Skip to main content
Springer Nature Link
Log in

Operational State Complexity Revisited: The Contribution of Monsters and Modifiers

  • Conference paper
  • First Online:
Descriptional Complexity of Formal Systems (DCFS 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13918))

Included in the following conference series:

  • 186 Accesses

Abstract

The state complexity of a language L is the number of states of its minimal automaton. The operational (state) complexity of an operation \(\otimes \) is the maximal state complexity of the languages \(\otimes (L_1,\cdots ,L_k)\) where \(L_1,\dots ,L_k\) are languages with given state complexities. We highlight two tools that have been developed in recent years and which theorise and summarise a large number of methods used since the 1970 s. A monster is an automaton where any function from states to states is represented by at least one letter. A modifier is a set of functions that allows us to transform a set of automata into an automaton. Thanks to these techniques, we revisit the state complexity of some language transformation operations. These tools allow to propose a unified vision on a large number of known results in a natural way and also to obtain new ones at lower cost. We detail two examples: the star of a boolean operation and the reversal of a boolean operation. By doing so, we directly obtain a new result when the boolean operation is the Xor.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Birget, J.-C.: Intersection and union of regular languages and state complexity. Inf. Process. Lett. 43(4), 185–190 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brzozowski, J.A.: In search of most complex regular languages. Int. J. Found. Comput. Sci. 24(6), 691–708 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Caron, P.: AG: A set of maple packages for manipulating automata and semigroups. Softw. Pract. Exper. 27(8), 863–884 (1997)

    Google Scholar 

  4. Caron, P., Hamel-De le court, E., Luque, J.B.: Algebraic and combinatorial tools for state complexity: Application to the star-xor problem. In: Leroux, J., Raskin, J.-F., (eds.) Proceedings Tenth International Symposium on Games, Automata, Logics, and Formal Verification, GandALF 2019, Bordeaux, France, 2–3rd September 2019, volume 305. EPTCS, pp. 154–168 (2019)

    Google Scholar 

  5. Caron, P., Hamel-de-le-court, E., Luque, J.-G.: A study of a simple class of modifiers: product modifiers. In: Jonoska, N., Savchuk, D. (eds.) DLT 2020. LNCS, vol. 12086, pp. 110–121. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-48516-0_9

    Chapter  Google Scholar 

  6. Caron, P., Hamel-De le Court, E., Luque, J.B.: Combination of roots and boolean operations: An application to state complexity. Inf. Comput. 289(Part), 104961 (2022)

    Google Scholar 

  7. Caron, P., Hamel-De le court, E., Luque, J.-G., Patrou, B.: New tools for state complexity. Discret. Math. Theor. Comput. Sci. 22(1) (2020)

    Google Scholar 

  8. Caron, P., Luque, J.-G., Mignot, L., Patrou, B.: State complexity of catenation combined with a boolean operation: A unified approach. Int. J. Found. Comput. Sci. 27(6), 675–704 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Caron, P., Luque, J.-G., Patrou, B.: State complexity of multiple catenations. Fundam. Inform. 160(3), 255–279 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Caron, P., Luque, J.-G., Patrou, B.: State complexity of combined operations involving catenation and binary boolean operations: beyond the brzozowski conjectures. Theor. Comput. Sci. 800, 15–30 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Caron, P., Luque, J.-G., Patrou, B.: A combinatorial approach for the state complexity of the shuffle product. J. Autom. Lang. Comb. 25(4), 291–320 (2020)

    MathSciNet  MATH  Google Scholar 

  12. Champarnaud, J.-M., Hansel, G.: Automate, a computing package for automata and finite semigroups. J. Symb. Comput. 12(2), 197–220 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cui, B., Gao, Y., Kari, L., Sheng, Yu.: State complexity of two combined operations: Catenation-union and catenation-intersection. Int. J. Found. Comput. Sci. 22(8), 1797–1812 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Davies, S.: A general approach to state complexity of operations: formalization and limitations. In: Hoshi, M., Seki, S. (eds.) DLT 2018. LNCS, vol. 11088, pp. 256–268. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-98654-8_21

