Reflexive Scott Domains are Not Complete for the Extensional Lambda Calculus

A longstanding open problem is whether there exists a model of the untyped lambda calculus in the category CPO of complete partial orderings and Scott continuous functions, whose theory is exactly the least lambda-theory lambda-beta or the least extensional lambda-theory lambda-beta-eta. In this paper we analyze the class of reflexive Scott domains, the models of lambda-calculus living in the category of Scott domains (a full subcategory of CPO). The following are the main results of the paper: (i) Extensional reflexive Scott domains are not complete for the beta-eta-calculus, i.e., there are equations not in lambda-beta-eta which hold in all extensional reflexive Scott domains.(ii) The order theory of an extensional reflexive Scott domain is never recursively enumerable. These results have been obtained by isolating among the reflexive Scott domains a class of webbed models arising from Scott's information systems, called iweb-models. The class of iweb-models includes all extensional reflexive Scott domains, all preordered coherent models and all filter models living in CPO. Based on a fine-grained study of an ``effective'' version of Scott's information systems, we have shown that there are equations not in lambda-beta (resp. lambda-beta-eta) which hold in all (extensional) iweb-models.