Christopher A. Stone and Robert Harper. Deciding Type Equivalence in a Language with Singleton Kinds. Technical Report CMU-CS-99-155, Department of Computer Science, Carnegie Mellon University, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....work. ML module systems include parametric modules, known as functors, for large scale software structuring and code reuse. In the single program world there are a number of subtle type equality issues, related to how generative functors are, and how one can express type sharing constraints [21, 17, 11, 28]. Our marshalling primitives should correctly reflect these subtleties in inter program communication. For example, the module SummedIntSet above, which explicitly references IntSet, might be re expressed in terms of a functor F which takes any argument structure U with interface IntSetSig and ....

....E , U :S Static typing of networks, n ok, simply means that all machines are well formed. We define compile time reductions m c m # of machines (performed after type checking) and run time reductions e e # and n n # for expressions and networks. 4.3. 1 Singleton kinds Following [17, 11, 28], we use singleton kinds to handle abstract and concrete signatures in a uniform way. We have two families of kinds: Type is the kind of all types; and, for any type T , Eq(T ) is the singleton kind of all types that are provably equal to T . A module consists of a structure [T0 , v . and a ....

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C. A. Stone and R. Harper. Deciding type equivalence in a language with singleton kinds. In Proc. 27th POPL, pages 214--227, 2000.

....ML given by Harper and Stone [9] currently the only formal account of an entire practical programming language in type theory) and manifest types are similarly employed (somewhat less formally) by Leroy [12] for Objective CAML. In this paper I consider a type theory based on singleton kinds [21], a variant of the translucent sum manifest type formalism. The singleton kind calculus differs from the standard accounts in that it separates the module system from the mechanisms for type abbreviations and focuses on the latter. This separation is appropriate, first, because the two issues are ....

....with the rule that if has kind S( 0 ) then = 0 . When using singleton kinds in practice, the question arises of how singleton kinds affect typechecking, given that they provide a new (and conceivably difficult to discover) way to show types to be equal. In fact, Harper and Stone [21] show that there exists a very simple algorithm for deciding equality of types in the presence of singleton kinds. Indeed, the algorithm is very nearly identical to the usual algorithm employed in the absence of singletons in practice (as opposed to the less efficient algorithms often considered ....

[Article contains additional citation context not shown here]

Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In Twenty-Seventh ACM Symposium on Principles of Programming Languages, Boston, January 2000. Extended version published as CMU

....for an early version of MC 2 [Cou97b] Although MC 2 has much more satisfying metatheoretical properties with respect to reduction, the complexity of its rules is quite similar to the systems of Leroy and Harper and Lillibridge. Harper and Stone recently studied languages with singleton kinds [SH00] Singleton kinds are intended to model manifest types. Harper and Stone have a decidability result for equality in their language, but have not given a reduction semantics so far. We believe our work can help to de ne a reduction notion for their language, and we hope that the proof we gave for ....

Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In Thomas Reps, editor, Conference Record of the 27th Symposium on Principles of Programming Languages, pages 214-227, Boston, Masschusetts, January 2000. ACM Press. Available at http://www.cs.hmc.edu/ ~stone/papers/popl00-preprint.ps.

....ML given by Harper and Stone [9] currently the only formal account of an entire practical programming language in type theory) and manifest types are similarly employed (somewhat less formally) by Leroy [12] for Objective CAML. In this paper I consider a type theory based on singleton kinds [21], a variant of the translucent sum manifest type formalism. The singleton kind calculus differs from the standard accounts in that it separates the module system from the mechanisms for type abbreviations and focuses on the latter. This separation is appropriate, first, because the two issues are ....

....with the rule that if has kind S( 0 ) then = 0 . When using singleton kinds in practice, the question arises of how singleton kinds affect typechecking, given that they provide a new (and conceivably difficult to discover) way to show types to be equal. In fact, Harper and Stone [21] show that there exists a very simple algorithm for deciding equality of types in the presence of singleton kinds. Indeed, the algorithm is very nearly identical to the usual algorithm employed in the absence of singletons in practice (as opposed to the less efficient algorithms often considered ....

