Program Verification by Using DISCOVERER

Recent advances in program verification indicate that various verification problems can be reduced to semi-algebraic system (SAS for short) solving. An SAS consists of polynomial equations and polynomial inequalities. Algorithms for quantifier elimination of real closed fields are the general method for those problems. But the general method usually have low efficiency for specific problems. To overcome the bottleneck of program verification with symbolic approach, one has to combine special techniques with the general method. Based on the work of complete discrimination systems of polynomials [32, 30], we invented new theories and algorithms [31, 29, 33] for SAS solving and partly implemented them as a real symbolic computation tool in Maple named DISCOVERER. In this paper, we first summarize the results that we have done so far both on SAS-solving and program verification with DISCOVERER, and then discuss the future work in this direction, including SAS-solving itself, termination analysis and invariant generation of programs, and reachability computation of hybrid systems etc. K ̄ ey words: semi-algebraic systems, DISCOVERER, program verification, termination, invariant generation, reachability computation