Sequence Types and Infinitary Semantics

TL;DR

This paper presents a novel sequence-based representation of non-idempotent intersection types, enabling the extension of intersection type theory to infinitary lambda calculus and solving Klop's Problem.

Contribution

It introduces a new sequence-based approach to non-idempotent intersection types, facilitating the analysis of infinitary lambda calculus and hereditary head normalization.

Findings

Characterizes hereditary head normalization in infinitary lambda calculus

Provides a positive solution to Klop's Problem

Retrieves known results on infinitary terms using non-idempotent intersection

Abstract

We introduce a new representation of non-idempotent intersection types, using \textbf{sequences} (families indexed with natural numbers) instead of lists or multisets. This allows scaling up \textbf{intersection type} theory to the infinitary $\lambda$-calculus. We thus characterize hereditary head normalization, which gives a positive answer to a question known as \textbf{Klop's Problem}. On our way, we use \textbf{non-idempotent intersection} to retrieve some well-known results on infinitary terms.