Robert Seely - Semantics for various noncommutative linear logics

Since Lambek's work on the syntactic calculus in the 1950's, a considerable body of work interpreting logic without structure rules has been developed in terms of monoidal categories. Linear logic is perhaps the best known recent ``logic without structure rules'', but one structure rule remains in a state of flux in linear logic, namely the exchange rule. Several variants of noncommutative linear logic have been proposed, from the purest noncommutativity (with two variant notions of negation and two notions of implication), the cyclic logic of Yetter (with only one notion of negation, but still two implications), and most recently, a system with both cyclic and commutative connectives due to Abrusci and Ruet. In this talk we shall outline suitable notions of (categorical) semantics for these variants of noncommutative logic, in a modular fashion starting from the AND-OR (or TENSOR-PAR) fragment, based on the notions of linear bicategory and linear functor.

This talk is based on joint work with Robin Cockett, Rick Blute, and Jürgen Koslowski.