Abstract
AbstractRelational program verification is a variant of program verification where one can reason about two programs and as a special case about two executions of a single program on different inputs. Relational program verification can be used for reasoning about a broad range of properties, including equivalence and refinement, and specialized notions such as continuity, information flow security, or relative cost. In a higher-order setting, relational program verification can be achieved using relational refinement type systems, a form of refinement types where assertions have a relational interpretation. Relational refinement type systems excel at relating structurally equivalent terms but provide limited support for relating terms with very different structures. We present a logic, called relational higher-order logic (RHOL), for proving relational properties of a simply typed λ-calculus with inductive types and recursive definitions. RHOL retains the type-directed flavor of relational refinement type systems but achieves greater expressivity through rules which simultaneously reason about the two terms as well as rules which only contemplate one of the two terms. We show that RHOL has strong foundations, by proving an equivalence with higher-order logic, and leverage this equivalence to derive key meta-theoretical properties: subject reduction, admissibility of a transitivity rule, and set-theoretical soundness. Moreover, we define sound embeddings for several existing relational type systems such as relational refinement types and type systems for dependency analysis and relative cost, and we verify examples that were out of reach of prior work.
Highlights
Many important aspects of program behavior go beyond the traditional characterization of program properties as sets of traces [Alpern and Schneider 1985]
We show that Relational Higher Order Logic (RHOL) has strong foundations, by proving an equivalence with higher-order logic (HOL), and leverage this equivalence to derive key meta-theoretical properties: subject reduction, admissibility of a transitivity rule and set-theoretical soundness
Soundness of the logic can be proved through a standard model-theoretic argument; we provide an alternative proof based on a sound and complete embedding into Higher-Order Logic (HOL, ğ 3)
Summary
Many important aspects of program behavior go beyond the traditional characterization of program properties as sets of traces [Alpern and Schneider 1985]. Hyperproperties [Clarkson and Schneider 2008] generalize properties and capture a larger class of program behaviors, by focusing on sets of sets of traces. As an intermediate point in this space, relational properties are sets of pairs of traces. Relational properties encompass many properties of interest, including program equivalence and refinement, as well as more specific notions such as non-interference and continuity. Lang., Vol 1, No ICFP, Article 21.
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