Size-Preserving Translations from Order-(n+1) Word Grammars to Order-n Tree Grammars

Higher-order grammars have recently been studied actively in the context of automated verification of higher-order programs. Asada and Kobayashi have previously shown that, for any order-(n+ 1) word grammar, there exists an order-n grammar whose frontier language coincides with the language generated by the word grammar. Their translation, however, blows up the size of the grammar, which inhibited complexity-preserving reductions from decision problems on word grammars to those on tree grammars. In this paper, we present a new translation from order-(n+ 1) word grammars to order-n tree grammars that is size-preserving in the sense that the size of the output tree grammar is polynomial in the size of an input tree grammar. The new translation and its correctness proof are arguably much simpler than the previous translation and proof. 2012 ACM Subject Classification Theory of computation→ Formal languages and automata theory