Abstract
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 theorems for the structures that interpret bunched logics, starting with the weakest systems, recovering the familiar Boolean BI, and concluding with Separation Logic. Our results encompass all the known existing algebraic approaches to Separation Logic and prove them sound with respect to the standard store-heap semantics. We additionally recover soundness and completeness theorems of the specific truth-functional models of these logics as presented in the literature. This approach synthesises a variety of techniques from modal, substructural and categorical logic and contextualises 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 resource frame (generalizing the standard model of Separation Logic) and its associated completeness proof can easily be adapted to other non-classical predicate logics.
Highlights
Bunched logics, beginning with O’Hearn and Pym’s BI [36], have proved to be exceptionally useful tools in modelling and reasoning about computational and information-theoretic phenomena such as resources, the structure of complex systems, and access control [14,15,23]
We show that the semantics on hyperdoctrines and indexed resource frames are equivalent and strengthen this relationship to a dual equivalence of categories
We have given a systematic treatment of Stone-type duality for the structures that interpret bunched logics, starting with the weakest systems, recovering the familiar Boolean BI (BBI), and concluding with Separation Logic
Summary
Bunched logics, beginning with O’Hearn and Pym’s BI [36], have proved to be exceptionally useful tools in modelling and reasoning about computational and information-theoretic phenomena such as resources, the structure of complex systems, and access control [14,15,23]. From the logical point of view, Stone-type dualities strengthen the semantic equivalence of truth-functional (such as BI’s resource semantics or Kripke’s semantics for intuitionistic logic) and algebraic (such as BI algebras or Heyting algebras) models to a dual equivalence of categories We give a systematic account of resource semantics via a family of Stone-type duality theorems that encompass the range of systems from the layered graph logics, via Boolean BI, to Separation Logic. All of the structures given in existing algebraic approaches to Separation Logic — including [13], [24], [21], [8] and [25] — are instances of the structures used in the present work These approaches are all proved sound with respect to the standard semantics on store-heap pairs by the results of this paper. Proofs of the main results of the paper can be found in an extended research note [22]
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