Skip to main content
SpringerLink
Log in
Book cover

International Conference on Descriptional Complexity of Formal Systems

DCFS 2019: Descriptional Complexity of Formal Systems pp 46–56Cite as

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

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,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.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  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_7

    Chapter  Google Scholar 

  2. Brzozowski, J.A.: Regular expression techniques for sequential circuits. Ph.D. Dissertation, Department of Electrical Engineering, Princeton University, Princeton, June 1962

    Google Scholar 

  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_4

    Chapter  Google Scholar 

  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.2000.2889

    Article  MathSciNet  MATH  Google Scholar 

  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. Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading (1979)

    MATH  Google Scholar 

  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/S0129054111008933

    Article  MathSciNet  MATH  Google Scholar 

  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. 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/2018014

    Article  MathSciNet  Google Scholar 

  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. Nagy, B.: Union-free regular languages and 1-cycle-free-path-automata. Publ. Math. Debrecen 68, 183–197 (2006)

    MathSciNet  MATH  Google Scholar 

  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. 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_2

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, North Cyprus, Mersin-10, Turkey

    Benedek Nagy

Authors
  1. Benedek Nagy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Benedek Nagy .

Editor information

Editors and Affiliations

  1. Slovak Academy of Sciences, Košice, Slovakia

    Michal Hospodár

  2. Slovak Academy of Sciences, Košice, Slovakia

    Galina Jirásková

  3. Saint Mary's University, Halifax, NS, Canada

    Stavros Konstantinidis

Rights and permissions

Reprints and permissions

Copyright information

© 2019 IFIP International Federation for Information Processing

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Nagy, B. (2019). Union-Freeness, Deterministic Union-Freeness and Union-Complexity. In: Hospodár, M., Jirásková, G., Konstantinidis, S. (eds) Descriptional Complexity of Formal Systems. DCFS 2019. Lecture Notes in Computer Science(), vol 11612. Springer, Cham. https://doi.org/10.1007/978-3-030-23247-4_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-23247-4_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-23246-7

  • Online ISBN: 978-3-030-23247-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Societies and partnerships

Search

Navigation