Minimal lambda-theories by ultraproducts

TL;DR

This paper introduces a general method to identify minimal lambda-theories within various classes of models, including i-models and graph models, advancing understanding of models with specific lambda-theories.

Contribution

It provides a new tool to find minimal lambda-theories in broad classes of lambda models, including webbed, graph, Krivine, coherent, and filter models.

Findings

Constructed an i-model with the theory common to all i-models.

Applied the method to various classes of models, confirming its broad applicability.

Identified models whose theories match well-known lambda-theories like lambda-beta and H.

Abstract

A longstanding open problem in lambda calculus is whether there exist continuous models of the untyped lambda calculus whose theory is exactly the least lambda-theory lambda-beta or the least sensible lambda-theory H (generated by equating all the unsolvable terms). A related question is whether, given a class of lambda models, there is a minimal lambda-theory represented by it. In this paper, we give a general tool to answer positively to this question and we apply it to a wide class of webbed models: the i-models. The method then applies also to graph models, Krivine models, coherent models and filter models. In particular, we build an i-model whose theory is the set of equations satisfied in all i-models.