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Union-Freeness, Deterministic Union-Freeness and Union-Complexity

  • Benedek NagyEmail author
Conference paper
  • 145 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11612)

Abstract

Union-free expressions are regular expressions without using the union operation. Consequently, union-free languages are described by regular expressions using only concatenation and Kleene star. The language class is also characterised by a special class of finite automata: 1CFPAs have exactly one cycle-free accepting path from each of their states. Obviously such an automaton has exactly one accepting state. The deterministic counterpart of such class of automata defines the deterministic union-free languages. A regular expression is in union (disjunctive) normal form if it is a finite union of union-free expressions. By manipulating regular expressions, each of them has equivalent expression in union normal form. By the minimum number of union-free expressions needed to describe a regular language, its union-complexity is defined. For any natural number n there are languages such that their union complexity is n. However, there is not known any simple algorithm to determine the union-complexity of any language. Regarding the deterministic union-free languages, there are regular languages such that they cannot be written as a union of finitely many deterministic union-free languages.

References

  1. 1.
    Afonin, S., Golomazov, D.: Minimal union-free decompositions of regular languages. In: Dediu, A.H., Ionescu, A.M., Martín-Vide, C. (eds.) LATA 2009. LNCS, vol. 5457, pp. 83–92. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-00982-2_7CrossRefGoogle Scholar
  2. 2.
    Brzozowski, J.A.: Regular expression techniques for sequential circuits. Ph.D. Dissertation, Department of Electrical Engineering, Princeton University, Princeton, June 1962Google Scholar
  3. 3.
    Brzozowski, J.A., Davies, S.: Most complex deterministic union-free regular languages. In: Konstantinidis, S., Pighizzini, G. (eds.) DCFS 2018. LNCS, vol. 10952, pp. 37–48. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-94631-3_4CrossRefGoogle Scholar
  4. 4.
    Crvenković, S., Dolinka, I., Ésik, Z.: On equations for union-free regular languages. Inf. Comput. 164(1), 152–172 (2001).  https://doi.org/10.1006/inco.2000.2889MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Holzer, M., Kutrib, M.: Structure and complexity of some subregular language families. In: Konstantinidis, S., Moreira, N., Reis, R., Shallit, J. (eds.) The Role of Theory in Computer Science - Essays Dedicated to Janusz Brzozowski, pp. 59–82. World Scientific (2017). https://doi.org/10.1142/9789813148208_0003
  6. 6.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading (1979)zbMATHGoogle Scholar
  7. 7.
    Jirásková, G., Masopust, T.: Complexity in union-free regular languages. Int. J. Found. Comput. Sci. 22, 1639–1653 (2011).  https://doi.org/10.1142/S0129054111008933MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jirásková, G., Nagy, B.: On union-free and deterministic union-free languages. In: Baeten, J.C.M., Ball, T., de Boer, F.S. (eds.) TCS 2012. LNCS, vol. 7604, pp. 179–192. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-33475-7_13
  9. 9.
    Kutrib, M., Wendlandt, M.: Expressive capacity of subregular expressions. RAIRO ITA Theor. Inform. Appl. 52(2–3–4), 201–218 (2018).  https://doi.org/10.1051/ita/2018014MathSciNetCrossRefGoogle Scholar
  10. 10.
    Nagy, B.: A normal form for regular expressions. In: Calude, C., Calude, E., Dinnen, M.J. (eds.) Supplemental Papers for DLT 2004, pp. 51–60. CDMTCS Report 252, Auckland (2004)Google Scholar
  11. 11.
    Nagy, B.: Union-free regular languages and 1-cycle-free-path-automata. Publ. Math. Debrecen 68, 183–197 (2006)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Nagy, B.: On union-complexity of regular languages. In: Proceedings of the 11th IEEE International Symposium on Computational Intelligence and Informatics, pp. 177–182 (2010)Google Scholar
  13. 13.
    Yu, S.: Regular languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, pp. 41–100. Springer, Heidelberg (1997).  https://doi.org/10.1007/978-3-642-59136-5_2CrossRefGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Arts and SciencesEastern Mediterranean UniversityFamagustaTurkey

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