State-dependent representation independence

Abstract

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.

First steps in synthetic guarded domain theory: step-indexing in the topos of trees

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A Contextual Equivalence Checker for IMJ*

“…We evaluated CONEQCT on examples drawn from the literature around contextual equivalence for high-level languages [1,3,5,6,8,12,13], adapted to IMJ * syntax. (The website of the tool contains a more detailed listing.)…”

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“…Having given these semantic counterparts we then omit the formal definition of the interpretation − . What changes is the definition of the second component, subeff 1 . It is defined only for relations R ∈ R(ε ∪ ε ′ ), for which e 1 is also defined.…”

“…For this we rely on the morphism (e 2 ) 1 . In particular, we use (e 2 ) 1 on the pullback w 1 1 x 1 x 1 1 w 1 and obtain a pullback w 2 q 2 such that σ 2 andσ 2 are equal in its low point.…”