Engineering Confluent Computational Fields: from Functions to Rewrite Rules

Among the many models used to specify spatial computations in terms of (computational) fields, those based on some form of rewrite rules have been proposed for their ability to support the development of open and situated systems - e.g., on top of distributed tuple space infrastructures. However, due to lack of modularity abstractions, one such approach hardly tackles multi-level combination of different fields, as typically happens in non-trivial self-organisation patterns. In this paper we introduce a functional notation to express computational fields (and their combination), and define its semantics by a translation into a rewrite-based model. The proposed language, which is reminiscent of Proto, includes a somewhat narrow set of constructs, yet providing a good trade-off between expressiveness and formal tractability. In fact, we state for it the confluence property, namely, so-called "don't care non-determinism": this implies - among the others - that specifications satisfying certain monotonicity properties result in fields that stabilise in linear time to a state of easily predictable shape.