Nonuniform families of polynomial-size finite automata and pushdown automata respectively have strong connections to nonuniform-NL and nonuniform-LOGCFL. We examine the behaviors of unambiguous and co-nondeterministic computations produced by such families of automata operating multiple counters. As its consequences, we obtain various collapses of the complexity classes of families of promise problems solvable by finite and pushdown automata families when all valid instances are limited to either polynomially long strings or unary strings. A key technical ingredient of our proofs is an inductive counting of reachable vertices of each computation graph of finite and pushdown automata that operate multiple counters simultaneously.

翻译：非均匀多项式规模的有限自动机与下推自动机族分别与非均匀-NL和非均匀-LOGCFL存在密切联系。本文研究了此类带多重计数器的自动机族所产生的明确计算与余非确定计算行为。由此，当所有有效实例被限定为多项式长度字符串或一元字符串时，我们获得了有限自动机族与下推自动机族可解许诺问题类的若干复杂性塌缩结果。证明中的关键技术要素是对同时操作多重计数器的有限自动机与下推自动机的每个计算图中可达顶点进行归纳计数。