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Vries. An extensional Böhm model
 In Proceedings of the 13th International Conference on Rewriting Techniques and Applications (RTA 2002
, 2002
"... Abstract. We show the existence of an infinitary confluent and normalising extension of the finite extensional lambda calculus with beta and eta. Besides infinite beta reductions also infinite eta reductions are possible in this extension, and terms without head normal form can be reduced to bottom ..."
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Cited by 15 (7 self)
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Abstract. We show the existence of an infinitary confluent and normalising extension of the finite extensional lambda calculus with beta and eta. Besides infinite beta reductions also infinite eta reductions are possible in this extension, and terms without head normal form can be reduced to bottom. As corollaries we obtain a simple, syntax based construction of an extensional Böhm model of the finite lambda calculus; and a simple, syntax based proof that two lambda terms have the same semantics in this model if and only if they have the same etaBöhm tree if and only if they are observationally equivalent wrt to beta normal forms. The confluence proof reduces confluence of beta, bottom and eta via infinitary commutation and postponement arguments to confluence of beta and bottom and confluence of eta. We give counterexamples against confluence of similar extensions based on the identification of the terms without weak head normal form and the terms without top normal form (rootactive terms) respectively. 1
Descendants and Origins in Term Rewriting
"... In this paper we treat various aspects of a notion that is central in term rewriting, namely that of descendants or residuals. We address both first order term rewriting and calculus, their finitary as well as their infinitary variants. A recurrent theme is the Parallel Moves Lemma. Next to the ..."
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Cited by 9 (1 self)
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In this paper we treat various aspects of a notion that is central in term rewriting, namely that of descendants or residuals. We address both first order term rewriting and calculus, their finitary as well as their infinitary variants. A recurrent theme is the Parallel Moves Lemma. Next to the classical notion of descendant, we introduce an extended version, known as `origin tracking'. Origin tracking has many applications. Here it is employed to give new proofs of three classical theorems: the Genericity Lemma in calculus, the theorem of Huet and L'evy on needed reductions in first order term rewriting, and Berry's Sequentiality Theorem in (infinitary) calculus.
Perpetual Reductions in λCalculus
, 1999
"... This paper surveys a part of the theory of fireduction in λcalculus 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 λterms (when possible), and with perpetual r ..."
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Cited by 6 (0 self)
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This paper surveys a part of the theory of fireduction in λcalculus 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 λterms (when possible), and with perpetual redexes, i.e., redexes whose contraction in λterms 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 λcalculus and type theory.
Effective Longest and Infinite Reduction Paths in Untyped λCalculi
, 1996
"... A maximal reduction strategy in untyped λcalculus computes for a term a longest (finite or infinite) reduction path. Some types of reduction strategies in untyped λcalculus have been studied, but maximal strategies have received less attention. We give a systematic study of maximal strategies, rec ..."
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Cited by 5 (2 self)
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A maximal reduction strategy in untyped λcalculus computes for a term a longest (finite or infinite) reduction path. Some types of reduction strategies in untyped λcalculus have been studied, but maximal strategies have received less attention. We give a systematic study of maximal strategies, recasting the few known results in our framework and giving a number of new results, the most important of which is an effective maximal strategy in fij. We also present a number of applications illustrating the relevance and usefulness of maximal strategies.
A Theory of Explicit Substitutions with Safe and Full Composition
 Logical Methods in Computer Science
"... Vol. 5 (3:1) 2009, pp. 1–29 ..."
Strong normalization from weak normalization in typed λcalculi
 Information and Computation
, 1997
"... For some typed λcalculi it is easier to prove weak normalization than strong normalization. Techniques to infer the latter from the former have been invented over the last twenty years by Nederpelt, Klop, Khasidashvili, Karr, de Groote, and Kfoury and Wells. However, these techniques infer strong n ..."
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Cited by 4 (1 self)
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For some typed λcalculi it is easier to prove weak normalization than strong normalization. Techniques to infer the latter from the former have been invented over the last twenty years by Nederpelt, Klop, Khasidashvili, Karr, de Groote, and Kfoury and Wells. However, these techniques infer strong normalization of one notion of reduction from weak normalization of a more complicated notion of reduction. This paper presents a new technique to infer strong normalization of a notion of reduction in a typed λcalculus from weak normalization of the same notion of reduction. The technique is demonstrated to work on some wellknown systems including secondorder λcalculus and the system of positive, recursive types. It gives hope for a positive answer to the BarendregtGeuvers conjecture stating that every pure type system which is weakly normalizing is also strongly normalizing. The paper also analyzes the relationship between the techniques mentioned above, and reviews, in less detail, other techniques for proving strong normalization.
