3-642-59126-6

    Book  MATH  Google Scholar 

  23. Rozenberg, G.: Extension of tabled OL-systems and languages. Int. J. Comput. Inf. Sci. 2, 311–336 (1973)

    Article  MathSciNet  Google Scholar 

  24. Sidki, S.: Automorphisms of one-rooted trees: growth, circuit structure, and acyclicity. J. Math. Sci. (New York) 100(1), 1925–1943 (2000). https://doi.org/10.1007/BF02677504. Algebra, 12

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors wish to thank Claas Röver, Michal Ferov and Laura Ciobanu for helpful comments.

Author information

Authors and Affiliations

  1. School of Mathematical and Physical Sciences, University of Technology Sydney, Ultimo, NSW, 2007, Australia

    Alex Bishop & Murray Elder

Authors
  1. Alex Bishop
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Murray Elder
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Murray Elder .

Editor information

Editors and Affiliations

  1. Rovira i Virgili University, Tarragona, Spain

    Carlos Martín-Vide

  2. Saint Petersburg State University, St. Petersburg, Russia

    Alexander Okhotin

  3. Ariel University, Ariel, Israel

    Dana Shapira

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Bishop, A., Elder, M. (2019). Bounded Automata Groups are co-ET0L. In: Martín-Vide, C., Okhotin, A., Shapira, D. (eds) Language and Automata Theory and Applications. LATA 2019. Lecture Notes in Computer Science(), vol 11417. Springer, Cham. https://doi.org/10.1007/978-3-030-13435-8_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-13435-8_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-13434-1

  • Online ISBN: 978-3-030-13435-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics