Effective lambda-models vs recursively enumerable lambda-theories

TL;DR

This paper explores whether models of the untyped lambda-calculus can have recursively enumerable theories, concluding that effective models in various semantics cannot have r.e. theories, and providing insights into the structure of graph models.

Contribution

It introduces the notion of effective models for lambda-calculus and proves their theories are not recursively enumerable, advancing understanding of lambda-models' theoretical limitations.

Findings

Effective models' order theories are never r.e.

No effective model in stable or strongly stable semantics has an r.e. equational theory.

Among graph models, some have minimal equational theories and the class satisfies a Lowenheim-Skolem property.

Abstract

A longstanding open problem is whether there exists a non syntactical model of 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 the equational/order theory of a model of the (untyped) lambda-calculus can be recursively enumerable (r.e. for brevity). We introduce a notion of effective model of lambda-calculus calculus, which covers in particular all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows that its equational theory cannot be l-beta or l-beta-eta. We then show 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 minimum among all theories of graph models. Finally, we show that the class of graph models enjoys a kind of downwards Lowenheim-Skolem theorem.