Stone-Type Dualities for Separation Logics

Stone-type duality theorems, which relate algebraic and relational/topological models, are important tools in logic because — in addition to elegant abstraction — they strengthen soundness and completeness to a categorical equivalence, yielding a framework through which both algebraic and topological methods can be brought to bear on a logic. We give a systematic treatment of Stone-type duality for the structures that interpret bunched logics, starting with the weakest systems, recovering the familiar BI and Boolean BI, and extending to both classical and intuitionistic Separation Logic. The structures we consider are the most general of all known existing algebraic approaches to Separation Logic and thus encompass them all. We demonstrate the uniformity of this analysis by additionally capturing the bunched logics obtained by extending BI and BBI with multiplicative connectives corresponding to disjunction, negation and falsum: De Morgan BI, Classical BI, and the sub-classical family of logics extending Bi-intuitionistic (B)BI. We additionally recover soundness and completeness theorems for the specific truth-functional models of these logics as presented in the literature, with new results given for DMBI, the sub-classical logics extending BiBI and a new bunched logic, CKBI, inspired by the interpretation of Concurrent Separation Logic in concurrent Kleene algebra. This approach synthesises a variety of techniques from modal, substructural and categorical logic and contextualizes the ‘resource semantics’ interpretation underpinning Separation Logic amongst them. As a consequence, theory from those fields — as well as algebraic and topological methods — can be applied to both Separation Logic and the systems of bunched logics it is built upon. Conversely, the notion of indexed frame (generalizing the standard memory model of Separation Logic) and its associated completeness proof can easily be adapted to other non-classical predicate logics.