Neurosymbolic AI for Reasoning on Graph Structures: A Survey

Neurosymbolic AI is an increasingly active area of research which aims to combine symbolic reasoning methods with deep learning to generate models with both high predictive performance and some degree of human-level comprehensibility. As knowledge graphs are becoming a popular way to represent heterogeneous and multi-relational data, methods for reasoning on graph structures have attempted to follow this neurosymbolic paradigm. Traditionally, such approaches have utilized either rule-based inference or generated representative numerical embeddings from which patterns could be extracted. However, several recent studies have attempted to bridge this dichotomy in ways that facilitate interpretability, maintain performance, and integrate expert knowledge. Within this article, we survey a breadth of methods that perform neurosymbolic reasoning tasks on graph structures. To better compare the various methods, we propose a novel taxonomy by which we can classify them. Specifically, we propose three major categories: (1) logically-informed embedding approaches, (2) embedding approaches with logical constraints, and (3) rule-learning approaches. Alongside the taxonomy, we provide a tabular overview of the approaches and links to their source code, if available, for more direct comparison. Finally, we discuss the applications on which these methods were primarily used and propose several prospective directions toward which this new field of research could evolve.

Constrained Training of Neural Networks via Theorem Proving

We introduce a theorem proving approach to the specification and generation of temporal logical constraints for training neural networks. We formalise a deep embedding of linear temporal logic over finite traces (LTL$_f$) and an associated evaluation function characterising its semantics within the higher-order logic of the Isabelle theorem prover. We then proceed to formalise a loss function $\mathcal{L}$ that we formally prove to be sound, and differentiable to a function $d\mathcal{L}$. We subsequently use Isabelle's automatic code generation mechanism to produce OCaml versions of LTL$_f$, $\mathcal{L}$ and $d\mathcal{L}$ that we integrate with PyTorch via OCaml bindings for Python. We show that, when used for training in an existing deep learning framework for dynamic movement, our approach produces expected results for common movement specification patterns such as obstacle avoidance and patrolling. The distinctive benefit of our approach is the fully rigorous method for constrained training, eliminating many of the risks inherent to ad-hoc implementations of logical aspects directly in an "unsafe" programming language such as Python.