Nondeterminism and Observable Sequentiality

Abstract

We give operational, intensional and extensional characterizations of a class of higher-order functionals which may be computed sequentially but nondeterministically.

Sequential algorithms on concrete data structures have been shown to correspond to (deterministic) “observably sequential functionals”, which can be computed in observably sequential PCF (SPCF), and in fact, in an affine version of SPCF in which there are no nested or recursively defined functions.

In this work, we extend these results to a setting with nondeterminism. The main new step is to define notions of concrete data structure in which the sets of cells, values and events are ordered. The nondeterministic states over an ordered CDS form a biorder in (essentially) the sense of Berry, and we show that co-stable and continuous functions, and stable and continuous functions on these biorders each correspond to states on a function-space concrete data structure (non-deterministic sequential algorithms), proving Cartesian closure for the corresponding categories.

We use these results to define a category of “convex sequential algorithms” which combine both stable and co-stable states, and use these give a model of SPCF extended with non-deterministic choice, for which we prove universality at finite types, and thus full abstraction.