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Planar Digraphs for Automatic Complexity

  • Achilles A. Beros
  • Bjørn Kjos-HanssenEmail author
  • Daylan Kaui Yogi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11436)

Abstract

We show that the digraph of a nondeterministic finite automaton witnessing the automatic complexity of a word can always be taken to be planar. In the case of total transition functions studied by Shallit and Wang, planarity can fail.

Let \(s_q(n)\) be the number of binary words x of length n having nondeterministic automatic complexity \(A_N(x)=q\). We show that \(s_q\) is eventually constant for each q and that the eventual constant value of \(s_q\) is computable.

Keywords

Automatic complexity Planar graph Möbius function Nondeterministic finite automata 

References

  1. 1.
    Bender, E.A., Goldman, J.R.: On the applications of Möbius inversion incombinatorial analysis. Am. Math. Mon. 82(8), 789–803 (1975).  https://doi.org/10.2307/2319793CrossRefzbMATHGoogle Scholar
  2. 2.
    Choi, J.S.: Counts of unique periodic binary strings of length n, September 2011. http://oeis.org/A152061
  3. 3.
    Hyde, K.K., Kjos-Hanssen, B.: Nondeterministic automatic complexity of overlap-free and almost square-free words. Electron. J. Combin. 22(3), 18 (2015). paper 3.22MathSciNetzbMATHGoogle Scholar
  4. 4.
  5. 5.
    Kjos-Hanssen, B.: Complexity option game. http://math.hawaii.edu/wordpress/bjoern/complexity-option-game/
  6. 6.
    Lyndon, R.C.: On Burnside’s problem. Trans. Am. Math. Soc. 77, 202–215 (1954).  https://doi.org/10.2307/1990868MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Shallit, J., Wang, M.W.: Automatic complexity of strings. J. Autom. Lang. Comb. 6(4), 537–554 (2001). 2nd Workshop on Descriptional Complexity of Automata, Grammars and Related Structures, London, ON (2000)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Sipser, M.: A complexity theoretic approach to randomness. In: Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, STOC 1983, pp. 330–335. ACM, New York (1983).  https://doi.org/10.1145/800061.808762
  9. 9.
    Witt, E.: Treue Darstellung Liescher Ringe. J. Reine Angew. Math. 177, 152–160 (1937).  https://doi.org/10.1515/crll.1937.177.152MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of Hawai‘i at MānoaHonoluluUSA

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