It is usual to assume some properties of the domain, such as the existence of limits of chains (see cpo) and a bottom element.
Various additional properties are often reasonable and helpful: the article on domain theory has more details.
A fundamental theorem of computational domain theory is that if progressionS is ω-continuous then DenoteS will exist.
The Actor model provides a modern and very general way the compositionality of programs can be analyzed.
Thus there arose the problem of how to provide modular denotational semantics for concurrent programming languages.
When an environment is sent a Lookup message with the address of an identifier x, it returns the latest (lexical) binding of x.
As an example of how this works consider the lambda expression
The denotational compositional semantics presented above is very general and can be used for functional, imperative, concurrent, logic, etc.
For example, it easily provides denotation semantics for constructs that are difficult to formalize using other approaches such as delays and futures.
In his doctoral dissertation, Will Clinger developed the first denotation semantics for the Actor model.
The advantage of the domain Timed Diagrams model is that it is physically motivated and the resulting computations have the desired property of ω-completeness (therefore unbounded nondeterminism) which provides guarantee of service.