Böhm Trees as Higher-Order Recursive Schemes

Andrzej Murawski

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 lambdaY-calculus, consisting of ground-type terms whose free variables have types of the form o->...->o (with o as a special case).

In this paper, we show that any lambdaY-term (with no restrictions on term type or the types of free variables) can actually be represented by a HORS. That is, for any lambda Y-term M, there exists a HORS generating a tree that faithfully represents M's 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 lambdaY-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.

This is joint work with Pierre Clairambault (University of Cambridge).