## Technical Report Softness of Hypercoherences and MALL Full Completeness (2003)

### BibTeX

@MISC{Blute03technicalreport,

author = {Richard Blute and Masahiro Hamano and Philip Scott},

title = {Technical Report Softness of Hypercoherences and MALL Full Completeness},

year = {2003}

}

### OpenURL

### Abstract

We prove a full completeness theorem for multiplicative-additive linear logic (i.e. MALL) using a double gluing construction applied to Ehrhard’s ∗-autonomous category of hypercoherences. This is the first non-game-theoretic full completeness theorem for this fragment. Our main result is that every dinatural transformation between definable functors arises from the denotation of a cut-free MALL proof. Our proof consists of three steps. We show: • Dinatural transformations on this category satisfy Joyal’s softness property for products and coproducts. • Softness, together with multiplicative full completeness, guarantees that every dinatural transformation corresponds to a Girard MALL proof-structure. • The proof-structure associated to any dinatural transformation is a MALL proofnet, hence a denotation of a proof. This last step involves a detailed study of cycles in additive proof structures.