Skip to main content
SpringerLink
Log in

State Complexity of GF(2)-Concatenation and GF(2)-Inverse on Unary Languages

  • Conference paper
  • First Online:
Descriptional Complexity of Formal Systems (DCFS 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11612))

Included in the following conference series:

Abstract

The paper investigates the state complexity of two operations on regular languages, known as GF(2)-concatenation and GF(2)-inverse (Bakinova et al., “Formal languages over GF(2)”, LATA 2018), in the case of a one-symbol alphabet. The GF(2)-concatenation is a variant of the classical concatenation obtained by replacing Boolean logic in its definition with the GF(2) field; it is proved that GF(2)-concatenation of two unary languages recognized by an m-state and an n-state DFA is recognized by a DFA with 2mn states, and this number of states is necessary in the worst case, as long as m and n are relatively prime. This operation is known to have an inverse, and the state complexity of the GF(2)-inverse operation over a unary alphabet is proved to be exactly \(2^{n-1}+1\).

Supported by Russian Science Foundation, project 18-11-00100.

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

References

  1. Bakinova, E., Basharin, A., Batmanov, I., Lyubort, K., Okhotin, A., Sazhneva, E.: Formal languages over GF(2). In: Klein, S.T., Martín-Vide, C., Shapira, D. (eds.) LATA 2018. LNCS, vol. 10792, pp. 68–79. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-77313-1_5

    Chapter  Google Scholar 

  2. Brzozowski, J.A.: Quotient complexity of regular languages. J. Autom. Lang. Comb. 15(1/2), 71–89 (2010). https://doi.org/10.25596/jalc-2010-071

    Article  MathSciNet  MATH  Google Scholar 

  3. Chrobak, M.: Finite automata and unary languages. Theor. Comput. Sci. 47(3), 149–158 (1986). https://doi.org/10.1016/0304-3975(86)90142-8

    Article  MathSciNet  MATH  Google Scholar 

  4. Daley, M., Domaratzki, M., Salomaa, K.: Orthogonal concatenation: language equations and state complexity. J. UCS 16(5), 653–675 (2010). https://doi.org/10.3217/jucs-016-05-0653

    Article  MathSciNet  MATH  Google Scholar 

  5. Geffert, V., Mereghetti, C., Pighizzini, G.: Converting two-way nondeterministic unary automata into simpler automata. Theor. Comput. Sci. 295, 189–203 (2003). https://doi.org/10.1016/S0304-3975(02)00403-6

    Article  MathSciNet  MATH  Google Scholar 

  6. Jirásková, G., Okhotin, A.: State complexity of unambiguous operations on deterministic finite automata. In: Konstantinidis, S., Pighizzini, G. (eds.) DCFS 2018. LNCS, vol. 10952, pp. 188–199. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94631-3_16

    Chapter  Google Scholar 

  7. Kunc, M., Okhotin, A.: Describing periodicity in two-way deterministic finite automata using transformation semigroups. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 324–336. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22321-1_28

    Chapter  MATH  Google Scholar 

  8. Makarov, V., Okhotin, A.: On the expressive power of GF(2)-grammars. In: Catania, B., Královič, R., Nawrocki, J., Pighizzini, G. (eds.) SOFSEM 2019. LNCS, vol. 11376, pp. 310–323. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-10801-4_25

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

    MATH  Google Scholar 

  10. Mereghetti, C., Pighizzini, G.: Optimal simulations between unary automata. SIAM J. Comput. 30(6), 1976–1992 (2001). https://doi.org/10.1137/S009753979935431X

    Article  MathSciNet  MATH  Google Scholar 

  11. Okhotin, A.: Unambiguous finite automata over a unary alphabet. Inf. Comput. 212, 15–36 (2012). https://doi.org/10.1016/j.ic.2012.01.003

    Article  MathSciNet  MATH  Google Scholar 

  12. Pighizzini, G., Shallit, J.: Unary language operations, state complexity and Jacobsthal’s function. Int. J. Found. Comput. Sci. 13(1), 145–159 (2002). https://doi.org/10.1142/S012905410200100X

    Article  MathSciNet  MATH  Google Scholar 

  13. Yu, S., Zhuang, Q., Salomaa, K.: The state complexities of some basic operations on regular languages. Theor. Comput. Sci. 125(2), 315–328 (1994). https://doi.org/10.1016/0304-3975(92)00011-F

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg State University, 7/9 Universitetskaya Nab., Saint Petersburg, 199034, Russia

    Alexander Okhotin & Elizaveta Sazhneva

Authors
  1. Alexander Okhotin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elizaveta Sazhneva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Okhotin .

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

Okhotin, A., Sazhneva, E. (2019). State Complexity of GF(2)-Concatenation and GF(2)-Inverse on Unary Languages. 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_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-23247-4_19

  • 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