Barrier Certificates Revisited

A barrier certificate can separate the state space of a considered hybrid system (HS) into safe and unsafe parts according to the safety property to be verified. Therefore this notion has been widely used in the verification of HSs. A stronger condition on barrier certificates means that less expressive barrier certificates can be synthesized. On the other hand, synthesizing more expressive barrier certificates often means high complexity. In [9], Kong et al considered how to relax the condition of barrier certificates while still keeping their convexity so that one can synthesize more expressive barrier certificates efficiently using semi-definite programming (SDP). In this paper, we first discuss how to relax the condition of barrier certificates in a general way, while still keeping their convexity. Particularly, one can then utilize different weaker conditions flexibly to synthesize different kinds of barrier certificates with more expressiveness efficiently using SDP. These barriers give more opportunities to verify the considered system. We also show how to combine two functions together to form a combined barrier certificate in order to prove a safety property under consideration, whereas neither of them can be used as a barrier certificate separately, even according to any relaxed condition. Another contribution of this paper is that we discuss how to discover certificates from the general relaxed condition by SDP. In particular, we focus on how to avoid the unsoundness because of numeric error caused by SDP with symbolic checking.