Separation logics are a family of extensions of Hoare logic for reasoning about programs that mutate memory. These logics are abstract because they are independent of any particular concrete memory model. Their assertion languages, called propositional abstract separation logics, extend the logic of (Boolean) Bunched Implications (BBI) in various ways. We develop a modular proof theory for various propositional abstract separation logics using cut-free labelled sequent calculi. We first extend the cut-fee labelled sequent calculus for BBI of Hou et al to handle Calcagno et als original logic of separation algebras by adding sound rules for partial-determinism and cancellativity, while preserving cut-elimination. We prove the completeness of our calculus via a sound intermediate calculus that enables us to construct counter-models from the failure to find a proof. We then capture other propositional abstract separation logics by adding sound rules for indivisible unit and disjointness, while maintaining completeness. We present a theorem prover based on our labelled calculus for these propositional abstract separation logics.
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