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New Notions of Reduction and NonSemantic Proofs of βStrong Normalization in Typed λCalculi
, 1994
"... Two new notions of reduction for terms of the λcalculus are introduced and the question of whether a λterm is βstrongly normalizing is reduced to the question of whether a λterm is merely normalizing under one of the new notions of reduction. This leads to a new way to provestrong normalization ..."
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Two new notions of reduction for terms of the λcalculus are introduced and the question of whether a λterm is βstrongly normalizing is reduced to the question of whether a λterm is merely normalizing under one of the new notions of reduction. This leads to a new way to provestrong normalization for typedcalculi. Instead of the usual semantic proof style based on Girard's "candidats de reductibilite", termination can be proved using a decreasing metric over a wellfounded ordering in a style more common in the eld of term rewriting. This new proof method is applied to the simplytyped λcalculus and the system of intersection types.
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.
Typing untyped Aterms, or Reducibility strikes again!
, 1994
"... Typing untyped Lambdaterms, or Reducibility strikes again! 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 c ..."
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Typing untyped Lambdaterms, or Reducibility strikes again! 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 DΩ. The proofs use variants of the method of reducibility. In this paper, we present a uniform approach for proving several metatheorems relating properties of λterms and their typability in the systems D and DΩ. 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 also characterize the terms that have weak headnormal forms, which appears to be new. We conclude by stating a number of challenging open problems
11. Applications to the SecondOrder Typed ACalculus Preliminary Version
, 1993
"... Realizability, Covers, and Sheaves II. Applications to the SecondOrder LambdaCalculus We present a general method for proving properties of typed λterms. This method is obtained by introducing a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory ..."
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Realizability, Covers, and Sheaves II. Applications to the SecondOrder LambdaCalculus We present a general method for proving properties of typed λterms. This method is obtained by introducing a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory, a cover algebra being a Grothendieck topology in the case of a preorder). For this, we introduce 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. Applying this theorem to the special case of the term model, yields a general theorem for proving properties of typed λterms, in particular, strong normalization and confluence. This approach clarifies the reducibility method by showing that the closure conditions on candidates of reducibility can be viewed as sheaf conditions. Part II of this paper applies the above approach to the secondorder (polymorphic) λcalculus λ→,∀2 (with types → and ∀).