Modes of Convergence for Term Graph Rewriting

Abstract

Term graph rewriting provides a simple mechanism to
finitely represent restricted forms of infinitary term
rewriting. The correspondence between infinitary term rewriting and
term graph rewriting has been studied to some extent. However, this
endeavour is impaired by the lack of an appropriate counterpart of
infinitary rewriting on the side of term graphs. We aim to fill this
gap by devising two modes of convergence based on a partial order
resp. a metric on term graphs. The thus obtained structures
generalise corresponding modes of convergence that are usually
studied in infinitary term rewriting. We argue that this yields a
common framework in which both term rewriting and term graph
rewriting can be studied. In order to substantiate our claim, we
compare convergence on term graphs and on terms. In particular, we
show that the resulting infinitary calculi of term graph rewriting
exhibit the same correspondence as we know it from term rewriting:
Convergence via the partial order is a conservative extension of the
metric convergence.