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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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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.
Parallel Reduction in Type Free λµCalculus
 KYUSHU UNIVERSITY
, 2000
"... Typed λµcalculus is known to be strongly normalizing and weakly ChurchRosser, and hence confluent. In fact, Parigot formulated a parallel reduction to prove confluency of typed λµcalculus by "TaitandMartinLöf" method. However, the diamond property does not hold for his parallel reduc ..."
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Typed λµcalculus is known to be strongly normalizing and weakly ChurchRosser, and hence confluent. In fact, Parigot formulated a parallel reduction to prove confluency of typed λµcalculus by "TaitandMartinLöf" method. However, the diamond property does not hold for his parallel reduction. The confluency for typefree λµcalculus cannot be derived from that of typed λµcalculus and is not known. We analyzed granualities of the reduction rules. We consider a renaming and consecutive structural reductions as one step parallel reduction, and show that the new formulation of parallel reduction has the diamond property, which yields the correct proof of confluency of type free λµcalculus. The diamond property of new parallel reduction is also shown for the callbyvalue version of λµcalculus contains the symmetric structural reduction rule.
Confluence of Untyped Lambda Calculus Via Simple Types
"... We present a new proof of confluence of the untyped lambda calculus by embedding untyped lambda terms into simply typed lambda terms. This embedding allows us to define a reduction on all lambda terms, whose transitive closure is the betareduction, using an auxiliary reduction and the betareductio ..."
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We present a new proof of confluence of the untyped lambda calculus by embedding untyped lambda terms into simply typed lambda terms. This embedding allows us to define a reduction on all lambda terms, whose transitive closure is the betareduction, using an auxiliary reduction and the betareduction on simply typed lambda terms. The confluence of the auxiliary reduction makes explicit the joining of the sets of redexes to be reduced. This embedding allows us to use the confluence of betareduction on simply typed lambda terms and thus prove the confluence of the reduction defined. As a consequence we obtain the confluence of betareduction in the untyped lambda calculus.
Notes on the Simply Typed Lambda Calculus
, 1998
"... Contents 1 Deduction 11 1.1 Inference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.2 Adding extra axioms . . . . . . . . . . . . . . . . . . . . . . . 12 1.1.3 Semantics for Inference System ..."
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Contents 1 Deduction 11 1.1 Inference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.2 Adding extra axioms . . . . . . . . . . . . . . . . . . . . . . . 12 1.1.3 Semantics for Inference Systems . . . . . . . . . . . . . . . . 12 1.1.4 Formal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.1.5 Rules of Inference . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2 Intuitionistic Implication . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.1 A Hilbertstyle formal system, H . . . . . . . . . . . . . . . . 14 1.2.2 Natural Deduction . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.3 Sequent Formulation, ND, of Natural Deduction . . . . . . . 17 1.2.4 Normal ND treeproofs . . . . . . . . . . . . . . . . . . . . . 17 1.2.5 Sequent Calculus SC . . . . . . . . . . . .
A New Formulation of the Catch/Throw Mechanism
 Second Fuji International Workshop on Functional and Logic Programming
, 1997
"... The catch/throw mechanism in Common Lisp gives a simple control structure for nonlocal exits. Nakano[7, 9] and Sato[13] proposed intuitionistic calculi with inference rules which give logical interpretations of the catch/throwconstructs. Although the calculi are theoretically wellfounded, we c ..."
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The catch/throw mechanism in Common Lisp gives a simple control structure for nonlocal exits. Nakano[7, 9] and Sato[13] proposed intuitionistic calculi with inference rules which give logical interpretations of the catch/throwconstructs. Although the calculi are theoretically wellfounded, we cannot use the catch/throw mechanism for handling runtime errors in a meaningful way, because of the sidecondition of the implicationintroduction rule (the formulation rule of the abstract). This deficiency is critical if we use higherorder functions with the catch/throw mechanism. In this paper, we propose a new formulation of catch/throw calculi, which has no sidecondition on the implicationintroduction rule. By restricting the types of thrown terms to data types (nonfunctional types) instead, we obtain a strongly normalizing calculus for the catch/throw mechanism where we can write higherorder functions which handles runtime errors. 1. Introduction Recently, control st...
A Proof Theoretical Account of Continuation Passing Style
 In CSL ’02: Proceedings of the 16th International Workshop and 11th Annual Conference of the EACSL on Computer Science Logic
, 2002
"... We study "classical proofs as programs" paradigm in CallBy Value (CBV) setting. Specifically, we show the CBV normalization for CND (Parigot 92) can be simulated by the cutelimination procedure for LKQ (DanosJoinetSchellinx 93), namely the qprotocol. We use proofterm assignment s ..."
