Affine

To: lineaire <linear@CS.Stanford.EDU>

Subject: Affine

From: Jean-Yves GIRARD <girard@iml.univ-mrs.fr>

Date: Tue, 1 Jul 97 14:27:03 +0100

Delivery-Date: Fri, 04 Jul 1997 10:40:26 -0500

Sender: lincoln@CS.Stanford.EDU

What is affine in "affine logic" ? Take the viewpoint of denotational semantics and consider the intuitionistic version of linear logic, in terms of coherent spaces or Banach spaces. Then the weakening rule introduces fake dependencies, eg constant functions, and when we combine it with additive features, we get more generally affine functions. That's it. However, this is slightly problematic : i) we need to modify the interpretation of the tensor, ie define E @ F by 1 & (E @ F) = (1&E) \otimes (1&F) ; however this connective will hardly distribute over \oplus ii) in presence of the classical symmetry, we need to consider weakening on both sides, and the situation is rather complex. So, to sum up, "affine" means that, if we define a denotational semantics, then it has to be done with affine functions ; but unfortunately, "affine logic" does not quite live at the level of denotational semantics... what is loosened up is denotational semantics, which is rather unpleasant.