A categorical framework for concurrent, anticipatory systems

Abstract

A categorical semantic domain is constructed for Petri nets which satisfies the diagonal compositionality requirement with respect to anticipations, i.e., Petri nets are equipped with a compositional anticipation mechanism (vertical compositionality) that distributes through net combinators (horizontal compositionality). The anticipation mechanism is based on graph transformations (single pushout approach). A finitely bicomplete category of partia! Petri nets and partia! morphisms is introduced. Classes of transformations stand for anticipations. The composition of anticipations (i.e., composition of pushouts) is defined, leading to a category of nets and anticipations which is also complete and cocomplete. Since the anticipation operation composes, the vertical compositionality requirement of Petri nets is achieved. Then, it is proven that the anticipation also satisfies the horizontal compositionality requirement. A specification grammar stands for a system specification and the corresponding induced subcategory of nets and anticipation's stands for ali possible dynamic anticipation's of the system (objects) and their relationship (morphims).

Nautilus: A Concurrent Anticipatory Programming Language

“…The composition of both functors from NAut into LCat(CMon) leads to an endofunctor, called transitive closure. The composition of anticipations of nonsequential automata is defined using Kleisli categories (see [9]). In fact, the adjunction above induces a monad which defines a Kleisli category.…”

“…The main purpose of this paper is to present Nautilus as a concurrent anticipatory programming language, i.e., its semantic is an anticipatory system that involves "its future", inspired by [5] and based on [8,9]. An anticipatory system [5] is a system for which the present behavior is based on past and/or present events but also on future events built from these past, present and future events.…”

“…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.…”