Bi - intuitionistic Boolean Bunched Logic June , 2014

We formulate and investigate a bi-intuitionistic extension, BiBBI, of the well known bunched logic Boolean BI (BBI), obtained by combining classical logic with full intuitionistic linear logic as considered by Hyland and de Paiva (as opposed to standard multiplicative intuitionistic linear logic). Thus, in addition to the multiplicative conjunction ∗ with its adjoint implication —∗ and unit ⊤∗, which are provided by BBI, our logic also features an intuitionistic multiplicative disjunction ∗ ∨, with its adjoint co-implication ∗ \ and unit ⊥∗. “Intuitionism” for the multiplicatives means here that disjunction and conjunction are related by a weak distribution principle, rather than by De Morgan equivalence. We formulate a Kripke semantics for BiBBI in which all the above multiplicatives are given an intuitionistic reading in terms of resource operations. Our main theoretical result is that validity according to this semantics exactly coincides with provability in our logic, given by a standard Hilbert-style axiomatic proof system. In particular, we isolate the Kripke frame conditions corresponding to various natural logical principles of FILL, which allows us to present soundness and completeness results that are modular with respect to the inclusion or otherwise of these axioms in the logic. Completeness follows by embedding BiBBI into a suitable modal logic and employing the famous Sahlqvist completeness theorem. We also investigate the Kripke models of BiBBI in some detail, chiefly in the hope that BiBBI might be used (like BBI) to underpin program verification applications based on separation logic. Interestingly, it turns out that the heap-like memory models of separation logic are also models of BiBBI, in which disjunction can be interpreted using a natural notion of heap intersection.