Modular Labelled Sequent Calculi for Abstract Separation Logics

Abstract

separation logics are a family of extensions of Hoare logic for reasoning about programs that manipulate resources such as memory locations. These logics are "abstract" because they are independent of any particular concrete resource model. Their assertion languages, called propositional abstract separation logics (PASLs), extend the logic of (Boolean) Bunched Implications (BBI) in various ways. In particular, these logics contain the connectives * and − * , denoting the composition and extension of resources respectively. This added expressive power comes at a price since the resulting logics are all undecidable. Given their wide applicability, even a semi-decision procedure for these logics is desirable. Although several PASLs and their relationships with BBI are discussed in the literature, the proof theory of, and automated reasoning for, these logics were open problems solved by the conference version of this paper, which developed a modular proof theory for various PASLs using cut-free labelled sequent calculi. This paper non-trivially improves upon this previous work by giving a general framework of calculi on which any new axiom in the logic satisfying a certain form corresponds to an inference rule in our framework, and the completeness proof is generalised to consider such axioms.Our base calculus handles Calcagno et al. 's original logic of separation algebras by adding sound rules for partial-determinism and cancellativity, while preserving cut-elimination. We then show that many important properties in separation logic, such as indivisible unit, disjointness, splittability, and cross-split, can be expressed in our general axiom form. Thus our framework offers inference rules and completeness for these properties for free. Finally, we show how our calculi reduce to calculi with global label substitutions, enabling more efficient implementation. emp in some literature), and separating implication − * , also called magic wand, from the logic of Bunched Implications (BI) [50]. Moreover, the assertion language introduces the points-to predicate E → E ′ on expressions, along with the usual quantifiers and predicates of first-order logic with equality and arithmetic. The additive connectives may be either intuitionistic, as for BI, or classical, as for the logic of Boolean Bunched Implications (BBI). Classical additives are more expressive as they support reasoning about non-monotonic commands such as memory de-allocation, and assertions such as "the heap is empty" [36]. In this paper we consider classical additives only.The concrete memory model for SL is given in terms of heaps, where a heap is a finite partial function from addresses to values. A heap satisfies P * Q iff it can be partitioned into heaps satisfying P and Q respectively; it satisfies ⊤ * iff it is empty; it satisfies P − * Q iff any extension with a heap that satisfies P must then satisfy Q; and it satisfies E → E ′ iff it is a singleton map sending the address specified by the expression E to the value specified by the expression E ′ . Wh...

A Complete Axiomatisation for Quantifier-Free Separation Logic

“…Besides, in [HCGT18] labelled sequent calculi are designed for several abstract separation logics by considering different sets of properties. The sequents contain labelled formulae (a formula prefixed by a label to be interpreted as an abstract heap) as well as relational atoms to express relationships between abstract heaps.…”

“…The sequents contain labelled formulae (a formula prefixed by a label to be interpreted as an abstract heap) as well as relational atoms to express relationships between abstract heaps. Though the framework in [HCGT18] is modular and very general to handle abstract separation logics, it is not tailored to separation logics with concrete semantics, see [HCGT18, Section 7] for possible future directions. In contrast, as explained already, the paper [HGT15] deals with first-order separation logic with concrete semantics and presents a sound labelled sequent calculus for it.…”

“…Of course, the calculus cannot be complete but more importantly in the context of the current paper, completeness is not established for the quantifier-free fragment. In [HGT15], the sequents contain labelled formulae and relational atoms, similarly to [HCGT18] (see also [Hóu15]). Hence, this does not meet our requirements to have a pure axiomatisation in which only logical formulae from quantifier-free separation logic are allowed.…”

“…[ABM01]) extends substantially the object language. Other frameworks to axiomatise classes of abstract separation logics can be found in [DP18,Doc19] and in [HCGT18], respectively with labelled tableaux calculi and with sequent-style proof systems.…”

“…Aiming at internal calculi is a non-trivial task as the general frameworks for abstract separation logics make use of labels, see e.g. [DP18,HCGT18]. We cannot rely on label-free calculi for BI, see e.g.…”