The Algebraic View of Computation: Implementation, Interpretation and Time

Computational implementations are special relations between what is computed and what computes it. Though the word “isomorphism” appears in philosophical discussions about the nature of implementations, it is used only metaphorically. Here we discuss computation in the precise language of abstract algebra. The capability of emulating computers is the defining property of computers. Such a chain of emulation is ultimately grounded in an algebraic object, a full transformation semigroup. Mathematically, emulation is defined by structure preserving maps (morphisms) between semigroups. These are systematic, very special relationships, crucial for defining implementation. In contrast, interpretations are general functions with no morphic properties. They can be used to derive semantic content from computations. Hierarchical structure imposed on a computational structure plays a similar semantic role. Beyond bringing precision into the investigation, the algebraic approach also sheds light on the interplay between time and computation.