Abstract. We introduce Modular Markovian Logic (MML) for compositional continuous-time and continuous-space Markov processes. MML combines operators specific to stochastic logics with operators that reflect the modular structure of the semantics, similar to those used by spatial and separation logics. We present a complete Hilbert-style axiomatization for MML, prove the small model property and analyze the relation between the stochastic bisimulation and the logical equivalence relation induced by MML on models. 1

In this paper we study the Continuous Markovian Logic (CML), a multimodal logic that expresses quantitative and qualitative properties of continuous-space and continuous-time labelled Markov processes(CMPs). The modalities of CML approximates the rates of the exponentially distributed random variables that characterize the duration of labeled transitions. We propose a soundcomplete Hilbert-style axiomatization for CML against the CMP-semantics and prove the small model property. It is known, from the similar case of probabilistic systems, that such a logic characterizes bisimulation and supports the definition of a distance between a model and a formula that quantifies the satisfiability relation; only that this distance is not always computable. We prove that in our case it can be approximated, within a given error, by using a distance between logical formulas that we define relying on the small model property of CML.

Abstract. We present the experience gained from implementing a new decision procedure for both graded and probabilistic modal logic. While our approach uses standard tableaux for propositional connectives, modal rules are given by linear constraints on the arguments of operators. The implementation uses binary decision diagrams for propositional connectives and a linear programming library for the modal rules. We compare our implementation, for graded modal logic, with other tools, showing average performance. Due to lack of other implementations, no comparison is provided for probabilistic modal logic, the main new feature of our implementation.

Abstract. 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”.