Solving Temporal Problems using SMT: Strong Controllability Alessandro Cimatti, Andrea Micheli, and Marco Roveri {cimatti,amicheli,roveri}@fbk.eu Fondazione Bruno Kessler — Irst Abstract. Many applications, such as scheduling and temporal plan- ning, require the solution of Temporal Problems (TP’s) representing con- straints over the timing of activities. A TP with uncertainty (TPU) is characterized by activities with uncontrollable duration. Depending on the Boolean structure of the constraints, we have simple (STPU), con- straint satisfaction (TCSPU), and disjunctive (DTPU) temporal prob- lems with uncertainty. In this work we tackle the problem of strong controllability, i.e. finding an assignment to all the controllable time points, such that the constraints are fulfilled under any possible assignment of uncontrollable time points. We work in the framework of Satisfiability Modulo Theory (SMT), where uncertainty is expressed by means of universal quantifiers. We obtain the first practical and comprehensive solution for strong controllability: the use of quantifier elimination techniques leads to quantifier-free encodings, which are in turn solved with efficient SMT solvers. We provide a detailed experimental evaluation of our approach over a large set of benchmarks. The results clearly demonstrate that the pro- posed approach is feasible, and outperforms the best state-of-the-art competitors, when available. 1 Introduction Many applications require the scheduling of a set of activities over time, subject to constraints of various nature. Scheduling is often expressed as a Temporal Problem (TP), where each activity is associated with two time points, repre- senting the start time and the end time, and with a duration, all subject to constraints. Several kinds of temporal problems have been identified, depending on the nature and structure of the constraints. If the constraints are expressible as a simple conjunction of constraints over distances of time points, then we have the so-called Simple Temporal Problem (STP). A more complex class is Temporal Constraint Satisfaction Problem (TCSP), where a distance between time points can be constrained to a list of disjoint intervals. Constraints in TCSP’s can be seen as a restricted form of Boolean combinations. When arbitrary Boolean combinations are allowed, we have a Disjunctive Temporal Problem (DTP). A temporal problem is said to be consistent if there exists an assignment for the time points, such that all the constraints are satisfied [1]. Such an assignment is This is a pre-print version of the homonymous paper appearing in CP 2012. Copyright (c) 2012 belongs to Springer.