A Noncommutative Full Completeness Theorem (Extended Abstract)

AbstractWe present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic known as cyclic linear logic (CyLL), first defined by Yetter. The semantics is obtained by considering dinatural transformations on a category of topological vector spaces which are invariant under certain actions of a noncocommutative Hopf algebra, called the shuffle algebra. Multiplicative sequents are assigned a vector space of such dinaturals, and we show that the space has the denotations of cut-free proofs in CyLL+MIX as a basis.This work is a natural extension of the authors' previous work, “Linear Läuchli Semantics”, where a similar theorem is obtained for the commutative logic. In that paper, we consider dinaturals which are invariant under certain actions of the additive group of integers. The passage from groups to Hopf algebras corresponds to the passage from commutative to noncommutative logic.This is an extended abstract. A full version of this paper, with complete proofs, is in preparation