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27
CutElimination in the Strict Intersection Type Assignment System is Strongly Normalising
 NOTRE DAME J. OF FORMAL LOGIC
, 2004
"... This paper defines reduction on derivations (cutelimination) in the Strict Intersection Type Assignment System of [1] and shows a strong normalisation result for this reduction. Using this result, new proofs are given for the approximation theorem and the characterisation of normalisability of term ..."
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Cited by 15 (12 self)
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This paper defines reduction on derivations (cutelimination) in the Strict Intersection Type Assignment System of [1] and shows a strong normalisation result for this reduction. Using this result, new proofs are given for the approximation theorem and the characterisation of normalisability of terms, using intersection types.
Normalisation, Approximation, and Semantics for Combinator Systems
 Theoretical Computer Science
, 2003
"... This paper studies normalization of typeable terms and the relation between approximation semantics and filter models for Combinator Systems. It presents notions of approximants for terms, intersection type assignment, and reduction on type derivations; the last will be proved to be strongly normali ..."
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Cited by 12 (11 self)
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This paper studies normalization of typeable terms and the relation between approximation semantics and filter models for Combinator Systems. It presents notions of approximants for terms, intersection type assignment, and reduction on type derivations; the last will be proved to be strongly normalizable. With this result, it is shown that, for every typeable term, there exists an approximant with the same type, and a characterization of the normalization behaviour of terms using their assignable types is given. Then the two semantics are defined and compared, and it is shown that the approximants semantics is fully abstract but the filter semantics is not.
Type Assignment and Termination of Interaction Nets
"... Interaction nets have proved to be a useful tool for the study of computational aspects of different formalisms (e.g. calculus, term rewriting systems), but they are also a programming paradigm in themselves, and this is actually how they were introduced by Lafont. In this paper we consider semisi ..."
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Cited by 7 (4 self)
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Interaction nets have proved to be a useful tool for the study of computational aspects of different formalisms (e.g. calculus, term rewriting systems), but they are also a programming paradigm in themselves, and this is actually how they were introduced by Lafont. In this paper we consider semisimple interaction nets as a programming language, and present a type assignment system using intersection types. First we show that interactions preserve types (i.e. the system enjoys subject reduction), and we compare this type assignment system with the intersection systems for calculus and term rewriting systems. Then we define a recursion scheme that ensures termination of all interaction sequences. By relaxing the scheme and using the type assignment system, we derive another sufficient condition for termination of interaction nets. Finally, we show that although the type system based on general intersection types is not decidable, its restriction to rank 2 types is, and we give an algo...
Strict Intersection Types for the Lambda Calculus
, 2010
"... This paper will show the usefulness and elegance of strict intersection types for the Lambda Calculus; these are strict in the sense that they are the representatives of equivalence classes of types in the BCDsystem [15]. We will focus on the essential intersection type assignment; this system is a ..."
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This paper will show the usefulness and elegance of strict intersection types for the Lambda Calculus; these are strict in the sense that they are the representatives of equivalence classes of types in the BCDsystem [15]. We will focus on the essential intersection type assignment; this system is almost syntax directed, and we will show that all major properties hold that are known to hold for other intersection systems, like the approximation theorem, the characterisation of (head/strong) normalisation, completeness of type assignment using filter semantics, strong normalisation for cutelimination and the principal pair property. In part, the proofs for these properties are new; we will briefly compare the essential system with other existing systems.
Approximation Semantics and Expressive Predicate Assignment for ObjectOriented Programming (Extended Abstract)
"... Abstract. We consider a semantics for a classbased objectoriented calculus based upon approximation; since in the context of LC such a semantics enjoys a strong correspondence with intersection type assignment systems, we also define such a system for our calculus and show that it is sound and com ..."
