On Completeness of Logical Relations for Monadic Types

Abstract

Abstract. Software security can be ensured by specifying and verifying security properties of software using formal methods with strong theoretical bases. In particular, programs can be modeled in the framework of lambda-calculi, and interesting properties can be expressed formally by contextual equivalence (a.k.a. observational equivalence). Furthermore, imperative features, which exist in most real-life software, can be nicely expressed in the so-called computational lambdacalculus. Contextual equivalence is difficult to prove directly, but we can often use logical relations as a tool to establish it in lambda-calculi. We have already defined logical relations for the computational lambda-calculus in previous work. We devote this paper to the study of their completeness w.r.t. contextual equivalence in the computational lambda-calculus.

Fully abstract models for effectful λ-calculi via category-theoretic logical relations

“…The closest work to our own is that of Goubault-Larrecq et al [2004], who show that a certain logical relation is sound and complete for contextual equivalence for a version of Moggi's monadic metalanguage with cryptographic primitives and name generation. This builds on previous work [Lasota et al 2007], which also relates the monadic lifting of Goubault-Larrecq et al [2008] to logical relations, and hence a form of full abstraction, but only for specific effects and types up to first-order.…”

“…We expect that, subject to natural assumptions on the monad and its underlying category, the definability predicate and contextual equivalence predicate (cf. Lasota et al [2007]; Power and Robinson [2000]) should both satisfy logical relations conditions, so that full abstraction at ground types lifts to all higher types. Second, generalising the construction of the hull functor, perhaps using the comprehension categories or subset types of Jacobs [1993Jacobs [ , 1999.…”

Program equivalence in linear contexts

“…But it is known that completeness of (strict) logical relations are often hard to achieve, especially for higher-order types. It is even worse for monadic types, particularly when non-determinism is present [17].…”