  All computation is physically embedded. Reflecting this, a growing body of
results embraces rate equations as the underlying mechanics of thermodynamic
computation and biological information processing. Strictly applying the
implied continuous-time Markov chains, however, excludes a universe of natural
computing. We show that expanding the toolset to continuous-time hidden Markov
chains substantially removes the constraints. The general point is made
concrete by our analyzing two eminently-useful computations that are impossible
to describe with a set of rate equations over the memory states. We design and
analyze a thermodynamically-costless bit flip, providing a first counterexample
to rate-equation modeling. We generalize this to a costless Fredkin gate---a
key operation in reversible computing that is computation universal. Going
beyond rate-equation dynamics is not only possible, but necessary if stochastic
thermodynamics is to become part of the paradigm for physical information
processing.
