Transformers Can Be Expressed In First-Order Logic with Majority

Characterizing the implicit structure of the computation within neural networks is a foundational problem in the area of deep learning interpretability. Can the inner decision process of neural networks be captured symbolically in some familiar logic? We show that any fixed-precision transformer neural network can be translated into an equivalent fixed-size $\mathsf{FO}(\mathsf{M})$ formula, i.e., a first-order logic formula that, in addition to standard universal and existential quantifiers, may also contain majority-vote quantifiers. The proof idea is to design highly uniform boolean threshold circuits that can simulate transformers, and then leverage known theoretical connections between circuits and logic. Our results reveal a surprisingly simple formalism for capturing the behavior of transformers, show that simple problems like integer division are "transformer-hard", and provide valuable insights for comparing transformers to other models like RNNs. Our results suggest that first-order logic with majority may be a useful language for expressing programs extracted from transformers.