Refinement mapping for general (discrete event) systems theory

Abstract

Abstract. A categoria! semantic domain for general (discrete event) systems based on labeled transition systems with full concurrency is constructed, where synchronization and hiding are functorial. Moreover, we claim that, within the proposed framework , a class of mappings stands for refinement. 1l1en we prove that refinement satisfies the diagonal compositionality requirement, i.e. , refinements compose (vertical) and distribute over system composition (horizontal). IntroductionWe construct a semantic domain for interacting systems which satisfies the diagonal compositionality requirement, i.e., refmements compose (vettically) , reflecting the stepwise description of systems, involving severa! leveis of abstraction, and distributes through combinators (horizontally), meaning that the refinement of a composite system is the composition of the refmement of its parts. Taking into consideration the developments in Petri net theory (mainly with seminal papers like [17], [11] and [15]) it was clear that nets rnight be good candidates. However, most of net-based models such as Petri nets in the sense of [14] and labeled n·ansition systems (see [12]) lack composition operations (modularity) and absn·action mechanisms in their original defmitions. This motivate the use of the category theory: the approach in [17] provides the former, where categorical constructions such as product and coproduct stand for system composition, and the approach in [11] provides the later for Pen·i nets where a special kind of net morphism conesponds to the notion of irnplementation. Also, category theory provides powelful techniques to unify different categories of models (i.e., classes of models categorically sn·uctured) through adjunctions (usually reflections and coreflections) expressing the relation of their semantics as in [15].We introduce the concept of (nonsequential) automaton as a kind of automaton structured on states and n·ansitions. Structured states are "bags" of local states like tokens in Pen·i nets and structured transitions specify a concwTency relationship between component transitions in the sense of [3] and [7]. In [9] we show that nonsequential automata are more concrete then Petri nets (in fact, categories of Petri nets are isomorphic to subcategories of nonsequential automata) extending the approach in [15], where a formal framework for classification of models for concunency is set.The resulting category is bicomplete where the categoria! product and coproduct stand for (system) composition. Synchronization and hiding are functorial operations. A Refinernent Mapping for General (Discrete E vent) Systems Theory

Nautilus, a Concurrent Diagrammatic Specification and Programming Language

“…This domain is based on labeled transition systems and inspired by Meseguer and Montanari's work [13]. Adjunctions between models for concurrency were provided in [8][9][10] extending the approach of Winskel and Nielsen [20]. The steps of abstraction involved in moving between models have shown that nonsequential automata are more concrete than Petri nets.…”

“…A refinement implements an automaton over sequential or concurrent computations of another, i.e., maps transitions into transactions reflecting the implementation of an automaton on top of another. Therefore, a refinement mapping is viewed as a special automaton morphism (a kind of implementation morphism [10]) where the target object is closed under computation, i.e., the target (more concrete) automaton is enriched with all the conceivable sequential and nonsequential computations that can be split into permutations of original transitions.…”

“…Nautilus is a high-level object based and oriented language with abstraction facilities inspired by its semantic domain (a categorial model named Nonsequential Automata [10,14,15]). Among the main features of the language we highlight the following: concurrency and non-determinism-the language is naturally concurrent and non-deterministic; refinement-abstraction mechanism which implements the concept of transaction, in the sense that we can define high-level (finite and atomic) actions as the composition of other sequential and/or concurrent actions; aggregation-it allows the definition of combined behavior of a set of objects.…”

Nautilus: A Concurrent Anticipatory Programming Language

“…An anticipation maps transitions into transactions reflecting the implementation of an automaton on top of another. Therefore, an anticipation mapping is viewed as a special automaton morphism (a kind of implementation morphism) [7] where the target object is closed under computation, i.e., the target (more concrete) automaton is enriched with all the conceivable sequential and nonsequential computations that can be split into permutations of original transitions. Accordingly, the anticipation of an object is specified over an existing object (an action may be mapped into a complex action of the target object).…”

“…Nautilus [8,2,3] is a general purpose concurrent object-based language, originally based on the language Gnome [10,11,14], and introduces some special features inspired by the semantic domain such as anticipation. A semantics for Nautilus is given by Nonsequential Automata [8,7,9,6], which constitute a categorial [1] semantic domain based on labeled transition system with full concurrency, where a class of morphisms stands for anticipation. It is a model which satisfies the diagonal compositionality requirement, i.e., anticipations compose and distribute over system combinators.…”