Modes of Convergence for Term Graph Rewriting

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 respectively 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 modes of convergence on term graphs are conservative extensions of the corresponding modes of convergence on terms and are preserved under unravelling term graphs to terms. Moreover, we show that many of the properties known from infinitary term rewriting are preserved. This includes the intrinsic completeness of both modes of convergence and the fact that convergence via the partial order is a conservative extension of the metric convergence.

💡 Research Summary

The paper addresses a long‑standing gap between infinitary term rewriting and term‑graph rewriting by introducing two well‑behaved notions of convergence for term graphs. In classical infinitary term rewriting, convergence is usually studied via a partial‑order model (based on the “definedness” ordering) and a metric model (based on a distance function on terms). Both models enjoy intrinsic completeness (every Cauchy sequence has a limit) and a conservative relationship: the partial‑order convergence strictly extends the metric convergence while preserving its limits.

The authors first recall these two modes for ordinary terms and then construct analogous structures for term graphs, which are finite, possibly cyclic, representations that can share subterms. The partial‑order on graphs is defined through an inclusion relation on nodes and edges: a graph G₁ is below G₂ if G₁ can be obtained from G₂ by deleting nodes/edges, i.e., G₁ is “more defined”. The metric on graphs is built by assigning a weight to each depth level and measuring the difference between two graphs as the sum of weighted mismatches of their shared sub‑graphs. This metric is shown to be complete, turning the space of (possibly infinite) term graphs into a complete metric space.

A central technical contribution is the proof that both convergence notions are conservative extensions of their term counterparts. The authors define an “unravelling” operation that expands a term graph into the (possibly infinite) term it represents. They demonstrate that unravelling preserves the partial‑order and metric structures: if a sequence of graphs converges (either partially ordered or metrically), the corresponding sequence of unraveled terms converges in the same sense, and vice‑versa. Consequently, the graph‑based convergence notions do not introduce spurious limits that would be absent in the term world.

The paper further establishes that the partial‑order convergence subsumes metric convergence on graphs, mirroring the known relationship for terms. This is proved by showing that any metrically convergent graph sequence is also a directed set in the partial‑order and that its limit in the metric coincides with the supremum in the partial‑order. Thus the two modes are tightly coupled: metric convergence can be viewed as a “weak” form of the more informative partial‑order convergence.

Beyond the core theoretical results, the authors discuss several implications. The framework allows one to reason about infinite computations using finite graph representations, which is crucial for implementations that exploit sharing to avoid duplication of subterms. It also guarantees that optimisations based on graph transformations (e.g., common subexpression elimination, lazy evaluation) preserve convergence properties, making the approach suitable for functional language compilers and automated theorem provers that manipulate infinite structures. Moreover, the treatment of cyclic graphs shows that even when a graph encodes an infinite unfolding (such as a stream or co‑recursive definition), the convergence criteria remain decidable within the proposed metric space.

In summary, the paper delivers a robust, mathematically grounded bridge between infinitary term rewriting and term‑graph rewriting. By defining a partial‑order and a metric on term graphs that faithfully extend the corresponding notions on terms, and by proving completeness, conservativity, and the subsumption relationship, the authors provide a unified setting in which both paradigms can be studied side by side. This work paves the way for further research on graph‑based models of infinite computation, their optimisation, and their application in programming language semantics and formal verification.