Tropical Fourier-Motzkin Elimination, with an Application to Real-Time Verification

We introduce a generalization of tropical polyhedra able to express both strict and non-strict inequalities. Such inequalities are handled by means of a semiring of germs (encoding infinitesimal perturbations). We develop a tropical analogue of Fourier-Motzkin elimination from which we derive geometrical properties of these polyhedra. In particular, we show that they coincide with the tropically convex union of (non-necessarily closed) cells that are convex both classically and tropically. We also prove that the redundant inequalities produced when performing successive elimination steps can be dynamically deleted by reduction to mean payoff game problems. As a complement, we provide a coarser (polynomial time) deletion procedure which is enough to arrive at a simply exponential bound for the total execution time. These algorithms are illustrated by an application to real-time systems (reachability analysis of timed automata). Fil: Allamigeon, Xavier. École Polytechnique. French National Institute for Computer Science and Applied Mathematics; Francia. Ecole Polytechnique. Centre de Mathematiques Appliquees; FranciaFil: Fahrenberg, Uli. Institut National de Recherche en Informatique et en Automatique; FranciaFil: Gaubert, Stéphane. Ecole Polytechnique. Centre de Mathematiques Appliquees; Francia. École Polytechnique. French National Institute for Computer Science and Applied Mathematics; FranciaFil: Katz, Ricardo David. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y Sistemas; ArgentinaFil: Legay, Axel. Institut National de Recherche en Informatique et en Automatique; Franci