Partial Order Infinitary Term Rewriting and Böhm Trees

Abstract

We investigate an alternative model of infinitary term
rewriting. Instead of a metric, a partial order on terms is employed
to formalise (strong) convergence. We compare this partial order
convergence of orthogonal term rewriting systems to the usual metric
convergence of the corresponding B{"o}hm extensions. The B{"o}hm extension
of a term rewriting system contains additional rules to equate
so-called root-active terms. The core result we present is that
reachability w.r.t. partial order convergence coincides with
reachability w.r.t. metric convergence in the B{"o}hm extension. This
result is used to show that, unlike in the metric model, orthogonal
systems are infinitarily confluent and infinitarily normalising in the
partial order model. Moreover, we obtain, as in the metric model, a
compression lemma. A corollary of this lemma is that reachability
w.r.t. partial order convergence is a conservative extension of
reachability w.r.t. metric convergence.