Böhm Trees as Higher-Order Recursive Schemes

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 λY -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 λY -term (with no restrictions on term type or the types of free variables) can actually be represented by a HORS. More precisely, for any λY -termM , there exists a HORS generating a tree that faithfully representsM ’s (η-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 λY -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 modelchecking of Böhm trees, despite the general undecidability of the problem. 1998 ACM Subject Classification F.3.3, F.4.1