1. VC-dimensions of nondeterministic finite automata for words of equal length
- Author
-
Kjos-Hanssen, Bjørn, Felix, Clyde James, Kim, Sun Young, Lamb, Ethan, and Takahashi, Davin
- Subjects
Computer Science - Formal Languages and Automata Theory ,Computer Science - Machine Learning - 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 $NFA_2(q)\cap B_n$., Comment: ISAIM 2020 (International Symposium on Artificial Intelligence and Mathematics), Fort Lauderdale, FL. January 6--8, 2020. Accepted for publication in Annals of Mathematics and Artificial Intelligence
- Published
- 2020