On the Existence of Universal Simulators of Attention

Abstract: Prior work on the learnability of transformers has established its capacity to approximate specific algorithmic patterns through training under restrictive architectural assumptions. Fundamentally, these arguments remain data-driven and therefore can only provide a probabilistic guarantee. Expressivity, on the contrary, has theoretically been explored to address the problems \emph{computable} by such architecture. These results proved the Turing-completeness of transformers, investigated bounds focused on circuit complexity, and formal logic. Being at the crossroad between learnability and expressivity, the question remains: \emph{can transformer architectures exactly simulate an arbitrary attention mechanism, or in particular, the underlying operations?} In this study, we investigate the transformer encoder's ability to simulate a vanilla attention mechanism. By constructing a universal simulator $\mathcal{U}$ composed of transformer encoders, we present algorithmic solutions to identically replicate attention outputs and the underlying elementary matrix and activation operations via RASP, a formal framework for transformer computation. Our proofs, for the first time, show the existence of an algorithmically achievable data-agnostic solution, previously known to be approximated only by learning.