When substitutions and bindings interact, there is a risk of undesired side effects if the substitution is applied naïvely. The λ-calculus captures this phenomenon concretely, as β-reduction may require the renaming of bound variables to avoid variable capture. In this paper we introduce α-paths as an estimation for α-avoidance, roughly expressing that α-conversions are not required to prevent variable capture. These paths provide a novel method to analyse and predict the potential need for α in different calculi. In particular, we show how α-path characterises α-avoidance for several sub-calculi of the λ-calculus like (i) developments, (ii) affine/linear λ-calculi, (iii) the weak λ-calculus, (iv) μ-unfolding and (iv) finally the safe λ-calculus. Furthermore, we study the unavoidability of α-conversions in untyped and simply-typed λ-calculi and prove undecidability of the need of α-conversions for (leftmost-outermost reductions) in the untyped λ-calculus. To ease the work with α-paths, we have implemented the method and the tool is publicly available.

We present the Z-property and instantiate it to various rewrite systems: associativity, positive braids, self-distributivity, the lambda-calculus, lambda-calculi with explicit substitutions, orthogonal TRSs, .... The Z-property is proven equivalent to Takahashi’s angle property by means of a syntax-free notion of development. We show that several classical consequences of having developments such as confluence, normalisation, and recurrence, can be regained in a syntax-free way, and investigate how the notion corresponds to the classical syntactic notion of development in term rewriting.

We introduce layer systems for proving generalizations of the modularity of confluence for first-order rewrite systems. Layer systems specify how terms can be divided into layers. We establish structural conditions on those systems that imply confluence. Our abstract framework covers known results like modularity, many-sorted persistence, layer-preservation, and currying. We present a counterexample to an extension of persistence to order-sorted rewriting and derive new sufficient conditions for the extension to hold. All our proofs are constructive.

We present and compare some well-founded proof orders for decreasing diagrams. These proof orders order a conversion above another conversion if the latter is obtained by filling any peak in the former by a (locally) decreasing diagram. Therefore each such proof order entails the decreasing diagrams technique for proving confluence. The proof orders differ with respect to monotonicity and complexity. Our results are developed in the setting of involutive monoids. We extend these results to obtain a decreasing diagrams technique for confluence modulo.

We introduce a process, dubbed triangulation, turning any rewrite relation into a confluent one. It is more direct than usual completion, in the sense that objects connected by a peak are directly related rather than their normal forms. We investigate conditions under which this process preserves desirable properties such as termination.

We show that, contrary to the situation in first-order term rewriting, almost none of the usual properties of rewriting are modular for higher-order rewriting, irrespective of the higher-order rewriting format. We show that for the particular format of simply typed applicative term rewriting systems modularity of confluence, normalization, and termination can be recovered by imposing suitable linearity constraints.

We present some contributions to the theory of infinitary rewriting for weakly orthogonal term rewrite systems, in which critical pairs may occur provided they are trivial. We show that the infinitary unique normal form property (UNinf) fails by a simple example of a weakly orthogonal TRS with two collapsing rules. By translating this example, we show that UNinf also fails for the infinitary lambda-beta-eta-calculus. As positive results we obtain the following: Infinitary confluence, and hence UNinf, holds for weakly orthogonal TRSs that do not contain collapsing rules. To this end we refine the compression lemma. Furthermore, we consider the triangle and diamond properties for infinitary developments in weakly orthogonal TRSs, by refining an earlier cluster-analysis for the finite case.

Realising Optimal Sharing

We describe several technical tools that prove to be efficient for investigating the rewrite systems associated with an equational specification. These tools consist of introducing a monoid of partial maps, listing the monoid relations corresponding to the local confluence diagrams of the rewrite system, introducing the group presented by these relations, and, finally, replacing the initial rewrite system with an internal process entirely sitting in this group. When the approach can be completed, one typically obtains a practical method for constructing algebras satisfying prescribed equations and for solving the associated word problem. The above techniques have been developed by the first author in a context of general algebra. The goal of this paper is to bring them to the attention of the rewrite system community. We hope that these techniques may be useful for more general rewrite systems.

The criterion for confluence using decreasing diagrams is a generalization of several well-known confluence criteria in abstract rewriting, such as the strong confluence lemma. We give a new proof of the decreasing diagram theorem based on a geometric study of infinite reduction diagrams, arising from unsuccessful attempts to obtain a confluent diagram by tiling with elementary diagrams.