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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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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.
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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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.
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
On an Open Problem of Amadio and Curien: the Finite Antichain Condition 1 Abstract
"... More than a dozen years ago, Amadio [1] (see Amadio and Curien [2] as well) raised the question of whether the category of stable bifinite domains of AmadioDroste [1,6,7] is the largest cartesian closed full subcategory of the category of ωalgebraic meetcpos with stable functions. A solution to ..."
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More than a dozen years ago, Amadio [1] (see Amadio and Curien [2] as well) raised the question of whether the category of stable bifinite domains of AmadioDroste [1,6,7] is the largest cartesian closed full subcategory of the category of ωalgebraic meetcpos with stable functions. A solution to this problem has two major steps: (1) Show that for any ωalgebraic meetcpo D, if all higherorder stable function spaces built from D are ωalgebraic, then D is finitary (i.e., it satisfies the socalled axiom I); (2) Show that for any ωalgebraic meetcpo D, if D violates MI ∞ , then [D → D] violates either M or I. We solve the first part of the problem in this paper, i.e., for any ωalgebraic meetcpo D, if the stable function space [D → D] satisfies M, then D is finitary. Our notion of (mub, meet)closed set, which is introduced for step 1, will also be used for treating some example cases in step 2. 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.
Weakly Distributive Domains ⋆
"... Abstract. In our previous work [17] we have shown that for anyωalgebraic meetcpo D, if all higherorder stable function spaces built from D areωalgebraic, then D is finitary. This accomplishes the first of a possible, twostep process in solving the problem raised in [1, 2]: whether the category ..."
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Abstract. In our previous work [17] we have shown that for anyωalgebraic meetcpo D, if all higherorder stable function spaces built from D areωalgebraic, then D is finitary. This accomplishes the first of a possible, twostep process in solving the problem raised in [1, 2]: whether the category of stable bifinite domains of AmadioDrosteGöbel [1, 6] is the largest cartesian closed full subcategory within the category ofωalgebraic meetcpos with stable functions. This paper presents results on the second step, which is to show that for any ωalgebraic meetcpo D satisfying axioms M and I to be contained in a cartesian closed full subcategory usingωalgebraic meetcpos with stable functions, it must not violate MI ∞. We introduce a new class of domains called weakly distributive domains and show that for these domains to be in a cartesian closed category usingωalgebraic meetcpos, property MI ∞ must not be violated. We further demonstrate that principally distributive domains (those for which each principle ideal is distributive) form a proper subclass of weakly distributive domains, and Birkhoff’s M3 and N5 [5] are weakly distributive (but nondistributive). We introduce also the notion of meetgenerators in constructing stable functions and show that if anωalgebraic meetcpo D contains an infinite number of meetgenerators, then [D→D] fails I. However, the original problem of Amadio and Curien remains open. 1
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.