The following theorem of Huet and L'evy forms the basis of all results on optimal reduction strategies for orthogonal term rewriting systems: every term not in normal form contains a needed redex, and repeated contraction of needed redexes results in the normal form, if the term under consideration has one. We generalize this theorem to computations to root-stable form and we argue that the resulting notion of root-neededness is more fundamental than (other variants of) neededness when it comes to infinitary normalization. 1 Introduction In this paper we are concerned with reduction strategies for term rewriting systems. A reduction strategy is called normalizing if repeated contraction of the redexes selected by the strategy leads to normal form. O'Donnell [13] showed that the parallel-outermost strategy, which contracts all outermost redexes in parallel, is normalizing for orthogonal term rewriting systems. Parallel-outermost is not an optimal reduction strategy since many of the r...

. We define Higher Order Recursive Program Schemes (HRPSs) by allowing metasubstitutions (as in the -calculus) in righthand sides of function and quantifier definitions. A study of several kinds of similarity of redexes makes it possible to lift properties of (first order) Recursive Program Schemes to the higher order case. The main result is the decidability of weak normalization in HRPSs, which immediately implies that HRPSs do not have full computational power. We analyze the structural properties of HRPSs and introduce several kinds of persistent expression reduction systems (PERSs) that enjoy similar properties. Finally, we design an optimal evaluation procedure for PERSs. 1 Introduction Higher Order Recursive Program Schemes (HRPSs) are recursive definitions of functions, predicates, and quantifiers, considered as rewriting systems. Similar definitions are used when one extends a theory by introducing new symbols [16]. 9aA , (øa(A)=a)A and 9!aA , 9aA 8a8b(A (b=a)A ) a = b) a...

We consider reductions in Orthogonal Expression Reduction Systems (OERS), that is, Orthogonal Term Rewriting Systems with bound variables and substitutions, as in the -calculus. We design a strategy that for any given term t constructs a longest reduction starting from t if t is strongly normalizable, and constructs an infinite reduction otherwise. The Conservation Theorem for OERSs follows easily from the properties of the strategy. We develop a method for computing the length of a longest reduction starting from a strongly normalizable term. We study properties of pure substitutions and several kinds of similarity of redexes. We apply these results to construct an algorithm for computing lengths of longest reductions in strongly persistent OERSs that does not require actual transformation of the input term. As a corollary, we have an algorithm for computing lengths of longest developments in OERSs. 1 Introduction A strategy is perpetual if, given a term t, it constructs an infinit...

. 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...

. We study reductions in orthogonal (left-linear and non-ambiguous) Expression Reduction Systems, a formalism for Term Rewriting Systems with bound variables and substitutions. To generalise the normalization theory of Huet and L'evy, we introduce the notion of neededness with respect to a set of reductions \Pi or a set of terms S so that each existing notion of neededness can be given by specifying \Pi or S. We imposed natural conditions on S, called stability, that are sufficient and necessary for each term not in S-normal form (i.e., not in S) to have at least one S-needed redex, and repeated contraction of S-needed redexes in a term t to lead to an S-normal form of t whenever there is one. Our relative neededness notion is based on tracing (open) components, which are occurrences of contexts not containing any bound variable, rather than tracing redexes or subterms. 1 Introduction Since a normalizable term, in a rewriting system, may have an infinite reduction, it is important to...

. We prove a version of the Standardization Theorem and the Discrete Normalization (DN) Theorem in stable Deterministic Residual Structures, which are Abstract Reduction Systems with axiomatized residual relation, and model orthogonal rewrite systems. The latter theorem gives a strategy for construction of reductions L'evy-equivalent (or permutation-equivalent) to a given, finite or infinite, regular (or semi-linear) reduction, based on the neededness concept of Huet and L'evy. This and other results of this paper add to the understanding of L'evy-equivalence of reductions, and consequently, L'evy's reduction space. We demonstrate how elements of this space can be used to give denotational semantics to known functional languages in an abstract manner. 1 Introduction Long ago, Curry and Feys [CuFe58] proved that repeated contraction of leftmostoutermost redexes in any normalizable -term eventually yields its normal form, even if the term possesses infinite reductions as well. The reaso...

. We design a strategy that for any given term t in an Orthogonal Term Rewriting System (OTRS) constructs a longest reduction starting from t if t is strongly normalizable, and constructs an infinite reduction otherwise. For some classes of OTRSs the strategy is easily computable. We develop a method for finding the least upper bound of lengths of reductions starting from a strongly normalizable term. We give also some applications of our results. 1 Introduction It is shown in O'Donnell [12] that the innermost strategy is perpetual for orthogonal term rewriting systems (OTRSs). That is, contraction of innermost redexes gives an infinite reduction of a given term whenever such a reduction exists. In fact, a strategy that only contracts redexes that do not erase any other redex is perpetual. Moreover, one can even reduce redexes whose erased arguments are strongly normalizable (Klop [10]). For the lambda-calculus, a more subtle perpetual strategy was invented in Barendregt et al. [1]. H...

This paper surveys a part of the theory of fi-reduction in -calculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from -terms (when possible), and with perpetual redexes, i.e., redexes whose contraction in -terms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in -calculus and type theory. 1. Introduction Considerable attention has been devoted to classification of reduction strategies in type-free -calculus [4, 6, 7, 15, 38, 44, 81]---see also [2, Ch. 13]. We are concerned with strategies differing in the length of reduction paths. This paper draws on several sources. In late 1994, van Raamsdonk and Severi [59] and Srensen [66, 67] independently developed ...

We present two characterizations of perpetual redexes, which are redexes whose contractions retain the possibility of infinite reductions. These characterizations generalize and strengthen existing criteria for the perpetuality of redexes in orthogonal Term Rewriting Systems and the -calculus due to Bergstra and Klop, and others. To unify our results with those in the literature, we introduce Context-sensitive Conditional Expression Reduction Systems (CCERSs) and prove confluence for orthogonal CCERSs. We then define a perpetual one-step reduction strategy which enables one to construct minimal (w.r.t. Levy's permutation ordering on reductions) infinite reductions in orthogonal CCERSs. We then prove (1) perpetuality (in a specific context) of a redex whose contraction does not erase potentially infinite arguments, which are possibly finite (i.e., strongly normalizable) arguments that may become infinite after a number of outside steps, and (2) perpetuality (in every con...

. In previous papers, the authors studied normalization relative to desirable sets S of `partial results', where it is shown that such sets must be stable. For example, the sets of normal forms, head-normal-forms, and weak head-normal-forms in the -calculus, are all stable. They showed that, for any stable S, S-needed reductions are S-normalizing. This paper continues the investigation into the theory of relative normalization. In particular, we prove existence of minimal normalizing reductions for regular stable sets of results. All the above mentioned sets are regular. We give a sufficient and necessary criterion for a normalizing reduction (w.r.t. a regular stable S) to be minimal. Finally, we establish a relationship between relative minimal and optimal reductions, revealing a conflict between minimality and optimality: for regular stable sets of results, a term need not possess a reduction that is minimal and optimal at the same time. 1 Introduction The Normalization Theorem in ...