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HigherOrder Rewriting
 12th Int. Conf. on Rewriting Techniques and Applications, LNCS 2051
, 1999
"... This paper will appear in the proceedings of the 10th international conference on rewriting techniques and applications (RTA'99). c flSpringer Verlag. ..."
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Cited by 20 (1 self)
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This paper will appear in the proceedings of the 10th international conference on rewriting techniques and applications (RTA'99). c flSpringer Verlag.
Comparing Combinatory Reduction Systems and HigherOrder Rewrite Systems
, 1993
"... In this paper two formats of higherorder rewriting are compared: Combinatory Reduction Systems introduced by Klop [Klo80] and Higherorder Rewrite Systems defined by Nipkow [Nipa]. Although it always has been obvious that both formats are closely related to each other, up to now the exact relations ..."
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Cited by 18 (3 self)
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In this paper two formats of higherorder rewriting are compared: Combinatory Reduction Systems introduced by Klop [Klo80] and Higherorder Rewrite Systems defined by Nipkow [Nipa]. Although it always has been obvious that both formats are closely related to each other, up to now the exact relationship between them has not been clear. This was an unsatisfying situation since it meant that proofs for much related frameworks were given twice. We present two translations, one from Combinatory Reduction Systems into HigherOrder Rewrite Systems and one vice versa, based on a detailed comparison of both formats. Since the translations are very `neat' in the sense that the rewrite relation is preserved and (almost) reflected, we can conclude that as far as rewrite theory is concerned, Combinatory Reduction Systems and HigherOrder Rewrite Systems are equivalent, the only difference being that Combinatory Reduction Systems employ a more `lazy' evaluation strategy. Moreover, due to this result...
Finite Family Developments
"... Associate to a rewrite system R having rules l → r, its labelled version R ω having rules l ◦ m+1 → r • , for any natural number m m ∈ ω. These rules roughly express that a lefthand side l carrying labels all larger than m can be replaced by its righthand side r carrying labels all smaller than o ..."
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Cited by 13 (6 self)
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Associate to a rewrite system R having rules l → r, its labelled version R ω having rules l ◦ m+1 → r • , for any natural number m m ∈ ω. These rules roughly express that a lefthand side l carrying labels all larger than m can be replaced by its righthand side r carrying labels all smaller than or equal to m. A rewrite system R enjoys finite family developments (FFD) if R ω is terminating. We show that the class of higher order pattern rewrite systems enjoys FFD, extending earlier results for the lambda calculus and first order term rewrite systems.
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 ...
Upper Bounds for Standardizations and an Application
 The Journal of Symbolic Logic
, 1996
"... We first present a new proof for the standardization theorem, a fundamental theorem in calculus. Since our proof is largely built upon structural induction on lambda terms, we can extract some bounds for the number of fireduction steps in the standard fireduction sequences obtained from transfor ..."
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Cited by 7 (1 self)
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We first present a new proof for the standardization theorem, a fundamental theorem in calculus. Since our proof is largely built upon structural induction on lambda terms, we can extract some bounds for the number of fireduction steps in the standard fireduction sequences obtained from transforming any given fireduction sequences. This result sharpens the standardization theorem and establishes a link between lazy and eager evaluation orders in the context of computational complexity. As an application, we establish a superexponential bound for the number of fireduction steps in fireduction sequences from any given simply typed terms. 1 Introduction The standardization theorem of Curry and Feys [CF58] is a very useful result, stating that if u reduces to v for terms u and v, then there is a standard fireduction from u to v. Using this theorem, we can readily prove the normalization theorem, i.e., a term has a normal form if and only if the leftmost fireduction sequence f...
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.
Evaluation under λAbstraction
, 1996
"... In light of the usual definition of values [15] as terms in weak head normal form (WHNF), a abstraction is regarded as a value, and therefore no expressions under abstraction can get evaluated and the sharing of computation under has to be achieved through program transformations such as lifting ..."
