Partial Cad I : the Lifting Phase ( Extended Technical Report )

PARTIAL CAD I: THE LIFTING PHASE (EXTENDED TECHNICAL REPORT) GRANT OLNEY PASSMORE AND PAUL B. JACKSON LFCS, Edinburgh and Clare Hall, Cambridge, 10 Crichton Street, Edinburgh EH8 9AB, UK e-mail address: [email protected] LFCS, Edinburgh, 10 Crichton Street, Edinburgh EH8 9AB, UK e-mail address: [email protected] Abstract. Though decidable, the theory of real closed fields (RCF) is fundamentally Though decidable, the theory of real closed fields (RCF) is fundamentally infeasible. This is unfortunate, as automatic proof methods for nonlinear real arithmetic are crucially needed in both formalised mathematics and the verification of real-world cyber-physical systems. Consequently, many researchers have proposed fast, sound but incomplete RCF proof procedures which are useful in various practical applications. We show how such practically useful, sound but incomplete RCF proof methods may be systematically utilised in the context of a complete RCF proof method without sacrificing its completeness. In particular, we present an extension of the RCF quantifier elimination method Partial CAD (P-CAD) which uses incomplete ∃ RCF proof procedures to “short-circuit” expensive computations during the lifting phase of P-CAD. We present the theoretical framework as well as preliminary experiments with an implementation we have undertaken in the open-source computer algebra system SAGE. These experiments include the use of RealPaver, a high-performance interval constraint solver, to short-circuit expensive computations during P-CAD construction.