## Section: New Results

### Temporal Logic Constraint Solving

Participants : François Fages, Aurélien Rizk.

Temporal logics and model-checking are at the core of BIOCHAM to express biological properties of complex biochemical systems and automatically verify their satisfaction in both qualitative and quantitative models. In particular, linear time logic constraints (QFLTL( )) are used to formalize the global behavior of the system known from biological experiments, and infer the unknown parameter values of the model.

In [16] we have studied the abstract properties of this approach and have given a general constraint solving algorithm for branching time logics (CTL) over arbitrary computation domains D . We have shown that the QFCTL constraint satisfiability problem is decidable in finite Kripke structures over an arbitrary computation domain with a decidable language of constraints, i.e. that any constraint solver can be lifted to a temporal logic constraint solver over finite Kripke structures. We have presented a generic QFCTL constraint solver which computes validity domains for the free variables of a formula, in quadratic time in the number of states. We show that when D is a metric space, this allows us to define a continuous degree of satisfaction for a temporal logic formula in a given Kripke structure, opening up the field of model-checking to optimization, in a very general setting.