[fadmmatt]

* Does the transfer function not need to take some input other than the state?

It depends on how you're encoding things. You might want to supply the I in I/O as an extra parameter to the transfer function.

Or, you could kick off the execution with the state containing a (possibly infinite) sequence representing all of the I/O events it will encounter during execution.

For the most part, transfer functions operate on just states.

* Can you convert a denotational semantics to an operational semantics (and vice versa)? Is there any value in doing this?

Yes, Olivier Danvy has a strong research program in this area. He has a battery of techniques that allow you to (almost) mechanically calculate one kind of semantics from another.

In many cases, calculating an operational semantics from a denotational semantics is the easiest way to prove their equivalence.