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 beta-reduction, using an auxiliary reduction and the beta-reduction 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 beta-reduction on simply typed lambda terms and thus prove the confluence of the reduction defined. As a consequence we obtain the confluence of beta-reduction in the untyped lambda calculus.

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 fi-reduction steps in the standard fi-reduction sequences obtained from transforming any given fi-reduction 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 fi-reduction steps in fi-reduction 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 fi-reduction 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 fi-reduction sequence f...

This paper surveys a part of the theory of fi-reduction in λ-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 λ-terms (when possible), and with perpetual redexes, i.e., redexes whose contraction in λ-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 λ-calculus and type theory.

The catch/throw mechanism in Common Lisp gives a simple control structure for non-local exits. Nakano[7, 9] and Sato[13] proposed intuitionistic calculi with inference rules which give logical interpretations of the catch/throw-constructs. Although the calculi are theoretically well-founded, we cannot use the catch/throw mechanism for handling run-time errors in a meaningful way, because of the side-condition of the implication-introduction rule (the formulation rule of the -abstract). This deficiency is critical if we use higher-order functions with the catch/throw mechanism. In this paper, we propose a new formulation of catch/throw calculi, which has no side-condition on the implication-introduction rule. By restricting the types of thrown terms to data types (non-functional types) instead, we obtain a strongly normalizing calculus for the catch/throw mechanism where we can write higher-order functions which handles run-time errors. 1. Introduction Recently, control st...

Contents 1 Deduction 1-1 1.1 Inference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1 1.1.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1 1.1.2 Adding extra axioms . . . . . . . . . . . . . . . . . . . . . . . 1-2 1.1.3 Semantics for Inference Systems . . . . . . . . . . . . . . . . 1-2 1.1.4 Formal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3 1.1.5 Rules of Inference . . . . . . . . . . . . . . . . . . . . . . . . 1-3 1.2 Intuitionistic Implication . . . . . . . . . . . . . . . . . . . . . . . . . 1-4 1.2.1 A Hilbert-style formal system, H . . . . . . . . . . . . . . . . 1-4 1.2.2 Natural Deduction . . . . . . . . . . . . . . . . . . . . . . . . 1-5 1.2.3 Sequent Formulation, ND, of Natural Deduction . . . . . . . 1-7 1.2.4 Normal ND tree-proofs . . . . . . . . . . . . . . . . . . . . . 1-7 1.2.5 Sequent Calculus SC . . . . . . . . . . . .

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 cut-elimination procedure for LKQ (Danos-Joinet-Schellinx 93), namely the q-protocol. We use proof-term 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 let-construct. 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 CPS-translation without translation on types.

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 Abadi-Cardelli 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 embedding-projection 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...

this paper, we construct a domain theoretic model of

Data Types and Existential-Types : : : : : : : : : : : : : : : : 108 5.2.1 The Existential Model of Pierce and Turner : : : : : : : : : : : : 110 5.2.2 Methods and Object-Types : : : : : : : : : : : : : : : : : : : : : 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 Recursive-Types : : : : : : : : : : : : : : : : : : : : : : : 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 Self-References : : : : : : : 129 6.2 Static Type System : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 131 6.2.1 Static and Strong Typing : : : : : : : : : : : : : : : : : : : : : : 131 6.2.2 Message-Send and Method Specialization : : : : : : : : : : : : : : 131 6.2.3 Operational Equivalence and Objects-Types : : : : : : : : : : : : 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 ...

This article is a brief review of the type free λ-calculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λ-calculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the λ-calculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn's notation for the λ-calculus is introduced and some of its advantages are outlined.