On Natural Non-dcpo Domains

Abstract

As Dag Normann has recently shown, the fully abstract model for PCF of hereditarily sequential functionals is not ω-complete and therefore not continuous in the traditional terminology (in contrast to the old fully abstract continuous dcpo model of Milner). This is also applicable to a wider class of models such as the recently constructed by the author fully abstract (universal) model for ${\bf PCF}^+={\bf PCF} + \textrm{parallel } {\bf if}$. Here we will present an outline of a general approach to this kind of “natural” domains which, although being non-dcpos, allow considering “naturally” continuous functions (with respect to existing directed “pointwise”, or “natural” least upper bounds) and also have appropriate version of “naturally” algebraic and “naturally” bounded complete “natural” domains. This is the non-dcpo analogue of the well-known concept of Scott domains, or equivalently, the complete f-spaces of Ershov. In fact, the latter version of natural domains, if considered under “natural” Scott topology, exactly corresponds to the class of f-spaces, not necessarily complete.