The Geometry of Orthogonal Reduction Spaces

We investigate mutual dependencies of subexpressions of a computable expression, in orthogonal rewrite systems, and identify conditions for their concurrent independent computation. To this end, we introduce concepts familiar from ordinary Euclidean Geometry (such as basis, projection, distance, etc.) for reduction spaces. We show how a basis for an expression can be constructed so that any reduction starting from that expression can be decomposed as the sum of its projections on the axes of the basis. To make the concepts more relevant computationally, we relativize them w.r.t. stable sets of results, and show that an optimal concurrent computation of an expression w.r.t. S consists of optimal computations of its S-independent subexpressions. All these results are obtained for Stable Deterministic Residual Structures, Abstract Reduction Systems with an axiomatized residual relation, which model all orthogonal rewrite systems.