Bounds on sizes of finite bisimulations of pfaffian dynamical systems

Abstract

We study finite bisimulations of dynamical systems in ℝn defined by Pfaffian maps. The pure existence of finite bisimulations for a more general class of o-minimal systems was shown in Brihaye et al. (Lecture Notes in Comput. Sci. 2993, 219–233, 2004), Davoren (Theor. Inf. Appl. 33(4/5), 357–382, 1999), Lafferriere et al. (Math. Control Signals Syst. 13, 1–21, 2000). In Lecture Notes in Comput. Sci. 3210, 2004, the authors proved a double exponential upper bound on the size of a bisimulation in terms of the size of description of the dynamical system. In the present paper we improve it to a single exponential upper bound, and show that this bound is tight, by exhibiting a parameterized class of systems on which it is attained.