Decreasing Diagrams and Relative Termination

Abstract

In this paper we use the decreasing diagrams technique to show that a left-linear term rewrite system R is confluent if all its critical pairs are joinable and the critical pair steps are relatively terminating with respect to R. We further show how to encode the rule-labeling heuristic for decreasing diagrams as a satisfiability problem. Experimental data for both methods are presented.

Certification of Classical Confluence Results for Left-Linear Term Rewrite Systems

“…To make the concepts that are hard to formalize in a proof assistant, e.g. measuring the amount of overlap between two multisteps or the descendants of a multistep, Hirokawa and Middeldorp [9] suggested to use proof terms to obtain a rigorous proof (and at the same time extended the result to commutation). This is a step forward but more is needed to obtain a formalized proof, also for the extension to higher-order systems.…”

“…This is a step forward but more is needed to obtain a formalized proof, also for the extension to higher-order systems. In particular, we anticipate the extensive use of sets of positions (in [9]) to be problematic without alternative notions. We plan to employ residual theory [20,Section 8.7] and to develop a notion of overlap for multisteps similar to Definition 5 to close the gap.…”

Confluence by Critical Pair Analysis Revisited

“…to range over steps. Taking their denotation yields the usual multistep [31,15] and step ARSs • −→ and → underlying a TRS R. These can be alternatively obtained by first applying src and tgt (of which only the former is guaranteed to yield a multipattern, by left-linearity) and then taking denotations: Φ src = Φ src and Φ tgt = Φ tgt . Pattern-and body-sizes of multipatterns are compositional.…”

“…The (proof of the) lemma allows to freely switch between viewing multisteps and multipatterns as let-expressions and as sets of sets of positions, and to reason about (non-)overlap of multipatterns and multisteps in lattice-theoretic terms. We show any multistep Φ can be decomposed horizontally as φ followed by Φ/φ for any step φ ∈ Φ [15,28], and vertically as some vector Φ substituted in a prefix Φ 0 of Φ, and that peaks can be decomposed correspondingly.…”

“…In the last decade various classical confluence results for term rewrite systems have been factored through the decreasing diagrams method [27,29] for proving confluence of abstract rewrite systems, often leading to generalisations along the way: e.g. Felgenhauer's multistep labelling [12] generalises Okui's simultaneous closedness [26], the layer framework [11] generalises Toyama's modularity [32], critical pair systems [15] generalise both orthogonality [30] and Knuth and Bendix' criterion [19], and Jouannaud and Liu generalise, among others [20], parallel closedness, but in a way we do not know how to generalise to development closedness [28]. This paper fits into this line of research.…”