Metric Automata Theory: AUnifying Theory of RNNs

We propose Metric Automata Theory, an elegant generalisation of classic Automata Theory to continuous dynamical systems, that constitutes a unifying theory of all kinds of Recurrent Neural Networks (RNNs), including widely-adopted architectures such as xLSTM and State Space Models (SSMs). The theory allows one to analyse RNNs both in the finite and unbounded precision settings seamlessly, while utilising fundamental results of Automata Theory. It also provides a novel notion of robustness that guarantees numerical stability and contributes to stability of learning. We employ the theory to prove a comprehensive set of expressivity results for widely-adopted RNNs, with a focus on robustness and finite-precision. Notably, we contrast the capabilities of xLSTM and SSMs for robustly modelling all star-free regular languages--xLSTM can do so, while SSMs cannot robustly recognize the FLIP-FLOP language.