The paper below is available by anonymous ftp from the following sites: triples.math.mcgill.ca, in the directory: pub/blute, theory.doc.ic.ac.uk, in the directory: papers/Scott. ftp.csi.uottawa.ca , in the directory: pub/papers/PhilScott The file is called: lauchli.ps.Z. Any comments would be greatly appreciated. Cheers, Philip Scott P. S. Of course, you may also contact either of the authors for a hard copy: R. F. Blute & P. J. Scott Dept. of Mathematics University of Ottawa 585 King Edward Ottawa, Ont. K1N 6N5 Canada ---------------------------------------------- LINEAR LAUCHLI SEMANTICS R. F. Blute P. J. Scott We introduce a linear analogue of Lauchli's semantics for intuitionistic logic. In fact, our result is a strengthening of Lauchli's work to the level of proofs, rather than provability. This is obtained by considering continuous actions of the additive group of integers on a category of topological vector spaces. The semantics, based on functorial polymorphism, consists of dinatural transformations which are equivariant with respect to all such actions. Such dinatural transformations are called uniform. To any sequent in Multiplicative Linear Logic (MLL), we associate a vector space of ``diadditive'' uniform transformations. We then show that this space is generated by denotations of cut-free proofs of the sequent in the theory MLL+MIX. Thus we obtain a full completeness theorem in the sense of Abramsky and Jagadeesan, although our result differs from theirs in the use of dinatural transformations. As corollaries, we show that these dinatural transformations compose, and obtain a conservativity result: diadditive dinatural transformations which are uniform with respect to actions of the additive group of integers are also uniform with respect to the actions of arbitrary cocommutative Hopf algebras. Finally, we discuss several possible extensions of this work to noncommutative logic. It is well known that the intuitionistic version of Lauchli's semantics is a special case of the theory of logical relations, due to Plotkin and Statman. Thus, our work can also be viewed as a first step towards developing a theory of logical relations for linear logic and concurrency.