Foreword: Special issue on numerical software verification

Abstract

Numerical abstractions are key to software verification. The verification community has actively pursued technologies such as Presburger arithmetic, convex polyhedra (linear arithmetic), bit vectors, SAT solvers and quantifier elimination to handle numerical abstractions. However, the application of this technology to verification is spread across many communities including verification, programming languages, embedded systems and software engineering. This special issue on numerical software verification contains a variety of papers centered around applications to the theory and practice of numerical abstractions for software verification. The papers that appear in this issue are based on work presented by the authors at the first workshop on Numerical Software Verification, held jointly with CAV 2008 at Princeton, NJ. The paper titled “Why does Astree scale up?” by Cousot et al. presents many valuable details on some of the implementation details behind the static analysis tool Astree that have not appeared elsewhere. The Astree tool has been successfully applied inside Airbus to verify command and control software running on Airbus 380. The paper by Martel titled “Enhancing the implementation of mathematical formulas for fixed-point and floating-point arithmetics” discusses techniques for finding an ideal evaluation order for complex arithmetic formulas that yields the best numerical quality. Such techniques can be useful in embedded control software, wherein evaluation of such formulas under error prone fixed and floating-point arithmetics is key. The paper by Bagnara, Ricci and Zaffanella titled “Weakly relational shapes for numeric abstractions: improved algorithms and proofs of correctness” provides techniques for managing the gap between syntactic representations of arithmetic formulae in so-called weakly relational domains and the geometric objects that they represent. They provide algorithms so