Conclusion

Abstract

Motivated by the need of solver technology for formal verification of hybrid discretecontinuous systems, we have investigated algorithms for solving formulae which are quantifier-free Boolean combinations of arithmetic constraints over the reals. Our contributions to the state of the art in the field are, in brief, as follows:

• We have demonstrated that acceleration techniques employed in modern propositional SAT solvers, in particular lazy clause evaluation, learning, and backjumping, generalize smoothly to DPLL-like procedures for solving conjunctions of pseudo-Boolean constraints, a much more succinct language for expressing Boolean functions than CNF.

• We have investigated how to efficiently couple a DPLL-based SAT solver with a linear programming routine in a DPLL(T) framework in order to obtain a solver which is tailored for BMC of hybrid systems with linear continuous dynamics. To this end, we took advantage of BMC-specific optimizations previously only employed in pure propositional solvers, and demonstrated that such optimizations, in particular sharing and isomorphic copying of conflict clauses, are even more effective in solvers with support for real-valued arithmetic, since the computational costs for conflict analysis are much higher in this domain.

• We have conceived a tight integration of the DPLL procedure for Boolean SAT solving with interval constraint solving. The resulting algorithm, called iSAT, generalizes the DPLL routine and is capable of solving Boolean combinations of nonlinear arithmetic constraints which may even involve transcendental functions. We have demonstrated that our approach can deal with formulae involving some ten thousands of Boolean and real-valued variables.