Many type systems for programming languages include a notion of subtyping. Subtyping is often defined syntactically by a formal system, but this gets increasingly complex when union, intersection, and negation types are introduced.

In the semantic subtyping approach, instead, types are interpreted as sets and subtyping is defined in terms of set containment. Then, an algorithm is derived from the definition. While the algorithm is complex, the interpretation of types serves as a fairly simple specification. This approach also ensures that union and intersection on types behave as the corresponding operations on sets.

I will give an introduction to this approach and show how to define subtyping semantically for types including arrows, union, intersection, and negation, following [Frisch et al., 2008]. Then, we will look at ongoing work on adapting this approach (originally studied for call-by-value languages) to lazy semantics.

[Frisch et al., 2008] A. Frisch, G. Castagna, and V. Benzaken, Semantic subtyping, JACM, 2008.