Skip to main content
SpringerLink
Log in

M-ambiguity Sequences for Parikh Matrices and Their Periodicity Revisited

  • Original Article
  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

The introduction of Parikh matrices by Mateescu et al. in 2001 has sparked numerous new investigations in the theory of formal languages by various researchers, among whom is Şerbănuţă. Recently, a decade-old conjecture by Şerbǎnuţǎ on the M-ambiguity of words has disproved, leading to new possibilities in the study of such words. In this paper, we investigate how selective repeated duplications of letters in a word affect the M-ambiguity of the resulting words. The corresponding M-ambiguity of each of those words is then presented in sequences, which we term as M-ambiguity sequences. We show that nearly all patterns of M-ambiguity sequences are attainable. Finally, by employing certain algebraic approach and some underlying theory in integer programming, we show that repeated periodic duplications of letters of the same type in a word result in an M-ambiguity sequence that is ultimately periodic.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. See Theorem A.1 in [16].

  2. It will be clear why each \(u_i\) is a letter instead of a word when the reader reaches the second sentence of the next paragraph.

References

  1. Atanasiu, A.: Binary amiable words. Int. J. Found. Comput. Sci. 18(2), 387–400 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atanasiu, A., Atanasiu, R., Petre, I.: Parikh matrices and amiable words. Theor. Comput. Sci. 390(1), 102–109 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Atanasiu, A., Teh, W.C.: A new operator over Parikh languages. Int. J. Found. Comput. Sci. 27(6), 757–769 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Mahalingam, K., Bera, S., Subramanian, K.G.: Properties of Parikh matrices of words obtained by an extension of a restricted shuffle operator. Int. J. Found. Comput. Sci. 29(3), 403–3413 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Mateescu, A., Salomaa, A.: Matrix indicators for subword occurrences and ambiguity. Int. J. Found. Comput. Sci. 15(2), 277–292 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Mateescu, A., Salomaa, A., Salomaa, K., Yu, S.: A sharpening of the Parikh mapping. Theor. Inform. Appl. 35(6), 551–564 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Parikh, R.J.: On context-free languages. J. Assoc. Comput. Mach. 13, 570–581 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  8. Poovanandran, G., Teh, W.C.: Strong \(2\cdot t\) and strong \(3\cdot t\) transformations for strong M-equivalence. Int. J. Found. Comput. Sci. 29(1), 123–137 (2018)

    Article  MATH  Google Scholar 

  9. Poovanandran, G., Teh, W.C.: Elementary matrix equivalence and core transformation graphs for Parikh matrices. Discrete Appl. Math. 251, 276–289 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Poovanandran, G., Teh, W.C.: On M-equivalence and strong M-equivalence for Parikh matrices. Int. J. Found. Comput. Sci. 29(1), 123–137 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Poovanandran, G., Teh, W.C.: Parikh matrices and M-ambiguity sequence. In: Journal of Physics: Conference Series, vol. 1132, p. 012012 (2018)

  12. Salomaa, A.: Criteria for the matrix equivalence of words. Theor. Comput. Sci. 411(16), 1818–1827 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Salomaa, A., Yu, S.: Subword occurrences, Parikh matrices and Lyndon images. Int. J. Found. Comput. Sci. 21(1), 91–111 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Hoboken (1998)

    MATH  Google Scholar 

  15. Şerbănuţă, V.N.: On Parikh matrices, ambiguity, and prints. Int. J. Found. Comput. Sci. 20(1), 151–165 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Şerbănuţă, V.N., Şerbănuţă, T.F.: Injectivity of the Parikh matrix mappings revisited. Fundam. Inform. 73(1), 265–283 (2006)

    MathSciNet  MATH  Google Scholar 

  17. Teh, W.C.: Parikh matrices and Parikh rewriting systems. Fundam. Inform. 146, 305–320 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Teh, W.C., Atanasiu, A.: On a conjecture about Parikh matrices. Theor. Comput. Sci. 628, 30–39 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Teh, W.C., Atanasiu, A., Poovanandran, G.: On strongly M-unambiguous prints and Şerbănuţă’s conjecture for Parikh matrices. Theor. Comput. Sci. 719, 86–93 (2018)

    Article  MATH  Google Scholar 

  20. Teh, W.C., Kwa, K.H.: Core words and Parikh matrices. Theor. Comput. Sci. 582, 60–69 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Teh, W.C., Subramanian, K.G., Bera, S.: Order of weak M-relation and Parikh matrices. Theor. Comput. Sci. 743, 83–92 (2018)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees for their careful reading of the original version of this paper. Their comments and suggestions have significantly improved the clarity of the work presented here. The authors also gratefully acknowledge support for this research by a Research University Grant No. 1001/PMATHS/8011019 of Universiti Sains Malaysia. This study is an extension of the work in [11].

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Gelugor, Malaysia

    Ghajendran Poovanandran & Wen Chean Teh

  2. School of Mathematics, Actuarial and Quantitative Studies, Asia Pacific University of Technology & Innovation, Technology Park Malaysia, 57000, Bukit Jalil, Kuala Lumpur, Malaysia

    Ghajendran Poovanandran

Authors
  1. Ghajendran Poovanandran
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wen Chean Teh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wen Chean Teh.

Additional information

Communicated by Rosihan M. Ali.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Poovanandran, G., Teh, W.C. M-ambiguity Sequences for Parikh Matrices and Their Periodicity Revisited. Bull. Malays. Math. Sci. Soc. 43, 3305–3321 (2020). https://doi.org/10.1007/s40840-019-00867-w

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-019-00867-w

Keywords

Mathematics Subject Classification

Search

Navigation