[Article contains additional citation context not shown here]

Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In Twenty-Seventh ACM Symposium on Principles of Programming Languages, Boston, January 2000. Extended version published as CMU technical report CMU-CS-99-155.

....formalisms similar to singleton kinds are explored in detail by Harper and Lillibridge [4] and Leroy [7] Despite providing a very elegant solution to the type propagation problem, singleton kinds can substantially complicate type checking, for reasons discussed in Section 2. Stone and Harper [15] have recently shown that, despite these complications, type checking is decidable in the presence of singleton kinds, and indeed is decidable by a practical algorithm. However, the correctness proof for their algorithm is somewhat complicated. As discussed above, a principal advantage of ....

....correctness theorem. Section 4 is dedicated to the proof of the correctness theorem, and concluding remarks appear in Section 5. This paper assumes familiarity with type systems with higher order type constructors and dependent types. The correctness proof draws from the work of Stone and Harper [15] showing decidability of type equivalence in the presence of singleton kinds, but we will use their results almost entirely off the shelf, so familiarity with their paper is not required. 2 A Singleton Kind Calculus We begin by formalizing the singleton calculus that is the subject of this ....

[Article contains additional citation context not shown here]

Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In Twenty-Seventh ACM Symposium on Principles of Programming Languages, Boston, January 2000. To appear. Extended version published as CMU technical report CMUCS -99-155.

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Christopher A. Stone and Robert Harper. Deciding Type Equivalence in a Language with Singleton Kinds. Technical Report CMU-CS-99-155, Department of Computer Science, Carnegie Mellon University, 1999.

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Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In POPL '00.

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Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In Twenty-Seventh ACM Symposium on Principles of Programming Languages, pages 214{ 227, Boston, January 2000.

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Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In POPL '00.

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Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In Twenty-Seventh ACM Symposium on Principles of Programming Languages, pages 214--227, Boston, January 2000.

No context found.

Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In Twenty-Seventh ACM Symposium on Principles of Programming Languages, pages 214{ 227, Boston, January 2000.

....mechanisms that, when combined, yield a flexible, expressive, and implementable type system for modules. Specifically, the following mechanisms are crucial. Singletons Propagation of type sharing is handled by singleton signatures, a variant of Aspinall s and Stone and Harper s singleton kinds [33, 32, 1]. Singletons provide a simple, orthogonal treatment of sharing that captures the full equational theory of types in a higher order module system with subtyping. No previous module system has provided both abstraction and the full equational theory supported by singletons, and consequently none ....

....are considered statically equivalent if they are equal modulo term components; that is, type fields must agree but term fields may differ. Singletons at signatures other than [ T ] are not provided primitively because they can be defined using the basic singleton, as described by Stone and Harper [33]. The definition of s # (M) the signature containing only modules equal to M at signature #) is given in Figure 5. signature SIG = sig type s type t = s int structure S : sig type u val f : u s end val g : t S.u . is compiled as . #s: T ] #t:s( Typ s int] ....

[Article contains additional citation context not shown here]

Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In TwentySeventh ACM Symposium on Principles of Programming Languages, pages 214--227, Boston, January 2000.

No context found.

Christopher A. Stone and Robert Harper. Decid- ing Type Equivalence in a Language with Singleton Kinds. Technical Report CMU-CS-99-155, Department of Computer Science, Carnegie Mellon University, 1999.

....mechanisms that, when combined, yield a flexible, expressive, and implementable type system for modules. Specifically, the following mechanisms are crucial. Singletons Propagation of type sharing is handled by singleton signatures, a variant of Aspinall s and Stone and Harper s singleton kinds [30, 29, 1]. Singletons provide a simple, orthogonal treatment of sharing that captures the full equational theory of types in a higher order module system with subtyping. No previous module system has provided both abstraction and the full equational theory supported by singletons, consequently none has ....

