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A longstanding open problem is whether there exists a non syntactical modelof the untyped lambda-calculus whose theory is exactly the least lambda-theory(l-beta). In this paper we investigate the more general question of whether theequational/order theory of a model of the (untyped) lambda-calculus can berecursively enumerable (r.e. for brevity). We introduce a notion of effectivemodel of lambda-calculus calculus, which covers in particular all the modelsindividually introduced in the literature. We prove that the order theory of aneffective model is never r.e.; from this it follows that its equational theorycannot be l-beta or l-beta-eta. We then show that no effective model living inthe stable or strongly stable semantics has an r.e. equational theory.Concerning Scott's semantics, we investigate the class of graph models andprove that no order theory of a graph model can be r.e., and that there existsan effective graph model whose equational/order theory is minimum among alltheories of graph models. Finally, we show that the class of graph modelsenjoys a kind of downwards Lowenheim-Skolem theorem.

Effective lambda-models vs recursively enumerable lambda-theories