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.

翻译：我们提出代数机器推理这一新型推理框架，该框架特别适用于抽象推理任务。实际上，代数机器推理将创新问题求解的复杂过程简化为常规代数计算。其核心代数研究对象是适当初始化的多项式环的理想。我们将阐述如何将Raven递进矩阵（RPMs）的求解实现为代数中的计算问题——该过程整合了多种经典代数子程序，包括：计算理想的Gröbner基、检验理想包含关系等。关键在于，理想满足的代数结构使其能够实现超越集合论运算的更丰富的理想操作。我们的代数机器推理框架不仅能从给定答案集中选择正确答案，还能仅根据问题矩阵直接生成正确答案。在I-RAVEN数据集上的实验取得了93.2%的整体准确率，这显著超越了当前最优的77.0%准确率，并超过了人类84.4%准确率的表现水平。