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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)

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.

Keywords

Model Check Decision Procedure Computable Function Interval Extension Bound Model Check 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© 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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