The Computational Content of Isomorphisms

Abstract

Abstract models of computation, such as Turing machines, λ-calculus and logic gates, allow us to express computation without being concerned about the underlying technology that realizes them in the physical world. These models embrace a classical worldview wherein computation is essentially irreversible. From the perspective of quantum physics however, the physical world is one where every fundamental interaction is essentially reversible and various quantities such as energy, mass, angular momentum are conserved. Thus the irreversible abstractions we choose as the basis of our most primitive models of computing are at odds with the underlying reversible physical reality and hence our thesis: By embracing irreversible physical primitives, models of computation have also implicitly included a class of computational effects which we call information effects.

To make this precise, we develop an information preserving model of computation (in the sense of Shannon entropy) wherein the process of computing does not gain or lose information. We then express information effects in this model using an arrow meta-language, in much the same way that we model computational effects in the λ-calculus using a monadic metalanguage. A consequence of this careful treatment of information, is that we effectively capture the gap between reversible computation and irreversible computation using a type-and-effect system.

The treatment of information effects has a parallel with open and closed systems in physics. Closed physical systems conserve mass and energy and are the basic unit of study in physics. Open systems interact with their environment, possibly exchanging matter or energy. These interactions may be thought of as effects that modify the conservation properties of the system. Computations with information effects are much like open systems and they can be converted into pure computations by making explicit the surrounding information environment that they interact with.

Finally, we show how conventional irreversible computation such as the λ-calculus can be embedded into this model, such that the embedding makes the implicit information effects of the λ-calculus explicit.