. 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...

. 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 ...

) Zurab Khasidashvili 1 and John Glauert 2 1 NTT Basic Research Laboratories, Atsugi, Kanagawa, 243-01, Japan zurab@theory.brl.ntt.co.jp 2 School of Information Systems, UEA, Norwich NR4 7TJ England jrwg@sys.uea.ac.uk ??? Abstract. We describe an event-style (or poset) semantics for conflict-free rewrite systems, such as the -calculus, and other stable transition systems with a residual relation. Our interpretation is based on considering redexfamilies as events. It treats permutation-equivalent reductions as representing the same concurrent computation. Due to erasure of redexes, Event Structures are inadequate for such an interpretation. We therefore extend the Prime Event Structure model by axiomatizing permutation-equivalence on finite configurations in two different ways, for the conflict-free case, and show that these extended models are equivalent to known stable transition models with axiomatized residual and family relations. 1 Introduction The goal of this paper is to...

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 Reduction 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 Prime Event Structures with an extra relation expressing (in)essentiality of events. A version of this paper appears in Proc. of CAAP'96 [GlKh96]. c fl J. Glauert & Z. Khasidashvili, UEA Norwich, 1996 1 Supported by the Engineering and Physical Sciences Research Council of Great Britain under grant GR/H 41300 1 Introduction A normalizable term, i...

We study normalization relative to desirable sets S of `normal forms' by generalizing Huet&L'evy theory of `normalization by neededness'. We impose 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. Further, we prove existence of minimal normalizing reductions for regular stable sets of normal forms. For example, the sets of normal forms, head-normal-forms, and weak head-normal-forms, in the -calculus, are all stable and regular. Finally, we generalize L'evy's Optimality theorem to the case of all stable sets of normal forms, and establish a relationship between relative minimal and optimal reductions, revealing a conflict between minimality and optimality of a reduction -- for regular stable sets of normal forms, a term need not posses a reduction that is minim...