This paper investigates the possibility of approximating multiple mathematical operations in latent space for expression derivation. To this end, we introduce different multi-operational representation paradigms, modelling mathematical operations as explicit geometric transformations. By leveraging a symbolic engine, we construct a large-scale dataset comprising 1.7M derivation steps stemming from 61K premises and 6 operators, analysing the properties of each paradigm when instantiated with state-of-the-art neural encoders. Specifically, we investigate how different encoding mechanisms can approximate equational reasoning in latent space, exploring the trade-off between learning different operators and specialising within single operations, as well as the ability to support multi-step derivations and out-of-distribution generalisation. Our empirical analysis reveals that the multi-operational paradigm is crucial for disentangling different operators, while discriminating the conclusions for a single operation is achievable in the original expression encoder. Moreover, we show that architectural choices can heavily affect the training dynamics, structural organisation, and generalisation of the latent space, resulting in significant variations across paradigms and classes of encoders.

翻译：本文探讨了在潜空间中逼近多种数学运算以实现表达式推导的可能性。为此，我们引入了不同的多运算表示范式，将数学运算建模为显式的几何变换。通过利用符号引擎，我们构建了一个大规模数据集，包含源自61K个前提和6个运算符的170万个推导步骤，分析了每种范式在使用最先进神经编码器实例化时的属性。具体而言，我们研究了不同编码机制如何在潜空间中逼近方程推理，探索了学习不同运算符与在单个运算内特化之间的权衡，以及支持多步推导和分布外泛化的能力。我们的实证分析表明，多运算范式对于解耦不同运算符至关重要，而通过原始表达式编码器即可实现对单个运算结论的区分。此外，我们发现架构选择会显著影响潜空间的训练动态、结构组织和泛化能力，从而导致各范式与编码器类别之间存在显著差异。