. Tile logic extends rewriting logic by taking into account sideeffects and rewriting synchronization. These aspects are very important when we model process calculi, because they allow us to express the dynamic interaction between processes and "the rest of the world". Since rewriting logic is the semantic basis of several language implementation efforts, an executable specification of tile systems can be obtained by mapping tile logic back into rewriting logic, in a conservative way. However, a correct rewriting implementation of tile logic requires the development of a metalayer to control rewritings, i.e., to discard computations that do not correspond to any deduction in tile logic. We show how such methodology can be applied to term tile systems that cover and extend a wide-class of SOS formats for the specification of process calculi. The well-known case-study of full CCS, where the term tile format is needed to deal with recursion (in the form of the replicator operator), is di...

Tile systems offer a general paradigm for modular descriptions of concurrent systems, based on a set of rewriting rules with side-effects. Monoidal double categories are a natural semantic framework for tile systems, because the mathematical structures describing system states and synchronizing actions (called configurations and observations, respectively, in our terminology) are monoidal categories having the same objects (the interfaces of the system). In particular, configurations and observations based on net-process-like and term structures are usually described in terms of symmetric monoidal and cartesian categories, where the auxiliary structures for the rearrangement of interfaces correspond to suitable natural transformations. In this paper we discuss the lifting of these auxiliary structures to double categories. We notice that the internal construction of double categories produces a pathological asymmetric notion of natural transformation, which is fully exploited in one dimension only (for example, for configurations or for observations, but not for both). Following Ehresmann (1963), we overcome this biased definition, introducing the notion of generalized natural transformation between four double functors (rather than two). As a consequence, the concepts of symmetric monoidal and cartesian (with consistently chosen products) double categories arise in a natural way from the corresponding ordinary versions, giving a very good relationship between the auxiliary structures of configurations and observations. Moreover, the Kelly–Mac Lane coherence axioms can be lifted to our setting without effort, thanks to the characterization of two suitable diagonal categories that are always present in a double category. Then, symmetric monoidal and cartesian double categories are shown to offer an adequate semantic setting for process and term tile systems.

this paper we discuss the lifting of these auxiliary structures to double categories. We notice that the internal construction of double categories produces a pathological asymmetric notion of natural transformation, which is fully exploited in one dimension only (e.g., for configurations or for effects, but not for both). Following Ehresmann (1963), we overcome this biased definition, introducing the notion of generalized natural transformation between four

2-categories and double categories are respectively the natural semantic ground for rewriting logic (rl) and tile logic (tl). Since 2-categories can be regarded as a special case of double categories, then rl can be easily embedded into tl, where also rewriting synchronization is considered. Since rl is the semantic basis of several existing languages, it is useful to map tl back into rl to have an executable framework for tile specifications. We extend the results of a previous work of two of the authors, focusing on tile systems where the algebraic structures for configurations and observations rely on some common auxiliary structure (e.g., for pairing, projecting, etc.). The new model theory required to relate the categorical models of the two logics is an extended version of the theory of 2-categories, and is defined using partial membership equational logic. More concretely, this semantic mapping yields a rewriting logic implementation of tile logic, where a meta-layer is requir...