## Parametric Temporal Logic for Model Measuring

Rajeev Alur, Kousha Etessami, Salvatore La Torre, and Doron Peled

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 {\em parametric temporal logic\/}, PLTL, which allows statements such as a request p is followed in at most x steps by a response q'', where x is a free variable. We show how one can, given a formula phi(x_1, ...x_k) of PLTL and a system model K, not only determine whether there exists a valuation of x_1, ... x_k under which the system K satisfies the property phi, 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. These 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.

Automata, Languages, and Programming: Proceedings of the 26th International Conference, (ICALP'99), Lecture Notes in Computer Science 1644, Springer, pp. 159--168, 1999.