Böhm Trees as Higher-Order Recursive Schemes

Abstract

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.