AI 中文总结
本文针对有限域上向量空间的封闭子集，提出基于二分划分范式的高效枚举算法，可解决封闭空间枚举、最大封闭空间枚举及最大维封闭空间求解问题，较暴力方法实现二次加速。
AI 中文摘要
本文研究给定集合中封闭向量空间的若干问题。设V是基数为c的有限域上的向量空间，S⊆V是一个向量集合，封闭于S的空间是V中同时包含于S的向量子空间W，即W⊆S。我们聚焦于枚举问题，任务是列出所有解，首先提供一种枚举所有封闭于S的空间的算法，该算法还可适配解决另外两个问题：（包含意义下）最大封闭空间的枚举，以及寻找最大维数封闭空间的问题。后者问题产生于布尔函数的正则性检测语境，也可视为著名线性张成的对偶版本：线性张成是包含给定向量集合S的最小维数向量空间，是线性代数的基础概念。我们提出的算法基于二分划分范式，总时间复杂度为$e^{\frac{1}{2\ln c}\ln^2 n - \Theta(\log n \log \log n)}$，其中$n=|S|$。用于枚举所有封闭空间的首个版本还实现了O(n)的延迟（连续输出之间的时间间隔）。与暴力方法相比，我们的算法实现了二次加速，尽管在布尔向量空间的实验评估中，加速效果更为显著。
英文摘要
In this paper, we address several problems concerning vector spaces enclosed in a given set. Let V be a vector space over a finite field of cardinality c, and let $S \subseteq V$ be a set of vectors. A space enclosed in S is a vector subspace W of V that is also contained in S: $W \subseteq S$. We focus on enumeration problems, where the task is to list all solutions, and we first provide an algorithm to enumerate all spaces that are enclosed in S. Our algorithm is further adapted to solve two more problems: the enumeration of (inclusion-)maximal enclosed spaces, and the problem of finding an enclosed space of maximum dimension. The latter problem arises in the context of Boolean functions' regularity detection. It can also be seen as a dual version of the well-known linear span: indeed, the span is the minimum-dimension vector space that contains a given set of vectors S, and it is a fundamental concept in linear algebra. Our proposed algorithms are based on the binary partition paradigm, and have total time complexity $e^{\frac{1}{2\ln c}\ln^2 n - Θ(\log n \log \log n)}$, where $n= |\inputset|$. The first version, for enumerating all enclosed spaces, also achieves a delay (time between consecutive outputs) of O(n). Our algorithms provide a quadratic speed-up with respect to a brute-force approach, although the speed-up appears even greater in our experimental evaluation on boolean vector spaces.
