Term Rewriting Systems play an important role in various areas, such as abstract data type specifications, implementations of functional programming languages and automated deduction. In this chapter we introduce several of the basic comcepts and facts for TRS's. Specifically, we discuss Abstract Reduction Systems

In this article we propose an extension of term rewriting techniques to automate the deduction in monotone pre-order theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a bi-rewrite system 〈R⊆, R⊇〉, that is, a pair of rewrite relations −−− → R ⊆ and −−− → R ⊇ , and seek a common term c such that a −−−→ R ⊆ c and b −−−→

We investigate restricted termination and confluence properties of term rewriting systems, in particular weak termination and innermost termination, and their interrelation. New criteria are provided which are sufficient for the equivalence of innermost / weak termination and uniform termination of term rewriting systems. These criteria provide interesting possibilities to infer completeness, i.e. termination plus confluence, from restricted termination and confluence properties. Using these basic results we are also able to prove some new results about modular termination of rewriting. In particular, we show that termination is modular for some classes of innermost terminating and locally confluent term rewriting systems, namely for non-overlapping and even for overlay systems. As an easy consequence this latter result also entails a simplified proof of the fact that completeness is a decomposable property of so-called constructor systems. Furthermore we show how to obtain similar re...

In this paper we present the algebraic--cube, an extension of Barendregt's -cube with first- and higherorder algebraic rewriting. We show that strong normalization is a modular property of all systems in the algebraic--cube, provided that the first-order rewrite rules are non-duplicating and the higher-order rules satisfy the general schema of Jouannaud and Okada. This result is proven for the algebraic extension of the Calculus of Constructions, which contains all the systems of the algebraic--cube. We also prove that local confluence is a modular property of all the systems in the algebraic--cube, provided that the higher-order rules do not introduce critical pairs. This property and the strong normalization result imply the modularity of confluence. 1 Introduction Many different computational models have been developed and studied by theoretical computer scientists. One of the main motivations for the development This research was partially supported by ESPRIT Basic Research Act...

. We design a normalizing strategy for orthogonal term rewriting systems (OTRSs), which is a generalization of the call-by-need strategy of Huet-L'evy [4]. The redexes contracted in our strategy are essential in the sense that they have "descendants" under any reduction of a given term. There is an essential redex in any term not in normal form. We further show that contraction of the innermost essential redexes gives an optimal reduction to normal form, if it exists. We classify OTRSs depending on possible kinds of redex creation as non-creating, persistent, inside-creating, non-left-absorbing, etc. All these classes are decidable. TRSs in these classes are sequential, but they do not need to be strongly sequential. For non-creating and persistent OTRSs, we show that our optimal strategy is efficient as well. 1 Introduction In this paper, we study correct and optimal computations in Orthogonal Term Rewriting Systems (OTRSs). We only consider one-step rewriting strategies, which selec...

We propose an extension of rewriting techniques to derive inclusion relations $a \subseteq b$ between terms built from monotonic operators. Instead of using only a rewriting relation $\REa$ and rewriting $a$ to $b$, we use another rewriting relation $\REb$ as well and seek a common expression $c$ such that $a \REa^* c$ and $b \REb^* c$. Each component of the bi-rewriting system $\pair{\REa}{\REb}$ is allowed to be a subset of the corresponding inclusion $\subseteq$ or $\superseteq$. In order to assure the decidability and completeness of the proof procedure we study the commutativity of $\REa$ and $\REb$. We also extend the existing techniques of rewriting modulo equalities to bi-rewriting modulo a set of inclusions. We present the canonical bi-rewriting system corresponding to the theory of non-distributive lattices.

A notion of type assignment on Curryfied Term Rewriting Systems is introduced that uses Intersection Types of Rank 2, and in which all function symbols are assumed to have a type. Type assignment will consist of specifying derivation rules that describe how types can be assigned to terms, using the types of function symbols. Using a modified unification procedure, for each term the principal pair (of basis and type) will be defined in the following sense: from these all admissible pairs can be generated by chains of operations on pairs, consisting of the operations substitution, copying, and weakening. In general, given an arbitrary typeable CuTRS, the subject reduction property does not hold. Using the principal type for the left-hand side of a rewrite rule, a sufficient and decidable condition will be formulated that typeable rewrite rules should satisfy in order to obtain this property. Introduction In the recent years, several paradigms have been investigated for the implementatio...

In this paper we introduce Curryfied Term Rewriting Systems, and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts and !. Three operations on types -- substitution, expansion, and lifting -- are used to define type assignment, and are proved to be sound. With this result the system is proved closed for reduction. Using a more liberal approach to recursion, we define a general scheme for recursive definitions and prove that, for all systems that satisfy this scheme, every term typeable without using the type-constant ! is strongly normalizable. We also show that, under certain restrictions, all typeable terms have a (weak) head-normal form, and that terms whose type does not contain ! are normalizable.

Introduction Interest in the theory and applications of rewriting has been growing rapidly, as evidenced in part by four conference proceedings #including this one# #15, 26, 41,66#; three workshop proceedings #33, 47, 77#; #ve special journal issues #5,88, 24, 40, 67#; more than ten surveys #2,7,27, 28, 44, 56,57,76, 82, 81#; one edited collection of papers #1#; four monographs #3, 12,55,65#; and seven books #four of them still in progress# #8,9, 35, 54, 60,75, 84#. To encourage and stimulate continued progress in this area, wehave collected #with the help of colleagues# a number of problems that appear to us to be of interest and regarding whichwe do not know the answer. Questions on rewriting and other equational paradigms have been included; manyhave not aged su#ciently to be accorded the appellation #open problem". Wehave limited ourselves to theoretical questions, though there are certainly many additional interesting questions relating to applications and implementation

Rules provide the functionality for constraint enforcement and view maintenance. A provably correct implementation of both issues based on rules, requires confluent and terminating behaviour of the rule set. However, little work has been done so far on the static detection of these properties. In this paper, a design theory for rule sets in an OODBMS is developed. This paper introduces two predicates, viz., Terminate(n) and Independent, which imply respectively termination and confluency. Both predicates are characterised under two kinds of rule execution semantics: set and instance based. The decidability of the predicates is proven and it is shown that set and instance based semantics coincide whenever the rule set is independent and terminates. Moreover, sufficient conditions of low algorithmic complexity for both Terminate(n) and Independent under both kinds of semantics are given. 1991 CR Categories: H.2.1[Information Systems]: Logical Design- data models; H.2.3[Informat...