A Coalgebraic View of Characteristic Formulas in Equational Modal Fixed Point Logics

Abstract

The literature on process theory and structural operational semantics abounds with various notions of behavioural equivalence and, more generally, simulation preorders. An important problem in this area from the point of view of logic is to find formulas that characterize states in finite transition systems with respect to these various relations. Recent work by Aceto et al. shows how such characterizing formulas in equational modal fixed point logics can be obtained for a wide variety of behavioural preorders using a single method. In this paper, we apply this basic insight from the work by Aceto et al. to Baltag's "logics for coalgebraic simulation" to obtain a general result that yields characteristic formulas for a wide range of relations, including strong bisimilarity, simulation, as well as bisimulation and simulation on Markov chains and more. Hence this paper both generalizes the work of Aceto et al. and makes explicit the coalgebraic aspects of their work.

A cottage industry of lax extensions

“…These results, generalized in [2, Corollary IV.3.2.3], stem from a different point of view as the one taken here, as they allow for a "change-base-base", namely passing from V = P + to V = 2 by modifying the underlying monad, whereas the current work fixes the quantale V . Second, the study of lax extensions is rooted in topology and order theory, but current trends show that it is in no way limited to these fields, as the cited references show, but also [6] and [9].…”