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E.: The Intensional Lambda Calculus
 LFCS
, 2007
"... Abstract. We introduce a natural deduction formulation for the Logic of Proofs, a refinement of modal logic S4 in which the assertion ✷A is replaced by [s]A whose intended reading is “s is a proof of A”. A term calculus for this formulation yields a typed lambda calculus λ I that internalises intens ..."
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Cited by 8 (2 self)
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Abstract. We introduce a natural deduction formulation for the Logic of Proofs, a refinement of modal logic S4 in which the assertion ✷A is replaced by [s]A whose intended reading is “s is a proof of A”. A term calculus for this formulation yields a typed lambda calculus λ I that internalises intensional information on how a term is computed. In the same way that the Logic of Proofs internalises its own derivations, λ I internalises its own computations. Confluence and strong normalisation of λ I is proved. This system serves as the basis for the study of type theories that internalise intensional aspects of computation. 1
Oostrom, Uniform normalisation beyond orthogonality
 Proceedings of the Twelfth International Conference on Rewriting Techniques and Applications (RTA ’01), Lecture Notes in Computer Science (2001
, 2001
"... Abstract. A rewrite system is called uniformly normalising if all its steps are perpetual, i.e. are such that if s → t and s has an infinite reduction, then t has one too. For such systems termination (SN) is equivalent to normalisation (WN). A wellknown fact is uniform normalisation of orthogonal ..."
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Cited by 4 (0 self)
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Abstract. A rewrite system is called uniformly normalising if all its steps are perpetual, i.e. are such that if s → t and s has an infinite reduction, then t has one too. For such systems termination (SN) is equivalent to normalisation (WN). A wellknown fact is uniform normalisation of orthogonal nonerasing term rewrite systems, e.g. the λIcalculus. In the present paper both restrictions are analysed. Orthogonality is seen to pertain to the linear part and nonerasingness to the nonlinear part of rewrite steps. Based on this analysis, a modular proof method for uniform normalisation is presented which allows to go beyond orthogonality. The method is shown applicable to biclosed first and secondorder term rewrite systems as well as to a λcalculus with explicit substitutions. 1
Stable Results and Relative Normalization
 Journal of Logic and Computation 10(3), Special Issue: Type Theory and Term Rewriting
, 2000
"... In orthogonal expression reduction systems, a common generalization of term rewriting and #calculus, we extend the concepts of normalization and needed reduction by considering, instead of the set of normal forms, a set S of "results". When S satisfies some simple axioms, we prove the correspondin ..."
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Cited by 3 (2 self)
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In orthogonal expression reduction systems, a common generalization of term rewriting and #calculus, we extend the concepts of normalization and needed reduction by considering, instead of the set of normal forms, a set S of "results". When S satisfies some simple axioms, we prove the corresponding generalizations of some fundamental theorems: the existence of needed redexes, that needed reduction is normalizing, the existence of minimal normalizing reductions, and the optimality theorem. 1 Introduction Since a normalizable term in a rewriting system may have an infinite reduction, it is important to have a normalizing strategy which enables one to construct reductions to normal form. It is well known that the leftmostoutermost strategy is normalizing in the #calculus [10]. Normalization by Needed Reduction For Orthogonal Term Rewriting Systems (OTRSs), a general normalizing strategy, called the needed strategy, was found by Huet and Levy in [18]. The needed strategy always contr...
Conservation and Uniform Normalization in Lambda Calculi With Erasing Reductions
, 2002
"... For a notion of reduction in a #calculus one can ask whether a term satises conservation and uniform normalization. Conservation means that singlestep reductions of the term preserve innite reduction paths from the term. Uniform normalization means that either the term will have no reduction path ..."
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For a notion of reduction in a #calculus one can ask whether a term satises conservation and uniform normalization. Conservation means that singlestep reductions of the term preserve innite reduction paths from the term. Uniform normalization means that either the term will have no reduction paths leading to a normal form, or all reduction paths will lead to a normal form.