Abstract
Suppose that L is a nonempty language over A, \(k\in \mathbb {N}^0\) and \(m\in \mathbb {N}\). If L satisfies \((LA^k)^mL\)\(\cap A^+(LA^k)^{m-1}LA^+=\emptyset \), then L is a code and is called a (k, m)-comma code. If L is a (k, m)-comma code, then it is known that L is an infix code when \(m=1\) and L is a bifix code when \(m\ge 1\). In this paper, we extend the study and find that the class of (k, m)-comma codes and the class of infix codes are incomparable but they are not disjoint, so we first characterize the (k, m)-comma codes with \(m\ge 2\) which are infix codes. We also describe the solid codes with minimum lengths greater than k and the 2-(k, 1)-comma codes by different approaches. Finally, the class of bifix codes will be classified into disjoint union of some subclasses and a criterion to determine the subclass membership of a given bifix code will be given.
Similar content being viewed by others
References
Berstel, J., Perrin, D., Reutenauer, C.: Codes and Automata. Cambridge University Press, Cambridge (2010)
Cao, C.H., Liu, H.Y., Yang, D.: Characterizations of \(k\)-comma codes and \(k\)-comma intercodes. Acta Inf. 53, 23–33 (2016)
Cui, B., Kari, L., Seki, S.: \(k\)-Comma codes and their generalizations. Fund. Inf. 107, 1–18 (2011)
Delbr, M., Welch, L.R., Golomb, S.W.: Construction and properties of comma-free codes. Matematika 23, 137–160 (1960)
Golomb, S.W., Gordon, B.L., Welch, R.: Comma-free codes. Can. J. Math. 10, 202–209 (1958)
Han, Y.S., Salomaa, K., Wood, D.: Intercode regular languages. Fund. Inf. 76, 113–128 (2007)
Hesieh, C.Y., Hsu, S.C., Shyr, H.J.: Some algebraic properties of comma-free codes. RIMS Kenkyuoku. 697, 57–66 (1989)
Ito, M., Jürgensen, H., Shyr, H.J., Thierrin, G.: Outfix and infix codes and related classes of languages. J. Comput. System. Sci. 43, 484–508 (1991)
Jürgensen, H., Yu, S.S.: Solid codes. J. Inform. Process Cyberne. EIK. 26, 563–574 (1990)
Jürgensen, H., Salomaa, K., Yu, S.: Decidability of the intercode property. Elektronische Informationsverarbeitung und Kybernetik 29(6), 375–380 (1993)
Jürgensen, H., Konstantinidis, S.: Codes. In: Rozenberg, G., Salomaa, A. (eds.) Word, language, grammar, volume 1 of handbook of formal languages, pp 511–607. Springer-Verlag, New York (1997)
Li, Z.Z., Tsai, Y.S.: Classification of bifix codes. Int. J. Comput. Math. 87, 2625–2643 (2010)
Liu, Y., Guo, Y.Q., Tsai, Y.S.: Solid codes and the uniform density of fd-domains. Sci. China Ser. A: Math. 50, 1026–1034 (2007)
Ries, C.M.: Intercodes and the semigroups they generated. Intern. J. Computer. Math. 51, 7–13 (1994)
Romanov, O.T.: Invariant decoding automata without look-ahead. Problemy Kibernetiki 17, 233–236 (1966). (in Russian)
Schützenberger, M.P.: Une théorie algébrique du codage. Séminaire Dubreil-Pisot 15, 1955–1956 (1955)
Shyr, H.J., Yu, S.S.: Intercodes and related properties. Soochow. J. Math. 16, 95–107 (1990)
Shyr, H.J., Yu, S.S.: Solid codes and disjunctive domains. Semigroup Forum. 41, 23–37 (1990)
Shyr, H.J.: Free Monoids and Languages, 3rd edn. Hon Min Book Company, Taichung (2001)
Watson, J.: Genes, Girls and Gamow: After the Double Helix. Oxford University Press, Oxford (2001)
Yu, S.S.: A characterization of intercodes. Int. J. Comput. Math. 36, 39–48 (1990)
Zhang, D., Guo, Y.Q., Shum, K.P.: On some decompositions of r-disjunctive languages. Bull. Malays. Math. Sci. Soc. 37, 727–746 (2014)
Zhang, D., Guo, Y.Q., Shum, K.P.: Some results in r-disjunctive languages and related topics. Soft. Comput. 21, 2477–2483 (2016)
Acknowledgements
The authors would like to thank the referees for their reading the manuscript carefully and for providing some valuable revising suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hamid Reza Ebrahimi Vishki.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported by National Natural Science Foundation of China #11861051.
Rights and permissions
About this article
Cite this article
Liu, H., Shum, K.P. & Guo, Y. Some Results on \({(\varvec{k,m})}\)-Comma Codes. Bull. Iran. Math. Soc. 46, 1143–1162 (2020). https://doi.org/10.1007/s41980-019-00318-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41980-019-00318-z