Exploring the relation between Intuitionistic BI and Boolean BI: an unexpected embedding

Abstract

The logic of Bunched Implications, through both its intuitionistic version (BI) and one of its classical versions, called BooleanBI(BBI), serves as a logical basis to spatial or separation logic frameworks. InBI, the logical implication is interpreted intuitionistically whereas it is generally interpreted classically in spatial or separation logics, as inBBI. In this paper, we aim to give some new insights into the semantic relations betweenBIandBBI. Then we propose a sound and complete syntactic constraints based framework for the Kripke semantics of bothBIandBBI, a sound labelled tableau proof system forBBI, and a representation theorem relating the syntactic models ofBIto those ofBBI. Finally, we deduce as our main, and unexpected, result, a sound and faithful embedding ofBIintoBBI.