Context logic as modal logic

Abstract

Separation Logic, Ambient Logic and Context Logic are based on a similar style of reasoning about structured data. They each consist of a structural (separating) composition for reasoning about disjoint subdata, and corresponding structural adjoint(s) for reasoning hypothetically about data. We show how to interpret these structural connectives as modalities in Modal Logic and prove completeness results. The structural connectives are essential for describing properties of the underlying data, such as weakest preconditions for Hoare reasoning for Separation and Context Logic, and security properties for Ambient Logic. In fact, we introduced Context Logic to reason about tree update, precisely because the structural connectives of the Ambient Logic did not have enough expressive power. Despite these connectives being essential, first Lozes then Dawar, Gardner and Ghelli proved elimination results for Separation Logic and Ambient Logic (without quantifiers). In this paper, we solve this apparent contradiction. We study parametric inexpressivity results, which demonstrate that the structural connectives are indeed fundamental for this style of reasoning.