Identifying Axiomatic Mathematical Transformation Steps Using Tree-Structured Pointer Networks

Abstract

The classification of mathematical relations has become a new area of research in deep learning. A major focus lies on determining mathematical equivalence. While previous work has simply approached the task as a binary classification without providing further insight into the underlying decision, we aim to iteratively find a sequence of necessary steps to transform a mathematical expression into an arbitrary equivalent form. Each step in this sequence is specified by an axiom together with its position of application. We denote this task as Stepwise Equation Transformation Identification (SETI) task. To solve the task efficiently, we further propose TreePointerNet, a novel architecture which exploits the inherent tree structure of mathematical equations and consists of three key building blocks: (i) a transformer model tailored to work on hierarchically tree-structured equations, making use of (ii) a copy-pointer mechanism to extract the exact location of a transformation in the tree and finally (iii) custom embeddings that map distinguishable occurrences of the same token type to a common embedding. In addition, we introduce new datasets of equations for the SETI task. We benchmark our model against various baselines and perform an ablation study to quantify the influence of our custom embeddings and the copy-pointer component. Furthermore, we test the robustness of our model on data of unseen complexity. Our results clearly show that incorporating the hierarchical structure, embeddings and copy-pointer into a single model is highly beneficial for solving the SETI task