Nominal rewriting systems (Fernandez, Gabbay, Mackie, 2004; Fernandez, Gabbay, 2007) have been introduced as a new framework of higher-order rewriting systems based on the nominal approach (Gabbay, Pitts, 2002; Pitts, 2003), which deals with variable binding via permutations and freshness conditions on atoms. Confluence of orthogonal nominal rewriting systems has been shown in (Fernandez, Gabbay, 2007). However, their definition of (non-trivial) critical pairs has a serious weakness so that the orthogonality does not actually hold for most of standard nominal rewriting systems in the presence of binders. To overcome this weakness, we divide the notion of overlaps into the self-rooted and proper ones, and introduce a notion of alpha-stability which guarantees alpha-equivalence of peaks from the self-rooted overlaps. Moreover, we give a sufficient criterion for uniformity and alpha-stability. The new definition of orthogonality and the criterion offer a novel confluence condition effectively applicable to many standard nominal rewriting systems. We also report on an implementation of a confluence prover for orthogonal nominal rewriting systems based on our framework.

We give a method to prove confluence of term rewriting systems that contain non-terminating rewrite rules such as commutativity and associativity. Usually, confluence of term rewriting systems containing such rules is proved by treating them as equational term rewriting systems and considering E-critical pairs and/or termination modulo E. In contrast, our method is based solely on usual critical pairs and usual termination. We first present confluence criteria for term rewriting systems whose rewrite rules can be partitioned into terminating part and possibly non-terminating part. We then give a reduction-preserving completion procedure so that the applicability of the criteria is enhanced. In contrast to the well-known Knuth-Bendix completion procedure which preserves the equivalence relation of the system, our completion procedure preserves the reduction relation of the system, by which confluence of the original system is inferred from that of the completed system.

Huet and Lang (1978) presented a framework of automated program transformation based on lambda calculus in which programs are transformed according to a given program transformation template. They introduced a second-order matching algorithm of simply-typed lambda calculus to verify whether the input program matches the template. They also showed how to validate the correctness of the program transformation using the denotational semantics.We propose in this paper a framework of program transformation by templates based on term rewriting. In our new framework, programs are given by term rewriting systems. To automate our program transformation, we introduce a term pattern matching problem and present a sound and complete algorithm that solves this problem.We also discuss how to validate the correctness of program transformation in our framework. We introduce a notion of developed templates and a simple method to construct such templates without explicit use of induction. We then show that in any program transformation by developed templates the correctness of the transformation can be verified automatically. In our framework the correctness of the program transformation is discussed based on the operational semantics. This is a sharp contrast to Huet and Lang's framework.

A term rewriting system is called complete if it is confluent and terminating.We prove that completeness of TRSS is a "modular" property (meaning that it stays preserved under direct sums), provided the constituent TRSS are left-linear.Here, the direct sum RO S3R ~is the union of TRSS R., RI with disjoint signature.The proof hinges crucially upon the (non)deterministic collapsing behavior of terms from the sum TRS.

G. Huet and J.J. Levy (INRIA Rep. 359, 1979) showed that for every strongly sequential orthogonal (i.e., left-linear and non-overlapping) term rewriting system, index reduction strategy is normalizing. Their result is extended to overlapping term rewriting systems. It is shown that index reduction is normalizing for the class of strongly sequential left-linear term rewriting systems in which every critical pair can be joined with root balanced reductions. This class includes all weakly orthogonal left-normal systems, for which a leftmost-outermost reduction strategy is normalizing.>

The direct sum of two term rewriting systems is the union of systems having disjoint sets of function symbols. It is shown that if two term rewriting systems both have the Chruch-Rosser property, then the direct sum of these systems also has this property.