Proof-theoretic Semantics for the Logic of Bunched Implications

The logic of bunched implications (BI) can be seen as the free combination of intuitionistic propositional logic (IPL) and intuitionistic multiplicative linear logic (IMLL). We present here a base-extension semantics (B-eS) for BI in the spirit of Sandqvist's B-eS for IPL, deferring an analysis of proof-theoretic validity, in the sense of Dummett and Prawitz, to another occasion. Essential to BI's formulation in proof-theoretic terms is the concept of a `bunch' of hypotheses that is familiar from relevance logic. Bunches amount to trees whose internal vertices are labelled with either the IMLL context-former or the IPL context-former and whose leaves are labelled with propositions or units for the context-formers. This structure presents significant technical challenges in setting up a base-extension semantics for BI. Our approach starts from the B-eS for IPL and the B-eS for IMLL and provides a systematic combination. Such a combination requires that base rules carry bunched structure, and so requires a more complex notion of derivability in a base and a correspondingly richer notion of support in a base. One reason why BI is a substructural logic of interest is that the `resource interpretation' of its semantics, given in terms of sharing and separation and which gives rise to Separation Logic in the field of program verification, is quite distinct from the `number-of-uses' reading of the propositions of linear logic as resources. This resource reading of BI provides useful intuitions in the formulation of its proof-theoretic semantics. We discuss a simple example of the use of the given B-eS in security modelling.