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25
Subtyping Dependent Types
, 2000
"... The need for subtyping in typesystems with dependent types has been realized for some years. But it is hard to prove that systems combining the two features have fundamental properties such as subject reduction. Here we investigate a subtyping extension of the system *P, which is an abstract versio ..."
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Cited by 68 (6 self)
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The need for subtyping in typesystems with dependent types has been realized for some years. But it is hard to prove that systems combining the two features have fundamental properties such as subject reduction. Here we investigate a subtyping extension of the system *P, which is an abstract version of the type system of the Edinburgh Logical Framework LF. By using an equivalent formulation, we establish some important properties of the new system *P^, including subject reduction. Our analysis culminates in a complete and terminating algorithm which establishes the decidability of typechecking.
Positive Subtyping
 Information and Computation
, 1994
"... The statement S T in a calculus with subtyping is traditionally interpreted as a semantic coercion function of type [[S]]![[T ]] that extracts the "T part" of an element of S. If the subtyping relation is restricted to covariant positions, this interpretation may be enriched to include both the ..."
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Cited by 51 (8 self)
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The statement S T in a calculus with subtyping is traditionally interpreted as a semantic coercion function of type [[S]]![[T ]] that extracts the "T part" of an element of S. If the subtyping relation is restricted to covariant positions, this interpretation may be enriched to include both the coercion and an overwriting function put[S; T ] 2 [[S]]![[T ]]![[S]] that updates the T part of an element of S.
On subtyping and matching
 In Proceedings ECOOP '95
, 1995
"... Abstract. A relation between recursive object types, called matching, has been proposed as a generalization of subtyping. Unlike subtyping, matching does not support subsumption, but it does support inheritance of binary methods. We argue that matching is a good idea, but that it should not be regar ..."
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Cited by 45 (3 self)
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Abstract. A relation between recursive object types, called matching, has been proposed as a generalization of subtyping. Unlike subtyping, matching does not support subsumption, but it does support inheritance of binary methods. We argue that matching is a good idea, but that it should not be regarded as a form of Fbounded subtyping (as was originally intended). We show that a new interpretation of matching as higherorder subtyping has better properties. Matching turns out to be a thirdorder construction, possibly the only one to have been proposed for general use in programming.
Decidability of HigherOrder Subtyping with Intersection Types
 University of Edinburgh, LFCS
, 1994
"... The combination of higherorder subtyping with intersection types yields a typed model of objectoriented programming with multiple inheritance [11]. The target calculus, F ! , a natural generalization of Girard's system F ! with intersection types and bounded polymorphism, is of independent i ..."
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Cited by 40 (11 self)
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The combination of higherorder subtyping with intersection types yields a typed model of objectoriented programming with multiple inheritance [11]. The target calculus, F ! , a natural generalization of Girard's system F ! with intersection types and bounded polymorphism, is of independent interest, and is our subject of study. Our main contribution is the proof that subtyping in F ! is decidable. This yields as a corollary the decidability of subtyping in F ! , its intersection free fragment, because the F ! subtyping system is a conservative extension of that of F ! . The calculus presented in [8] has no reductions on types. In the F ! subtyping system the presence of ficonversion  an extension of ficonversion with distributivity laws  drastically increases the complexity of proving the decidability of the subtyping relation. Our proof consists of, firstly, defining an algorithmic presentation of the subtyping system of F ! , secondly, proving that th...
Once Upon a Polymorphic Type
, 1998
"... We present a sound typebased `usage analysis' for a realistic lazy functional language. Accurate information on the usage of program subexpressions in a lazy functional language permits a compiler to perform a number of useful optimisations. However, existing analyses are either adhoc and approxim ..."
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Cited by 37 (5 self)
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We present a sound typebased `usage analysis' for a realistic lazy functional language. Accurate information on the usage of program subexpressions in a lazy functional language permits a compiler to perform a number of useful optimisations. However, existing analyses are either adhoc and approximate, or defined over restricted languages. Our work extends the Once Upon A Type system of Turner, Mossin, and Wadler (FPCA'95). Firstly, we add type polymorphism, an essential feature of typed functional programming languages. Secondly, we include general Haskellstyle userdefined algebraic data types. Thirdly, we explain and solve the `poisoning problem', which causes the earlier analysis to yield poor results. Interesting design choices turn up in each of these areas. Our analysis is sound with respect to a Launchburystyle operational semantics, and it is straightforward to implement. Good results have been obtained from a prototype implementation, and we are currently integrating the system into the Glasgow Haskell Compiler.
B.C.: On decidability of nominal subtyping with variance
"... or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must ..."
