A General Proof System for Modalities in Concurrent Constraint Programming

The combination of timed, spatial, and epistemic information is often needed in the specification of modern concurrent systems. We propose the proof system SELL$^\Cap$, which extends linear logic with subexponentials with quantifiers over subexponentials, therefore allowing for an arbitrary number of modalities. We then show how a proper structure of the subexponential signature in SELL$^\Cap$ allows for the specification of concurrent systems with timed, spatial, and epistemic modalities. In the context of Concurrent Constraint Programming (CCP), a declarative model of concurrency, we illustrate how the view of subexponentials as specific modalities is general enough to modularly encode into SELL$^\Cap$ variants of CCP with these three modalities, thus providing a proof-theoretic foundations for those calculi.