The complexity class LOGCFL (resp., LOGDCFL) consists of all languages that are many-one reducible to context-free (resp., deterministic context-free) languages using logarithmic space. These complexity classes have been studied over five decades in connection to parallel computation since they are located between Nick's classes $\mathrm{NC}^1$ and $\mathrm{NC}^2$. In contrast, the state complexity of nonuniform finite-automaton families was first discussed in the 1970s and it has been extensively explored lately for various finite-automata families. We extend this old subject to the stack-state complexity (i.e., the total number of inner states plus simultaneously pushable stack symbol series) of nonuniform families of various pushdown automata. We introduce reasonable "parameterizations" of LOGCFL and LOGDCFL and apply them as a technical tool to establish a close connection between the LOGCFL$\subseteq$LOGDCFL/poly question and the polynomial stack-state complexity of nonuniform families of two-way pushdown automata. We also discuss the precise computational complexity of polynomial-size one-way pushdown automata.

中文翻译：

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对数空间约简的参数化、下推自动机非均匀族的堆栈状态复杂性，以及通往 LOGCFL$\subseteq$LOGDCFL/poly 问题的道路

复杂性类 LOGCFL（相应的，LOGDCFL）由使用对数空间可以多一还原为上下文无关（相应的，确定性上下文无关）语言的所有语言组成。由于这些复杂性类位于 Nick 的类 $\mathrm{NC}^1$ 和 $\mathrm{NC}^2$ 之间，因此这些复杂性类已经与并行计算相关的研究了五十多年。相比之下，非均匀有限自动机系列的状态复杂性在 1970 年代首次被讨论，并且最近已针对各种有限自动机系列进行了广泛的探索。我们将这个古老的主题扩展到各种下推自动机的非均匀族的堆栈状态复杂性（即，内部状态的总数加上可同时推送的堆栈符号系列）。我们引入了合理的“参数化” LOGCFL 和 LOGDCFL，并将它们用作技术工具，以在 LOGCFL$\subseteq$LOGDCFL/poly 问题与双向下推自动机的非均匀族的多项式堆栈状态复杂性之间建立紧密联系。我们还讨论了多项式大小的单向下推自动机的精确计算复杂度。