Parametric completeness for separation theories

Abstract

In this paper, we close the logical gap between provability in the logic BBI, which is the propositional basis for separation logic, and validity in an intended class of separation models, as employed in applications of separation logic such as program verification. An intended class of separation models is usually specified by a collection of axioms describing the specific model properties that are expected to hold, which we call a separation theory. Our main contributions are as follows. First, we show that several typical properties of separation theories are not definable in BBI. Second, we show that these properties become definable in a suitable hybrid extension of BBI, obtained by adding a theory of naming to BBI in the same way that hybrid logic extends normal modal logic. The binder-free extension HyBBI captures most of the properties we consider, and the full extension HyBBI(↓) with the usual ↓ binder of hybrid logic covers all these properties. Third, we present an axiomatic proof system for our hybrid logic whose extension with any set of "pure" axioms is sound and complete with respect to the models satisfying those axioms. As a corollary of this general result, we obtain, in a parametric manner, a sound and complete axiomatic proof system for any separation theory from our considered class. To the best of our knowledge, this class includes all separation theories appearing in the published literature.

The Effects of Adding Reachability Predicates in Propositional Separation Logic

“…A natural question is how to combine these extensions, and which separation logic fragment that allows Boolean connectives, magic wand and generalised recursive predicates can be decided with some adequate restrictions. As already advocated in [8,21,28,34,36], dealing with the separating implication − * is a desirable feature for program verification and several semi-automated or automated verification tools support it in some way, see e.g. [21,28,34,36].…”

Modular Labelled Sequent Calculi for Abstract Separation Logics

“…We run BBeye in an iterative deepening way, and the time counted for BBeye is the total time it spends. Formulae (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) are used by Park et al to test their prover BBeye for BBI [51]. We can see that for formulae (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) the performance of Separata is comparable with the heuristic based prover for FVLS BBI .…”

“…This also implies that h 2 = ϵ whenever h 1 • h 2 = ϵ. Indivisible unit can be axiomatised by the formula ⊤ * ∧ (A * B) → A [13].…”

“…Splittability can be axiomatised as the formula ¬⊤ * → (¬⊤ * * ¬⊤ * ) [13]. We give the following structural rule for this property with the side-condition that x, y do not occur in the conclusion:…”

“…However completeness does not follow so easily; in [35] the completeness of LS BBI was shown by mimicking derivations in the Hilbert axiomatisation of BBI. This avenue is no longer viable for PASL because partial-determinism and cancellativity are not axiomatisable in BBI [13]. That is, there can be no Hilbert calculus in the language of BBI which is sound and complete with respect to separation algebras.…”