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A proof of strong normalisation using domain theory
 In LICS’06
, 2006
"... U. Berger, [11] significantly simplified Tait’s normalisation proof for bar recursion [27], see also [9], replacing Tait’s introduction of infinite terms by the construction of a domain having the property that a term is strongly normalizing if its semantics is. The goal of this paper is to show tha ..."
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Cited by 13 (1 self)
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U. Berger, [11] significantly simplified Tait’s normalisation proof for bar recursion [27], see also [9], replacing Tait’s introduction of infinite terms by the construction of a domain having the property that a term is strongly normalizing if its semantics is. The goal of this paper is to show that, using ideas from the theory of intersection types [2, 6, 7, 21] and MartinLöf’s domain interpretation of type theory [18], we can in turn simplify U. Berger’s argument in the construction of such a domain model. We think that our domain model can be used to give modular proofs of strong normalization for various type theory. As an example, we show in some details how it can be used to prove strong normalization for MartinLöf dependent type theory extended with bar recursion, and with some form of proofirrelevance. 1
On the ubiquity of certain total type structures
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2007
"... It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel co ..."
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Cited by 4 (2 self)
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It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel continuous functionals, its effective substructure C eff, and the type structure HEO of the hereditarily effective operations. However, the proofs of the relevant equivalences are often nontrivial, and it is not immediately clear why these particular type structures should arise so ubiquitously. In this paper we present some new results which go some way towards explaining this phenomenon. Our results show that a large class of extensional collapse constructions always give rise to C, C eff or HEO (as appropriate). We obtain versions of our results for both the “standard” and “modified” extensional collapse constructions. The proofs make essential use of a technique due to Normann. Many new results, as well as some previously known ones, can be obtained as instances of our theorems, but more importantly, the proofs apply uniformly to a whole family of constructions, and provide strong evidence that the above three type structures are highly canonical mathematical objects.
A Bargain for Intersection Types: A Simple Strong Normalization Proof
"... This pearl gives a discount proof of the folklore theorem that every strongly #normalizing #term is typable with an intersection type. (We consider typings that do not use the empty intersection # which can type any term.) The proof uses the perpetual reduction strategy which finds a longest path. ..."
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This pearl gives a discount proof of the folklore theorem that every strongly #normalizing #term is typable with an intersection type. (We consider typings that do not use the empty intersection # which can type any term.) The proof uses the perpetual reduction strategy which finds a longest path. This is a simplification over existing proofs that consider any longest reduction path. The choice of reduction strategy avoids the need for weakening or strengthening of type derivations. The proof becomes a bargain because it works for more intersection type systems, while being simpler than existing proofs.
Using Models to ModelCheck Recursive Schemes
"... Abstract. We propose a modelbased approach to the model checking problem for recursive schemes. Since simply typed lambda calculus with the fixpoint operator, λYcalculus, is equivalent to schemes, we propose the use a model of λY to discriminate the terms that satisfy a given property. If a model ..."
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Abstract. We propose a modelbased approach to the model checking problem for recursive schemes. Since simply typed lambda calculus with the fixpoint operator, λYcalculus, is equivalent to schemes, we propose the use a model of λY to discriminate the terms that satisfy a given property. If a model is finite in every type, this gives a decision procedure. We provide a construction of such a model for every property expressed by automata with trivial acceptance conditions and divergence testing. Such properties pose already interesting challenges for model construction. Moreover, we argue that having models capturing some class of properties has several other virtues in addition to providing decidability of the modelchecking problem. As an illustration, we show a very simple construction transforming a scheme to a scheme reflecting a property captured by a given model. 1
Urzyczyn and Loader are Logically Related
"... Abstract. In simply typed λcalculus with one ground type the following theorem due to Loader holds. (i) Given the full model F over a finite set, the question whether some element f ∈ F is λdefinable is undecidable. In the λcalculus with intersection types based on countably many atoms, the follo ..."
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Abstract. In simply typed λcalculus with one ground type the following theorem due to Loader holds. (i) Given the full model F over a finite set, the question whether some element f ∈ F is λdefinable is undecidable. In the λcalculus with intersection types based on countably many atoms, the following is proved by Urzyczyn. (ii) It is undecidable whether a type is inhabited. Both statements are major results presented in [3]. We show that (i) and (ii) follow from each other in a natural way, by interpreting intersection types as continuous functions logically related to elements of F. From this, and a result by Joly on λdefinability, we get that Urzyczyn’s theorem already holds for intersection types with at most two atoms.