Stable Bistructure Models of PCF

Abstract

Stable bistructures are a generalisation of event structures to represent spaces of functions at higher types; the partial order of causal dependency is replaced by two orders, one associated with input and the other output in the behaviour of functions. They represent Berry's bidomains. The representation can proceed in two stages. Bistructures form a categorical model of Girard's linear logic consisting of a linear category together with a comonad. The comonad has a co-Kleisli category which is equivalent to a cartesian-closed full subcategory of Berry's bidomains. A main motivation for bidomains came from the full abstraction problem for Plotkin's functional language PCF. However, although the bidomain model incorporates both the Berry stable order and the Scott pointwise order, its PCF theory (those inequalities on terms which hold in the bidomain model) does not include that of the Scott model. With a simple modification we can obtain a new model of PCF, combining the Berry and Scott orders, which does not have this inadequacy.