Enhancing Group Search with Graph Techniques

群论是数学中的一个领域，研究被称为群的代数结构，它可以用来描述任何物体的对称性。因此，群论在病毒学、量子计算、网络安全和通信理论中有着广泛的应用。群还用于查找和减少搜索问题中的对称性，这在现代人工智能中至关重要。因此，尽可能快地发现群体、它们的属性和计算基本的群体操作，是英国许多世界领先研究领域的核心，并将对经济成功产生广泛的影响。在群论算法的研究中将会有一个继承的循环改进，因为用于寻找子群、子群的性质或计算操作的计算技术之一本身就是一种搜索算法。因此，改进分组搜索也将改进对其他应用程序的搜索。当前的群组搜索技术很先进，但仍在努力解决规模问题。我们建议使用图搜索问题中常用的搜索技术，并将其应用于群理论搜索算法。这些技术是(1)在搜索过程中（习得的）nogood子句，然后将通知进一步的搜索步骤，(2)以明确的间隔从顶部重新开始搜索，以及(3)在搜索决策中使用加权排序。我们发现，在图问题中使用这些技术，搜索速度至少提高了两个数量级。我们希望在小组问题中，性能的改善是相同的，甚至更好。此外，这些技术允许以一种简单的方式并行当前的顺序算法，而不需要设计/设计一个全新的算法。加速解决群体问题的算法，并使它们易于并行，将进一步影响病毒学等应用，在病毒学中，群论被用来描述病毒的结构和几何特性。

Group theory is a field in mathematics which investigates algebraic structures called groups, which can be used to describe the symmeties of any object. Thus, group theory has wide-reaching applications in virology, quantum computing, cyber security and communications theory. Groups are also used in finding and reducing symmetries within search problems, which is vital in modern Artificial Intelligence. So finding groups, their properties and computing fundamental group operations as fast as possible, is at the core of many UK world leading research areas and will have wide reaching impact in economic successes.There is an inherit circular improvement that will come from the research in algorithms in group theory, as one of the computational techniques used in finding subgroups, their properties or computing operations is a type of search algorithm itself. Thus, improving search in groups will improve the search for other applications as well.The current search techniques in groups are advanced but are still struggling with scaling issues. We are proposing to use search techniques commonly used in graph search problems and apply them to group theoretical search algorithms. Such techniques are(1) (learned) nogood clauses during search which then will inform further search steps, (2) restarting the search from the top at well defined intervals, and (3) using weighted orderings on the search decisions.We have found that using these techniques in graph problems sped up the search by at least two orders of magnitude. We expect the performance improvements to be the same or better across the board in group problems.Further, these techniques allow for a simple way of parallelising the current sequential algorithms, without the need to design/engineer a whole new algorithm. Speeding up algorithms which solve group problems and enabling them to be easily parallelisable will further impact applications such as virology, where group theory is used to describe the structural and geometrical properties of viruses.