Process and Term Tile Logic

In a similar way as 2-categories can be regarded as a special case of double categories, rewriting logic (in the unconditional case) can be embedded into the more general tile logic, where also side-effects and rewriting synchronization are considered. Since rewriting logic is the semantic basis of several language implementation efforts, it is useful to map tile logic back into rewriting logic in a conservative way, to obtain executable specifications of tile systems. We extend the results of earlier work by two of the authors, focusing on some interesting cases where the mathematical structures representing configurations (\ie, states) and effects (\ie, observable actions) are very similar, in the sense that they have in common some auxiliary structure (\eg, for tupling, projecting, etc.). In particular, we give in full detail the descriptions of two such cases where (net) process-like and usual term structures are employed. Corresponding to these two cases, we introduce two categorical notions, namely, symmetric strict monoidal double category and cartesian double category with consistently chosen products, which seem to offer an adequate semantic setting for process and term tile systems. The new model theory of 2EVH-categories required to relate the categorical models of tile logic and rewriting logic is presented making use of a recently developed framework, called partial membership equational logic, particularly suitable to deal with categorical structures. Consequently, symmetric strict monoidal and cartesian classes of double categories and 2-categories are compared through their embedding in the corresponding versions of 2EVH-categories. As a result of this comparison, we obtain a correct rewriting implementation of tile logic. This implementation uses a meta-layer to control the rewritings, so that only tile proofs are accepted. Making use of the reflective capabilities of the Maude language, some (general) internal strategies are then defined to implement the mapping from tile systems into rewriting systems, and some interesting applications related to the implementation of concurrent process calculi are presented.