Pumping Lemma for Higher-order Languages

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.