Relational semantics for a fragment of linear logic

Relational semantics, given by Kripke frames, play an essential role in the study of modal and intuitionistic logic. In [4] it is shown that the theory of relational semantics is also available in the more general setting of substructural logic, at least in an algebraic guise. Building on these ideas, in [5] a type of frames is described which generalize Kripke frames and provide semantics for substructural logics in a purely relational form. We will extend the work in [4, 5] and use their approach to obtain relational semantics for multiplicative additive linear logic. Hereby we illustrate the strength of using canonical extensions to retrieve relational semantics: it allows a modular and uniform treatment of additional operations and axioms. Traditionally, so-called phase spaces are used to describe semantics for linear logic [8]. These have the drawback that, contrary to our approach, they do not allow a modular treatment of additional axioms. However, the two approaches are related, as we will explain.