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35
Complete restrictions of the intersection type discipline
 Theoretical Computer Science
, 1992
"... In this paper the intersection type discipline as defined in [Barendregt et al. ’83] is studied. We will present two different and independent complete restrictions of the intersection type discipline. The first restricted system, the strict type assignment system, is presented in section two. Its m ..."
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Cited by 103 (40 self)
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In this paper the intersection type discipline as defined in [Barendregt et al. ’83] is studied. We will present two different and independent complete restrictions of the intersection type discipline. The first restricted system, the strict type assignment system, is presented in section two. Its major feature is the absence of the derivation rule (≤) and it is based on a set of strict types. We will show that these together give rise to a strict filter lambda model that is essentially different from the one presented in [Barendregt et al. ’83]. We will show that the strict type assignment system is the nucleus of the full system, i.e. for every derivation in the intersection type discipline there is a derivation in which (≤) is used only at the very end. Finally we will prove that strict type assignment is complete for inference semantics. The second restricted system is presented in section three. Its major feature is the absence of the type ω. We will show that this system gives rise to a filter λImodel and that type assignment without ω is complete for the λIcalculus. Finally we will prove that a lambda term is typeable in this system if and only if it is strongly normalizable.
The Essence of Principal Typings
 In Proc. 29th Int’l Coll. Automata, Languages, and Programming, volume 2380 of LNCS
, 2002
"... Let S be some type system. A typing in S for a typable term M is the collection of all of the information other than M which appears in the final judgement of a proof derivation showing that M is typable. For example, suppose there is a derivation in S ending with the judgement A M : # meanin ..."
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Cited by 86 (12 self)
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Let S be some type system. A typing in S for a typable term M is the collection of all of the information other than M which appears in the final judgement of a proof derivation showing that M is typable. For example, suppose there is a derivation in S ending with the judgement A M : # meaning that M has result type # when assuming the types of free variables are given by A. Then (A, #) is a typing for M .
Principal type schemes for the strict type assignment system
 Logic and Computation
, 1993
"... We study the strict type assignment system, a restriction of the intersection type discipline [6], and prove that it has the principal type property. We define, for a term, the principal pair (of basis and type). We specify three operations on pairs, and prove that all pairs deducible for can be obt ..."
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Cited by 35 (19 self)
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We study the strict type assignment system, a restriction of the intersection type discipline [6], and prove that it has the principal type property. We define, for a term, the principal pair (of basis and type). We specify three operations on pairs, and prove that all pairs deducible for can be obtained from the principal one by these operations, and that these map deducible pairs to deducible pairs.
Principality and Type Inference for Intersection Types Using Expansion Variables
, 2003
"... Principality of typings is the property that for each typable term, there is a typing from which all other typings are obtained via some set of operations. Type inference is the problem of finding a typing for a given term, if possible. We define an intersection type system which has principal typ ..."
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Cited by 26 (12 self)
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Principality of typings is the property that for each typable term, there is a typing from which all other typings are obtained via some set of operations. Type inference is the problem of finding a typing for a given term, if possible. We define an intersection type system which has principal typings and types exactly the strongly normalizable #terms. More interestingly, every finiterank restriction of this system (using Leivant's first notion of rank) has principal typings and also has decidable type inference.
System E: Expansion variables for flexible typing with linear and nonlinear types and intersection types
 IN PROGRAMMING LANGUAGES & SYSTEMS, 13TH EUROPEAN SYMP. PROGRAMMING
, 2004
"... Types are often used to control and analyze computer programs. ..."
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Cited by 25 (15 self)
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Types are often used to control and analyze computer programs.
Strong normalization and typability with intersection types
 Notre Dame Journal of Formal Logic
, 1996
"... Abstract A simple proof is given of the property that the set of strongly normalizing lambda terms coincides with the set of lambda terms typable in certain intersection type assignment systems. 1Introduction Intersection type assignment systems were introduced and developed in the 1980s by Barendre ..."