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Cited by 11 (0 self)
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or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior
Semicontinuous sized types and termination
 In Zoltán Ésik, editor, Computer Science Logic, 20th International Workshop, CSL 2006, 15th Annual Conference of the EACSL
"... Abstract. Some typebased approaches to termination use sized types: an ordinal bound for the size of a data structure is stored in its type. A recursive function over a sized type is accepted if it is visible in the type system that recursive calls occur just at a smaller size. This approach is onl ..."
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Cited by 10 (5 self)
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Abstract. Some typebased approaches to termination use sized types: an ordinal bound for the size of a data structure is stored in its type. A recursive function over a sized type is accepted if it is visible in the type system that recursive calls occur just at a smaller size. This approach is only sound if the type of the recursive function is admissible, i.e., depends on the size index in a certain way. To explore the space of admissible functions in the presence of higherkinded data types and impredicative polymorphism, a semantics is developed where sized types are interpreted as functions from ordinals into sets of strongly normalizing terms. It is shown that upper semicontinuity of such functions is a sufficient semantic criterion for admissibility. To provide a syntactical criterion, a calculus for semicontinuous functions is developed. 1.
Subtyping for Object Type Constructors
 In FOOL 6. Foundations of ObjectOriented Languages
, 1999
"... Object type constructors have been introduced as an approach to adding container object types to a language with type inference. Useful subtyping for object type constructors requires a flexible subtype rule for type constructors that is not simply the pointwise extension of subtyping for types. At ..."
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Cited by 10 (2 self)
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Object type constructors have been introduced as an approach to adding container object types to a language with type inference. Useful subtyping for object type constructors requires a flexible subtype rule for type constructors that is not simply the pointwise extension of subtyping for types. At the same time, subtyping should avoid requiring runtime type checks, as in the Java subtype rule for arrays. An extension of object type constructors is considered to allow this subtyping, extending the kinds of object type constructors with polarities that allow this subtyping without jeopardizing soundness. 1 Introduction Recent years have seen much attention paid to the foundations of typed object oriented programming languages. Less attention has been paid to the types of container objects (for example, vectors, lists and arrays). Such objects provides interfaces that are parameterized by one or more type variables (representing the element type, in the aforesaid examples). A formal ca...
Type Destructors
, 1998
"... We study a variant of System F that integrates and generalizes several existing proposals for calculi with structural typing rules. To the usual type constructors (!, \Theta, All, Some, Rec) we add a number of type destructors, each internalizing a useful fact about the subtyping relation. For exa ..."
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Cited by 7 (0 self)
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We study a variant of System F that integrates and generalizes several existing proposals for calculi with structural typing rules. To the usual type constructors (!, \Theta, All, Some, Rec) we add a number of type destructors, each internalizing a useful fact about the subtyping relation. For example, in F with products every closed subtype of a product S\ThetaT must itself be a product S 0 \ThetaT 0 with S 0 !:S and T 0 !:T. We internalise this observation by introducing type destructors .1 and .2 and postulating an equivalence T = j T.1\ThetaT.2 whenever T !: U\ThetaV (including, for example, when T is a variable). In other words, every subtype of a product type literally is a product type, modulo jconversion. Adding type destructors provides a clean solution to the problem of polymorphic update without introducing new term formers, new forms of polymorphism, or quantification over type operators. We illustrate this by giving elementary presentations of two wellknown e...
Fixed points of type constructors and primitive recursion
 Computer Science Logic, 18th International Workshop, CSL 2004, 13th Annual Conference of the EACSL, Karpacz, Poland, September 2024, 2004, Proceedings, volume 3210 of Lecture Notes in Computer Science
, 2004
"... Our contribution to CSL 04 [AM04] contains a little error, which is easily corrected by 2 elementary editing steps (replacing one character and deleting another). Definition of wellformed contexts (fifth page). Typing contexts should, in contrast to kinding contexts, only contain type variable decla ..."
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Cited by 7 (3 self)
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Our contribution to CSL 04 [AM04] contains a little error, which is easily corrected by 2 elementary editing steps (replacing one character and deleting another). Definition of wellformed contexts (fifth page). Typing contexts should, in contrast to kinding contexts, only contain type variable declarations without variance information. Hence, the second rule is too liberal; we must insist on p = ◦. The corrected set of rules is then: ⋄ cxt ∆ cxt ∆, X ◦κ cxt ∆ cxt ∆ ⊢ A: ∗ ∆, x:A cxt Definition of welltyped terms (immediately following). Since wellformed typing contexts ∆ contain no variance information, hence ◦ ∆ = ∆, we might drop the “◦ ” in the instantiation rule (fifth rule). The new set of rules is consequently, (x:A) ∈ ∆ ∆ cxt ∆ ⊢ x: A ∆, X ◦κ ⊢ t: A ∆ ⊢ t: ∀X κ. A ∆, x:A ⊢ t: B ∆ ⊢ λx.t: A → B ∆ ⊢ t: ∀X κ. A ∆ ⊢ F: κ