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

Abstract

Let NFAb(q)NFA_b(q) denote the set of languages accepted by nondeterministic finite automata with qq states over an alphabet with bb letters. Let BnB_n denote the set of words of length nn. We give a quadratic lower bound on the VC dimension of \[ NFA_2(q)\cap B_n = \{L\cap B_n \mid L \in NFA_2(q)\} \] as a function of qq. Next, the work of Gruber and Holzer (2007) gives an upper bound for the nondeterministic state complexity of finite languages contained in BnB_n, which we strengthen using our methods. Finally, we give some theoretical and experimental results on the dependence on nn of the VC dimension and testing dimension of NFA2(q)BnNFA_2(q)\cap B_n.

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