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Cited by 21 (8 self)
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Abstract A simple proof is given of the property that the set of strongly normalizing lambda terms coincides with the set of lambda terms typable in certain intersection type assignment systems. 1Introduction Intersection type assignment systems were introduced and developed in the 1980s by Barendregt, Coppo, DezaniCiancaglini and Venneri (see [2], [3], and [4]). They are meant to be extensions of Curry’s basic functional theory which will provide types for a larger class of lambda terms. On the one hand this aim was fulfilled, and on the other hand they became of interest for their other properties as well. We shall deal with four intersection type assignment systems: the original ones D and D � introduced in [3] and [4]and their extensions D ≤ and D� ≤ with the rule (≤), which involves partial ordering on types. The problem of typability in a type system is whether there is a type for a given term. The problem of typability in the full intersection type assignment system D�≤ is trivial, since every lambda term is typable by the type ω. For the same reasons typability in D � is trivial as well. This property changes essentially when the (ω)rule is left out. It turns out that all strongly normalizing lambda terms are typable in D ≤ and D, and they are the only terms typable in these systems (see Krivine [9] and van Bakel [15]). The idea that strongly normalizing lambda terms are exactly the terms typable in the intersection type assignment systems without the (ω)rule first appeared in [4], Pottinger [11], and Leivant [10]. Further, this subject is treated in [15], [9], and Ronchi della Rocca et al. [12], with different approaches. We shall present a modified proof of this property and compare it with the proofs mentioned above. Section 2 is an overview of the systems considered. In Section 3 we shall present a proof àlaTait of strong normalization for D and D ≤ based on the proof of strong
Expansion: the Crucial Mechanism for Type Inference with Intersection Types: Survey and Explanation
 In: (ITRS ’04
, 2005
"... The operation of expansion on typings was introduced at the end of the 1970s by Coppo, Dezani, and Venneri for reasoning about the possible typings of a term when using intersection types. Until recently, it has remained somewhat mysterious and unfamiliar, even though it is essential for carrying ..."
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Cited by 17 (7 self)
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The operation of expansion on typings was introduced at the end of the 1970s by Coppo, Dezani, and Venneri for reasoning about the possible typings of a term when using intersection types. Until recently, it has remained somewhat mysterious and unfamiliar, even though it is essential for carrying out compositional type inference. The fundamental idea of expansion is to be able to calculate the effect on the final judgement of a typing derivation of inserting a use of the intersectionintroduction typing rule at some (possibly deeply nested) position, without actually needing to build the new derivation.
Simple easy terms
 Intersection Types and Related Systems, volume 70 of Electronic Notes in Computer Science
, 2002
"... Dipartimento di Informatica Universit`a di Venezia ..."
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Cited by 12 (4 self)
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Dipartimento di Informatica Universit`a di Venezia
Type Inference with Expansion Variables and Intersection Types in System E and an Exact Correspondence with βReduction
 In Proc. 6th Int’l Conf. Principles & Practice Declarative Programming
"... System E is a recently designed type system for the # calculus with intersection types and expansion variables. During automatic type inference, expansion variables allow postponing decisions about which nonsyntaxdriven typing rules to use until the right information is available and allow imple ..."
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Cited by 11 (4 self)
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System E is a recently designed type system for the # calculus with intersection types and expansion variables. During automatic type inference, expansion variables allow postponing decisions about which nonsyntaxdriven typing rules to use until the right information is available and allow implementing the choices via substitution.
Intersection Types and Domain Operators
, 2003
"... We use intersection types as a tool for obtaining λmodels. Relying on the notion of easy intersection type theory we successfully build a λmodel in which the interpretation of an arbitrary simple easy term is any filter which can be described by a continuous predicate. This allows us ..."
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Cited by 6 (3 self)
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We use intersection types as a tool for obtaining λmodels. Relying on the notion of easy intersection type theory we successfully build a λmodel in which the interpretation of an arbitrary simple easy term is any filter which can be described by a continuous predicate. This allows us to prove two results. The first gives a proof of consistency of the λtheory where the λterm (λx.xx)(λx.xx) is forced to behave as the join operator. This result has interesting consequences on the algebraic structure of the lattice of λtheories. The second result is that for any simple easy term there is a λmodel where the interpretation of the term is the minimal fixed point operator.