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

• Published in: Annals of mathematics and artificial intelligence Vol. 90; no. 1; pp. 93 - 105
• Main Authors: Kjos-Hanssen, Bjørn, Felix, Clyde James, Kim, Sun Young, Lamb, Ethan, Takahashi, Davin
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.01.2022; Springer; Springer Nature B.V
• Subjects: Artificial Intelligence; Automata theory; Complex Systems; Computer Science; Construction; Languages; Lower bounds; Mathematics; Robots; Upper bounds; Venture capital; Words (language); 68Q68; 68Q45; Vapnik-Chervonenkis dimension; Finite automata; 68T05; Nondeterminism; State complexity; Testing dimension
• ISSN: 1012-2443; 1573-7470
• DOI: 10.1007/s10472-021-09769-9

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 .