Abstract
The study of hairpin-free words has been initiated in the context of DNA computing. DNA strands that, theoretically speaking, are finite strings over the alphabet {A, G, C, T} are used in DNA computing to encode information. Due to the fact that A is complementary to T and G to C, DNA single strands that are complementary can bind to each other or to themselves in either intended or unintended ways. One of the structures that is usually undesirable for biocomputation, since it makes the affected DNA string unavailable for future interactions, is the hairpin: if some subsequences of a DNA single string are complementary to each other, the string will bind to itself forming a hairpin-like structure. This paper continues the theoretical study of hairpin-free languages. We study algebraic properties of hairpin-free words and hairpins. We also give a complete characterization of the syntactic monoid of the language consisting of all hairpin-free words over a given alphabet and illustrate it with an example using the DNA alphabet.
Similar content being viewed by others
References
Berstel J. and Perrin D. (1985). Theory of Codes. Academic, Orlando
Bonizzoni P., Defelice C., Mauri G. and Zizza R. (2006). Linear splicing and syntatctic monoid. Discrete Appl. Math. 15(3): 452–470
Haines H. (1969). On free monoids partially ordered by embedding. J. Comb. Theory 6: 94–98
Howie J.M. (1995). Funadamentals of Semigroup Theory. Oxford Science Publications, New York
Ito M., Júrgensen H., Shyr H.J. and Thierrin G. (1991). Outfix and infix codes and related classes of languages. J. Comput. Syst. Sci. 43: 484–508
Jonoska N., Mahalingam K. and Chen J. (2005). Involution codes: with application to DNA coded languages. Nat. Comput. 4(2): 141–162
Jürgensen H. (1999). Syntactic monoid of codes. Acta Cybern. 14: 117–133
Jürgensen, H., Konstantinidis, S.: Codes. In: Rozenberg, G., Salomaa A. (eds.) Handbook of Formal Languages, vol. 1, chap. 8, pp. 511–608 (1997)
Jürgensen, H., Thierrin, G.: Infix codes. In: Proceedings of the 4th Hungarian Computer Science Conference, pp. 25–29 (1985)
Kari L., Konstantinidis S., Losseva E. and Wozniak G. (2003). Sticky-free and overhang-free DNA languages. Acta Inf. 40: 119–157
Kari, L., Konstantinidis, S., Losseva, E., Sosik, P., Thierrin, G.: Hairpin structures in DNA words. In: Proceedings of the 11th International Meeting on DNA Computing, LNCS 3892, pp. 158–170 (2005)
Lallement G. (1995). Semigroups and Combinatorial Dynamics. Wiley/Interscience, New York
Luca A.D. and Restivo A. (1980). A characterization of strictly locally testable languages and its application to subsemigroups of a free semigroup. Inf. Control 44: 300–319
Moriya, T.: Closure under composition and syntactic monoids of some codes. Report 00-04, Department of Mathematics and Computer Science, Martin-Luther-Universität Halle-Wittenberg (1999)
Paun G., Rozenberg G. and Yokomori T. (2001). Hairpin languages. Int. J. Found. Comput. Sci. 12(6): 849–857
Petrich M. and Reis C.M. (1995). The syntactic monoid of the semigroup generated by a comma-free code. Proc. R. Soc. Edinb. 125: 165–179
Petrich M. and Thierrin G. (1990). The syntactic monoid of an infix code. Proc. Am. Math. Soc. 109(4): 865–873
Petrich M., Reis C.M. and Thierrin G. (1996). The syntactic monoid of the semigroup generated by a maximal prefix code. Proc. Am. Math. Soc. 124(3): 655–663
Pin J.E. (1986). Varieties of Formal Languages. Plenum Press, New York
Shyr H.J. (2001). Free Monoids and Languages. Hon Min Book Company, Taichung
Simon, I.: Piecewise testable events. In: Automata Theory and Formal Languages, Proceedings of the 2nd GI Conference, LNCS 33, pp. 214–222. Springer, Berlin (1975)
Thierrin, G.: Convex languages. In: Proceedings of IRIA Symposium, pp. 481–492. North Holland, Amsterdam (1972)
Thierrin G. (1973). The syntactic monoid of a hypercode. Semigroup Forum 6: 227–231
Yu S.S. (2005). Languages and codes. Lecture Notes, Department of Computer Science. National Chung-Hsing University, Taichung, Taiwan
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kari, L., Mahalingam, K. & Thierrin, G. The syntactic monoid of hairpin-free languages. Acta Informatica 44, 153–166 (2007). https://doi.org/10.1007/s00236-007-0041-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00236-007-0041-4