Synthesizing Basic Semialgebraic Invariants by Semidefinite Programming

Invariant generation plays a central role in the verification of programs and hybrid systems. In this paper, we propose an approach to synthesize basic semialgebraic invariants using semidefinite programming (SDP) that combine advantages of both symbolic constraint solving and numeric constraint solving. The advantages of our approach is threefold: first, it is powerful enough to be applied to arbitrary templates as symbolic computation based techniques; second, it uses semidefinite programming instead of time-consuming symbolic subroutines and is therefore efficient enough as other numeric computation based techniques; lastly, there are some (although weaker) theoretical guarantees of completeness of it, which previously can only be provided by symbolic computation based techniques. In addition, we discuss how to generalize our approach to the case when templates are semialgebraic and the case when non-polynomial functions such as trigonometric functions, logarithmic functions, exponential functions, rational functions are present in programs.

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