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Term graph rewriting provides a simple mechanism to finitely representrestricted forms of infinitary term rewriting. The correspondence betweeninfinitary term rewriting and term graph rewriting has been studied to someextent. However, this endeavour is impaired by the lack of an appropriatecounterpart of infinitary rewriting on the side of term graphs. We aim to fillthis gap by devising two modes of convergence based on a partial orderrespectively a metric on term graphs. The thus obtained structures generalisecorresponding modes of convergence that are usually studied in infinitary termrewriting. We argue that this yields a common framework in which both termrewriting and term graph rewriting can be studied. In order to substantiate ourclaim, we compare convergence on term graphs and on terms. In particular, weshow that the modes of convergence on term graphs are conservative extensionsof the corresponding modes of convergence on terms and are preserved underunravelling term graphs to terms. Moreover, we show that many of the propertiesknown from infinitary term rewriting are preserved. This includes the intrinsiccompleteness of both modes of convergence and the fact that convergence via thepartial order is a conservative extension of the metric convergence.

Modes of Convergence for Term Graph Rewriting