Introduction: complete partial orders

Having looked at the abstract definition of ‘model’ in the last two chapters, let us now study one particular model in detail. It will be a variant of Dana Scott's D∞, which was the first non-trivial model invented, and has been a dominant influence on the semantics of λ-calculus and programming languages ever since.

Actually, D∞ came as quite a surprise to all workers in λ − even to Scott. In autumn 1969 he wrote a paper which argued vigorously that an interpretation of all untyped λ-terms in set theory was highly unlikely, and that those who were interested in making models of λ should limit themselves to the typed version. (For that paper, see [Sco93].) The paper included a sketch of a new interpretation of typed terms. Then, only a month later, Scott realized that, by altering this new interpretation only slightly, he could make it into a model of untyped λ; this was D∞.

D∞ is a model of both CLw and λβ, and is also extensional. The description below will owe much to accounts by Dana Scott and Gordon Plotkin, and to the well-presented account in [Bar84], but it will give more details than these and will assume the reader has a less mathematical background.

The construction of D∞ involves notions from topology. These will be defined below. They are very different from the syntactical techniques used in this book so far, but they are standard tools in the semantics of programming languages.