# Rank 3 Inhabitation of Intersection Types Revisited (Extended Version)

## TL;DR

This paper strengthens the undecidability results of rank 3 intersection type inhabitation by showing it remains undecidable at order 3 and clarifies the principles needed for simulating Turing machines directly, offering a more concise construction.

## Contribution

It proves that intersection type inhabitation is undecidable for types of rank 3 and order 3, and provides a clearer, more direct simulation of Turing machine computation.

## Key findings

- Undecidability of rank 3 intersection type inhabitation at order 3
- A more concise construction for simulating Turing machines
- Insights into the expressiveness of intersection type inhabitation

## Abstract

We revisit the undecidability result of rank 3 intersection type inhabitation (Urzyczyn 2009) in pursuit of two goals. First, we strengthen the previous result by showing that intersection type inhabitation is undecidable for types of rank 3 and order 3, i.e. it is not necessary to introduce new functional dependencies (new instructions) during proof search. Second, we pinpoint the principles necessary to simulate Turing machine computation directly, whereas previous constructions used a highly parallel and non-deterministic computation model. Since our construction is more concise than existing approaches taking no detours, we believe that it is valuable for a better understanding of the expressiveness of intersection type inhabitation.