Obsessional Cliques: A Semantic Characterization of Bounded Time Complexity

We give a semantic characterization of bounded complexity proofs. We introduce the notion of obsessional clique in the relational model of linear logic and show that restricting the morphisms of the category REL to obsessional cliques yields models of ELL and SLL. Conversely, we prove that these models are relatively complete: an LL proof whose interpretation is an obsessional clique is always an ELL/SLL proof. These results are achieved by introducing a system of ELL/SLL untyped proof-nets, which is both correct and complete with respect to elementary/ polynomial time