Higher-order recursive schemes (HORS) are schematic representations of functional programs. They generate possibly infinite ranked labelled trees and, in that respect, are known to be equivalent to a restricted fragment of the lambda-Y-calculus consisting of ground-type terms whose free variables have types of the form o -> ... -> o (with o being a special case). In this paper, we show that any lambda-Y-term (with no restrictions on term type or the types of free variables) can actually be represented by a HORS. More precisely, for any lambda-Y-term M, there exists a HORS generating a tree that faithfully represents M's (eta-long) Böhm tree. In particular, the HORS captures higher-order binding information contained in the Böhm tree. An analogous result holds for finitary PCF. As a consequence, we can reduce a variety of problems related to the lambda-Y-calculus or finitary PCF to problems concerning higher-order recursive schemes. For instance, Böhm tree equivalence can be reduced to the equivalence problem for HORS. Our results also enable MSO model-checking of Böhm trees, despite the general undecidability of the problem.
@InProceedings{clairambault_et_al:LIPIcs.FSTTCS.2013.91, author = {Clairambault, Pierre and Murawski, Andrzej S.}, title = {{B\"{o}hm Trees as Higher-Order Recursive Schemes}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {91--102}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-64-4}, ISSN = {1868-8969}, year = {2013}, volume = {24}, editor = {Seth, Anil and Vishnoi, Nisheeth K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.91}, URN = {urn:nbn:de:0030-drops-43644}, doi = {10.4230/LIPIcs.FSTTCS.2013.91}, annote = {Keywords: Lambda calculus, B\"{o}hm trees, Recursion Schemes} }
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