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A Fresh Look at Separation Algebras and Share Accounting ⋆
"... Abstract. Separation Algebras serve as models of Separation Logics; Share Accounting allows reasoning about concurrentread/exclusivewrite resources in Separation Logic. In designing a Concurrent Separation Logic and in mechanizing proofs of its soundness, we found previous axiomatizations of separ ..."
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Cited by 26 (8 self)
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Abstract. Separation Algebras serve as models of Separation Logics; Share Accounting allows reasoning about concurrentread/exclusivewrite resources in Separation Logic. In designing a Concurrent Separation Logic and in mechanizing proofs of its soundness, we found previous axiomatizations of separation algebras and previous systems of share accounting to be useful but imperfect. We adjust the axioms of separation algebras; we demonstrate an operator calculus for constructing new separation algebras; we present a more powerful system of share accounting with a new, simple model; and we provide a reusable Coq development. 1
Undecidability of propositional separation logic and its neighbours
"... Separation logic has proven an adequate formalism for the analysis of programs that manipulate memory (in the form of pointers, heaps, stacks, etc.). In this paper, we consider the purely propositional fragment of separation logic, as well as a number of closely related substructural logical systems ..."
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Cited by 8 (4 self)
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Separation logic has proven an adequate formalism for the analysis of programs that manipulate memory (in the form of pointers, heaps, stacks, etc.). In this paper, we consider the purely propositional fragment of separation logic, as well as a number of closely related substructural logical systems. We show that, surprisingly, all of these propositional logics are undecidable. In particular, we solve an open problem by establishing the undecidability of Boolean BI. 1
A unified display proof theory for bunched logic
 in Proceedings of MFPS26
"... We formulate a unified display calculus proof theory for the four principal varieties of bunched logic by combining display calculi for their component logics. Our calculi satisfy cutelimination, and are sound and complete with respect to their standard presentations. We show that the standard sequ ..."
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Cited by 6 (3 self)
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We formulate a unified display calculus proof theory for the four principal varieties of bunched logic by combining display calculi for their component logics. Our calculi satisfy cutelimination, and are sound and complete with respect to their standard presentations. We show that the standard sequent calculus for BI can be seen as a reformulation of its display calculus, and argue that analogous sequent calculi for the other varieties of bunched logic seem very unlikely to exist.
A cutfree proof theory for Boolean BI (via display logic). Unpublished note; available from http://www.doc.ic.ac. uk/~jbrother
, 2009
"... We give a display calculus proof system for Boolean BI (BBI) based on Belnap’s general display logic. We show that cutelimination holds in our system and that it is sound and complete with respect to the usual notion of validity for BBI. We then show how to constrain proof search in the system (wit ..."
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Cited by 4 (1 self)
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We give a display calculus proof system for Boolean BI (BBI) based on Belnap’s general display logic. We show that cutelimination holds in our system and that it is sound and complete with respect to the usual notion of validity for BBI. We then show how to constrain proof search in the system (without loss of generality) by means of a series of proof transformations. By doing so, we gain some insight into the problem of decidability for BBI. 1
Bunched Logics Displayed
, 2010
"... We formulate a unified display calculus proof theory for the four principal varieties of bunched logic by combining display calculi for their component logics. Our calculi satisfy cutelimination, and are sound and complete with respect to their standard presentations. We show how to constrain app ..."
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Cited by 2 (1 self)
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We formulate a unified display calculus proof theory for the four principal varieties of bunched logic by combining display calculi for their component logics. Our calculi satisfy cutelimination, and are sound and complete with respect to their standard presentations. We show how to constrain applications of displayequivalence in our calculi in such a way that an exhaustive proof search needbe only finitely branching, and establish a full deduction theorem for the bunched logics with classical additives, BBI and CBI. We also show that the standard sequent calculus for BI can be seen as a reformulation of its display calculus, and argue that analogous sequent calculi for the other varieties of bunched logic are very unlikely to exist.
CLASSICAL BI: ITS SEMANTICS AND PROOF THEORY
"... Abstract. We present Classical BI (CBI), a new addition to the family of bunched logics which originates in O’Hearn and Pym’s logic of bunched implications BI. CBI differs from existing bunched logics in that its multiplicative connectives behave classically rather than intuitionistically (including ..."
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Cited by 2 (2 self)
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Abstract. We present Classical BI (CBI), a new addition to the family of bunched logics which originates in O’Hearn and Pym’s logic of bunched implications BI. CBI differs from existing bunched logics in that its multiplicative connectives behave classically rather than intuitionistically (including in particular a multiplicative version of classical negation). At the semantic level, CBIformulas have the normal bunched logic reading as declarative statements about resources, but its resource models necessarily feature more structure than those for other bunched logics; principally, they satisfy the requirement that every resource has a unique dual. At the prooftheoretic level, a very natural formalism for CBI is provided by a display calculus à la Belnap, which can be seen as a generalisation of the bunched sequent calculus for BI. In this paper we formulate the aforementioned model theory and proof theory for CBI, and prove some fundamental results about the logic, most notably completeness of the proof theory with respect to the semantics. 1.
Boolean BI is Decidable (via Display Logic)
"... We give a display calculus proof system for Boolean BI (BBI) based on Belnap’s general display logic. We show that cutelimination holds in our system and that it is sound and complete with respect to the usual notion of validity for BBI. We then demonstrate that proof search in the system can be re ..."
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We give a display calculus proof system for Boolean BI (BBI) based on Belnap’s general display logic. We show that cutelimination holds in our system and that it is sound and complete with respect to the usual notion of validity for BBI. We then demonstrate that proof search in the system can be restricted to a finitely bounded space (without loss of generality). Thus provability in our display calculus is decidable, and consequently so too is validity in BBI. 1.
1 Nondeterministic Phase Semantics and the Undecidability of Boolean BI
"... We solve the open problem of the decidability of Boolean BI logic (BBI), which can be considered as the “core ” of Separation and Spatial Logics. For this, we define a complete phase semantics suitable for BBI and characterize it as trivial phase semantics. We deduce an embedding between trivial pha ..."
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We solve the open problem of the decidability of Boolean BI logic (BBI), which can be considered as the “core ” of Separation and Spatial Logics. For this, we define a complete phase semantics suitable for BBI and characterize it as trivial phase semantics. We deduce an embedding between trivial phase semantics for intuitionistic linear logic (ILL) and Kripke semantics for BBI. We single out the elementary fragment of ILL which is both undecidable and complete for trivial phase semantics. Thus, we obtain the undecidability of BBI.
On Model Checking Boolean BI
"... Abstract. The logic of bunched implications (BI), introduced by O’Hearn and Pym, is a substructural logic which freely combines additive and multiplicative implications. Boolean BI (BBI) denotes BI with classical interpretation of additives and its model is the commutative monoid. We show that when ..."
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Abstract. The logic of bunched implications (BI), introduced by O’Hearn and Pym, is a substructural logic which freely combines additive and multiplicative implications. Boolean BI (BBI) denotes BI with classical interpretation of additives and its model is the commutative monoid. We show that when the monoid is finitely generated and propositions are recursively defined, or the monoid is infinitely generated and propositions are restricted to generator propositions, the model checking problem is undecidable. In the case of finitely related monoid and generator propositions, the model checking problem is EXPSPACEcomplete. 1