A Stone-type Duality Theorem for Separation Logic Via its Underlying Bunched Logics

A categorical duality for algebras of partial functions

“…In representable {;, A, R, }-algebras, the prime and maximal filters coincide. 5 Proof. Take first a prime filter P .…”

“…Variants and extensions of Stone duality are pervasive in logic and computer science: for example in modal [14], intuitionistic [8], substructural [7], and many-valued [13] logic, and in semantics [2], formal language theory [11], and logics for static analysis [5]. In its most basic form-between Boolean algebras and Stone spaces-it provides a duality for the isomorphs of algebras of unary relations equipped with the union and complement operations.…”

A substructural epistemic resource logic: theory and modelling applications

“…Given such structures, the logic BI of bunched implications -see, for example, [21,30,34,36] -which freely combines intuitionistic propositional additives with intuitionistic propositional multiplicatives -has its Kripke semantics given by the following satisfaction relation, where V is an interpretation of propositional letters in ℘(R), in the usual way: r |= p iff r ∈ V (p) r |= ⊥ never r |= always r |= ¬φ iff r |= φ r |= φ ∨ ψ iff r |= φ or r |= ψ r |= φ ∧ ψ iff r |= φ and r |= ψ r |= φ → ψ iff for all r s, if s |= φ, then s |= ψ r |= I iff e r r |= φ * ψ iff there exist r 1 , r 2 ∈ R s.t. r 1 • r 2 ↓, r r 1 • r 2 , and r 1 |= φ and r 2 |= ψ r |= φ − * ψ iff for all r ∈ R, if r • r ↓ and r |= φ, then r • r |= ψ This resource semantics for BI -that is, the interpretation of BI's semantics in terms of resources -underpins its applications to Separation Logic -and its family of derivatives; see [18,19] for an extensive discussion -and is mainly concerned with sharing and separation.…”

“…Let us note that ↓ denotes definedness of the composition. Given such structures, the logic BI of bunched implications -see, for example, [21,30,34,36] -which freely combines intuitionistic propositional additives with intuitionistic propositional multiplicatives -has its Kripke semantics given by the following satisfaction relation, where V is an interpretation of propositional letters in ℘(R), in the usual way: This resource semantics for BI -that is, the interpretation of BI's semantics in terms of resources -underpins its applications to Separation Logic -and its family of derivatives; see [18,19] for an extensive discussion -and is mainly concerned with sharing and separation.…”

“…u φ : x) ∈ F , then ∀y ∈ Λ r , xλ(r) u y ∈ C implies (Tφ : y) ∈ F19. If (FL r u φ : x) ∈ F , then ∃y ∈ Λ r , xλ(r) u y ∈ C and (Fφ : y) ∈ F 20.…”