Lambda theories of effective lambda models

Abstract. A longstanding open problem is whether there exists a nonsyntactical model of the untyped λ-calculus whose theory is exactly the least λ-theory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λcalculus can be recursively enumerable (r.e. for brevity). We introduce a notion of effective model of λ-calculus, which covers in particular all the models individually introduced in the literature, and prove that the equational theory of an effective model cannot be λβ, λβη. In other results of the paper we show that the order theory of an effective model cannot be r.e. and that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott’s semantics, we investigate the class of graph models and prove that no order theory of a graph model can be r.e., and that there exists an effective graph model whose equational/order theory is the minimum one. Finally, we show that the class of graph models enjoys a kind of downwards Löwenheim-Skolem theorem