Parametric temporal logic for “model measuring”

We extend the standard model checking paradigm of linear temporal logic, LTL, to a “model measuring” paradigm where one can obtain more quantitative information beyond a “Yes/No” answer. For this purpose, we define a <italic>parametric temporal logic</italic>, PLTL, which allows statements such as “a request <italic>p</italic> is followed in at most <italic>x</italic> steps by a response <italic>q</italic>,” where <italic>x</italic> is a free variable. We show how one can, given a formula ***(<italic>x<subscrpt>1</subscrpt>...,x<subsccrpt>k</subscrpt></italic>) of PLTL and a system model <italic>K</italic> satisfies the property ***, but if so find valuations which satisfy various optimality criteria. In particular, we present algorithms for finding valuations which minimize (or maximize) the maximum (or minimum) of all parameters. Theses algorithms exhibit the same PSPACE complexity as LTL model checking. We show that our choice of syntax for PLTL lies at the threshold of decidability for parametric temporal logics, in that several natural extensions have undecidable “model measuring” problems.