Chapter Two Embracing the laws of physics: Three reversible models of computation

Our main models of computation (the Turing Machine and the RAM) and most modern computer architectures make fundamental assumptions about which primitive operations are realizable on a physical computing device. The consensus is that these primitive operations include logical operations like conjunction, disjunction and negation, as well as reading and writing to a large collection of memory locations. This perspective conforms to a macro-level view of physics and indeed these operations are realizable using macro-level devices involving thousands of electrons. This point of view is however incompatible with computation realized using quantum devices or analyzed using elementary thermodynamics as both these fundamental physical theories imply that information is a conserved quantity of physical processes and hence of primitive computational operations. Our aim is to redevelop foundational computational models in a way that embraces the principle of conservation of information. We first define what information is and what its conservation means in a computational setting. We emphasize the idea that computations must be reversible transformations on data. One can think of data as modeled using topological spaces and programs as modeled by reversible deformations of these spaces. We then illustrate this idea using three notions of data and their associated reversible computational models. The first instance only assumes unstructured finite data, i.e., discrete topological spaces. The corresponding notion of reversible computation is that of permutations. We show how this simple model subsumes conventional computations on finite sets. We then consider a modern structured notion of data based on the Curry–Howard correspondence between logic and type theory. We develop the corresponding notion of reversible deformations using a sound and complete programming language for witnessing type isomorphisms and proof terms for commutative semirings. We then “move up a level” to examine spaces that treat programs as data, which is a crucial notion for any universal model of computation. To derive the corresponding notion of reversible programs between programs, i.e., reversible program equivalences, we look at the “higher dimensional” analog to commutative semirings: symmetric rig groupoids. The coherence laws for these groupoids turn out to be exactly the sound and complete reversible program equivalences we seek. We conclude with some possible generalizations inspired by homotopy type theory and survey several open directions for further research.