Minimal Work Required for Arbitrary Computation

Recent studies have analyzed the minimal thermodynamic work required for a given logical map to be implemented on any physical system. These studies have focused on maps whose output does not depend on the input, e.g., bit erasure in a digital computer. In addition, they have considered physical systems whose design varies depending on the distribution of inputs to the map. However very often we are interested in implementing a map whose output depends on its input. In addition, we often want our system to implement the same map even if the system's environment changes, so that the distribution over map inputs changes. Here I introduce a thermodynamic engine that satisfies both of these desiderata. I then calculate how much work it requires, deriving an additive correction to the ``generalized Landauer bound" of previous studies. I also calculate the Bayes-optimal engine for any given distribution over environments. I end with a short discussion on how these results relate the free energy flux incident on an organism / robot / biosphere to the maximal amount of (noisy) computation that the organism / robot / biosphere can do per unit time.