A Graph Model for Imperative Computation

TL;DR

This paper introduces a new graph-based model for imperative computation that incorporates order of observations, enabling a fully abstract interpretation of higher-order imperative languages with state, building on and extending previous models.

Contribution

It develops a relation-based graph model using sequences instead of sets, capturing imperative features and providing a universal domain for Reddy's object spaces model.

Findings

Model is fully abstract for Reynolds's Syntactic Control of Interference

Equivalent to Reddy's object spaces model at a concrete level

Extends Scott's graph model to ordered sequences for imperative features

Abstract

Scott's graph model is a lambda-algebra based on the observation that continuous endofunctions on the lattice of sets of natural numbers can be represented via their graphs. A graph is a relation mapping finite sets of input values to output values. We consider a similar model based on relations whose input values are finite sequences rather than sets. This alteration means that we are taking into account the order in which observations are made. This new notion of graph gives rise to a model of affine lambda-calculus that admits an interpretation of imperative constructs including variable assignment, dereferencing and allocation. Extending this untyped model, we construct a category that provides a model of typed higher-order imperative computation with an affine type system. An appropriate language of this kind is Reynolds's Syntactic Control of Interference. Our model turns out to be fully abstract for this language. At a concrete level, it is the same as Reddy's object spaces model, which was the first "state-free" model of a higher-order imperative programming language and an important precursor of games models. The graph model can therefore be seen as a universal domain for Reddy's model.