Generalized Reversible Computing

Landauer's Principle that information loss from a computation implies entropy increase can be rigorously proved from mathematical physics. However, carefully examining its detailed formulation reveals that the traditional identification of logically reversible computational operations with bijective transformations of the full digital state space is actually not the correct logical-level characterization of the full set of classical computational operations that can be carried out physically with asymptotically zero energy dissipation. To find the correct logical conditions for physical reversibility, we must account for initial-state probabilities when applying the Principle. The minimal logical-level requirement for the physical reversibility of deterministic computational operations is that the subset of initial states that exhibit nonzero probability in a given statistical operating context must be transformed one-to-one into final states. Thus, any computational operation is conditionally reversible relative to any sufficiently-restrictive precondition on its initial state, and the minimum dissipation required for any deterministic operation by Landauer's Principle asymptotically approaches 0 when the probability of meeting any preselected one of its suitable preconditions approaches 1. This realization facilitates simpler designs for asymptotically thermodynamically reversible computational hardware, compared to designs that are restricted to using only fully-bijective operations such as Toffoli type operations. Thus, this more general framework for reversible computing provides a more effective theoretical foundation to use for the design of practical reversible computers than does the more restrictive traditional model of reversible logic. In this paper, we formally develop the theoretical foundations of the generalized model, and briefly survey some of its applications.