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Relational semantics for full linear logic - Radboud Universiteit

Relational semantics for full linear logic - Radboud Universiteit

Relational semantics for full linear logic - Radboud

Relational semantics for full linear logic Dion Coumans IMAPP, Radboud Universiteit Nijmegen, the Netherlands coumans@math.ru.nl Mai Gehrke LIAFA, Université Paris Diderot - Paris 7 and CNRS, France Mai.Gehrke@liafa.jussieu.fr Lorijn van Rooijen LaBRI, Université Bordeaux 1, France lorijn.vanrooijen@labri.fr February 24, 2012 Abstract Relational semantics, given by Kripke frames, play an essential role in the study of modal and intuitionistic logic. In [DGP05] 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 [Geh06] a type of frames is described which generalise Kripke frames and provide semantics for substructural logics in a purely relational form. In this paper we study full linear logic from an algebraic point of view. The main additional hurdle is the exponential. We analyse this operation algebraically and use canonical extensions to obtain relational semantics. Thus, we extend the work in [DGP05, Geh06] and use their approach to obtain relational semantics for full linear logic. Hereby we illustrate the strength of using canonical extension to retrieve relational semantics: it allows a modular and uniform treatment of additional operations and axioms. Traditionally, so-called phase semantics are used as models for (provability in) linear logic [Gir87]. 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. 1 Introduction Relational semantics, given by Kripke frames, play an essential role in the study of modal and intuitionistic logic [BdRV01]. They provide an intuitive interpretation of the logic and a means to obtain information about it. The possibility of applying semantical techniques to obtain information about a logic motivates the search for relational semantics in a more general setting. Many logics are closely related to corresponding classes of algebraic structures which provide algebraic semantics for the logics. The algebras associated to classical modal logic are Boolean algebras with an additional operator (BAOs). Kripke frames arise naturally from the duality 1

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