A complete realisability semantics for intersection types and arbitrary expansion variables

TL;DR

This paper develops a comprehensive realisability semantics for intersection types with arbitrary expansion variables, overcoming previous limitations by allowing multiple E-variables and the universal type, and proving soundness and completeness.

Contribution

It introduces a novel semantics for intersection type systems with unlimited E-variables and the universal type, resolving longstanding challenges in the field.

Findings

Semantics is sound for systems with multiple E-variables.

Semantics is complete including the universal type ω.

Allows arbitrary (possibly infinite) number of expansion variables.

Abstract

Expansion was introduced at the end of the 1970s for calculating principal typings for $\lambda$-terms in intersection type systems. Expansion variables (E-variables) were introduced at the end of the 1990s to simplify and help mechanise expansion. Recently, E-variables have been further simplified and generalised to also allow calculating other type operators than just intersection. There has been much work on semantics for intersection type systems, but only one such work on intersection type systems with E-variables. That work established that building a semantics for E-variables is very challenging. Because it is unclear how to devise a space of meanings for E-variables, that work developed instead a space of meanings for types that is hierarchical in the sense of having many degrees (denoted by indexes). However, although the indexed calculus helped identify the serious problems of giving a semantics for expansion variables, the sound realisability semantics was only complete when one single E-variable is used and furthermore, the universal type $\omega$ was not allowed. In this paper, we are able to overcome these challenges. We develop a realisability semantics where we allow an arbitrary (possibly infinite) number of expansion variables and where $\omega$ is present. We show the soundness and completeness of our proposed semantics.