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Cited by 4 (2 self)
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In light of the usual definition of values [15] as terms in weak head normal form (WHNF), a abstraction is regarded as a value, and therefore no expressions under abstraction can get evaluated and the sharing of computation under has to be achieved through program transformations such as lifting and supercombinators. In this paper we generalise the notion of head normal form (HNF) and introduce the definition of generalised head normal form (GHNF). We then define values as terms in GHNF with flexible heads, and study a callbyvalue calculus v hd corresponding to this new notion of values. After establishing a version of normalisation theorem in v hd , we construct an evaluation function eval v hd for v hd which evaluates under  abstraction. We prove that a program can be evaluated in v hd to a term in GHNF if and only if it can be evaluated in the usual calculus to a term in HNF. We also present an operational semantics for v hd via a SECD machine. We argue that l...
A syntactic account of singleton types via hereditary substitution
, 2009
"... We give a syntactic proof of decidability and consistency of equivalence for the singleton type calculus, which lies at the foundation of modern module systems such as that of ML. Unlike existing proofs, which work by constructing a model, our syntactic proof makes few demands on the underlying proo ..."
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Cited by 3 (2 self)
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We give a syntactic proof of decidability and consistency of equivalence for the singleton type calculus, which lies at the foundation of modern module systems such as that of ML. Unlike existing proofs, which work by constructing a model, our syntactic proof makes few demands on the underlying proof theory and mathematical foundation. Consequently, it can be — and has been — entirely formulated in the Twelf metalogic, and provides an important piece of a Twelfformalized typesafety proof for Standard ML. The proof works by translation of the singleton type calculus into a canonical presentation, adapted from work on logical frameworks, in which equivalent terms are written identically. Canonical forms are not preserved under standard substitution, so we employ an alternative definition of substitution called hereditary substitution, which contracts redices that arise during substitution. 1
An Induction Measure on λTerms and Its Applications
, 1996
"... This paper presents a useful induction measure on terms. Combining leftmost reduction with the proper subterm relation, we introduce a new notion called Hrelation for untyped calculus. We then prove the equivalence between strong normalisability and Hnormalisability. Exploiting the new notion, w ..."
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This paper presents a useful induction measure on terms. Combining leftmost reduction with the proper subterm relation, we introduce a new notion called Hrelation for untyped calculus. We then prove the equivalence between strong normalisability and Hnormalisability. Exploiting the new notion, we present some simplified proofs for several fundamental theorems such as finiteness of developments, the conservation theorem for Kcalculus, and the strong normalisation theorem for simply typed calculus. Also a simplified proof of the characterisation theorem on perpetual redexes in [BK82] is included. Compared with other proofs in the literature, all presented proofs are quite concise and perspicuous. Finally, we give a brief comparison between Hrelation and other methods such as perpetual strategies. 1 1. Introduction In calculus and some other rewriting systems, an induction measure on terms usually plays a pivotal role in the proofs of various theorems related to strong normalis...
Separating Developments in λCalculus
, 1996
"... We introduce a proof technique in calculus which can facilitate inductive reasoning on terms by separating certain fidevelopments from other fireductions. We present proofs based on this technique for several fundamental theorems in calculus such as the ChurchRosser theorem, the standardisatio ..."
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We introduce a proof technique in calculus which can facilitate inductive reasoning on terms by separating certain fidevelopments from other fireductions. We present proofs based on this technique for several fundamental theorems in calculus such as the ChurchRosser theorem, the standardisation theorem, the conservation theorem and the normalisation theorem. The appealing features of these proofs lie in their inductive styles and perspicuities. 1. Introduction Proofs based on structural inductions have certain desirable features. They usually enhance comprehensibility, yield more on the meaning of the proven theorems, and can be formalised relatively easily. Unfortunately, many theorems in calculus cannot be proven via structural induction on terms. This is mainly due to the fact that fireduction is not compositional, namely, a fireduction sequence from Mfx := Ng usually cannot be viewed as the composition of some fireduction sequences from M and N since new firedexes may...