Abstract
We introduce the notion of “δ-complete decision procedures” for solving SMT problems over the real numbers, with the aim of handling a wide range of nonlinear functions including transcendental functions and solutions of Lipschitz-continuous ODEs. Given an SMT problem ϕ and a positive rational number δ, a δ-complete decision procedure determines either that ϕ is unsatisfiable, or that the “δ-weakening” of ϕ is satisfiable. Here, the δ-weakening of ϕ is a variant of ϕ that allows δ-bounded numerical perturbations on ϕ. We establish the existence and complexity of δ-complete decision procedures for bounded SMT over reals with functions mentioned above. We propose to use δ-completeness as an ideal requirement for numerically-driven decision procedures. As a concrete example, we formally analyze the DPLL〈ICP〉 framework, which integrates Interval Constraint Propagation in DPLL(T), and establish necessary and sufficient conditions for its δ-completeness. We discuss practical applications of δ-complete decision procedures for correctness-critical applications including formal verification and theorem proving.
This research was sponsored by the National Science Foundation grants no. DMS1068829, no. CNS0926181, and no. CNS0931985, the GSRC under contract no. 1041377 (Princeton University), the Semiconductor Research Corporation under contract no. 2005TJ1366, General Motors under contract no. GMCMUCRLNV301, and the Office of Naval Research under award no. N000141010188.
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Gao, S., Avigad, J., Clarke, E.M. (2012). δ-Complete Decision Procedures for Satisfiability over the Reals. In: Gramlich, B., Miller, D., Sattler, U. (eds) Automated Reasoning. IJCAR 2012. Lecture Notes in Computer Science(), vol 7364. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31365-3_23
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