Coalgebraic Modal Logic for Dynamic Systems with Uncertainty
Andrés Gallardo, Ignacio Viglizzo
TL;DR
The paper develops a unifying coalgebraic modal logic for dynamic systems under uncertainty by introducing upper probability polynomial functors and a multi-sorted language with semantics grounded in $[\![\cdot]\!]$ and $(p,q)$ bounds. It provides a complete deduction system through canonical coalgebras built from Lindenbaum maximal sets, and proves soundness and completeness for the base upper-probability setting, with extensions to probability measures, plausibility/belief, and possibility/necessity measures via additional functors $\Delta$, $\Delta_{Pl}$, and $\Delta_{Ps}$. The authors construct canonical measures and morphisms to show finality and to establish that satisfiability coincides with derivability. This framework yields a rigorous, extensible foundation for reasoning about uncertainty in state-based systems, accommodating diverse representations of uncertainty within a single coalgebraic apparatus.
Abstract
In this paper we define a class of polynomial functors suited for constructing coalgebras representing processes in which uncertainty plays an important role. In these polynomial functors we include upper and lower probability measures, finitely additive probability measures, plausibilty measures (and their duals, belief functions), and possibility measures. We give axioms and inference rules for the associated system of coalgebraic modal logic, and construct the canonical coalgebras to prove a completeness result.