Concurrency of manipulations in multidimensional information structures

Abstract

Given a sequence of manipulation rules together with dependence relations (allowing later rules to depend on the effects of earlier rules), we construct a single "concurrent manipulation rule" with the following property: Each application of the sequence of rules to a multidimensional information structure — such that the dependence relations are respected — can be performed in a single step applying the concurrent manipulation rule to the same structure. Moreover this becomes a bijective correspondence between such manipulation sequences and "concurrent manipulations". This "Concurrency Theorem" is formulated and proved in the framework of the algebraic theory of graph grammars using new pushout and pullback lemmas for the 3- and 4-dimensional cube. As corollaries we obtain a recently published Church-Rosser-Theorem for graph derivations and a theorem reducing the strong to the weak Church-Rosser-property.