Pumping Lemma for Higher-order Languages

Authors Kazuyuki Asada, Naoki Kobayashi

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Kazuyuki Asada
Naoki Kobayashi

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Kazuyuki Asada and Naoki Kobayashi. Pumping Lemma for Higher-order Languages. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 97:1-97:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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.
  • pumping lemma
  • higher-order grammars
  • Kruskal's tree theorem


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