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14
Typing untyped λterms, or Reducibility strikes again!
, 1995
"... It was observed by Curry that when (untyped) λterms 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 ..."
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It was observed by Curry that when (untyped) λterms 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 headnormal forms, can be characterized in some systems D and D. The proofs use variants of the method of reducibility. In this paper, we presenta uniform approach for proving several metatheorems relating properties ofterms 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 easily rederived. We alsocharacterize the terms that have weak headnormal forms, which appears to be new. We conclude by stating a number of challenging open problems regarding possible generalizations of the realizability method.
On the Correspondence Between Proofs and λTerms
 Cahiers Du Centre de Logique
, 1995
"... Abstract. The correspondence between natural deduction proofs and λterms is presented and discussed. A variant of the reducibility method is presented, and a general theorem for establishing properties of typed (firstorder) λterms is proved. As a corollary, we obtain a simple proof of the Church ..."
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Abstract. The correspondence between natural deduction proofs and λterms is presented and discussed. A variant of the reducibility method is presented, and a general theorem for establishing properties of typed (firstorder) λterms is proved. As a corollary, we obtain a simple proof of the ChurchRosser property, and of the strong normalization property, for the typed λcalculus associated with the system of (intuitionistic) firstorder natural deduction, including all the connectors
Embedding developments into simply typed λcalculus
"... By using an infinity of extra variables every λterm with indexed redexes is interpreted into a term in the simply typed lambda calculus à la Curry. A development becomes a usual βreduction in the simply typed lambda calculus and the corresponding properties of developments come out from the corres ..."
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Cited by 1 (1 self)
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By using an infinity of extra variables every λterm with indexed redexes is interpreted into a term in the simply typed lambda calculus à la Curry. A development becomes a usual βreduction in the simply typed lambda calculus and the corresponding properties of developments come out from the corresponding properties (strong normalization and ChurchRosser) holding in this system. In this way we obtain a complete simulation of the notion of development into the system of simply typed lambda calculus. Keywords: developments, strong normalization, ChurchRosser property, simple types 1
Proving Properties of Typed λTerms Using Realizability, Covers, and Sheaves
, 1995
"... 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 ..."
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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 preapplicative 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 metatheorem 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 typedterms, 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 simplytypedcalculus (with types!,,+,and?), and to the secondorder (polymorphic)calculus (with types! and 82), for which it yields a new theorem.
IOS Press On Realisability Semantics for Intersection Types with Expansion Variables
"... Abstract. Expansion is a crucial operation for calculating principal typings in intersection type systems. Because the early definitions of expansion were complicated, Evariables were introduced in order to make the calculations easier to mechanise and reason about. Recently, Evariables have been ..."
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Abstract. Expansion is a crucial operation for calculating principal typings in intersection type systems. Because the early definitions of expansion were complicated, Evariables were introduced in order to make the calculations easier to mechanise and reason about. Recently, Evariables have been further simplified and generalised to also allow calculating other type operators than just intersection. There has been much work on semantics for type systems with intersection types, but none whatsoever before our work, on type systems with Evariables. In this paper we expose the challenges of building a semantics for Evariables and we provide a novel solution. Because it is unclear how to devise a space of meanings for Evariables, we develop instead a space of meanings for types that is hierarchical. First, we index each type with a natural number and show that although this intuitively captures the use of Evariables, it is difficult to index the universal type ω with this hierarchy and it is not possible to obtain completeness of the semantics if more than one Evariable is used. We then move to a more complex semantics where each type is associated with a list of natural numbers and establish that both ω and an arbitrary number of Evariables can be represented without losing any of the desirable properties of a realisability semantics. Keywords: Realisability semantics, expansion variables, intersection types, completeness
Printed in Greece c ○ Greek Mathametical Society PROPERTIES OF DEVELOPMENTS VIA SIMPLE TYPES
"... By using an infinity of extra constants every λterm with indexed redexes is interpreted into a term in the simply typed λcalculus à la Curry. A development becomes a usual βreduction in the simply typed lambda calculus and the corresponding properties of developments come out from the correspondi ..."
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By using an infinity of extra constants every λterm with indexed redexes is interpreted into a term in the simply typed λcalculus à la Curry. A development becomes a usual βreduction in the simply typed lambda calculus and the corresponding properties of developments come out from the corresponding properties (strong normalization and ChurchRosser) holding in this system. In this way we obtain a complete simulation of the notion of development into the system of simply typed lambda calculus. Keywords: developments, strong normalization, ChurchRosser property, simple types 1.
Reducibility proofs in λcalculi with intersection types
, 2008
"... Reducibility has been used to prove a number of properties in the λcalculus and is well known to offer on one hand very general proofs which can be applied to a number of instantiations, and on the other hand, to be quite mysterious and inflexible. In this paper, we look at two related but differen ..."
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Reducibility has been used to prove a number of properties in the λcalculus and is well known to offer on one hand very general proofs which can be applied to a number of instantiations, and on the other hand, to be quite mysterious and inflexible. In this paper, we look at two related but different results in λcalculi with intersection types. We show that one such result (which aims at giving reducibility proofs of ChurchRosser, standardisation and weak normalisation for the untyped λcalculus) faces serious problems which break the reducibility method and then we provide a proposal to partially repair the method. Then, we consider a second result whose purpose is to use reducibility for typed terms to show ChurchRosser of βdevelopments for untyped terms (without needing to use strong normalisation), from which ChurchRosser of βreduction easily follows. We extend the second result to encompass both βI and βηreduction rather than simply βreduction. 1