VC-dimensions of nondeterministic finite automata for words of equal length - Kjos-Hanssen, Bjørn; Felix, Clyde James; Kim, Sun Young; Lamb, Ethan; Takahashi, Davin (Annals of mathematics and artificial intelligence, 2022)
Let NFA b ( q ) denote the set of languages accepted by nondeterministic finite automata with q states over an alphabet with b letters. Let B n denote the set of words of length n . We give a quadratic lower bound on the VC dimension of NFA 2 ( q ) ∩ B n = { L ∩ B n ∣ L ∈ 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 B n , 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 NFA 2 ( q ) ∩ B n .