Skip to main content
SpringerLink
Log in
SpringerLink
Log in

More on Deterministic and Nondeterministic Finite Cover Automata

Extended Abstract

  • Conference paper
  • First Online:
Implementation and Application of Automata (CIAA 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9223))

Included in the following conference series:

  • 505 Accesses

Abstract

Finite languages are an important sub-regular language family, which were intensively studied during the last two decades in particular from a descriptional complexity perspective. An important contribution to the theory of finite languages are the deterministic and the recently introduced nondeterministic finite cover automata (DFCAs and NFCAs, respectively) as an alternative representation of finite languages by ordinary finite automata. We compare these two types of cover automata from a descriptional complexity point of view, showing that these devices have a lot in common with ordinary finite automata. In particular, we study how to adapt lower bound techniques for nondeterministic finite automata to NFCAs such as, e.g., the biclique edge cover technique, solving an open problem from the literature. Moreover, the trade-off of conversions between DFCAs and NFCAs as well as between finite cover automata and ordinary finite automata are investigated. Finally, we present some results on the average size of finite cover automata.

Part of the work was done while the first author was at Institut für Informatik, Ludwig-Maximilians-Universität München, Oettingenstraße 67, 80538 München, Germany and the second author was at Institut für Informatik, Technische Universität München, Boltzmannstraße 3, 85748 Garching bei München, Germany.

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

Access this chapter

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

Institutional subscriptions

Notes

  1. 1.

    For a language \(L\subseteq \varSigma ^*\) define the Myhill-Nerode relation \(\equiv _L\) on words as follows: for \(u,v\in \varSigma ^*\) let \(u\equiv _Lv\) if and only if \(uw\in L\iff vw\in L\), for all \(w\in \varSigma ^*\).

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Bassino, F., Giambruno, L., Nicaud, C.: The average state complexity of rational operations on finite languages. Internat. J. Found. Comput. Sci. 21(4), 495–516 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brzozowski, J.A.: Canonical Regular Expressions and Minimal State Graphs for Definite Events. Mathematical Theory of Automata, MRI Symposia Series. Polytechnic Press, New York (1962)

    Google Scholar 

  4. Câmpeanu, C.: Non-deterministic finite cover automata. Sci. Ann. Comput. Sci. 25(1), 3–28 (2015)

    Article  MathSciNet  Google Scholar 

  5. Câmpeanu, C., Ho, W.H.: The maximum state complexity for finite languages. J. Autom. Lang. Comb. 9(2–3), 189–202 (2004)

    MathSciNet  MATH  Google Scholar 

  6. Câmpeanu, C., Kari, L., Păun, A.: Results on Transforming NFA into DFCA. Fundam. Inform. 64(1–4), 53–63 (2005)

    MATH  Google Scholar 

  7. Câmpeanu, C., Moreira, N., Reis, R.: Expected compression ratio for DFCA: experimental average case analysis. Universidade do Porto (2011), Technical report DCC-2011-07

    Google Scholar 

  8. Câmpeanu, C., Păun, A., Smith, J.R.: Tight bounds for the state complexity of deterministic cover automata. In: Leung, H., Pighizzini, G. (eds.) Proceedings of the 8th Workshop on Descriptional Complexity of Formal Systems, pp. 223–231, Las Cruces (2006), Computer Science Technical report NMSU-CS-2006-001

    Google Scholar 

  9. Câmpeanu, C., Păun, A., Yu, S.: An efficient algorithm for constructing minimal cover automata for finite languages. Internat. J. Found. Comput. Sci. 13(1), 83–97 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Câmpeanu, C., Sântean, N., Yu, S.: Minimal cover-automata for finite languages. Theoret. Comput. Sci. 267(1–2), 3–16 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Champarnaud, J.M., Guingne, F., Hansel, G.: Similarity relations and cover automata. RAIRO-Informatique théorique et Appl. Theor. Inform. Appl. 39(1), 115–123 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Champarnaud, J.M., Pin, J.E.: A maxmin problem on finite automata. Discrete Appl. Math. 23, 91–96 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Domaratzki, M., Kisman, D., Shallit, J.: On the number of distinct languages accepted by finite automata with n states. J. Autom. Lang. Comb. 7(4), 469–486 (2002)