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We study "classical proofs as programs" paradigm in CallBy Value (CBV) setting. Specifically, we show the CBV normalization for CND (Parigot 92) can be simulated by the cutelimination procedure for LKQ (DanosJoinetSchellinx 93), namely the qprotocol. We use proofterm assignment system to prove this fact. The term calculus for CND we use follows Parigot's #calculus with new CBV normalization procedure. A new term calculus for LKQ is presented as a variant of #calculus with a letconstruct. We then define a translation from CND into LKQ and prove simulation theorem. We also show the translation we use can be thought of a familiar CBV CPStranslation without translation on types.
Reviewing the classical and the de Bruijn notation for λcalculus and pure type systems
 Logic and Computation
, 2001
"... This article is a brief review of the type free λcalculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λcalculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentat ..."
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This article is a brief review of the type free λcalculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λcalculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the λcalculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn's notation for the λcalculus is introduced and some of its advantages are outlined.
A Computationally Adequate Model for Overloading Via DomainValued Functors
, 1993
"... this paper, we construct a domain theoretic model of ..."
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this paper, we construct a domain theoretic model of
Operational Subsumption, an Ideal Model of Subtyping
, 1998
"... In a previous paper we have defined a semantic preorder called operational subsumption, which compares terms according to their error generation behaviour. Here we apply this abstract framework to a concrete language, namely the AbadiCardelli object calculus. Unlike most semantic studies of objects ..."
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In a previous paper we have defined a semantic preorder called operational subsumption, which compares terms according to their error generation behaviour. Here we apply this abstract framework to a concrete language, namely the AbadiCardelli object calculus. Unlike most semantic studies of objects, which deal with typed equalities and therefore require explicitly typed languages, we start here from a untyped world. Type inference is introduced in a second step, together with an ideal model of types and subtyping. We show how this approach flexibly accommodates for several variants, and finally propose a novel semantic interpretation of structural subtyping as embeddingprojection pairs. 1 Introduction In a previous paper [10] we have defined a semantic preorder called operational subsumption, which compares terms according to their error generation behaviour. Together with the technical device of labeled reductions, used as a syntactic characterization of finite approximations, thi...
Type Assigment Systems for Lambda Calculi and for the Lambda Calculus of Objects
, 1996
"... Data Types and ExistentialTypes : : : : : : : : : : : : : : : : 108 5.2.1 The Existential Model of Pierce and Turner : : : : : : : : : : : : 110 5.2.2 Methods and ObjectTypes : : : : : : : : : : : : : : : : : : : : : 111 5.2.3 Methods and Objects : : : : : : : : : : : : : : : : : : : : : : : : : 1 ..."
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Data Types and ExistentialTypes : : : : : : : : : : : : : : : : 108 5.2.1 The Existential Model of Pierce and Turner : : : : : : : : : : : : 110 5.2.2 Methods and ObjectTypes : : : : : : : : : : : : : : : : : : : : : 111 5.2.3 Methods and Objects : : : : : : : : : : : : : : : : : : : : : : : : : 112 5.2.4 Methods and Message Send : : : : : : : : : : : : : : : : : : : : : 112 5.2.5 Classes and Inheritance : : : : : : : : : : : : : : : : : : : : : : : : 113 5.2.6 Conclusions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 115 5.3 The Primitive Object Calculus of Abadi and Cardelli : : : : : : : : : : : 116 5.3.1 Syntax and Operational Semantics : : : : : : : : : : : : : : : : : 117 5.3.2 The Type System: a Survey : : : : : : : : : : : : : : : : : : : : : 119 5.3.3 Adding Subtyping : : : : : : : : : : : : : : : : : : : : : : : : : : : 121 5.3.4 Adding RecursiveTypes : : : : : : : : : : : : : : : : : : : : : : : 122 CONTENTS 3 6 The Lambda Calculus of Objects 125 6.1 The obj : Syntax and Semantics : : : : : : : : : : : : : : : : : : : : : : : 126 6.1.1 Syntax of the Core Language : : : : : : : : : : : : : : : : : : : : 126 6.1.2 The Operational Semantics of obj : : : : : : : : : : : : : : : : : 127 6.1.3 Examples of Objects, Inheritance and SelfReferences : : : : : : : 129 6.2 Static Type System : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 131 6.2.1 Static and Strong Typing : : : : : : : : : : : : : : : : : : : : : : 131 6.2.2 MessageSend and Method Specialization : : : : : : : : : : : : : : 131 6.2.3 Operational Equivalence and ObjectsTypes : : : : : : : : : : : : 132 6.2.4 Syntax of the Type System : : : : : : : : : : : : : : : : : : : : : 133 6.2.5 Analysis of the Main Typing Rules : : : : : : : : : : : : : : : : : 134 6.2.6 Example of ...