Mathematical Reasoning in Latent Space (1909.11851v1)

Abstract: We design and conduct a simple experiment to study whether neural networks can perform several steps of approximate reasoning in a fixed dimensional latent space. The set of rewrites (i.e. transformations) that can be successfully performed on a statement represents essential semantic features of the statement. We can compress this information by embedding the formula in a vector space, such that the vector associated with a statement can be used to predict whether a statement can be rewritten by other theorems. Predicting the embedding of a formula generated by some rewrite rule is naturally viewed as approximate reasoning in the latent space. In order to measure the effectiveness of this reasoning, we perform approximate deduction sequences in the latent space and use the resulting embedding to inform the semantic features of the corresponding formal statement (which is obtained by performing the corresponding rewrite sequence using real formulas). Our experiments show that graph neural networks can make non-trivial predictions about the rewrite-success of statements, even when they propagate predicted latent representations for several steps. Since our corpus of mathematical formulas includes a wide variety of mathematical disciplines, this experiment is a strong indicator for the feasibility of deduction in latent space in general.

• The paper shows that by embedding formulas in vector space, neural networks can accurately predict successful rewrite operations across multiple reasoning steps.
• The study employs graph neural networks to map mathematical statements, achieving superior prediction accuracy compared to random or usage-based baselines.
• These findings hint at practical advancements for automated theorem proving and open new theoretical avenues in neural-based logical inference.

Mathematical Reasoning in Latent Space: An Analytical Exploration

The paper "Mathematical Reasoning in Latent Space" by Dennis Lee et al. examines the potential for neural networks to perform mathematical reasoning within a fixed dimensional latent space. This research challenges traditional theorem-proving methodologies that rely heavily on deterministic algorithms and interactive proof assistants such as HOL Light, shifting focus to a latent vector approach that leverages graph neural networks for performing logical deductions.

Research Objective and Approach

The central objective of this paper is to determine the suitability of neural networks for conducting multi-step mathematical reasoning by predicting latent representations, effectively bypassing traditional theorem proving processes. The authors propose that by embedding mathematical formulas in a vector space, reasoned transformations on those formulas can be predicted accurately without requiring explicit proof construction or algorithmic backtracking.

A corpus of mathematical statements sourced from diverse domains, including topology, multivariate calculus, and real and complex analysis, serves as the basis for experiments. Graph neural networks are tasked with mapping statements to latent spaces, and these representations are used to predict the success of rewrite operations governed by established theorems.

Key Findings

The experiments reveal that graph neural networks can effectively predict whether rewrite operations will succeed. Notably, they retain prediction accuracy across multiple latent steps, suggesting that meaningful semantic information can indeed be propagated in vector form. The authors tested this hypothesis over multiple rewrite operations, finding notable persistence in prediction capabilities despite increasing the number of steps.

Quantitatively, the paper presents strong numerical evidence for the feasibility of latent space deduction. Even after four consecutive steps in the latent space, predictions regarding rewrite success were superior to baselines such as random or usage-based predictions, indicating a robust capacity for reasoning despite potential degradation at higher steps.

Implications and Future Directions

The implications of these findings are twofold:

1. Practical Implications: This approach may lead to more flexible and efficient automated theorem-proving systems, potentially reducing dependencies on traditional proof assistants and enhancing capability in handling large and complex theorem databases.
2. Theoretical Insights: The understanding of mathematical reasoning as a process that can be abstracted in latent vector form may enrich theoretical research in both neural-based modeling and formal logic representation.

Future research can expand on these results by improving network architectures and training methodologies, potentially integrating the theorem prediction process to operate holistically within a single embedding space. Additionally, self-supervised learning paradigms might further validate and refine network performance across diverse reasoning tasks.

The exploration of reasoning in latent space carries significant promise in adjusting and enhancing traditional mathematics-oriented AI applications, underscoring the evolving relationship between deep learning and logical inference.