$\textsf {ML}_{\nu }$: A Distributed Real-Time Modal Logic

Abstract

Distributed Real-Time Systems (DRTS) can be characterized by several communicating components whose behavior depends on a large number of timing constraints and such components can basically be located at several computers spread over a communication network. Extensions of Timed Modal Logics (TML) such as, Timed Propositional Modal Logic (TPML), Timed Modal $\mu$-calculus and $\textsf {L}_{\nu }$ have been proposed to capture timed and temporal properties in real-time systems. However, these logics rely on a so-called mono-timed semantics for the underlying Timed Labelled Transition Systems (TLTS). This semantics does not capture complex interactions between components with their associated local clocks, thus missing possible action sequences. Based on Multi-Timed Labelled Transition Systems (MLTS), which are an extension of TLTS in order to cope with the notion of distributed clocks, we propose $\textsf {ML}_{\nu }$, an extension of $\textsf {L}_{\nu }$ that relies on a distributed semantics for Timed Automata (TA) instead of considering uniform clocks over the distributed systems, we let time vary independently in each TA. We define the syntax and the semantics of $\textsf {ML}_{\nu }$ over executions of MLTS with such a semantics and we show that its model checking problem against $\textsf {ML}_{\nu }$ is EXPTIME-complete.