Master equations are commonly used to model the dynamics of physical systems . Surprisingly, many deterministic maps x \rightarrow f(x) cannot be implemented by any master equation, even approximately . This raises the question of how they arise in real-world systems like digital computers. We show that any deterministic map over some "visible" states can be implemented with a master equation--but only if additional "hidden" states are dynamically coupled to those visible states . We also show that any master equation implementing a given map can be decomposed into a sequence of "hidden" timesteps, demarcated by changes in what transitions are allowed under the rate matrix. Often there is a real-world cost for each additional hidden state, and for each additional hidden timestep. We derive the associated " space/time " tradeoff between the numbers of hidden states and  of hidden timesteps needed to implement any given f(x) .
Master equations are commonly used to model the dynamics of physical systems , including systems that implement single-valued functions like a computer's update step. However, many such functions cannot be implemented by any master equation, even approximately , which raises the question of how they can occur in the real world. Here we show how any function over some "visible" states can be implemented with master equation dynamics--if the dynamics exploits additional, "hidden" states at intermediate times . We also show that any master equation implementing a function can be decomposed into a sequence of "hidden" timesteps, demarcated by changes in what state-to-state transitions have nonzero probability. In many real-world situations there is a cost both for more hidden states and for more hidden timesteps. Accordingly, we derive a " space-time " tradeoff between the number of hidden states and the number of hidden timesteps needed to implement any given function .
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