    MathSciNet  MATH  Google Scholar 

  14. Glaister, I., Shallit, J.: A lower bound technique for the size of nondeterministic finite automata. Inform. Process. Lett. 59, 75–77 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gramlich, G., Schnitger, G.: Minimizing nfa’s and regular expressions. J. Comput. System Sci. 73(6), 908–923 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gruber, H., Holzer, M.: Finding lower bounds for nondeterministic state complexity is hard. In: Ibarra, O.H., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 363–374. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  17. Gruber, H., Holzer, M.: Results on the average state and transition complexity of finite automata accepting finite languages. Theoret. Comput. Sci. 387(2), 155–166 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Harrison, M.A.: Introduction to Formal Language Theory. Addison-Wesley, Newyork (1978)

    MATH  Google Scholar 

  19. Holzer, M., Jakobi, S.: From equivalence to almost-equivalence, and beyond: minimizing automata with errors. Internat. J. Found. Comput. Sci. 24(7), 1083–1134 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lucchesi, C.L., Kowaltowski, T.: Applications of finite automata representing large vocabularies. Softw. Pract. Exper. 23(1), 15–30 (1993)

    Article  Google Scholar 

  21. Körner, H.: A time and space efficient algorithm for minimizing cover automata for finite languages. Internat. J. Found. Comput. Sci. 14(6), 1071–1086 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Leung, H.: Separating exponentially ambiguous finite automata from polynomially ambiguous finite automata. SIAM J. Comput. 27(4), 1073–1082 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lupanov, O.B.: Über den Vergleich zweier Typen endlicher Quellen. Probleme der Kybernetik 6, 328–335 (1966)

    MATH  Google Scholar 

  24. Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM J. Res. Dev. 3, 114–125 (1959)

    Article  MathSciNet  Google Scholar 

  25. Salomaa, K., Yu, S.: NFA to DFA transformation for finite language over arbitrary alphabets. J. Autom. Lang. Comb. 2(3), 177–186 (1997)

    MathSciNet  Google Scholar 

  26. Sgarbas, K.N., Fakotakis, N.D., Kokkinakis, G.K.: Incremental construction of compact acyclic NFAs. In: 39th Annual Meeting of the Association for Computational Linguistics, pp. 482–489. Association for Computational Linguistics (2001)

    Google Scholar 

  27. Shannon, C.E.: The synthesis of two-terminal switching circuits. Bell Sys. Tech. J. 28(1), 59–98 (1949)

    Article  MathSciNet  Google Scholar 

  28. Yu, S.: Cover automata for finite language. Bull. EATCS 92, 65–74 (2007)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Knowledgepark AG, Leonrodstr. 68, 80636, Munich, Germany

    Hermann Gruber

  2. Institut für Informatik, Universität Giessen, Arndtstr. 2, 35392, Giessen, Germany

    Markus Holzer & Sebastian Jakobi

Authors
  1. Hermann Gruber
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Markus Holzer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sebastian Jakobi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Markus Holzer .

Editor information

Editors and Affiliations

  1. Umeå University, Umeå, Sweden

    Frank Drewes

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Gruber, H., Holzer, M., Jakobi, S. (2015). More on Deterministic and Nondeterministic Finite Cover Automata. In: Drewes, F. (eds) Implementation and Application of Automata. CIAA 2015. Lecture Notes in Computer Science(), vol 9223. Springer, Cham. https://doi.org/10.1007/978-3-319-22360-5_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-22360-5_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22359-9

  • Online ISBN: 978-3-319-22360-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation