Abstract not found

We introduce an intersection type system which is a restriction of the intersection type discipline. This restriction leads to a principal type property for normal forms in the classical sense, while retaining the expressivity of the classical discipline. We characterize the structure of principal types of normal forms and give an algorithm that reconstructs normal forms from types. Having shown the equivalence between principal types and normal forms, we define an expansion operation on types which allows us to recover all possible types for any normalizable -term. The contribution of this work is a new and simpler presentation of the intersection type discipline through a purely syntactic and completely characterized notion of principal types.

ion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 80 III A complete example 83 12 ASL: A Small Language 85 12.1 ASL abstract syntax trees : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 85 12.2 Parsing ASL programs : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 86 13 Untyped semantics of ASL programs 91 13.1 Semantic values : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 91 13.2 Semantic functions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 92 13.3 Examples : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 93 14 Encoding recursion 95 14.1 Fixpoint combinators : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 95 14.2 Recursion as a primitive construct : : : : : : : : : : : : : : : : : : : : : : : : : : : : CONTENTS 3 15 Static typing, polymorphism and type synthesis 97 15.1 The type syst...

Abstract. The correspondence between natural deduction proofs and λ-terms is presented and discussed. A variant of the reducibility method is presented, and a general theorem for establishing properties of typed (first-order) λ-terms is proved. As a corollary, we obtain a simple proof of the Church-Rosser property, and of the strong normalization property, for the typed λ-calculus associated with the system of (intuitionistic) first-order natural deduction, including all the connectors

. It was observed by Curry that when (untyped) -terms can be assigned types, for example, simple types, these terms have nice properties (for example, they are strongly normalizing) . Coppo, Dezani, and Veneri, introduced type systems using conjunctive types, and showed that several important classes of (untyped) terms can be characterized according to the shape of the types that can be assigned to these terms. For example, the strongly normalizable terms, the normalizable terms, and the terms having head-normal forms, can be characterized in some systems D and D \Omega\Gamma The proofs use variants of the method of reducibility. In this paper, we present a uniform approach for proving several meta-theorems relating properties of -terms and their typability in the systems D and D Our proofs use a new and more modular version of the reducibility method. As an application of our metatheorems, we show how the characterizations obtained by Coppo, Dezani, Veneri, and Pottinger, can be easi...

. The correspondence between natural deduction proofs and -terms is presented and discussed. A variant of the reducibility method is presented, and a general theorem for establishing properties of typed (first-order) -terms is proved. As a corollary, we obtain a simple proof of the Church-Rosser property, and of the strong normalization property, for the typed -calculus associated with the system of (intuitionistic) first-order natural deduction, including all the connectors !, \Theta, +, 8, 9, and ? (falsity) (with or without j-like rules). This research was partially supported by ONR Grant NOOO14-88-K-0593. Contents 1 Introduction 3 2 Natural Deduction, Simply-Typed -Calculus 5 3 Adding Conjunction, Negation, and Disjunction 11 4 First-Order Quantifiers 14 5 P-Candidates for the Arrow Type Constructor ! 18 6 Adding Product and Sum Types \Theta and + 23 7 Adding the Absurdity Type ? 28 8 Adding First-Order Quantifiers 8 and 9 35 9 Adding j-like Reduction Rules 52 1 Introductio...

. The main purpose of this paper is to take apart the reducibility method in order to understand how its pieces fit together, and in particular, to recast the conditions on candidates of reducibility as sheaf conditions. There has been a feeling among experts on this subject that it should be possible to present the reducibility method using more semantic means, and that a deeper understanding would then be gained. This paper gives mathematical substance to this feeling, by presenting a generalization of the reducibility method based on a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory). A key technical ingredient is the introduction a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene's recursive realizability and a variant of Kreisel's modified realizability both fit into this...

These course notes represent an introduction to functional programming. They have been written while the author was teaching at the University of Salerno (Italy) during spring 1990. These notes can be used for teaching functional programming to students who already had a first contact with Computer Science. However, no prerequisite is strictly required. Through the study of the CAML functional programming, we present polymorphic type synthesis implemented by Milner's algorithm. We also present a simple execution model for functional languages: a simplified version of the Categorical Abstract Machine (CAM). These notes end by the complete prototyping of a simple functional language. ii Contents 1 Introduction 1 I Functional Programming 3 2 Functional languages 5 2.1 History of functional languages : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.2 The ML family : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.3 The Miranda family : :...

ion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 82 III A complete example 85 12 ASL: A Small Language 87 12.1 ASL abstract syntax trees : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 87 12.2 Parsing ASL programs : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 88 13 Untyped semantics of ASL programs 93 13.1 Semantic values : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 93 13.2 Semantic functions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 94 13.3 Examples : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 95 14 Encoding recursion 97 14.1 Fixpoint combinators : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 97 14.2 Recursion as a primitive construct : : : : : : : : : : : : : : : : : : : : : : : : : : : : 98 CONTENTS 3 15 Static typing, polymorphism and type synthesis 99 15.1 The type system : : ...

The main purpose of this paper is to take apart the reducibility method in order to understand how its pieces t together, and in particular, to recast the conditions on candidates of reducibility as sheaf conditions. There has been a feeling among experts on this subject that it should be possible to present the reducibility method using more semantic means, and that a deeper understanding would then be gained. This paper gives mathematical substance to this feeling, by presenting a generalization of the reducibility method based on a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory). A key technical ingredient is the introduction a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene's recursive realizability and a variant of Kreisel's modi ed realizability both t into this framework. We are then able to prove a meta-theorem which shows that if a property of realizers satis es some simple conditions, then it holds for the semantic interpretations of all terms. Applying this theorem to the special case of the term model, yields a general theorem for proving properties of typed-terms, in particular, strong normalization and con uence. This approach clari es the reducibility method by showing that the closure conditions on candidates of reducibility can be viewed as sheaf conditions. The above approach is applied to the simply-typed-calculus (with types!,,+,and?), and to the second-order (polymorphic)-calculus (with types! and 82), for which it yields a new theorem.