A Decidable Timeout based Extension of Propositional Linear Temporal Logic

Abstract

We develop a timeout based extension of propositional linear temporal logic (which we call TLTL) to specify timing properties of timeout based models of real time systems. TLTL formulas explicitly refer to a running global clock together with static timing variables as well as a dynamic variable abstracting the timeout behavior. We extend LTL with the capability to express timeout constraints. From the expressiveness view point, TLTL is not comparable with important known clock based real-time logics including TPTL, XCTL, and MTL, i.e., TLTL can specify certain properties, which cannot be specified in these logics (also vice-versa). We define a corresponding timeout tableau for satisfiability checking of the TLTL formulas. Also a model checking algorithm over timeout Kripke structure is presented. Further we prove that the validity checking for such an extended logic remains PSPACE-complete even in the presence of timeout constraints and infinite state models. Under discrete time semantics, with bounded timeout increments, the model-checking problem that if a TLTL-formula holds in a timeout Kripke structure is also PSPACE complete. We further prove that when TLTL is interpreted over discrete time, it can be embedded in the monadic second order logic with time, and when TLTL is interpreted over dense time without the condition of non-zenoness, the resulting logic becomes $Σ_1^1$-complete.