Reflection and Preservation of Properties in Coalgebraic (bi)Simulations
Ignacio Fábregas, Miguel Palomino, David de Frutos-Escrig
TL;DR
The paper investigates when coalgebraic simulations, in the sense of Hughes–Jacobs, preserve and reflect temporal properties expressed in Jacobs' generalized LTL (extended with Kurtz's atomic propositions). It first shows that bisimulations yield preservation/reflection for formulas without negation, then demonstrates that general simulations do not guarantee these properties. To regain them, the authors introduce two layers of restrictions: down-closed orders on the functorial structure and a polynomial-like class of functors (Order) that ensure preservation/reflection for standard temporal operators; they further extend the framework to atomic propositions via natural transformations and down-natural/up-natural ν-orders, obtaining reflection/preservation results under these refined conditions. The work provides a structured coalgebraic methodology for reasoning about property transfer across simulations and outlines directions for canonical simulations and probabilistic extensions.
Abstract
Our objective is to extend the standard results of preservation and reflection of properties by bisimulations to the coalgebraic setting, as well as to study under what conditions these results hold for simulations. The notion of bisimulation is the classical one, while for simulations we use that proposed by Hughes and Jacobs. As for properties, we start by using a generalization of linear temporal logic to arbitrary coalgebras suggested by Jacobs, and then an extension by Kurtz which includes atomic propositions too.