Recursive Polymorphic Types and Parametricity in an Operational Framework

Melliès, Paul-André; Vouillon, Jérôme

doi:10.1109/lics.2005.42

“…3 For a logical relation to be complete it must typically be what Pitts terms "equivalence-respecting." There are different ways to achieve this condition, such as -closure [20], biorthogonality [16], or working with contextual equivalence classes of terms [11]. Pitts' -closure neatly combines the equivalencerespecting property together with admissibility (or continuity, necessary for handling recursive functions) into one package.…”

Section: Related and Future Workmentioning

confidence: 99%

State-dependent representation independence

Ahmed

¹

,

DreyerDerek

²

,

RossbergAndreas

³

2009

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Mitchell's notion of representation independence is a particularly useful application of Reynolds' relational parametricity -two different implementations of an abstract data type can be shown contextually equivalent so long as there exists a relation between their type representations that is preserved by their operations. There have been a number of methods proposed for proving representation independence in various pure extensions of System F (where data abstraction is achieved through existential typing), as well as in Algol-or Java-like languages (where data abstraction is achieved through the use of local mutable state). However, none of these approaches addresses the interaction of existential type abstraction and local state. In particular, none allows one to prove representation independence results for generative ADTs -i.e., ADTs that both maintain some local state and define abstract types whose internal representations are dependent on that local state.In this paper, we present a syntactic, logical-relations-based method for proving representation independence of generative ADTs in a language supporting polymorphic types, existential types, general recursive types, and unrestricted ML-style mutable references. We demonstrate the effectiveness of our method by using it to prove several interesting contextual equivalences that involve a close interaction between existential typing and local state, as well as some well-known equivalences from the literature (such as Pitts and Stark's "awkward" example) that have caused trouble for previous logical-relations-based methods.The success of our method relies on two key technical innovations. First, in order to handle generative ADTs, we develop a possible-worlds model in which relational interpretations of types are allowed to grow over time in a manner that is tightly coupled with changes to some local state. Second, we employ a step-indexed stratification of possible worlds, which facilitates a simplified account of mutable references of higher type.

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“…Originally these ideas were developed in the context of (variants of) System F, but over the past two decades there has been a great deal of work on extending them to the setting of more realistic languages, such as those with recursive functions [20], general recursive types [16,1,11], selective strictness [29], etc. In these functional languages, data abstraction is achieved through the use of existential types.…”

Section: Introductionmentioning

confidence: 99%

State-dependent representation independence

Ahmed

¹

,

Dreyer

²

,

Rossberg

³

2009

Proceedings of the 36th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

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Mitchell's notion of representation independence is a particularly useful application of Reynolds' relational parametricity -two different implementations of an abstract data type can be shown contextually equivalent so long as there exists a relation between their type representations that is preserved by their operations. There have been a number of methods proposed for proving representation independence in various pure extensions of System F (where data abstraction is achieved through existential typing), as well as in Algol-or Java-like languages (where data abstraction is achieved through the use of local mutable state). However, none of these approaches addresses the interaction of existential type abstraction and local state. In particular, none allows one to prove representation independence results for generative ADTs -i.e., ADTs that both maintain some local state and define abstract types whose internal representations are dependent on that local state.In this paper, we present a syntactic, logical-relations-based method for proving representation independence of generative ADTs in a language supporting polymorphic types, existential types, general recursive types, and unrestricted ML-style mutable references. We demonstrate the effectiveness of our method by using it to prove several interesting contextual equivalences that involve a close interaction between existential typing and local state, as well as some well-known equivalences from the literature (such as Pitts and Stark's "awkward" example) that have caused trouble for previous logical-relations-based methods.The success of our method relies on two key technical innovations. First, in order to handle generative ADTs, we develop a possible-worlds model in which relational interpretations of types are allowed to grow over time in a manner that is tightly coupled with changes to some local state. Second, we employ a step-indexed stratification of possible worlds, which facilitates a simplified account of mutable references of higher type.

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“…Finally, besides step-indexed logical relations, a number of other logical relations methods have been proposed for languages with parametric polymorphism, recursion, and/or recursive types, e.g., [25,26,18,22,10,13]. One of the most important advances in this domain is the idea of ⊤⊤-closure (aka biorthogonality).…”

Section: Related Work and Conclusionmentioning

confidence: 99%

Logical Step-Indexed Logical Relations

Dreyer

¹

,

Ahmed

²

,

Birkedal

³

2011

Logical Methods in Computer Science

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Abstract. Appel and McAllester's "step-indexed" logical relations have proven to be a simple and effective technique for reasoning about programs in languages with semantically interesting types, such as general recursive types and general reference types. However, proofs using step-indexed models typically involve tedious, error-prone, and proofobscuring step-index arithmetic, so it is important to develop clean, high-level, equational proof principles that avoid mention of step indices.In this paper, we show how to reason about binary step-indexed logical relations in an abstract and elegant way. Specifically, we define a logic LSLR, which is inspired by Plotkin and Abadi's logic for parametricity, but also supports recursively defined relations by means of the modal "later" operator from Appel, Melliès, Richards, and Vouillon's "very modal model" paper. We encode in LSLR a logical relation for reasoning relationally about programs in call-by-value System F extended with general recursive types. Using this logical relation, we derive a set of useful rules with which we can prove contextual equivalence and approximation results without counting steps.

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Recursive Polymorphic Types and Parametricity in an Operational Framework

Cited by 35 publications

References 28 publications

State-dependent representation independence

State-dependent representation independence

State-dependent representation independence

Logical Step-Indexed Logical Relations

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