Upper and Lower Bounds on Sizes of Finite Bisimulations of Pfaffian Hybrid Systems

In this paper we study Pfaffian hybrid systems which were first introduced in [9]. Pfaffian hybrid systems are a sub-class of o-minimal hybrid systems which capture rich continuous dynamics and yet can be studied using finite bisimulations. The existence of finite bisimulations for ominimal hybrid systems has been shown by several authors (see e.g. [2, 3, 11]). The next natural question to investigate is how the sizes of such bisimulations can be bounded. First step in this direction was done in [9] where a double exponential upper bound was shown for Pfaffian hybrid systems. In this paper we improve this bound to a single exponential upper bound. Moreover we show that this bound is tight in general, by exhibiting a parameterized class of Pfaffian hybrid systems on which the exponential bound is attained. The bounds provide a basis for designing efficient algorithms for computing bisimulations, solving reachability and motion planning problems.