δ-Complete Decision Procedures for Satisfiability over the Reals

  • Sicun Gao
  • Jeremy Avigad
  • Edmund M. Clarke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7364)


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.


Model Check Decision Procedure Computable Function Interval Extension Bound Model Check 
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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sicun Gao
    • 1
  • Jeremy Avigad
    • 1
  • Edmund M. Clarke
    • 1
  1. 1.Carnegie Mellon UniversityPittsburghUSA

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