Modeling linear logic with implicit functions
Just as intuitionistic proofs can be modeled by functions, linear logic proofs, being symmetric in the inputs and outputs, can be modeled by relations (for example, cliques in coherence spaces). However generic relations do not establish any functional dependence between the arguments, and therefore it is questionable whether they can be thought as reasonable generalizations of functions. On the other hand, in some situations (typically in differential calculus) one can speak in some precise sense about an implicit functional dependence defined by a relation. It turns out that it is possible to model linear logic with implicit functions rather than general relations, an adequate language for such a semantics being (elementary) differential calculus. This results in a non-degenerate model enjoying quite strong completeness properties.