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Abstract. We consider a semantics for a classbased objectoriented calculus based upon approximation; since in the context of LC such a semantics enjoys a strong correspondence with intersection type assignment systems, we also define such a system for our calculus and show that it is sound and complete. We establish the link with between type (we use the terminology predicate here) assignment and the approximation semantics by showing an approximation result, which leads to a sufficient condition for headnormalisation and termination. We show the expressivity of our predicate system by defining an encoding of Combinatory Logic (and so also LC) into our calculus. We show that this encoding preserves predicateability and also that our system characterises the normalising and strongly normalising terms for this encoding, demonstrating that the great analytic capabilities of these predicates can be applied to OO. 1
Polymorphic Intersection Type Assignment for Rewite Systems with Intersection and betarule (Extended Abstract)
 IN TYPES’99. LNCS
, 2000
"... We define two type assignment systems for firstorder rewriting extended with application,abstraction, andreduction (TRS). The types used in these systems are a combination of (free) intersection and polymorphic types. The first system is the general one, for which we prove a subject reduction t ..."
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Cited by 4 (2 self)
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We define two type assignment systems for firstorder rewriting extended with application,abstraction, andreduction (TRS). The types used in these systems are a combination of (free) intersection and polymorphic types. The first system is the general one, for which we prove a subject reduction theorem and show that all typeable terms are strongly normalisable. The second is a decidable subsystem of the first, by restricting types to Rank 2. For this system we define, using an extended notion of unification, a notion of principal type, and show that type assignment is decidable.
Rank 2 Types for Term Graph Rewriting (Extended Abstract)
"... We define a notion of type assignment with polymorphic intersection types of rank 2 fora term graph rewriting language that expresses sharing and cycles. We show that type assignment is decidable through defining, using the extended notion of unification from [5],a notion of principal pair which gen ..."
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We define a notion of type assignment with polymorphic intersection types of rank 2 fora term graph rewriting language that expresses sharing and cycles. We show that type assignment is decidable through defining, using the extended notion of unification from [5],a notion of principal pair which generalizes ml's principal type property.
Rank 2 Types for Term Graph Rewriting
"... We define a notion of type assignment with polymorphic intersection types of rank 2 for a term graph rewriting language that expresses sharing and cycles. We show that type assignment is decidable through defining, using the extended notion of unification from [5], a notion of principal pair which ..."
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Cited by 2 (1 self)
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We define a notion of type assignment with polymorphic intersection types of rank 2 for a term graph rewriting language that expresses sharing and cycles. We show that type assignment is decidable through defining, using the extended notion of unification from [5], a notion of principal pair which generalizes ML's principal type property.
Completeness and Soundness results forX with Intersection and Union Types
"... This paper defines intersection and union type assignment for the sequent calculus X, a substitutionfree language that enjoys the CurryHoward correspondence with respect to Gentzen’s sequent calculus for classical logic. We show that this notion is complete (i.e. closed for subjectexpansion), and ..."
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This paper defines intersection and union type assignment for the sequent calculus X, a substitutionfree language that enjoys the CurryHoward correspondence with respect to Gentzen’s sequent calculus for classical logic. We show that this notion is complete (i.e. closed for subjectexpansion), and show that the nonlogical nature of both intersection and union types disturbs the soundness (i.e. closed for reduction) properties. This implies that this notion of intersectionunion type assignment needs to be restricted to satisfy soundness as well, making it unsuitable to define a semantics. We will look at two (confluent) notions of reduction, called CallbyName and CallbyValue, and prove soundness results for those.
Sound and Complete Typing for λµ
"... In this paper we define intersection and union type assignment for Parigot’s calculus λµ. We show that this notion is complete (i.e. closed under subjectexpansion), and show also that it is sound (i.e. closed under subjectreduction). This implies that this notion of intersectionunion type assignme ..."
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In this paper we define intersection and union type assignment for Parigot’s calculus λµ. We show that this notion is complete (i.e. closed under subjectexpansion), and show also that it is sound (i.e. closed under subjectreduction). This implies that this notion of intersectionunion type assignment is suitable to define a semantics.