有限语言的确定性带宽
Deterministic Bandwidth of Finite Languages
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中文总结 AI 辅助
本文研究有限语言的确定性带宽，证明其存在无限层次结构，给出带宽1语言的多项式时间判定算法，并推导最小DFA带宽的高效上下界，还找到上下界完全匹配的语言族。
中文摘要 AI 辅助
带宽限制了自动机中状态排序下转移的移动距离。虽然所有有限语言都存在带宽为2的非确定有限自动机（NFA）表示，但确定性场景的限制要严格得多。我们研究接受有限语言的部分确定有限自动机（partial DFA）的带宽，证明有界带宽对确定性表示施加了严格的结构限制，存在由确定性带宽定义的有限语言类的无限层次结构。作为受关注的特殊情况，我们考虑带宽为1的partial DFA接受的有限语言，证明它们具有位置特性，可得到多项式时间决策算法。我们进一步研究如何从最小DFA的结构估计最小DFA带宽，基于位置展开和可达残余状态的局部增长推导可计算的上下界，这些界可高效计算，且在最大单词长度上的差异最多为线性因子，还给出了一个上下界完全匹配的简单语言族。
英文摘要
Bandwidth restricts how far transitions can move under an ordering of the states in an automaton. While every finite language admits a bandwidth-2 NFA representation, the deterministic setting is substantially more restrictive. We investigate the bandwidth of partial DFAs accepting finite languages. We show that bounded bandwidth imposes strong structural restrictions on deterministic representations. We prove that there is an infinite hierarchy of classes of finite languages defined by deterministic bandwidth. As a special case of interest, we consider finite languages accepted by bandwidth-1 partial DFAs, and show that they admit a positional characterization, which yields a polynomial-time decision algorithm. We further study how the minimum DFA bandwidth can be estimated from the structure of the minimal DFA. We derive computable upper and lower bounds based on position-unfolding, and local growth of reachable residual states. These bounds can be computed efficiently and differ by at most a linear factor in the maximum word length. We also present a simple language family where the bounds match exactly.
