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

• URL: http://arxiv.org/abs/2001.02309v2
• Date: Wed, 4 Aug 2021 18:28:10 GMT
• Title: VC-dimensions of nondeterministic finite automata for words of equal length
• Authors: Bj{\o}rn Kjos-Hanssen, Clyde James Felix, Sun Young Kim, Ethan Lamb, Davin Takahashi
• Abstract summary: 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 $n$.
• Score: 2.099922236065961
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 \[ NFA_2(q)\cap B_n = \{L\cap B_n \mid L \in NFA_2(q)\} \] as a function of $q$. Next, the work of Gruber and Holzer (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 $NFA_2(q)\cap B_n$.

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