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


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.

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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).

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Correspondence to Bjørn Kjos-Hanssen.

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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).

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

Mathematics Subject Classification (2010)

  • 68Q45
  • 68Q68
  • 68T05