Non-strongly Stable Orders Also Define Interesting Simulation Relations
Ignacio Fábregas, David de Frutos-Escrig, Miguel Palomino
TL;DR
The paper investigates coalgebraic simulations defined via functorial preorders and shows that stability properties, notably right-stability, induce a directionality and composition-preserving behavior essential for robust simulations. It introduces two novel stable notions—covariant-contravariant simulations and conformance simulations—and demonstrates they can be realized as coalgebraic simulations by decomposing their orders into stable components. The authors provide decomposition-based proofs that reuse Rel(F) machinery, establishing stability for these notions and clarifying how to handle inverse simulations and dual directions. This work extends the coalgebraic simulation framework to key semantic notions in process theory and lays a scalable foundation for integrating these ideas with broader process semantics.
Abstract
We present a study of the notion of coalgebraic simulation introduced by Hughes and Jacobs. Although in their original paper they allow any functorial order in their definition of coalgebraic simulation, for the simulation relations to have good properties they focus their attention on functors with orders which are strongly stable. This guarantees a so-called "composition-preserving" property from which all the desired good properties follow. We have noticed that the notion of strong stability not only ensures such good properties but also "distinguishes the direction" of the simulation. For example, the classic notion of simulation for labeled transition systems, the relation "p is simulated by q", can be defined as a coalgebraic simulation relation by means of a strongly stable order, whereas the opposite relation, "p simulates q", cannot. Our study was motivated by some interesting classes of simulations that illustrate the application of these results: covariant-contravariant simulations and conformance simulations.