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Programming with Intersection Types and Bounded Polymorphism
, 1991
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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.
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.
Deriving algorithms from type inference systems: Application to strictness analysis
, 1994
"... The role of nonstandard type inference in static program analysis has been much studied recently. Early work emphasised the efficiency of type inference algorithms and paid little attention to the correctness of the inference system. Recently more powerful inference systems have been investigated b ..."
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Cited by 26 (8 self)
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The role of nonstandard type inference in static program analysis has been much studied recently. Early work emphasised the efficiency of type inference algorithms and paid little attention to the correctness of the inference system. Recently more powerful inference systems have been investigated but the connection with efficient inference algorithms has been obscured. The contribution of this paper is twofold: first we show how to transform a program logic into an algorithm and, second, we introduce the notion of lazy types and show how to derive an efficient algorithm for strictness analysis. 1 Introduction Two major formal frameworks have been proposed for static analysis of functional languages: abstract interpretation and type inference. A lot of work has been done to characterise formally the correctness and the power of abstract interpretation. However the development of algorithms has not kept pace with the theoretical developments. This is now a major barrier that is preven...
Normalization Results for Typeable Rewrite Systems
, 1997
"... In this paper we introduce Curryfied Term Rewriting Systems, and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts and !. Three operations on types  substitution, expansion, and lifting  are used to define type assignment, and are proved to be ..."
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Cited by 24 (23 self)
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In this paper we introduce Curryfied Term Rewriting Systems, and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts and !. Three operations on types  substitution, expansion, and lifting  are used to define type assignment, and are proved to be sound. With this result the system is proved closed for reduction. Using a more liberal approach to recursion, we define a general scheme for recursive definitions and prove that, for all systems that satisfy this scheme, every term typeable without using the typeconstant ! is strongly normalizable. We also show that, under certain restrictions, all typeable terms have a (weak) headnormal form, and that terms whose type does not contain ! are normalizable.
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...
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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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
Addendum to `New notions of reduction and nonsemantic proofs of βstrong normalization in typed λcalculi
, 1995
"... ..."
Partial Intersection Type Assignment in Applicative Term Rewriting Systems
 Proceedings of TLCA '93. International Conference on Typed Lambda Calculi and Applications, Utrecht, The Netherlands, volume 664 of Lecture Notes in Computer Science
, 1993
"... This paper introduces a notion of partial type assignment on applicative term rewriting systems that is based on a combination of an essential intersection type assignment system, and the type assignment system as defined for ML [16], both extensions of Curry's type assignment system [11]. Terms a ..."
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Cited by 17 (14 self)
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This paper introduces a notion of partial type assignment on applicative term rewriting systems that is based on a combination of an essential intersection type assignment system, and the type assignment system as defined for ML [16], both extensions of Curry's type assignment system [11]. Terms and rewrite rules will be written as trees, and type assignment will consists of assigning intersection types function symbols, and specifying the way in which types can be assigned to nodes and edges between nodes. The only constraints on this system are local: they are imposed by the relation between the type assigned to a node and those assigned to its incoming and outgoing edges. In general, given an arbitrary typeable applicative term rewriting system, the subject reduction property does not hold. We will formulate a sufficient but undecidable condition typeable rewrite rules should satisfy in order to obtain this property. Introduction In the recent years several paradigms hav...