    Chapter  Google Scholar 

  15. Domaratzki, M.: State complexity of proportional removals. J. Autom. Lang. Comb. 7(4), 455–468 (2002)

    MathSciNet  MATH  Google Scholar 

  16. Ésik, Z., Gao, Y., Liu, G., Sheng, Yu.: Estimation of state complexity of combined operations. Theor. Comput. Sci. 410(35), 3272–3280 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gao, Y., Moreira, N., Reis, R., Sheng, Yu.: A survey on operational state complexity. J. Autom. Lang. Comb. 21(4), 251–310 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Gao, Y., Salomaa, K., Sheng, Yu.: The state complexity of two combined operations: Star of catenation and star of reversal. Fundam. Inform. 83(1–2), 75–89 (2008)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  20. Jirásek, J., Jirásková, G.: The exact complexity of star-complement-star. In: Câmpeanu, C. (ed.) CIAA 2018. LNCS, vol. 10977, pp. 223–235. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94812-6_19

    Chapter  MATH  Google Scholar 

  21. Jirásek, J., Jirásková, G., Szabari, A.: State complexity of concatenation and complementation. Int. J. Found. Comput. Sci. 16(3), 511–529 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jirásková, G.: State complexity of some operations on binary regular languages. Theor. Comput. Sci. 330(2), 287–298 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jirásková, G., Okhotin, A.: On the state complexity of star of union and star of intersection. Fundam. Inform. 109(2), 161–178 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jirásková, G., Shallit, J.: The state complexity of star-complement-star. In: Yen, H.-C., Ibarra, O.H. (eds.) DLT 2012. LNCS, vol. 7410, pp. 380–391. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31653-1_34

    Chapter  Google Scholar 

  25. Liu, G., Martín-Vide, C., Salomaa, A., Sheng, Yu.: State complexity of basic language operations combined with reversal. Inf. Comput. 206(9–10), 1178–1186 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Maslov, A.N.: Estimates of the number of states of finite automata. Soviet Math. Dokl. 11, 1373–1375 (1970)

    MATH  Google Scholar 

  27. Sakoda, W.J., Sipser, M.: Nondeterminism and the size of two way finite automata. In Lipton, R.J., Burkhard, W.A., Savitch, W.J., Friedman, E.P., Aho, A.V., (eds.) Proceedings of the 10th Annual ACM Symposium on Theory of Computing, 1–3 May 1978, San Diego, California, USA, pp. 275–286. ACM (1978)

    Google Scholar 

  28. Salomaa, A., Salomaa, K., Sheng, Yu.: State complexity of combined operations. Theor. Comput. Sci. 383(2–3), 140–152 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Salomaa, A., Wood, D., Sheng, Yu.: On the state complexity of reversals of regular languages. Theor. Comput. Sci. 320(2–3), 315–329 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sheng, Yu., Zhuang, Q., Salomaa, K.: The state complexities of some basic operations on regular languages. Theor. Comput. Sci. 125(2), 315–328 (1994)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LITIS, Université de Rouen, Avenue de l’Université, 76801, Saint-Étienne du Rouvray Cedex, France

    Pascal Caron & Bruno Patrou

  2. GR2IF, Université de Rouen, Avenue de l’Université, 76801, Saint-Étienne du Rouvray Cedex, France

    Jean-Gabriel Luque

Authors
  1. Pascal Caron
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean-Gabriel Luque
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bruno Patrou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pascal Caron .

Editor information

Editors and Affiliations

  1. University of Potsdam, Potsdam, Germany

    Henning Bordihn

  2. Santa Clara University, Santa Clara, CA, USA

    Nicholas Tran

  3. University of Debrecen, Debrecen, Hungary

    György Vaszil

Rights and permissions

Reprints and permissions

Copyright information

© 2023 IFIP International Federation for Information Processing

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Caron, P., Luque, JG., Patrou, B. (2023). Operational State Complexity Revisited: The Contribution of Monsters and Modifiers. In: Bordihn, H., Tran, N., Vaszil, G. (eds) Descriptional Complexity of Formal Systems. DCFS 2023. Lecture Notes in Computer Science, vol 13918. Springer, Cham. https://doi.org/10.1007/978-3-031-34326-1_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-34326-1_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-34325-4

  • Online ISBN: 978-3-031-34326-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Societies and partnerships