Continuous Markovian Logics - Axiomatization and Quantified Metatheory

Continuous Markovian Logic (CML) is a multimodal logic that expresses quantitative and qualitative properties of continuous-time labelled Markov processes with arbitrary (analytic) state-spaces, henceforth called continuous Markov processes (CMPs). The modalities of CML evaluate the rates of the exponentially distributed random variables that characterize the duration of the labeled transitions of a CMP. In this paper we present weak and strong complete axiomatizations for CML and prove a series of metaproperties, including the finite model property and the construction of canonical models. CML characterizes stochastic bisimilarity and it supports the definition of a quantified extension of the satisfiability relation that measures the “compatibility” between a model and a property. In this context, the metaproperties allows us to prove two robustness theorems for the logic stating that one can perturb formulas and maintain “approximate satisfaction”.

💡 Research Summary

Continuous Markovian Logic (CML) is introduced as a multimodal formalism designed to capture both quantitative and qualitative aspects of continuous‑time labelled Markov processes (CMPs) with arbitrary analytic state spaces. The syntax extends classical propositional logic with a family of rate‑modalities ⟨a⟩≥r·φ, where a is a label, r∈ℝ≥0, and φ is a formula. Semantically, a CMP is a triple (M, Σ, λ) where M is a measurable state space, Σ assigns to each label a a σ‑finite transition measure λ_a(s,·), and the sojourn time of a transition labelled a from state s follows an exponential distribution with rate λ_a(s,M). The modality ⟨a⟩≥r·φ holds at s exactly when the total rate of a‑transitions leading to states satisfying φ is at least r.

The paper presents two axiom systems. The weak completeness system consists of standard propositional axioms, a K‑style axiom for the rate‑modalities, monotonicity (if r≤s then ⟨a⟩≥r·φ→⟨a⟩≥s·φ), and an additive axiom (⟨a⟩≥r·φ∧⟨a⟩≥s·φ↔⟨a⟩≥(r+s)·φ). The strong completeness system augments these with rules that guarantee every valid formula has a finite proof, using a canonical construction based on maximal consistent sets. Soundness of both systems follows directly from the semantics; completeness is proved by building a canonical model where each world is a maximal consistent set and the transition rates are extracted from the rate‑inequalities present in the set.

A series of metatheoretic results are then established. First, the finite model property is proved: if a formula is satisfiable, there exists a CMP with a finite state space that satisfies it. The proof relies on approximating real rates by rational numbers and collapsing bisimilar states into equivalence classes, thereby obtaining a finite quotient model. Second, a canonical (or “Henkin”) model is constructed, showing that the logic is strongly complete with respect to the class of all CMPs. Third, the authors demonstrate that CML characterises stochastic bisimilarity: two CMPs are bisimilar iff they satisfy exactly the same CML formulas. This establishes CML as a behavioural logic for continuous‑time stochastic systems.

Beyond the classic Boolean satisfaction relation, the paper defines a quantitative satisfaction function ⟦φ⟧_M(s)∈