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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 prove the existence of δcomplete decision procedures for bounded SMT over reals with functions mentioned above. For functions in Type 2 complexity class C, under mild assumptions, the bounded δ-SMT problem is in NP C . This stands in sharp contrast to the well-known undecidability results. δ-Complete decision procedures can exploit scalable numerical methods for handling nonlinearity, and we propose to use this notion 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 (ICP) 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. IntroductionGiven a first-order signature L and a structure M, the Satisfiability Modulo Theories (SMT) problem asks whether a quantifier-free L-formula is satisfiable over M, or equivalently, whether an existential L-sentence is true in M. Solvers for SMT problems have become the key enabling technology in formal verification and related areas. SMT problems over the real numbers are of particular interest, because of their importance in verification and design of hybrid systems, as well as in theorem proving. While efficient algorithms [10] exist for deciding SMT problems with only linear real arithmetic, practical problems normally ⋆

Given any collection F of computable functions over the reals, we show that there exists an algorithm that, given any sentence A containing only bounded quantifiers and functions in F, and any positive rational number delta, decides either "A is true", or "a delta-strengthening of A is false". Moreover, if F can be computed in complexity class C, then under mild assumptions, this "delta-decision problem" for bounded Sigma k-sentences resides in Sigma k(C). The results stand in sharp contrast to the well-known undecidability of the general first-order theories with these functions, and serve as a theoretical basis for the use of numerical methods in decision procedures for formulas over the reals.Index Terms-Decision procedures, first-order theories over the reals, computable analysis.

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Abstract. We describe a DPLL-based solver for the problem of quantified boolean formulas (QBF) in non-prenex, non-CNF form. We make two contributions. First, we reformulate clause/cube learning, extending it to non-prenex instances. We call the resulting technique game-state learning. Second, we introduce a propagation technique using ghost literals that exploits the structure of a non-CNF instance in a manner that is symmetric between the universal and existential variables. Experimental results on the QBFLIB benchmarks indicate our approach outperforms other state-of-the-art solvers on certain benchmark families, including the tipfixpoint and tipdiam families of model checking problems.

We propose new methods for learning control policies and neural network Lyapunov functions for nonlinear control problems, with provable guarantee of stability. The framework consists of a learner that attempts to find the control and Lyapunov functions, and a falsifier that finds counterexamples to quickly guide the learner towards solutions. The procedure terminates when no counterexample is found by the falsifier, in which case the controlled nonlinear system is provably stable. The approach significantly simplifies the process of Lyapunov control design, provides end-to-end correctness guarantee, and can obtain much larger regions of attraction than existing methods such as LQR and SOS/SDP. We show experiments on how the new methods obtain high-quality solutions for challenging control problems.

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