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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 104 (41 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 .
Programming with Intersection Types and Bounded Polymorphism
, 1991
"... representing the official policies, either expressed or implied, of the U.S. Government. ..."
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Cited by 67 (4 self)
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representing the official policies, either expressed or implied, of the U.S. Government.
Intersection Type Assignment Systems
 THEORETICAL COMPUTER SCIENCE
, 1995
"... This paper gives an overview of intersection type assignment for the Lambda Calculus, as well as compare in detail variants that have been defined in the past. It presents the essential intersection type assignment system, that will prove to be as powerful as the wellknown BCDsystem. It is essenti ..."
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Cited by 62 (34 self)
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This paper gives an overview of intersection type assignment for the Lambda Calculus, as well as compare in detail variants that have been defined in the past. It presents the essential intersection type assignment system, that will prove to be as powerful as the wellknown BCDsystem. It is essential in the following sense: it is an almost syntax directed system that satisfies all major properties of the BCDsystem, and the types used are the representatives of equivalence classes of types in the BCDsystem. The set of typeable terms can be characterized in the same way, the system is complete with respect to the simple type semantics, and it has the principal type property.
Principality and Decidable Type Inference for FiniteRank Intersection Types
 In Conf. Rec. POPL ’99: 26th ACM Symp. Princ. of Prog. Langs
, 1999
"... 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 typin ..."
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Cited by 52 (17 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. This is in contrast to System F where the finite rank restriction for every finite rank at 3 and above has neither principal typings nor decidable type inference. This is also in contrast to earlier presentations of intersection types where the status (decidable or undecidable) of these properties is unknown for the finiterank restrictions at 3 and above. Furthermore, the notion of principal typings for our system involves only one operation, substitution, rather than severa...
Programming With Intersection Types, Union Types, and Polymorphism
, 1991
"... Type systems based on intersection types have been studied extensively in recent years, both as tools for the analysis of the pure calculus and, more recently, as the basis for practical programming languages. The dual notion, union types, also appears to have practical interest. For example, by re ..."
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Cited by 50 (3 self)
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Type systems based on intersection types have been studied extensively in recent years, both as tools for the analysis of the pure calculus and, more recently, as the basis for practical programming languages. The dual notion, union types, also appears to have practical interest. For example, by refining types ordinarily considered as atomic, union types allow a restricted form of abstract interpretation to be performed during typechecking. The addition of secondorder polymorphic types further increases the power of the type system, allowing interesting variants of many common datatypes to be encoded in the "pure" fragment with no type or term constants. This report summarizes a preliminary investigation of the expressiveness of a programming language combining intersection types, union types, and polymorphism.
Intersection Types and Bounded Polymorphism
, 1996
"... this paper (Compagnoni, Intersection Types and Bounded Polymorphism 3 1994; Compagnoni, 1995) has been used in a typetheoretic model of objectoriented multiple inheritance (Compagnoni & Pierce, 1996). Related calculi combining restricted forms of intersection types with higherorder polymorphism ..."
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Cited by 37 (0 self)
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this paper (Compagnoni, Intersection Types and Bounded Polymorphism 3 1994; Compagnoni, 1995) has been used in a typetheoretic model of objectoriented multiple inheritance (Compagnoni & Pierce, 1996). Related calculi combining restricted forms of intersection types with higherorder polymorphism and dependent types have been studied by Pfenning (Pfenning, 1993). Following a more detailed discussion of the pure systems of intersections and bounded quantification (Section 2), we describe, in Section 3, a typed calculus called F ("Fmeet ") integrating the features of both. Section 4 gives some examples illustrating this system's expressive power. Section 5 presents the main results of the paper: a prooftheoretic analysis of F 's subtyping and typechecking relations leading to algorithms for checking subtyping and for synthesizing minimal types for terms. Section 6 discusses semantic aspects of the calculus, obtaining a simple soundness proof for the typing rules by interpreting types as partial equivalence relations; however, another prooftheoretic result, the nonexistence of least upper bounds for arbitrary pairs of types, implies that typed models may be more difficult to construct. Section 7 offers concluding remarks. 2. Background
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 36 (20 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.
Rank 2 Intersection Type Assignment in Term Rewriting Systems
, 1996
"... A notion of type assignment on Curryfied Term Rewriting Systems is introduced that uses Intersection Types of Rank 2, and in which all function symbols are assumed to have a type. Type assignment will consist of specifying derivation rules that describe how types can be assigned to terms, using the ..."
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Cited by 23 (15 self)
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A notion of type assignment on Curryfied Term Rewriting Systems is introduced that uses Intersection Types of Rank 2, and in which all function symbols are assumed to have a type. Type assignment will consist of specifying derivation rules that describe how types can be assigned to terms, using the types of function symbols. Using a modified unification procedure, for each term the principal pair (of basis and type) will be defined in the following sense: from these all admissible pairs can be generated by chains of operations on pairs, consisting of the operations substitution, copying, and weakening. In general, given an arbitrary typeable CuTRS, the subject reduction property does not hold. Using the principal type for the lefthand side of a rewrite rule, a sufficient and decidable condition will be formulated that typeable rewrite rules should satisfy in order to obtain this property. Introduction In the recent years, several paradigms have been investigated for the implementatio...