arXiv 2108.127512021-08-29cs.FLcs.CC

二阶有限自动机

Second-Order Finite Automata

Alexsander Andrade de Melo, Mateus de Oliveira Oliveira

备注：: 41 pages. This is an extended version of a paper with the same title that appeared at CSR 2020

摘要：

传统上，有限自动机理论一直被用来作为一个框架，表示可能无限的字符串集。在这项工作中，我们介绍了二阶有限自动机的概念，一种形式主义，结合有限自动机与有序决策图，其目的是表示可能无限{ \em集}的字符串。我们的主要结果表明，二阶有限自动机可以相对于它们所代表的二阶语言被规范化。使用这个标准化的结果，我们表明，由二阶有限自动机表示的字符串集合的集合是封闭的通常的布尔运算下，如联合，交叉，差异，甚至在一个合适的概念互补。此外，交集和包含的空性是可判定的。我们提供了二阶自动机的两个算法应用。首先，我们证明了几个宽度/大小最小化问题的确定性和不确定性的ODD是可解的固定参数易处理的时间时，参数化的输入ODD的宽度。特别是，我们的研究结果意味着FPT算法相应的宽度/大小最小化问题的有序二元决策图（OBDDs）与一个固定的变量排序。以前，只有在输入OBDD的大小上花费指数时间的算法才被称为宽度最小化，即使对于恒定宽度的OBDD也是如此。第二，我们表明，对于每个$k$和$w$可以计数的不同功能的数量可计算的ODD的宽度最多$w$和长度$k$在时间$h(|Σ|,w)\cdot k^{O(1)}$，为一个合适的$h:\mathbb{N}\times \mathbb{N}\rightarrow \mathbb{N}$。这在显式枚举所有这样的函数所需的时间上呈指数级地改进，这在ODD的宽度参数$w$和长度$k$两者上都是指数级的。

英文摘要：

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 $h(|Σ|,w)\cdot k^{O(1)}$, 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.

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摘要：

英文摘要：

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