Program Verification by Using DISCOVERER

Abstract

Abstract. 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 has low efficiency for specific problems. To overcome the bottleneck of program verification with a symbolic approach, one has to combine special techniques with the general method. Based on the work of complete discrimination systems of polynomials [33,31], we invented new theories and algorithms [32,30,35] 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.

Recent advances in program verification through computer algebra

“…Also, in Ref. [25], we showed how to use DISCOVERER to improve the efficiency of the computation of reachable sets of linear hybrid systems; etc.…”

“…[37], Xu et al gave a method on how to compute reachable sets of rational eigenvalue linear hybrid systems with rational eigenvalue; in Ref. [25], we proposed an off-line approach to generate sufficient termination conditions for linear loops. Also, in Ref.…”

“…In Ref. [25], we further developed the work of Ref. [29] by calculating symbolic (sufficient) conditions for the termination of the class through computer algebra tool, DISCOVERER.…”

“…In fact, recently more and more mathematical theories are found to be quite useful in program verification. For example, the theory of Gröbner Base is successfully applied to invariant generation of programs [19][20][21]; The first-order quantifier elimination techniques, mainly based on cylindrical algebra decomposition [22,23], have been used in reachability computation of linear hybrid systems [24,25] and program verification [26], and so on. 1…”

Automatic discovery of non-linear ranking functions of loop programs

“…In other words, for some program that may have ranking functions of the predefined form, it can not find them. [6] reduced nonlinear ranking function discovering for polynomial programs to semi-algebraic system solving by means of the symbolic computation tools, DISCOVERER [17,18,19,20].…”