An SMT-based approach to weak controllability for disjunctive temporal problems with uncertainty

The framework of temporal problems with uncertainty (TPU) is useful to express temporal constraints over a set of activities subject to uncertain (and uncontrollable) duration. In this work, we focus on the most general class of TPU, namely disjunctive TPU (DTPU), and consider the case of weak controllability, that allows one to model problems arising in practical scenarios (e.g. on-line scheduling). We first tackle the decision problem, i.e. whether there exists a schedule of the activities that, depending on the uncertainty, satisfies all the constraints. We propose a logical approach, based on the reduction to a problem of Satisfiability Modulo Theories (SMT), in the theory of Linear Real Arithmetic with Quantifiers. This results in the first implemented solver for weak controllability of DTPUs. Then, we tackle the problem of synthesizing control strategies for scheduling the activities. We focus on strategies that are amenable for efficient execution. We prove that linear strategies are not always sufficient, even in the sub-case of simple TPU (STPU), while piecewise-linear strategies, that are multiple conditionally-applied linear strategies, are always sufficient. We present several algorithms for the synthesis of linear and piecewise-linear strategies, in case of STPU and of DTPU. All the algorithms are implemented on top of SMT solvers. We provide experimental evidence of the scalability of the proposed techniques, with dramatic speed-ups in strategy execution compared to on-line reasoning.