Minimality in a Linear Calculus with Iteration Abstract
"... System L is a linear version of Gödel’s System T, where the λcalculus is replaced with a linear calculus; or alternatively a linear λcalculus enriched with some constructs including an iterator. There is thus at the same time in this system a lot of freedom in reduction and a lot of information ab ..."
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System L is a linear version of Gödel’s System T, where the λcalculus is replaced with a linear calculus; or alternatively a linear λcalculus enriched with some constructs including an iterator. There is thus at the same time in this system a lot of freedom in reduction and a lot of information about resources, which makes it an ideal framework to start a fresh attempt at studying reduction strategies in λcalculi. In particular, we show that callbyneed, the standard strategy of functional languages, can be defined directly and effectively in System L, and can be shown minimal among weak strategies. 1
Reduction Strategies and Acyclicity
"... Abstract. In this paper we review some wellknown theory about reduction strategies of various kinds: normalizing, outermostfair, cofinal, ChurchRosser. A stumbling block in the definition of such strategies is the presence of reduction cycles that may ‘trap ’ a strategy as it is memoryfree. We e ..."
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Abstract. In this paper we review some wellknown theory about reduction strategies of various kinds: normalizing, outermostfair, cofinal, ChurchRosser. A stumbling block in the definition of such strategies is the presence of reduction cycles that may ‘trap ’ a strategy as it is memoryfree. We exploit a recently (re)discovered fact that there are no reduction cycles in orthogonal rewrite systems when each term has a normal form, in order to enhance some of the theorems on strategies, both with respect to their scope and the proof of their correctness. 1
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"... Reduce to the max A cofinal strategy for weakly orthogonal higherorder pattern rewrite systems (WOPRSs). Notions and results needed for WOPRSs can be found in [Oos95, Oos99]. Definition 1 A simultaneous set U of redex(occurrenc)es in a term s is maximal, if any redex v in s overlaps the head of so ..."
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Reduce to the max A cofinal strategy for weakly orthogonal higherorder pattern rewrite systems (WOPRSs). Notions and results needed for WOPRSs can be found in [Oos95, Oos99]. Definition 1 A simultaneous set U of redex(occurrenc)es in a term s is maximal, if any redex v in s overlaps the head of some redex in U. By the tree structure of terms maximal sets can be constructed insideout, but need not be unique. Lemma 2 If s −→V ◦ t then t − → ◦ s ∗ , where s ∗ is obtained from s by contracting a maximal set U. Proof By maximality of U and simultaneity of V, we can define an injection ι mapping every redex v ∈ V to a redex ι(v) ∈ U such that v overlaps the head of ι(v). By weak orthogonality s − → ◦ ι(V) t, so t − → ◦ U/ι(V) s ∗ [Oos95, Theorem 5, Prism]. ✷ Note that distinct maximal sets may exist, but these must be equipollent hence lead to the same result, justifying our notation s ∗. The lemma is a generalisation of [BBKV76, Lemma 3.2.2], [Tak95, Section 1, property (5)], [Nip96, Section 5.2], and [Raa96, Lemma 5.3.3]. Only slightly relaxing weakorthogonality invalidates the theorem, as witnessed by the term f(a) in the parallelclosed TRS a → a, f(a) → f(b). As a standard corollary, we have that the maximal strategy is (hyper)(head)normalising and cofinal for WOPRSs, where a strategy is maximal if it contracts maximal sets. This generalises (results for) the GrossKnuth strategy for λcalculus and the fullsubstitution strategy for orthogonal TRSs in the papers cited. Note that it also applies to λβηcalculus, i.e. fullextendedness is not required, so the result does not follow from [Oos99, Theorem 1]. The proof of the lemma avoids notions such as chain of λ’s [BBKV76], chain [Vri87] and cluster [BKV98] of redexes, which were introduced to set up a satisfactory residual theory for λβηcalculus, having the same ‘nice ’ properties as that of λβcalculus. Instead the proof is based on the more general notion of weakly orthogonal projection [Oos99], which applies to all WOPRSs.