On Completeness of Logical Relations for Monadic Types

TL;DR

This paper investigates the completeness of logical relations in the computational lambda-calculus, which is crucial for verifying security properties of software modeled with lambda-calculi, especially with imperative features.

Contribution

It provides a detailed study of the completeness of logical relations for monadic types within the computational lambda-calculus, extending previous work.

Findings

Logical relations are shown to be complete for certain monadic types.

The results facilitate proving contextual equivalence in security-related lambda-calculi.

The study advances formal verification methods for software with imperative features.

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 lambda-calculus. 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.