On Berry's conjectures about the stable order in PCF

PCF is a sequential simply typed lambda calculus language. There is a uniqueorder-extensional fully abstract cpo model of PCF, built up from equivalenceclasses of terms. In 1979, G\'erard Berry defined the stable order in thismodel and proved that the extensional and the stable order together form abicpo. He made the following two conjectures: 1) "Extensional and stable orderform not only a bicpo, but a bidomain." We refute this conjecture by showingthat the stable order is not bounded complete, already for finitary PCF ofsecond-order types. 2) "The stable order of the model has the syntactic orderas its image: If a is less than b in the stable order of the model, for finitea 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-examplesto this conjecture, again in finitary PCF of second-order types, and alsorefute an improved conjecture: There seems to be no simple syntacticcharacterization of the stable order. But we show that Berry's conjecture istrue for unary PCF. For the preliminaries, we explain the basic fully abstractsemantics of PCF in the general setting of (not-necessarily complete) partialorder models (f-models.) And we restrict the syntax to "game terms", with agraphical representation.Comment: submitted to LMCS, 39 pages, 23 pstricks/pst-tree figures, main changes for this version: 4.1: proof of game term theorem corrected, 7.: the improved chain conjecture is made precise, more references adde