Diagram techniques for confluence

AbstractWe develop diagram techniques for proving confluence in abstract reductions systems. The underlying theory gives a systematic and uniform framework in which a number of known results, widely scattered throughout the literature, can be understood. These results include Newman's lemma, Lemma 3.1 of Winkler and Buchberger, the Hindley–Rosen lemma, the Request lemmas of Staples, the Strong Confluence lemma of Huet, the lemma of De Bruijn

Decreasing Diagrams and Relative Termination

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.Comment: v3: missing references adde

Confluence by Decreasing Diagrams -- Formalized

This paper presents a formalization of decreasing diagrams in the theorem prover Isabelle. It discusses mechanical proofs showing that any locally decreasing abstract rewrite system is confluent. The valley and the conversion version of decreasing diagrams are considered.Comment: 17 pages; valley and conversion version; RTA 201

Braids via term rewriting

We present a brief introduction to braids, in particular simple positive braids, with a double emphasis: first, we focus on term rewriting techniques, in particular, reduction diagrams and decreasing diagrams. The second focus is our employment of the colored braid notation next to the more familiar Artin notation. Whereas the latter is a relative, position dependent, notation, the former is an absolute notation that seems more suitable for term rewriting techniques such as symbol tracing. Artin's equations translate in this notation to simple word inversions. With these points of departure we treat several basic properties of positive braids, in particular related to the word problem, confluence property, projection equivalence, and the congruence property. In our introduction the beautiful diamond known as the permutohedron plays a decisive role

Formalising Strong Normalisation Proofs of the Explicit Substitution Calculi in ALF1

Explicit substitution calculi have become very fashionable in the last decade. The reason is that substitution calculi bridge theory and implementation and enable control over evaluation steps and strategies. Of the most important questions of explicit substitution calculi is that of the termination of the underlying calculus of substitution. For this reason, one finds with every new calculus of explicit substitution, a section devoted to the termination of substitutions. Those proofs of termination fall under two categories. Proofs that are easy because a decreasing measure can be established and proofs that are difficult because such a decreasing measure is not easy to establish. Another fashionable subject has been the checking of proofs using a proof checker. This is useful because some proofs can be intricate and hard to believe if they are not proof checked. This thesis investigates the methods to prove termination of explicit substitution calculi and their formalisations in the proof checker ALF. Two styles of explicit substitution calculi are chosen for this purpose, one is the calculus s whose termination is guaranteed by a decreasing weight, the other is the calculus a whose termination is extremely complex. Two new termination proofs of the calculus s are given. All termination proofs of both s and a presented in the thesis are formalised in ALF. During our process of formalisations we comment on what is needed to make a proof checkable during the checking process

Diagram Techniques for Confluence

We develop diagram techniques for proving confluence in abstract reductions systems. The underlying theory gives a systematic and uniform framework in which a number of known results, widely scattered throughout the literature, can be understood. These results include Newman's Lemma (1942), Lemma 3.1 of Winkler and Buchberger (1985), the Hindley-Rosen Lemma (1964), the Request Lemmas of Staples (1975), the Strong Confluence Lemma of Huet (1980) and the Lemma of De Bruijn (1978)