The Microcosm Principle and Compositionality of GSOS-Based Component Calculi

In the previous work by Jacobs, Sokolova and the author, synchronous parallel composition of coalgebras—yielding a coalgebra—and parallel composition of behaviors—yielding a behavior, where behaviors are identified with states of the final coalgebra—were observed to form an instance of the microcosm principle. The microcosm principle, a term by Baez and Dolan, refers to the general phenomenon of nested algebraic structures such as a monoid in a monoidal category. Suitable organization of these two levels of parallel composition led to a general compositionality theorem: the behavior of the composed system relies only on the behaviors of its constituent parts. In the current paper this framework is extended so that it accommodates any process operator—not restricted to parallel composition—whose meaning is specified by means of GSOS rules. This generalizes Turi and Plotkin’s bialgebraic modeling of GSOS, by allowing a process operator to act as a connector between components as coalgebras.