Length-Independent State Tracking Under a Parallel Scan

Julien Brandoit, Arthur Fyon, Thomas Braipson, Tom Clara, Florent De Geeter +3

Learning robust and scalable finite-state tracking is fundamental to sequence processing. While linear recurrent neural networks (RNNs), linear attention, and state space models enable scalable parallel training through affine recurrences, their theoretical expressivity guarantees assume idealized arithmetic and do not extend to finite precision, where the parallel scan that makes them fast is itself a source of perturbation. We formalize finite-state tracking at finite precision and characterize length independence: tracking that stays correct at every sequence length, at a precision cost that does not grow with the length. We show that length-independent state tracking requires two competing dynamics within a single map: contraction to suppress numerical perturbations and separation to keep distinct states apart. We prove that affine recurrences, which offer a single rate at each step to serve both roles, realize at most definite automata at finite precision. Instead of treating scan compatibility as a restriction on the update map, we reinterpret it as a computational budget and introduce the Neural Finite-State Machine (NFSM): a nonaffine, scan-compatible RNN built for…