Parametricity and Local Variables

Abstract

We propose that the phenomenon of local state may be understood in terms of Strachey's concept of parametric (i.e., uniform) polymorphism. The intuitive basis for our proposal is the following analogy: a non-local procedure is independent of locally-declared variables in the same way that a parametrically polymorphic function is independent of types to which it is instantiated.A connection between parametricity and representational abstraction was rst suggested by J.C. Reynolds. Reynolds used logical relations to formalize this connection in languages with type variables and user-de ned types. We use relational parametricity to construct a model for an Algol-like language in which interactions between local and non-local entities satisfy certain relational criteria. Reasoning about local variables essentially involves proving properties of polymorphic functions. The new model supports straightforward validations of all the test equivalences that have been proposed in the literature for local-variable semantics, and encompasses standard methods of reasoning about data representations. It is not known whether our techniques yield fully abstract semantics. A model based on partial equivalence relations on the natural numbers is also brie y examined.

Full abstraction for non-deterministic and probabilistic extensions of PCF I: The angelic cases

“…Our extensions of PCF with choice will be side-effect free, just like PCF. The questions of soundness, computational adequacy, and full abstraction, in the presence of side-effects (assignments and global stores, input/output for example) fall outside the scope of this paper, and already involve dealing with subtle issues such as snapback (see [35,36] or [39,33] for example).…”

Correctness of Data Representations Involving Heap Data Structures

“…The intuition for the definition of (σ, S) <: (ρ, R) is that S extends R by imposing additional conditions for new locations but does not alter R for the part of the heap that R deals with. This is the same intuition as that in [16,14] Having defined relational correspondences, we must specify how these are used to relate states. Note that the relation EQ ρ * R(ρ) relates heaps (or partial states with arbitrary domains).…”

“…We use the setting of reflexive graph categories [14,23,3] to explicate the categorical structure that we use.…”

“…(To deal with divergence and recursion, we must really use Cpo in place of Set. We omit the treatment of recursion in this version of the paper, but it can be treated the same way as in [14]. )…”

Reasoning about local variables with operationally-based logical relations

“…Research partially supported by the EU HCM Research Network on 'Lambda Calcul Typé '. state are such that, despite the considerable efforts of a number of researchers [5,11,10,16,8,20], no concrete denotational model of Algol has yet been constructed which exactly captures observational equivalence for n-th order procedures beyond n = 2 or 3 (depending upon how one counts orders).…”