....The algorithm consists of two parts: computation of principal signatures for modules, which we show exist, and checking of subsignature relationships. The latter aspect of the algorithm reduces to checking module equivalence, for which we rely on an extension of Stone and Harper s algorithm [30]. Second, we provide an e#ective elaboration algorithm from a general external language (with hidden types and the resulting avoidance problem) into our type system. 2 Technical Development We begin our technical development by presenting the syntax of our language in Figure 1. Our language ....

[Article contains additional citation context not shown here]

Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In Twenty-Seventh ACM Symposium on Principles of Programming Languages, pages 214--227, Boston, January 2000.

....mechanisms that, when combined, yield a flexible, expressive, and implementable type system for modules. Specifically, the following mechanisms are crucial. Singletons Propagation of type sharing is handled by singleton signatures, a variant of Aspinall s and Stone and Harper s singleton kinds [32, 31, 1]. Singletons provide a simple, orthogonal treatment of sharing that captures the full equational theory of types in a higher order module system with subtyping. No previous module system has provided both abstraction and the full equational theory supported by singletons, consequently none has ....

....are considered statically equivalent if they are equal modulo term components; that is, type fields must agree but term fields may di#er. Singletons at signatures other than [ T ] are not provided primitively because they can be defined using the basic singleton, as described by Stone and Harper [32]. The definition of s # (M) the signature containing only modules equal to M at signature #) is given in Figure 5. Modules The module syntax contains module variables (s) the atomic modules, and the usual introduction and elimination constructs for # and # signatures, except that # modules are ....

[Article contains additional citation context not shown here]

Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In Twenty-Seventh ACM Symposium on Principles of Programming Languages, pages 214--227, Boston, January 2000.

....are recursive types. It is not clear, though, how trails affect more complex type systems that contain type constructors of higher kind, such as Girard s F w [6] In addition to higher kinds, the MIL (Middle Intermediate Language) of TILT employs singleton kinds to model SML s type sharing [13], and the proof that MIL typechecking is decidable is rather delicate and involved. While we have implemented the above trailing algorithm in TILT for experimental purposes (see Section 5) the interaction of trails and singletons is not well understood. As for the remaining conflict between the ....

Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In TwentySeventh ACM Symposium on Principles of Programming Languages, pages 214--227, Boston, January 2000.

....mechanisms that, when combined, yield a flexible, expressive, and implementable type system for modules. Specifically, the following mechanisms are crucial. Singletons Propagation of type sharing is handled by singleton signatures, a variant of Aspinall s and Stone and Harper s singleton kinds [33, 32, 1]. Singletons provide a simple, orthogonal treatment of sharing that captures the full equational theory of types in a higher order module system with subtyping. No previous module system has provided both abstraction and the full equational theory supported by singletons, and consequently none ....

....are considered statically equivalent if they are equal modulo term components; that is, type fields must agree but term fields may differ. Singletons at signatures other than [ T ] are not provided primitively because they can be defined using the basic singleton, as described by Stone and Harper [33]. The definition of s s (M) the signature containing only modules equal to M at signature s) is given in Figure 5. signature SIG = sig type s type t = s int structure S : sig type u val f : u s end val g : t S.u . is compiled as . Ss: T ] St:s( Typ s int] ....

[Article contains additional citation context not shown here]

Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In TwentySeventh ACM Symposium on Principles of Programming Languages, pages 214--227, Boston, January 2000.

....Finally we discuss related work and conclude. Appendix A contains the full set of rules for the calculus. Due to space considerations, the proofs of our results have been condensed or omitted in this extended abstract. Full details can be found in the companion technical report [17]. 2 The calculus 2.1 Overview The syntax of is shown in Figure 1. The constants b i of kind T representbasetypes suchasint. As usual, weuse Contexts ;# Delta: ffl Emptycontext j ;#ff:K Context extension Kinds K#L : T Kind of types j S(A) Singleton kind j Piff:K1 :K2 ....

