denotational

A pure functional language can use denotational semantics to embed non-pure languages/programs/concepts, because the denotational semantics of non-pure concepts are still pure.

In the limit case, it can use denotational semantics to describe an emulation of a state-based program.

And that's what a monad is (or at least, that's what they look like to me): a DSL for denotational semantics.

Termination (really, non-bottom-ness vs bottom-ness) is a semantic (denotational) property, independent of the whims of any correct compiler.

The only available reasoning techniques used to be based on denotational semantics, following Scott's breakthrough domain-theoretic semantics of lambda-calculus.

The importance of nice denotational semantics is that they fully define the abstraction and make it easy to think about.

The author is the originator of FRP and has had a clear definition of it for almost two decades that included a written denotational semantics and modeling of continuous time.

One answer to this problem of opaqueness in abstractions is having a well defined denotational semantics.

Futhermore it only applies to a very specific ML-like language with a precisely defined denotational semantics.

It is not known to apply to Haskell as no one has even written down the denotational semantics for Haskell.

That said, the paper quickly went over my head after he started discussing denotational semantics!

The lambda calculus is much more useful for thinking about computation from a denotational perspective.

"Events" are included in the denotational model in order to consider signals which change "infinitely quickly".

The math backs this up as well in the sense that integration is exactly what lets us convert the world of mutation into the world of constant physics (FRP from the "continuous time denotational semantics" perspective of Conal Elliott not the usual way the term is abused).

The idea that monadic IO is a "pure" or "denotational" way to interact with the real world is arguably pretty weak, and that's the point of conal's post here: if you take the idea to its logical conclusion, we might end up just concluding that cpp is pure and denotational.

Denotational semantics does this in the most direct way, by giving me a mapping from programs to some abstract domain I can reason about and manipulate.

Denotational semantics are horrible in my opinion, but I really like operational semantics (especially structural operational semantics and natural semantics), they still consider a program as a single mathematical object, but like Hoare logic it also tracks the state before and after each statement.