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Contextsensitive Conditional Expression Reduction Systems
 In Proc. of the International Workshop on Graph Rewriting and Computation, SEGRAGRA'95
, 1995
"... We introduce Contextsensitive Conditional Expression Reduction Systems (CERS) by extending and generalizing the notion of conditional TRS to the higher order case. We justify our framework in two ways. First, we define orthogonality for CERSs and show that the usual results for orthogonal systems ..."
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Cited by 12 (4 self)
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We introduce Contextsensitive Conditional Expression Reduction Systems (CERS) by extending and generalizing the notion of conditional TRS to the higher order case. We justify our framework in two ways. First, we define orthogonality for CERSs and show that the usual results for orthogonal systems (finiteness of developments, confluence, permutation equivalence) carry over immediately. This can be used e.g. to infer confluence from the subject reduction property in several typed calculi possibly enriched with patternmatching definitions. Second, we express several proof and transition systems as CERSs. In particular, we give encodings of Hilbertstyle proof systems, Gentzenstyle sequentcalculi, rewrite systems with rule priorities, and the ßcalculus into CERSs. This last encoding is an (important) example of real contextsensitive rewriting. 1 Introduction A term rewriting system is a pair consisting of an alphabet and a set of rewrite rules. The alphabet is used freely to gene...
Perpetual Reductions in λCalculus
, 1999
"... This paper surveys a part of the theory of fireduction 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 red ..."
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Cited by 7 (0 self)
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This paper surveys a part of the theory of fireduction 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 typefree 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 ...
Perpetuality and Uniform Normalization in Orthogonal Rewrite Systems
 INFORMATION AND COMPUTATION
"... 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 ..."
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Cited by 7 (2 self)
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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 Contextsensitive Conditional Expression Reduction Systems (CCERSs) and prove confluence for orthogonal CCERSs. We then define a perpetual onestep 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...
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.
Oostrom, Uniform normalisation beyond orthogonality
 Proceedings of the Twelfth International Conference on Rewriting Techniques and Applications (RTA ’01), Lecture Notes in Computer Science (2001
, 2001
"... Abstract. A rewrite system is called uniformly normalising if all its steps are perpetual, i.e. are such that if s → t and s has an infinite reduction, then t has one too. For such systems termination (SN) is equivalent to normalisation (WN). A wellknown fact is uniform normalisation of orthogonal ..."
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Cited by 4 (0 self)
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Abstract. A rewrite system is called uniformly normalising if all its steps are perpetual, i.e. are such that if s → t and s has an infinite reduction, then t has one too. For such systems termination (SN) is equivalent to normalisation (WN). A wellknown fact is uniform normalisation of orthogonal nonerasing term rewrite systems, e.g. the λIcalculus. In the present paper both restrictions are analysed. Orthogonality is seen to pertain to the linear part and nonerasingness to the nonlinear part of rewrite steps. Based on this analysis, a modular proof method for uniform normalisation is presented which allows to go beyond orthogonality. The method is shown applicable to biclosed first and secondorder term rewrite systems as well as to a λcalculus with explicit substitutions. 1
Perpetuality and Uniform Normalization
 In Proc. of the 6 th International Conference on Algebraic and Logic Programming, ALP'97
, 1997
"... . We define a perpetual onestep reduction strategy which enables one to construct minimal (w.r.t. L'evy's ordering \Theta on reductions) infinite reductions in Conditional Orthogonal Expression Reduction Systems. We use this strategy to derive two characterizations of perpetual redexes, i.e., redex ..."
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Cited by 4 (2 self)
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. We define a perpetual onestep reduction strategy which enables one to construct minimal (w.r.t. L'evy's ordering \Theta on reductions) infinite reductions in Conditional Orthogonal Expression Reduction Systems. We use this strategy to derive two characterizations of perpetual redexes, i.e., redexes whose contractions retain the existence of infinite reductions. These characterizations generalize existing related criteria for perpetuality of redexes. We give a number of applications of our results, demonstrating their usefulness. In particular, we prove equivalence of weak and strong normalization (the uniform normalization property) for various restricted calculi, which cannot be derived from previously known perpetuality criteria. 1 Introduction The objective of this paper is to study sufficient conditions for uniform normalization, UN, of a term in an orthogonal (first or higherorder) rewrite system, and for the UN property of the rewrite system itself. Here a term is UN if ei...
Normalization of Typable Terms by Superdevelopments
 Computer Science Logic'98, Springer LNCS 1584
, 1999
"... . We define a class of hyperbalanced lterms by imposing syntactic constraints on the construction of lterms, and show that such terms are strongly normalizing. Furthermore, we show that for any hyperbalanced term, the total number of superdevelopments needed to compute its normal form can be stati ..."
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Cited by 2 (1 self)
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. We define a class of hyperbalanced lterms by imposing syntactic constraints on the construction of lterms, and show that such terms are strongly normalizing. Furthermore, we show that for any hyperbalanced term, the total number of superdevelopments needed to compute its normal form can be statically determined at the beginning of reduction. To obtain the latter result, we develop an algorithm that, in a hyperbalanced term M, statically detects all inessential (or unneeded)subterms which can be replaced by fresh variables without effecting the normal form of M; that is, full garbage collection can be performed before starting the reduction. Finally, we show that, modulo a restricted hexpansion, all simply typable lterms are hyperbalanced, implying importance of the class of hyperbalanced terms. 1 Introduction The termination of breduction for typed terms is one of the most studied topics in l calculus. After classical proofs of Tait [21] and Girard [8], many interesting proo...
Properties of Infinite Reduction Paths in Untyped λCalculus
"... this paper is to formalize the two ..."
Least Upper Bounds on the Size of ChurchRosser Diagrams in Term Rewriting and λCalculus
"... Abstract. We study the ChurchRosser property—which is also known as confluence—in term rewriting and λcalculus. Given a system R and a peak t ∗ ← s → ∗ t ′ in R, we are interested in the length of the reductions in the smallest corresponding valley t → ∗ s ′ ∗ ← t ′ as a function vsR(m, n) of ..."
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Abstract. We study the ChurchRosser property—which is also known as confluence—in term rewriting and λcalculus. Given a system R and a peak t ∗ ← s → ∗ t ′ in R, we are interested in the length of the reductions in the smallest corresponding valley t → ∗ s ′ ∗ ← t ′ as a function vsR(m, n) of the size m of s and the maximum length n of the reductions in the peak. For confluent term rewriting systems (TRSs), we prove the (expected) result that vsR(m, n) is a computable function. Conversely, for every total computable function ϕ(n) there is a TRS with a single term s such that vsR(s, n) ≥ ϕ(n) for all n. In contrast, for orthogonal term rewriting systems R we prove that there is a constant k such that vsR(m, n) is bounded from above by a function exponential in k and independent of the size of s. For λcalculus, we show that vsR(m, n) is bounded from above by a function contained in the fourth level of the Grzegorczyk hierarchy. 1