....of our logical relations proof is fairly standard, our formulation in particular, the equivalence relation involving two constructors, two kinds, and twoworlds appears novel, as is the extension to subkinding and singleton kinds. Full proofs can be found in the companion technical report [17]. We believe that our proof should generalize well to extensions of such as subtyping and power kinds like 10 ; p # B if ; p S(B) A1 , A2 : T if ; A1 p1, A2 p2 , and ; p1 p2 T ; A1 , A2 : S(B) always ; A1 , A2 : Piff:K if ;#ff:K A1 ff , A2 ff : ....

Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. Technical Report CMUCS -99-155, Department of Computer Science, Carnegie Mellon University, 1999.

....Finally we discuss related work and conclude. Appendix A contains the full set of rules for the calculus. Due to space considerations, the proofs of our results have been condensed or omitted in this extended abstract. Full details can be found in the companion technical report [17]. 2 The calculus 2.1 Overview The syntax of is shown in Figure 1. The constants b i of kind T represent base types such as int. As usual, we use Contexts Gamma; Delta : ffl Empty context j Gamma; ff:K Context extension Kinds K;L : T Kind of types j S(A) Singleton kind j ....

....of our logical relations proof is fairly standard, our formulation in particular, the equivalence relation involving two constructors, two kinds, and two worlds appears novel, as is the extension to subkinding and singleton kinds. Full proofs can be found in the companion technical report [17]. We believe that our proof should generalize well to extensions of such as subtyping and power kinds like 10 Gamma E[2hA1 ; A2i] E[A2 ] Gamma p ; B if Gamma p S(B) Gamma A1 , A2 : T if Gamma A1 p1 , Gamma A2 p2 , and Gamma p1 p2 T Gamma A1 , A2 : ....

Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. Technical Report CMUCS -99-155, Department of Computer Science, Carnegie Mellon University, 1999.

....s(c : T ) def = s(c) s(c : s(c # ) def = s(c) s(c : ##:# 1 .# 2 ) def = ##:# 1 . s(c # : # 2 ) s(c : 1) def = 1 s(c : ##:# 1 .# 2 ) def = s(c.1 : # 1 ) s(c.2 : # 2 [c. 1 #] Figure 2: Higher Order Singletons language of the TILT compiler for Standard ML [19] See Stone and Harper [25] for a more concise account of the motivation and practicality of singleton kinds. The recursive operators at the constructor and term levels take the form ##:#.c and fix (x:#.e) respectively. The type constructor ##:#.c denotes the unique fixed point of the constructor function ##:#.c and is ....

Christopher A. Stone and Robert Harper. Deciding type equivalence for a language with singleton kinds. In Twenty-Seventh ACM Symposium on Principles of Programming Languages, pages 214--227, Boston, MA, January 2000.

....University Pittsburgh, PA 15213 Abstract The TILT compiler for Standard ML represents programs internally using a predicative lambda calculus based on Girard s F . At the kind level, this language is notable for containing singleton kinds and dependent product and function kinds. Previous work [SH99] established the decidability of type equivalence for this language. This paper presents a typechecking algorithm for the full TILT internal language and discusses some of the more interesting features of the language. The particular use of intensional type analysis to handle arrays of unboxed ....

....that the necessary primitives for data representation optimizations be present at this level. 1.2 Overview This paper gives a detailed overview of the MIL largely as implemented in the TILT compiler. The major omission is that closure conversion and the typing of closures is not treated here. In [SH99] Stone and Harper present an algorithm for deciding type equivalence in a lambda calculus with singleton kinds. Section 2 of this paper describes the extension of this calculus to the full MIL language. Design issues motivating the extensions are discussed, and algorithms for typechecking are ....

[Article contains additional citation context not shown here]

Christopher A. Stone and Robert Harper. Deciding Type Equivalence in a Language with Singleton Kinds. Technical Report CMU-CS-99-155, Department of Computer Science, Carnegie Mellon University, 1999.

