VC Dimensions of Nondeterministic Finite Automata for Words of Equal Length
📝 Original Paper Info
- Title: VC-dimensions of nondeterministic finite automata for words of equal length- ArXiv ID: 2001.02309
- Date: 2021-08-06
- Authors: Bj{ o}rn Kjos-Hanssen, Clyde James Felix, Sun Young Kim, Ethan Lamb, Davin Takahashi
📝 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 $NFA_2(q)\cap B_n$.💡 Summary & Analysis
This paper delves into the study of VC dimensions for languages accepted by nondeterministic finite automata (NFA) over words of equal length. The authors provide a quadratic lower bound on the VC dimension for NFA with $q$ states and an alphabet of two letters, focusing specifically on sets of words of length $n$. They also strengthen previous upper bounds on the nondeterministic state complexity using their methods.The problem addressed revolves around understanding how NFAs can recognize certain languages and what factors determine this recognition’s complexity. Specifically, studying VC dimensions for word sets with equal lengths helps in both theoretical insights and practical applications.
Using detailed analysis and experimental results, the authors show that the VC dimension of NFA configurations is linked to the number of states ($q$) and the length of words ($n$). They provide evidence supporting a quadratic lower bound on this dimensionality. This insight not only contributes theoretically but also offers new bounds for nondeterministic state complexity, which can be crucial in developing efficient algorithms and understanding language recognition processes.
The significance lies in advancing our comprehension of NFAs’ capabilities and limitations, potentially impacting areas like computer science, information theory, and machine learning where language recognition plays a critical role. This research can aid in designing more effective computational models and improving the efficiency of recognizing patterns within languages or datasets.
📄 Full Paper Content (ArXiv Source)
📊 논문 시각자료 (Figures)
