A Graph Model for Imperative Computation

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

Polynomial approximations of the relational semantics of imperative programs

AbstractWe present a static analysis that approximates the relational semantics of imperative programs by systems of low-degree polynomial equalities. Our method is based on Abstract Interpretation in a lattice of polynomial pseudo ideals — finite-dimensional vector spaces of degree-bounded polynomials that are closed under degree-bounded products. For a fixed degree bound, the sizes of bases of pseudo ideals and the lengths of chains in the lattice of pseudo ideals are bounded by polynomials in the number of program variables. Despite the approximate nature of our analysis, for several programs taken from the literature on non-linear polynomial invariant generation our method produces results that are as precise as those produced by methods based on polynomial ideals and Gröbner bases