.... : T ) def = s(c) s(c : s(c 0 ) def = s(c) s(c : 1 : 2 ) def = 1 : s(c : 2 ) s(c : 1) def = 1 s(c : 1 : 2 ) def = s(c:1 : 1 ) s(c:2 : 2 [c:1= Figure 2: Higher Order Singletons language of the TILT compiler for Standard ML [19] See Stone and Harper [25] for a more concise account of the motivation and practicality of singleton kinds. The recursive operators at the constructor and term levels take the form : c and x (x: e) respectively. The type constructor : c denotes the unique xed point of the constructor function : c and is ....

Christopher A. Stone and Robert Harper. Deciding type equivalence for a language with singleton kinds. In Twenty-Seventh ACM Symposium on Principles of Programming Languages, pages 214-227, Boston, MA, January 2000.

....This yields the practical algorithm used in the TILT implementation. Finally we discuss related work and conclude. Due to space considerations, the proofs of our results have been condensed or omitted in this extended abstract. Full details can be found in the companion technical report [20]. 2 The S calculus 2.1 Overview The S calculus models the type constructors and kinds of the TILT intermediate language. Since the term level of this language does not enter into the model, S can be viewed in isolation as a dependently lambda calculus. It is solely ....

Christopher A. Stone and Robert Harper. Deciding Type Equivalence in a Language with Singleton Kinds. Technical Report CMU-CS-99-155, Department of Computer Science, Carnegie Mellon University, 1999.

....Finally we discuss related work and conclude. Appendix A contains the full set of rules for the Pi SigmaS calculus. Due to space considerations, the proofs of our results have been condensed or omitted in this extended abstract. Full details can be found in the companion technical report [17]. 2 The Pi SigmaS calculus 2.1 Overview The syntax of Pi SigmaS is shown in Figure 1. The constants b i of kind T represent base types such as int. As usual, we use Contexts Gamma; Delta : ffl Empty context j Gamma; ff:K Context extension Kinds K;L : T Kind of types j S(A) ....

....of our logical relations proof is fairly standard, our formulation in particular, the equivalence relation involving two constructors, two kinds, and two worlds appears novel, as is the extension to subkinding and singleton kinds. Full proofs can be found in the companion technical report [17]. We believe that our proof should generalize well to extensions of Pi SigmaS such as subtyping and power kinds like Weak head reduction Gamma E[ ff:K:A)A 0 ] E[fff7 A 0 gA] Gamma E[ 1hA1 ; A2i] E[A1 ] Gamma E[ 2hA1 ; A2i] E[A2 ] Gamma p ; B if Gamma p S(B) ....

Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. Technical Report CMUCS -99-155, Department of Computer Science, Carnegie Mellon University, 1999.

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C. Stone and R. Harper. Deciding type equivalence in a language with singleton kinds. In TwentySeventh ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 214--225, Boston, MA, USA, Jan. 2000.

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C. A. Stone and R. Harper. Deciding type equivalence in a language with singleton kinds. In Proc. 27th POPL, pages 214--227, 2000. http://www-2.cs.cmu.edu/rwh/papers. htm.

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Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In Conference Record of POPL'00: The 27th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 214--227, Boston, Massachusetts, January 19--21, 2000.

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Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In Proc. 27th POPL, pages 214--227, 2000.

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C. Stone and R. Harper. Deciding type equivalence in a language with singleton kinds. In TwentySeventh ACMSIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 214--225, Boston, MA, USA, Jan. 2000.

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Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In Conference Record of POPL'00: The 27th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 214--227, Boston, Massachusetts, January 19--21, 2000.

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Chris Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In Twenty-Seventh ACMSIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 214--225, Boston, MA, USA, January 2000.

No context found.

C. Stone and R. Harper. Deciding type equivalence in a language with singleton kinds. In Twenty-Seventh ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 214--225, Boston, MA, USA, Jan. 2000.

No context found.

Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In Proc. 27th POPL, pages 214--227, 2000.

No context found.

Christopher A. Stone and Robert Harper. Deciding type equivalence in a language with singleton kinds. In Twenty-Seventh ACM Symposium on Principles of Programming Languages, Boston, January 2000. Extended version published as CMU

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