On Berry's conjectures about the stable order in PCF

TL;DR

This paper investigates Berry's conjectures about the stable order in PCF's fully abstract model, providing counterexamples to some conjectures and confirming others in the unary case, thus clarifying the structure of the stable order.

Contribution

It refutes Berry's conjectures about the stable order forming a bidomain and having a simple syntactic characterization, and confirms the conjecture for unary PCF.

Findings

The stable order is not bounded complete in finitary PCF of second-order types.

Counterexamples show the stable order does not always correspond to the syntactic order.

Berry's conjecture holds true for unary PCF.

Abstract

PCF is a sequential simply typed lambda calculus language. There is a unique order-extensional fully abstract cpo model of PCF, built up from equivalence classes of terms. In 1979, G\'erard Berry defined the stable order in this model and proved that the extensional and the stable order together form a bicpo. He made the following two conjectures: 1) "Extensional and stable order form not only a bicpo, but a bidomain." We refute this conjecture by showing that the stable order is not bounded complete, already for finitary PCF of second-order types. 2) "The stable order of the model has the syntactic order as its image: If a is less than b in the stable order of the model, for finite a and b, then there are normal form terms A and B with the semantics a, resp. b, such that A is less than B in the syntactic order." We give counter-examples to this conjecture, again in finitary PCF of second-order types, and also refute an improved conjecture: There seems to be no simple syntactic characterization of the stable order. But we show that Berry's conjecture is true for unary PCF. For the preliminaries, we explain the basic fully abstract semantics of PCF in the general setting of (not-necessarily complete) partial order models (f-models.) And we restrict the syntax to "game terms", with a graphical representation.