Abstract. We present a labelled sequent calculus for Boolean BI (BBI), a classical variant of the logic of Bunched Implication. The calculus is simple, sound, complete, and enjoys cut-elimination. We show that all the structural rules in the calculus, i.e., those rules that manipulate labels and ternary relations, can be localised around applications of certain logical rules, thereby localising the handling of these rules in proof search. Based on this, we demonstrate a free variable calculus that deals with the structural rules lazily in a constraint system. We propose a heuristic method to quickly solve certain constraints, and show some experimental results to confirm that our approach is feasible for proof search. Additionally, we show that different semantics for BBI and some axioms in concrete models can be captured by adding extra structural rules.

This paper considers Reynolds's separation logic with all logical connectives but without arbitrary predicates. This logic is not recursively enumerable but is very useful in practice. We give a sound labelled sequent calculus for this logic. Using numerous examples, we illustrate the subtle deficiencies of several existing proof calculi for separation logic, and show that our rules repair these deficiencies. We extend the calculus with rules for linked lists and binary trees, giving a sound, complete and terminating proof system for a popular fragment called symbolic heaps. Our prover has comparable performance to Smallfoot, a prover dedicated to symbolic heaps, on valid formulae extracted from program verification examples; but our prover is not competitive on invalid formulae. We also show the ability of our prover beyond symbolic heaps, our prover handles the largest fragment of logical connectives in separation logic.

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 Hóu et al to handle Calcagno et al's 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 and cut-elimination. We present a theorem prover based on our labelled calculus for these logics.

Abstract. We present a labelled sequent calculus for Boolean BI (BBI), a classical variant of the logic of Bunched Implication. The calculus is simple, sound, complete, and enjoys cut-elimination. We show that all the structural rules in the calculus, i.e., those rules that manipulate labels and ternary relations, can be localised around applications of certain logical rules, thereby localising the handling of these rules in proof search. Based on this, we demonstrate a free variable calculus that deals with the structural rules lazily in a constraint system. We propose a heuristic method to quickly solve certain constraints, and show some experimental results to confirm that our approach is feasible for proof search. Additionally, we show that different semantics for BBI and some axioms in concrete models can be captured by adding extra structural rules.

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The SPARCv8 instruction set architecture (ISA) has been used in various processors for workstations, embedded systems, and space missions. However, there are no publicly available formal models for the SPARCv8 ISA. In this work, we give the first formal model for the integer unit of SPARCv8 ISA in Isabelle/HOL. We capture the operational semantics of the instructions using monadic definitions. Our model is a detailed model, which covers many features specific to SPARC processors, such as delayed-write for control registers, windowed general registers, and more complex memory access. Our model is also general, as we retain an abstract layer of the model which allows it to be instantiated to support all SPARCv8 compliant processors. We extract executable code from our formalisation, giving us the first systematically verified executable semantics for the SPARCv8 ISA. We have tested our model extensively against a LEON3 simulation board, covering both single-step executions and sequential execution of programs. We prove some important properties for our formal model, including a non-interference property for the LEON3 processor.

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