# Pumping Lemma for Higher-order Languages

## TL;DR

This paper extends the pumping lemma to higher-order languages, providing a tool to analyze their generative capacity and showing its validity for certain cases, with implications for language classification.

## Contribution

It introduces a pumping lemma for higher-order word/tree languages and proves its validity for order-2 cases, advancing understanding of higher-order language properties.

## Key findings

- Pumping lemma established for arbitrary order-words and trees, assuming a conjecture.
- Confirmed the conjecture for order-2, enabling the lemma's application to order-2 tree and order-3 word languages.
- Provides a new method to distinguish higher-order languages from simpler classes.

## Abstract

We study a pumping lemma for the word/tree languages generated by higher-order grammars. Pumping lemmas are known up to order-2 word languages (i.e., for regular/context-free/indexed languages), and have been used to show that a given language does not belong to the classes of regular/context-free/indexed languages. We prove a pumping lemma for word/tree languages of arbitrary orders, modulo a conjecture that a higher-order version of Kruskal's tree theorem holds. We also show that the conjecture indeed holds for the order-2 case, which yields a pumping lemma for order-2 tree languages and order-3 word languages.

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Source: https://tomesphere.com/paper/1705.10699