Combinatory Reduction Systems, or CRSs for short, were designed to combine the usual first-order format of term rewriting with the presence of bound variables as in pure -calculus and various typed -calculi. Bound variables are also present in many other rewrite systems, such as systems with simplification rules for proof normalization. The original idea of CRSs is due to Aczel, who introduced a restricted class of CRSs and, under the assumption of orthogonality, proved confluence. Orthogonality means that the rules are non-ambiguous (no overlap leading to a critical pair) and left-linear (no global comparison of terms necessary). We introduce the class of orthogonal CRSs, illustrated with many examples, discuss its expressive power, and give an outline of a short proof of confluence. This proof is a direct generalization of Aczel's original proof, which is close to the well-known confluence proof for -calculus by Tait and Martin-Lof. There is a well-known connection between the para...

We present an equational framework for term graph rewriting with cycles. The usual notion of homomorphism is phrased in terms of the notion of bisimulation, which is well-known in process algebra and concurrency theory. Specifically, a homomorphism is a functional bisimulation. We prove that the bisimilarity class of a term graph, partially ordered by functional bisimulation, is a complete lattice. It is shown how Equational Logic induces a notion of copying and substitution on term graphs, or systems of recursion equations, and also suggests the introduction of hidden or nameless nodes in a term graph. Hidden nodes can be used only once. The general framework of term graphs with copying is compared with the more restricted copying facilities embodied in the µ-rule, and translations are given between term graphs and µ-expressions. Using these, a proof system is given for µ-expressions that is complete for the semantics given by infinite tree unwinding. Next, orthogonal term graph rewrite ...

We introduce several approaches for solving higher-order equational problems by higher-order narrowing and give first completeness results. The results apply to higher-order functional-logic programming languages and to higher-order unification modulo a higher-order equational theory. We lift the general notion of first-order narrowing to so-called higher-order patterns and argue that the full higher-order case is problematic. Integrating narrowing into unification, called lazy narrowing, can avoid these problems and can be adapted to the full higher-order case. For the second-order case, we develop a version where the needed second-order unification remains decidable. Finally we discuss a method that combines both approaches by using narrowing on higher-order patterns with full higher-order constraints.

In this paper two formats of higher-order rewriting are compared: Combinatory Reduction Systems introduced by Klop [Klo80] and Higher-order Rewrite Systems defined by Nipkow [Nipa]. Although it always has been obvious that both formats are closely related to each other, up to now the exact relationship between them has not been clear. This was an unsatisfying situation since it meant that proofs for much related frameworks were given twice. We present two translations, one from Combinatory Reduction Systems into Higher-Order Rewrite Systems and one vice versa, based on a detailed comparison of both formats. Since the translations are very `neat' in the sense that the rewrite relation is preserved and (almost) reflected, we can conclude that as far as rewrite theory is concerned, Combinatory Reduction Systems and Higher-Order Rewrite Systems are equivalent, the only difference being that Combinatory Reduction Systems employ a more `lazy' evaluation strategy. Moreover, due to this result...

We introduce Context-sensitive Conditional Expression Reduction Systems (CERS) by extending and generalizing the notion of conditional TRS to the higher order case. We justify our framework in two ways. First, we define orthogonality for CERSs and show that the usual results for orthogonal systems (finiteness of developments, confluence, permutation equivalence) carry over immediately. This can be used e.g. to infer confluence from the subject reduction property in several typed -calculi possibly enriched with pattern-matching definitions. Second, we express several proof and transition systems as CERSs. In particular, we give encodings of Hilbert-style proof systems, Gentzen-style sequent-calculi, rewrite systems with rule priorities, and the ß-calculus into CERSs. This last encoding is an (important) example of real context-sensitive rewriting. 1 Introduction A term rewriting system is a pair consisting of an alphabet and a set of rewrite rules. The alphabet is used freely to gene...

Consider Adam and Eve. Count generations starting from them. Supposing that there will always be people, then it's true that for any generation X, eventually there will be people belonging to the next generation X + 1. In this paper the same result is established for the class of higher order pattern rewriting systems. 1 Introduction Consider a set of structures and a set of transformations on them specifying how a structure may be transformed into another one. Suppose the transformations are of the following form: first a structure is decomposed into substructures, next some substructure is replaced by another one, and finally the substructures are composed into a structure again. (destroy) The parts of the initial structure eliminated in the course of the transformation (i.e. the parts of the replaced substructure as well as the parts eliminated in the initial decomposition) can be thought of as being destroyed . (create) The parts of the final structure introduced in the cou...

. The class of orthogonal rewriting systems (rewriting systems where rewrite steps cannot depend on one another) is the main class of not-necessarily-terminating rewriting systems for which confluence is known to hold. Huet and Toyama have shown that for left-linear firstorder term rewriting systems (TRSs) the orthogonality restriction can be relaxed somewhat by allowing critical pairs (arising from maximally general ways of dependence between steps), but requiring them to be parallel closed. We extend these results by replacing the parallel closed condition by a development closed condition. This also permits to generalise them to higher-order term rewriting, yielding a confluence criterion for Klop's combinatory reduction systems (CRSs), Khasidashvili's expression reduction systems (ERSs), and Nipkow's higher-order pattern rewriting systems (PRSs). 1 Introduction This paper is concerned with a method to prove confluence of rewriting systems. It's an extension of some co...

. We define a class of hyperbalanced l-terms by imposing syntactic constraints on the construction of l-terms, and show that such terms are strongly normalizing. Furthermore, we show that for any hyperbalanced term, the total number of superdevelopments needed to compute its normal form can be statically determined at the beginning of reduction. To obtain the latter result, we develop an algorithm that, in a hyperbalanced term M, statically detects all inessential (or unneeded)subterms which can be replaced by fresh variables without effecting the normal form of M; that is, full garbage collection can be performed before starting the reduction. Finally, we show that, modulo a restricted h-expansion, all simply typable l-terms are hyperbalanced, implying importance of the class of hyperbalanced terms. 1 Introduction The termination of b-reduction for typed terms is one of the most studied topics in l- calculus. After classical proofs of Tait [21] and Girard [8], many interesting proo...

We introduce Context-sensitive Conditional Expression Reduction Systems (CERS) by extending and generalizing the notion of conditional TRS to the higher order case. We justify our framework in two ways. First, we define orthogonality for CERSs and show that the usual results for orthogonal systems (finiteness of developments, confluence, permutation equivalence) carry over immediately. This can be used e.g. to infer confluence from the subject reduction property in several typed λ-calculi possibly enriched with pattern-matching definitions. Second, we express several proof and transition systems as CERSs. In particular, we give encodings of Hilbert-style proof systems, Gentzen-style sequent-calculi, rewrite systems with rule priorities, and the π-calculus into CERSs. This last encoding is an important example of real context-sensitive rewriting. ○c

Abstract. To perform higher-order matching, we need to decide the βη-equivalence on λ-terms. The first way to do it is to use simply typed λ-calculus and this is the usual framework where higher-order matching is performed. Another approach consists in deciding a restricted equivalence. This restricted equivalence can be based on finite developments or more interestingly on finite superdevelopments. We consider higher-order matching modulo (super)developments over untyped λ-terms for which we propose terminating, sound and complete matching algorithms. This is in particular of interest since all second-order β-matches are matches modulo superdevelopments. We further propose a restriction to second-order matching that gives exactly all second-order matches. We finally apply these results in the context of higherorder rewriting. Contents 1. Normalization in the lambda-calculus 3 2. Matching modulo beta (and eta) 9 3. Matching modulo superdevelopments (and eta) 11