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 Boolean BI (BBI), serves as a logical basis to spatial or separation logic frameworks. In BI, the logical implication is interpreted intuitionistically whereas it is generally interpreted classically in spatial or separation logics, as in BBI. In this paper, we aim to give some new insights into the semantic relations between BI and BBI. Then we propose a sound and complete syntactic constraints based framework for the Kripke semantics of both BI and BBI, a sound labelled tableau proof system for BBI, and a representation theorem relating the syntactic models of BI to those of BBI. Finally, we deduce as our main, and unexpected, result, a sound and faithful embedding of BI into BBI.

The Undecidability of Boolean BI through Phase Semantics

“…Given a non-deterministic monoid (M, •, ) and an interpretation δ : Var −→ P(M) of propositional variables, we define the Kripke forcing relation by induction on the structure of formulae: In some papers, you might find BBI defined by nondeterministic monoidal Kripke semantics [1,5], in other papers it is defined by partial but deterministic monoidal Kripke semantics and generally separation logic models are particular instances of partial (deterministic) monoids. See [15] for a general discussion about these issues. Definition 2.4.…”

“…BBI PD can be proved sound and complete w.r.t. the semantic constraints based tableaux proof-system presented in [15] (although only the soundness proof is presented in that particular paper) and the adaptation of this tableaux system to BBI TD should be straightforward (contrary to BBI ND ).…”

“…But for some fundamental logical reasons explained in this paper, his attempt was bound to fail. On the other hand, the authors recently obtained a sound and faithful embedding of BI into BBI (both defined with their partial deterministic Kripke semantics), illustrating the counter-intuitive fact that Boolean BI is surprisingly more expressive than intuitionistic BI [15]. The result is based on the study of the specific properties of the counter-models generated by proof-search in labelled tableaux systems.…”

“…The additives of BI behave either intuitionistically or classically giving rise to intuitionistic BI or Boolean BI (denoted BBI). The language of BI, and in particular its composition operators * and − * , is at the heart of separation and spatial logics frameworks (see [15] for a discussion on these aspects).…”

A theorem prover for Boolean BI

“…Larchey-Wendling and Gamliche [28] formulate a labelled tableau calculus by extending the labelled tableau calculus for intuitionistic BI in [21], but only in order to investigate the relation between intuitionistic BI and Boolean BI. Brotherston [10] shows that a modular combination of display calculi for classical logic and intuitionistic linear logic gives rise to a display calculus DLBBI for Boolean BI, the first cut-free syntactic formulation of Boolean BI, and proves the cut elimination property by observing that its rules obey all the syntactic constraints given in [1].…”