First order and higher order tile models for open and mobile systems

There are two principal ways in which the dynamics of many calculi can be naturally modeled: (1) by means of labeled transition systems (LTS) where states are terms in the calculus and transition labels express some observation on the evolution step; or (2) by defining a reduction semantics over terms up to a suitable structural congruence (in the style of the chemical abstract machine [1]). While the first approach comes straightforwardly equipped with a variety of abstract semantics (e.g. bisimilarity, trace equivalence) whose compositionality can be guaranteed by specifying the LTS accordingly to suitable syntactic formats (e.g. [8, 10, 2]), the second approach is often easier to apply but provides less informative abstract equivalences w.r.t the dynamics and interactions of components [16, 12] (e.g. reduction to the same normal form). In both cases, it is enviable that the abstract semantics be a congruence, otherwise suitable techniques should be applied for extracting a congruence out of it (e.g. closing w.r.t. all contexts, using dynamic bisimilarity [14]). Another issue is related to the study of calculi with name hiding, passing and creation, whose features often require ad-hoc techniques when defining observational congruences (e.g. early vs. late bisimilarity). Moreover, in several cases, the abstract semantics is defined on ground terms only (representing closed systems), and it is then extended to open terms (i.e. contexts) by instantiating w.r.t. all possible ground instances. Tile logic [9, 3] attempts at reconciling the most interesting features of (1) and (2) and, at the same time, at dealing with ground and open terms in a uniform way. To this aim, transition labels become pairs, whose components are called triggers (expressing the interaction of a context with its arguments) and effect (representing the behavior offered to the rest of the system, i.e. a possible context). Tiles can be represented as rectangles where the horizontal dimension is devoted to the assembling of states and the vertical dimension is dedicated to the evolution of components. Thus, triggers and effects form the left and right sides of tiles, respectively. The vertices of tiles are called interfaces, connecting the input and output observations to the initial (before the step) and final (after the step) configurations. Thanks to the abstract notions of configuration and observation, tiles allow us to develop a theoretical framework parametric in such structures (e.g. graphs or hypergraphs or trees or λ-terms rather than terms), and able to capture analogies in the structures by means of suitable auxiliary tiles (e.g. This research has been partially supported by MURST project TOSCA as part of theme 3, task Tiles and theme 7, task TilesSuperiore. †We warmfully thank our co-authors David de Frutos-Escrig, Narciso Martı́-Oliet, Francesca Rossi, and Vladimiro Sassone for their contribution to the results presented here.