Transformers Implement First-Order Logic with Majority Quantifiers

Characterizing the implicit structure of the computation within neural networks is a foundational problem in the area of deep learning interpretability. Can their inner decision process be captured symbolically in some familiar logic? We show that any transformer neural network can be translated into an equivalent fixed-size first-order logic formula which may also use majority quantifiers. The idea is to simulate transformers with highly uniform threshold circuits and leverage known theoretical connections between circuits and logic. Our findings also reveal the surprising fact that the entire transformer computation can be reduced merely to the division of two (large) integers. While our results are most pertinent for transformers, they apply equally to a broader class of neural network architectures, namely those with a fixed-depth uniform computation graph made up of standard neural net components, which includes feedforward and convolutional networks.