# Relational Graph Models at Work

## TL;DR

This paper investigates relational graph models within lambda-calculus, establishing minimal and fully abstract models for key observational equivalences, and introduces new model classes like lambda-König and hyperimmune models.

## Contribution

It characterizes conditions for relational graph models to be fully abstract for important lambda-calculus theories, introducing lambda-König and hyperimmune models.

## Key findings

- Existence of a minimal lambda-theory induced by relational graph models.
- Relational graph models that are extensional and lambda-König are fully abstract for H+.
- Extensional and hyperimmune models fully capture H*.

## Abstract

We study the relational graph models that constitute a natural subclass of relational models of lambda-calculus. We prove that among the lambda-theories induced by such models there exists a minimal one, and that the corresponding relational graph model is very natural and easy to construct. We then study relational graph models that are fully abstract, in the sense that they capture some observational equivalence between lambda-terms. We focus on the two main observational equivalences in the lambda-calculus, the theory H+ generated by taking as observables the beta-normal forms, and H* generated by considering as observables the head normal forms. On the one hand we introduce a notion of lambda-K\"onig model and prove that a relational graph model is fully abstract for H+ if and only if it is extensional and lambda-K\"onig. On the other hand we show that the dual notion of hyperimmune model, together with extensionality, captures the full abstraction for H*.