Abstract
This article is concerned with automating the decreasing diagrams technique of van Oostrom for establishing confluence of term rewrite systems. We study abstract criteria that allow to lexicographically combine labelings to show local diagrams decreasing. This approach has two immediate benefits. First, it allows to use labelings for linear rewrite systems also for left-linear ones, provided some mild conditions are satisfied. Second, it admits an incremental method for proving confluence which subsumes recent developments in automating decreasing diagrams. The techniques proposed in the article have been implemented and experimental results demonstrate how, e.g., the rule labeling benefits from our contributions.
Highlights
Confluence is an important property of rewrite systems since it ensures unique normal forms
The decreasing diagrams technique of van Oostrom [16] is a complete method for showing confluence of countable abstract rewrite systems
In [17] van Oostrom presented the rule labeling that allows to conclude confluence of linear rewrite systems by checking decreasingness of the critical peaks
Summary
Confluence is an important property of rewrite systems since it ensures unique normal forms. Confluence of the TRS in Example 1 can be established automatically, e.g., by the rule labeling, while current approaches based on the decreasing diagrams technique [1, 11] as well as other confluence criteria like Knuth and Bendix’ criterion or orthogonality (and its refinements) fail This system is locally confluent since all its 34 critical pairs are joinable. Three of these constraints are based on relative termination while the fourth employs persistence. – Section 4, which studies parallel rewriting to make any weak LL-labeling applicable to showing confluence of left-linear systems without additional (relative termination) constraints The latter generalizes and incorporates recent findings from [6], which studies the rule labeling for parallel rewriting
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