Towards a Cut-free Sequent Calculus for Boolean BI

The logic of bunched implications (BI) of O’Hearn and Pym [5] is a substructural logic which freely combines additive connectives ⊃ , ∧, ∨ from propositional logic and multiplicative connectives −?, ? from linear logic. Because of its concise yet rich representation of states of resources, BI is regarded as a logic suitable for reasoning about resources. For example, by building a model for BI based on a monoid of heaps, we obtain separation logic [7] which extends Hoare logic to facilitate reasoning about imperative programs manipulating heap memory. Depending on its interpretation of additive connectives, BI comes with two flavors: intuitionistic BI and boolean BI. Intuitionistic BI interprets additive connectives intuitionistically, while boolean BI interprets additive connectives classically and admits such principles as the law of excluded middle. Both logics interpret multiplicative connectives intuitionistically and do not introduce multiplicative falsity or negation. Intuitionistic BI has a well-developed proof theory. It has a natural deduction system with the normalization property and also a cut-free sequent calculus. Because free combinations of additive connectives and multiplicative connectives are allowed, contexts in such proof-theoretic formulations are not unordered sets as usual, but bunches: trees whose internal nodes indicate whether subtrees are combined additively or multiplicatively. For example, the sequent calculus uses a sequent of the form Γ −→ A where Γ denotes a bunch and A is a formula. For boolean BI, however, no such well-behaved proof theory exists. Usually a naive extension of intuitionistic BI with the double negation principle is taken as the proof-theoretic definition of boolean BI, but no equivalent cut-free sequent calculus has been reported yet. Following the style of multiconclusioned sequent calculi, Pym [6] considers sequents of the form Γ −→ Γ (where both sides use bunches), but it is shown that the cut elimination property fails in the resultant sequent calculus. As such, previous work on boolean BI has mostly focused on its Kripke semantics and Hilbert-style system [3, 4] or a different style of proof theory based on display logic [1]. Our goal is to develop a proof-theoretic formulation for building an automated theorem prover for boolean BI (and ultimately for separation logic). Ideally a cut-free sequent calculus for boolean BI would be the best candidate, but such a sequent calculus is unlikely to exist, as already observed by Brotherston [2]. Following the conjecture that such a sequent calculus does not exist, we aim to develop a cut-free sequent calculus for another variant of BI that may be incompatible with boolean BI (in the sense that it proves some non-theorems in boolean BI and fails to prove some theorems in boolean BI), but still interprets all additive connectives classically and all multiplicative connectives intuitionistically. By identifying such a variant of boolean BI for which a cut-free sequent calculus exists, we may also be able to better understand why boolean BI is unlikely to have a well-behaved proof theory. The main obstacle is to identify a form of sequent such that additive connectives are interpreted classically (thereby admitting the law of excluded middle) while multiplicative connectives are interpreted intuitionistically. We find that sequents of the form Γ −→ A or Γ −→ Γ are inadequate, and introduce a new form of sequent, called world sequent, which expresses logical inconsistency in a tree-like structure of worlds. In order to interpret additive connectives classically, each world maintains not only true statements but also false statements.