We describe several technical tools that prove to be efficient for investigating the rewrite systems associated with a family of algebraic laws, and might be useful for more general rewrite systems. These tools consist in introducing a monoid of partial operators, listing the monoid relations expressing the possible local confluence of the rewrite system, then introducing the group presented by...

Deduction modulo is an extension of first-order predicate logic where axioms are replaced by rewrite rules. For instance, the axiom P ⇔ (Q ⇒ R) is replaced by the rule P −→ (Q ⇒ R). These rules define an equivalence relation and, in a proof, a proposition can be replaced by an equivalent one at any time. A presentation of Simple Type Theory in Deduction modulo, calledHOL, has been given in [3]....

The complexity of normal form rewrite sequences for Associativity. Abstract The complexity of a particular term-rewrite system is considered: the rule of associativity (x * y) * z ⊲ x * (y * z). Algorithms and exact calculations are given for the longest and shortest sequences of applications of ⊲ that result in normal form (NF). The shortest NF sequence for a term x is always n − d rm (x), whe...

Unfailing completion is a commonly used technique for equational reasoning. For equational problems with associative and commutative functions, unfailing completion often generates a large number of rewrite rules. By comparing it with a ground completion procedure, we show that many of the rewrite rules generated are redundant. A set of consistency constraints is formulated to detect redundant ...

We introduce parameterized rewrite systems for describing infinite families of finite string rewrite systems depending upon non-negative integer parameters, as well as ways to reason uniformly over these families. Unlike previous work, the vocabulary on which a rewrite system in the family is built depends itself on the integer parameters. Rewriting makes use of a toolkit for parameterized word...

In 1978, Klop demonstrated that a rewrite system constructed by adding the untyped lambda calculus, which has the Church-Rosser property, to a Church-Rosser rst-order algebraic rewrite system may not be Church-Rosser. In contrast, Breazu-Tannen recently showed that augmenting any Church-Rosser rst-order algebraic rewrite system with the simply-typed lambda calculus results in a Church-Rosser re...

In rewriting theory, termination of a rewrite system by Kruskal's theorem implies a theoretical upper bound on the complexity of the system. This bound is, however, far from having been reached by known examples of rewrite systems. All known orderings used to establish termination by Kruskal's theorem yield a multiply recursive bound. Furthermore, the study of the order types of such orderings ...

Associate to a rewrite system R having rules l → r, its labelled version R having rules l ◦ m+1 → r • m , for any natural number m ∈ ω. These rules roughly express that a left-hand side l carrying labels all larger than m can be replaced by its right-hand side r carrying labels all smaller than or equal to m. A rewrite system R enjoys finite family developments (FFD) if R is terminating. We sho...

KURE is a Haskell hosted Domain Specific Language (DSL) for writing transformation systems based on rewrite strategies. When writing transformation systems, a significant amount of engineering effort goes into setting up plumbing to make sure that specific rewrite rules can fire. Systems like Stratego and Strafunski provide most of this plumbing as infrastructure, allowing the DSL user to focus...