Confluence by decreasing diagrams

Decreasing Diagrams and Relative Termination

“…We conclude this preliminary section by recalling the decreasing diagrams technique for abstract rewrite systems (ARSs) from [25,27]. We write A, {→ α } α∈I to denote the ARS A, → where → is the union of → α for all α ∈ I.…”

“…The ARS A = A, {→ α } α∈I is locally decreasing if there exists a well-founded order > on I such that LD > (α, β) for all α, β ∈ I for all α, β ∈ I. Van Oostrom [27] obtained the following result.…”

“…This result can be viewed as a generalization of the two standard approaches for proving confluence of term rewrite systems: orthogonality and joinability of critical pairs for terminating systems. In the proof we use the conversion version [27] of decreasing diagrams with the predecessor labeling in which rewrite steps are labeled by a term that can be rewritten to the starting term of the step. For countable abstract rewrite systems, the decreasing diagrams technique of van Oostrom [25,27] subsumes all sufficient conditions for confluence.…”

“…In the proof we use the conversion version [27] of decreasing diagrams with the predecessor labeling in which rewrite steps are labeled by a term that can be rewritten to the starting term of the step. For countable abstract rewrite systems, the decreasing diagrams technique of van Oostrom [25,27] subsumes all sufficient conditions for confluence. To use this technique for term rewrite systems, a well-founded order on the rewrite steps has to be supplied such that rewrite peaks can be completed into so-called decreasing diagrams.…”

“…The second result of this article is the encoding of the rule-labeling heuristic of van Oostrom [27] for linear rewrite systems as a satisfiability problem. In this heuristic rewrite steps are labeled by the applied rewrite rule.…”