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.

翻译：暂无翻译