Asymptotic behaviour in temporal logic

Abstract

We study the "approximability" of unbounded temporal operators with time-bounded operators, as soon as some time bounds tend to ∞. More specifically, for formulas in the fragments PLTL⋄ and PLTL◻ of the Parametric Linear Temporal Logic of Alur et al., we provide algorithms for computing the limit entropy as all parameters tend to ∞. As a consequence, we can decide the problem whether the limit entropy of a formula in one of the two fragments coincides with that of its time-unbounded transformation, obtained by replacing each occurrence of a time-bounded operator into its time-unbounded version. The algorithms proceed by translation of the two fragments of PLTL into two classes of discrete-time timed automata and analysis of their strongly-connected components.