Traditionally, finite automata theory has been used as a framework for the representation of possibly infinite sets of strings. In this work, we introduce the notion of second-order finite automata, a formalism that combines finite automata with ordered decision diagrams, with the aim of representing possibly infinite {\em sets of sets} of strings. Our main result states that second-order finite automata can be canonized with respect to the second-order languages they represent. Using this canonization result, we show that sets of sets of strings represented by second-order finite automata are closed under the usual Boolean operations, such as union, intersection, difference and even under a suitable notion of complementation. Additionally, emptiness of intersection and inclusion are decidable. We provide two algorithmic applications for second-order automata. First, we show that several width/size minimization problems for deterministic and nondeterministic ODDs are solvable in fixed-parameter tractable time when parameterized by the width of the input ODD. In particular, our results imply FPT algorithms for corresponding width/size minimization problems for ordered binary decision diagrams (OBDDs) with a fixed variable ordering. Previously, only algorithms that take exponential time in the size of the input OBDD were known for width minimization, even for OBDDs of constant width. Second, we show that for each $k$ and $w$ one can count the number of distinct functions computable by ODDs of width at most $w$ and length $k$ in time $h(|\Sigma|,w)\cdot k^{O(1)}$, for a suitable $h:\mathbb{N}\times \mathbb{N}\rightarrow \mathbb{N}$. This improves exponentially on the time necessary to explicitly enumerate all such functions, which is exponential in both the width parameter $w$ and in the length $k$ of the ODDs.

中文翻译：

---

二阶有限自动机

传统上，有限自动机理论已被用作表示可能无限字符串集的框架。在这项工作中，我们引入了二阶有限自动机的概念，这是一种将有限自动机与有序决策图相结合的形式主义，旨在表示可能无限的 {\em 集合集} 字符串。我们的主要结果表明，二阶有限自动机可以根据它们所代表的二阶语言进行规范化。使用这个规范化结果，我们表明由二阶有限自动机表示的字符串集在通常的布尔运算下是封闭的，例如并集、交集、差集，甚至在合适的互补概念下。此外，交集和包含的空性是可判定的。我们为二阶自动机提供了两种算法应用。首先，我们表明，当通过输入 ODD 的宽度参数化时，确定性和非确定性 ODD 的几个宽度/大小最小化问题可以在固定参数易处理时间内解决。特别是，我们的结果暗示了 FPT 算法用于具有固定变量排序的有序二元决策图 (OBDD) 的相应宽度/大小最小化问题。以前，只有在输入 OBDD 的大小上花费指数时间的算法才知道宽度最小化，即使对于恒定宽度的 OBDD。其次，我们证明，对于每个 $k$ 和 $w$，可以计算在时间 $h(|\Sigma|,w)\ cdot k^{O(1)}$，对于合适的 $h:\mathbb{N}\times \mathbb{N}\rightarrow \mathbb{N}$。