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On Word and Frontier Languages of Unsafe Higher-Order Grammars Asada, Kazuyuki Kobayashi, Naoki intersection types higher-order grammars Higher-order grammars are an extension of regular and context-free grammars, where nonterminals may take parameters. They have been extensively studied in 1980's, and restudied recently in the context of model checking and program verification. We show that the class of unsafe order-(n+1) word languages coincides with the class of frontier languages of unsafe order-n tree languages. We use intersection types for transforming an order-(n+1) word grammar to a corresponding order-n tree grammar. The result has been proved for safe languages by Damm in 1982, but it has been open for unsafe languages, to our knowledge. Various known results on higher-order grammars can be obtained as almost immediate corollaries of our result. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Kazuyuki Asada and Naoki Kobayashi 2016 Is Part Of LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2016.111 urn:nbn:de:0030-drops-62469 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.111 eng https://creativecommons.org/licenses/by/3.0/legalcode