Hopfield attractor networks are robust distributed models of human memory. We propose construction rules such that an attractor network may implement an arbitrary finite state machine (FSM), where states and stimuli are represented by high-dimensional random bipolar vectors, and all state transitions are enacted by the attractor network's dynamics. Numerical simulations show the capacity of the model, in terms of the maximum size of implementable FSM, to be linear in the size of the attractor network. We show that the model is robust to imprecise and noisy weights, and so a prime candidate for implementation with high-density but unreliable devices. By endowing attractor networks with the ability to emulate arbitrary FSMs, we propose a plausible path by which FSMs may exist as a distributed computational primitive in biological neural networks.