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Developing neural architectures that are capable of logical reasoning hasbecome increasingly important for a wide range of applications (e.g., naturallanguage processing). Towards this grand objective, we propose a symbolicreasoning architecture that chains many join operators together to model outputlogical expressions. In particular, we demonstrate that such an ensemble ofjoin-chains can express a broad subset of ''tree-structured'' first-orderlogical expressions, named FOET, which is particularly useful for modelingnatural languages. To endow it with differentiable learning capability, weclosely examine various neural operators for approximating the symbolicjoin-chains. Interestingly, we find that the widely used multi-headself-attention module in transformer can be understood as a special neuraloperator that implements the union bound of the join operator in probabilisticpredicate space. Our analysis not only provides a new perspective on themechanism of the pretrained models such as BERT for natural languageunderstanding but also suggests several important future improvementdirections.

Join-Chain Network: A Logical Reasoning View of the Multi-head Attention in Transformer