Denotational semantics of the Actor model
The denotational semantics of the Actor model is the subject of denotational domain theory for Actors. The historical development of this subject is recounted in [Hewitt 2008b].
The denotational theory of computational system semantics is concerned with finding mathematical objects that represent what systems do. Collections of such objects are called domains. The Actor uses the domain of event diagram scenarios. 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 domain is typically a partial order, which can be understood as an order of definedness. For instance, given event diagram scenarios x and y, one might let "x≤y" mean that "y extends the computations x".
The mathematical denotation denoted by a system S is found by constructing increasingly better approximations from an initial empty denotation called ⊥S using some denotation approximating function progressionS to construct a denotation (meaning ) for S as follows:
It would be expected that progressionS would be monotone, i.e., if x≤y then progressionS(x)≤progressionS(y). More generally, we would expect that
This last stated property of progressionS is called ω-continuity.
A central question of denotational semantics is to characterize when it is possible to create denotations (meanings) according to the equation for DenoteS. A fundamental theorem of computational domain theory is that if progressionS is ω-continuous then DenoteS will exist.
It follows from the ω-continuity of progressionS that
The above equation motivates the terminology that DenoteS is a fixed point of progressionS.
Furthermore this fixed point is least among all fixed points of progressionS.
...
Wikipedia