Skip to main content

VC-dimensions of nondeterministic finite automata for words of equal length

Abstract

Let NFAb(q) denote the set of languages accepted by nondeterministic finite automata with q states over an alphabet with b letters. Let Bn denote the set of words of length n. We give a quadratic lower bound on the VC dimension of

$$ \text{NFA}_{2}(q)\cap B_{n} = \{L\cap B_{n} \mid L \in \text{NFA}_{2}(q)\} $$

as a function of q. Next, the work of Gruber and Holzer (Theoret. Comput. Sci. 387(2): 155–166, 2007) gives an upper bound for the nondeterministic state complexity of finite languages contained in Bn, which we strengthen using our methods. Finally, we give some theoretical and experimental results on the dependence on n of the VC dimension and testing dimension of NFA2(q) ∩ Bn.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Gruber, H., Holzer, M.: Results on the average state and transition complexity of finite automata accepting finite languages (extended abstract). In: Leung, H., Pighizzini, G. (eds.) 8th International Workshop on Descriptional Complexity of Formal Systems - DCFS 2006, Las Cruces, New Mexico, USA, June 21 - 23, 2006 Proceedings. New Mexico State University, Las Cruces, New Mexico, USA, pp 267–275 (2006)

  2. 2.

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

    MathSciNet  Article  Google Scholar 

  3. 3.

    Hyde, K.K., Kjos-Hanssen, B.: Nondeterministic automatic complexity of overlap-free and almost square-free words. Electron. J. Combin. 22(3), 3.22, 18 (2015)

    MathSciNet  Article  Google Scholar 

  4. 4.

    OEIS Foundation Inc: The on-line encyclopedia of integer sequences http://oeis.org/A005187 (2021)

  5. 5.

    Ishigami, Y., Sei’ichi, T.: The VC-Dimensions of Finite Automata with N States. In: Algorithmic Learning Theory (Tokyo, 1993), volume 744 of Lecture Notes in Comput Sci, pp 328–341. Springer, Berlin (1993)

  6. 6.

    Ishigami, Y., Tani, S.: VC-dimensions of finite automata and commutative finite automata with k letters and n states. Discrete Appl. Math. 74(2), 123–134 (1997)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Kjos-Hanssen, B., Takahashi, D.K.: Code for VC-dimensions of finite automata for words of equal length https://github.com/bjoernkjoshanssen/vc (2019)

  8. 8.

    Romanik, K.: Approximate testing and learnability. In: Proceedings of the Fifth Annual Workshop on Computational Learning Theory, COLT ’92, pp 327–332. Association for Computing Machinery, New York (1992)

  9. 9.

    Romanik, K: Approximate testing and its relationship to learning. Theoret. Comput. Sci. 188(1-2), 79–99 (1997)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Sauer, N.: On the density of families of sets. J. Combinatorial Theory Ser. A 13, 145–147 (1972)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Shallit, J., Wang, M.-W.: Automatic complexity of strings. In: 2nd Workshop on Descriptional complexity of automata grammars and related structures (London ON 2000), vol. 6, pp 537–554 (2001)

  12. 12.

    Shelah, S.: A combinatorial problem; stability and order for models and theories in infinitary languages. Pacific J. Math. 41, 247–261 (1972)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors were supported by Faculty Mentoring Grants for Summer Undergraduate Research and Creative Works, sponsored by the Undergraduate Research Opportunities Program (UROP) in the Office of the Vice Chancellor for Research, University of Hawai‘ i. This work was partially supported by grants from the Simons Foundation (#315188 and #704836 to Bjørn Kjos-Hanssen).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Bjørn Kjos-Hanssen.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kjos-Hanssen, B., Felix, C.J., Kim, S.Y. et al. VC-dimensions of nondeterministic finite automata for words of equal length. Ann Math Artif Intell (2021). https://doi.org/10.1007/s10472-021-09769-9

Download citation

Keywords

  • Vapnik-Chervonenkis dimension
  • Testing dimension
  • Finite automata
  • Nondeterminism
  • State complexity

Mathematics Subject Classification (2010)

  • 68Q45
  • 68Q68
  • 68T05