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

. Rewriting is a computational paradigm that specifies the actions, but not the control. We introduce a hierarchical structure representing, at a high level of abstraction, a form of control. Its application solves a specific problem arising in the design and implementation of inherently sequential, lazy, functional programming languages based on rewriting. For example, we show how to extend the expressive power of Log(F ) and how to improve the efficiency of an implementation of BABEL. Our framework provides a notion of degree of parallelism of an operation and shows that the elements of a necessary set of redexes are related by an and-or relation. Both concepts find application in parallel implementations of rewriting. In an environment in which computations can be executed in parallel we are able to detect sequential computations in order to minimize overheads and/or optimize execution. Conversely, we are able to detect when inherently sequential computations can be executed in para...

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...

Context-sensitive rewriting is a simple restriction of rewriting which is formalized by imposing fixed restrictions on replacements. Such a restriction is given on a purely syntactic basis: it is (explicitly or automatically) specified on the arguments of symbols of the signature and inductively extended to arbitrary positions of terms built from those symbols. Termination is not only preserved but usually improved and several methods have been developed to formally prove it. In this paper, we investigate the definition, properties, and use of context-sensitive rewriting strategies, i.e., particular, fixed sequences of context-sensitive rewriting steps. We study how to define them in order to obtain efficient computations and to ensure that context-sensitive computations terminate whenever possible. We give conditions enabling the use of these strategies for root-normalization, normalization, and infinitary normalization. We show that this theory is suitable for formalizing ...

Orthogonal term rewriting systems (also known as regular systems) provide an elegant framework for programming in equational logic. O'Donnell showed that the parallel-outermost strategy, which replaces all outermost redexes in each step, is complete for such systems. Many of the reductions performed by this strategy could be wasteful in general. A lazy normalization algorithm that completely eliminated these wasteful reductions by reducing only "needed redexes" was later developed by Huet and Levy. However, this algorithm required the input programs to be restricted to the subclass of strongly sequential systems. This is because needed redexes do not exist for all orthogonal programs, and even when they do, they may not be computable. It is therefore quite natural to ask whether it is possible to devise a complete normalization algorithm for the entire class that minimizes (rather than eliminate) the wasteful reductions. In this paper we propose a solution to this problem using the concept of a necessary set of redexes. In such a set, at least one of the redexes must be reduced to normalize a term. We devise an algorithm to compute a necessary set for any term not in normal form and show that a strategy that repeatedly reduces all redexes in such a set is complete for orthogonal programs. We also show that our algorithm is "optimal" among all normalization algorithms that are based on left-hand sides alone. This means that our algorithm is lazy (like Huet-Levy's) on strongly sequential parts of a program and "relaxes laziness minimally" to handle the other parts and thus does not sacrifice generality for the sake of efficiency.

. 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...

Kennaway proved the remarkable result that every (almost) orthogonal term rewriting system admits a computable sequential normalizing reduction strategy. In this paper we present a computable sequential reduction strategy similar in scope, but simpler and more general. Our strategy can be thought of as an outermost-fair-like strategy that is allowed to be unfair to some redex of a term when contracting the redex is useless for the normalization of the term. Unlike the strategy of Kennaway, our strategy does not rely on syntactic restrictions that imply conuence. On the contrary, it can easily be applied to any term rewriting system, and we show that the class of term rewriting systems for which our strategy is normalizing properly includes all (almost) orthogonal systems. Our strategy is more versatile; in case of (almost) orthogonal term rewriting systems, it can be used to detect certain cases of nontermination. Our normalization proof is more accessible than Kennaway's. W...

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

. This paper generalizes the Huet and L'evy theory of normalization by neededness to an abstract setting. We define Stable Deterministic Residual Structures (SDRS) and Deterministic Family Structures (DFS) by axiomatizing some properties of the residual relation and the family relation on redexes in an Abstract Rewriting System. We present two proofs of the Relative Normalization Theorem, one for SDRSs for regular stable sets, and another for DFSs for all stable sets of desirable `normal forms'. We further prove the Relative Optimality Theorem for DFSs. We extend this result to deterministic Computation Structures which are deterministic Event Structures with an extra relation expressing self-essentiality. 1 Introduction A normalizable term, in a rewriting system, may have an infinite reduction, so it is important to have a normalizing strategy which enables one to construct reductions to normal form. It is well known that the leftmost-outermost strategy is normalizing in the -calc...

. Confluent term rewriting systems can be seen as a model for functional computations, in which redexes corresponding to instances of left hand sides of rules are repeatedly replaced by their corresponding right hand side instance. Lazy sequential strategies reduce a given redex in a term if and only if this redex must be reduced by any other sequential strategy computing the normal form of the term. Lazy strategies always terminate when a normal form exist, and are indeed optimal. In a landmark paper, Huet and Levy showed that such a strategy always exists for left linear nonoverlapping rewrite systems that are strongly sequential, a property that they proved decidable for such systems. This paper generalises the result to the case of left-linear, possibly overlapping rewrite systems. 1 Introduction Confluent term rewriting systems can be seen as a model for functional computations, in which redexes corresponding to instances of left hand sides of rules are repeatedly replaced by the...