We introduce algebraic machine reasoning, a new reasoning framework that is well-suited for abstract reasoning. Effectively, algebraic machine reasoning reduces the difficult process of novel problem-solving to routine algebraic computation. The fundamental algebraic objects of interest are the ideals of some suitably initialized polynomial ring. We shall explain how solving Raven's Progressive Matrices (RPMs) can be realized as computational problems in algebra, which combine various well-known algebraic subroutines that include: Computing the Gr\"obner basis of an ideal, checking for ideal containment, etc. Crucially, the additional algebraic structure satisfied by ideals allows for more operations on ideals beyond set-theoretic operations. Our algebraic machine reasoning framework is not only able to select the correct answer from a given answer set, but also able to generate the correct answer with only the question matrix given. Experiments on the I-RAVEN dataset yield an overall $93.2\%$ accuracy, which significantly outperforms the current state-of-the-art accuracy of $77.0\%$ and exceeds human performance at $84.4\%$ accuracy.

介绍一种新的代数机器推理框架，将Raven渐进矩阵问题求解转化为代数计算，利用理想的Gröbner基计算等数学算法，能够从答案集合中选出正确答案并仅凭问题矩阵就生成正确答案，试验在I-RAVEN数据集上达到93.2%的准确率，超过当前最先进的77.0%和人类表现的84.4%准确率。

抽象视觉推理: 一种代数方法用于解决Raven渐进矩阵问题