Graph models of lambda-calculus at work, and variations
Résumé
The paper surveys the knowledge we have acquired these last ten years about the lattice LT of all lambda-theories (=equational extensions of untyped lambda-calculus), via the sets LC consisting of the lambda-theories which are representable in a uniform class C of lambda-models. This includes positive answers to several questions raised in [Berline, Theor. Comput. Sci. 249 (2000)] as well as several independent questions, the state of the art about the long-standing open questions concerning the representability of lambda-beta, lambda-beta-eta, and H as theories of models, a,d many open problems. We will focus on the class G of graph-models since almost all the existing semantic proofs on LT have been, or could be, more easily, obtained via graph models, or slight variations when needed. But in this paper we will also give some evidence that, for all uniform classes C,C' of proper lambda-models living in functional semantics, LC-LC' should have cardinality 2^omega, as soon as C is not included in C'.