Proofs of Unsatisfiability for mixed Boolean and Non-linear Arithmetic Constraint Formulae 1 Stefan Kupferschmid ∗ , Tino Teige † , Bernd Becker ∗ , Martin Fr¨ anzle † ∗ Albert-Ludwigs-Universit¨ at Freiburg im Breisgau, Germany {skupfers,becker}@informatik.uni-freiburg.de † Carl von Ossietzky Universit¨ at Oldenburg, Germany {teige,fraenzle}@informatik.uni-oldenburg.de Abstract. Symbolic methods in computer-aided verification rely on appropriate constraint solvers. Correctness and reliability of solvers are a vital requirement in the analysis of safety-critical sys- tems, e.g., in the automotive context. Satisfiability results of a solver can usually be checked by probing the computed solution. However, efficient validation of an uncertified unsatisfiability result for some constraint formula is nearly impossible. In this paper, we propose a certification method for unsatisfiability results for mixed Boolean and non-linear arithmetic constraint formu- lae. Such formulae arise in the analysis of hybrid discrete-continuous systems. 1 Introduction Over the last decade, computer-aided verification techniques have been intensively studied and are becoming increasingly accepted by the industry. Applications range from hardware verification of digital circuits to the analysis of embedded hybrid discrete-continuous systems in the car and avia- tion industry. Among the most successful approaches to the safety analysis of systems is bounded model checking (BMC) [9, 3]. The idea of BMC is to find errors in system designs. To do so, the system’s behavior for some bounded depth k of transitions and some safety property P are symboli- cally encoded as a formula ϕ k s.t. the system violates P within depth k iff ϕ k is satisfiable. To check the satisfiability of ϕ k , appropriate proof engines are employed, e.g. propositional satisfiability (SAT) solvers for the verification of finite-state systems, or satisfiability modulo theories (SMT) solvers (e.g., cf. [16, 15]) for systems involving continuous state subject to decidable arithmetic constraints like linear expressions or complex data-structures like arrays. To facilitate satisfiability checking also for mixed Boolean and non-linear arithmetic constraint formulae over the reals and integers, which naturally arise in the verification of hybrid discrete-continuous systems, the authors of [7] introduced the iSAT algorithm. If the constraint solver proves the satisfiability of the formula ϕ k then it usually delivers a solution of ϕ k , e.g. a satisfying value assignment to the variables of ϕ k . Thus, the BMC layer is able to verify the result of the solver by simply probing the suggested solution. However, if the solver returns that ϕ k is unsatisfiable without any certificate then the BMC layer cannot validate the result and has to trust correctness of the solver. This is dissatisfactory, as recent solvers are too complex pieces of software to guarantee correctness of the tool. However, guaranteed results are a necessity in the verification of industrial safety-critical systems. Thus, there is a vital industrial need to certify also the unsatisfiability of formulae. In this paper, we propose a simple rule-based calculus RC for unsatisfiable mixed Boolean and non-linear arithmetic constraint formulae ϕ over the reals and integers. Given an unsatisfiable formula ϕ, RC derives new implied clauses where deduction of the empty clause certifies unsatisfiability of ϕ. Then, all derived clauses connected with their implying clauses are a proof of unsatisfiability. To facilitate subsequent validations of unsatisfiability results of the iSAT tool [7], we enhance iSAT to produce unsatisfiability proofs. For validating the proofs, we implement an external proof checker 1 This work was partly supported by the German Research Council (DFG) as part of the Transregional Collabo- rative Research Center “Automatic Verification and Analysis of Complex Systems” (SFB/TR 14 AVACS). See www.avacs.org for more information.