Given a domain, how do we build a language whose denotation is the domain? Say we have an arbitrary domain $D$ with a countable basis $B$. Now, how do i build a &quot;language&quot; whose &quot;denotation&quot; lives in the domain?
My understanding is that Dana Scott initially built domains to get a model of typed lambda calculus. Then, people bgan studing domains in their own right. So if I know that something forms a domain, can I &quot;extract computation&quot; / &quot;extract a language&quot; out of it?
Perhaps said differently, does every domain $D$ [with more conditions as necessary] always come equipped with the structure of a closed cartesian category?
As it turns out, the OP is interested in the specific case of the interval domain. Martín Escardó's PhD thesis &quot;PCF extended with real numbers: a domain-theoretic approach to higher-order exact real number computation&quot; extends the programming language PCF with a datatype of reals whose denotation is the interval domain.