Expressiveness of probabilistic modal logics: A gradual approach

Expressivity of Quantitative Modal Logics : Categorical Foundations via Codensity and Approximation

“…In order to do so, each of the expressivity proofs in [14]- [18] uses some kind of "approximation." However, each of these arguments has a specialized, tailor-made flavor: Stone-Weierstrass-like arguments for metric spaces [17], the unique structure theorem for analytic spaces [15], and so on. It does not seem easy to distill the essence that is common to different quantitative expressivity proofs.…”

“…On top of our fibrational notion of approximating family, we establish a general expressivity framework, which is the first to unify existing quantitative expressivity results including [15]- [17]. In our unified framework, we have two proof principles for expressivity-Knaster-Tarski (Thm.…”

“…We demonstrate our general framework with three examples: expressivity for the Kantorovich bisimulation metric (from [17], §V); that for Markov process bisimilarity (from [15], §VI); and that for the so-called bisimulation uniformity ( §VII). Both the Knaster-Tarski and Kleene principles are used for proofs.…”

“…We note that the role of the notion of approximating family is as a useful axiomatization: it tells us what key lemma to prove in an expressivity proof, but it does not tell how to prove the key lemma. The proof of this key lemma is where the technical hardcore lies in existing expressivity proofs (by a Stone-Weierstrass-like result in [17], by the unique structure theorem in [15], etc.). For the new instance of bisimulation uniformity ( §VII), the general axiomatization of approximating family allowed us to discover a result we need in a paper [21] that is seemingly unrelated to modal logic.…”

“…• The framework is instantiated to two known expressivity results [15], [17] and one new result ( §VII). • The framework is given an abstract and fully fibrational recap ( §VIII) that exposes further fibrational structures.…”