[ProfHewitt]

The Actor (which cannot be implemented by a nondeterministic TM) sends the 'stop' message to itself. However, just as there can be an arbitrarily long amount of time between two steps of a computation, there can be a arbitrarily long amount of time for a message to be delivered.

[pron]

> The Actor (which cannot be implemented by a nondeterministic TM)

As you now describe it, it cannot be implemented (physically realized) at all. Or, conversely, any physically implementable refinement of it (which will not exhibit the entire range of behaviors) will be simulatable by a TM.

There are many abstract machines that cannot be implemented by TMs -- e.g. various oracle TMs. There is nothing special, surprising or new about that.

There are many formalisms that are more or less convenient abstractions for various kinds of systems. There is nothing special, surprising or new about that, either. In fact, some formalisms that can describe non-computable behaviors are commonly used to specify either software or hardware as they're convenient (like Lamport's TLA).

But you're making a claim about the Church-Turing thesis, which, as commonly interpreted today (as the physical Church-Turing thesis), is the claim that any mechanism that can be implemented by a physical mechanism can be simulated by a TM. Unless you show how to build a physical system that cannot be simulated by a TM, your claim is no refutation of the thesis; it has nothing to do with it. Your claim that arbiters in digital circuits cannot be simulated has not been established and is not recognized by scientific consensus.

> However, just as there can be an arbitrarily long amount of time between two steps of a computation, there can be a arbitrarily long amount of time for a message to be delivered.

This is a completely different use of "arbitrary". In TMs, the fact that an arbitrary amount of time can pass between steps means that any device, with any finite amount of time between steps, can produce the full range of TM behaviors. In your actor case, to get non-computable behavior, you need to show that the device can delay the message by every amount of time. You need to show that such a physical device can exist.

Put simply, it's one thing to propose a non-computable abstraction that's convenient to model some systems, and another thing altogether to claim that there are realizable physical systems that cannot be simulated by a Turing machine. The former is useful but mundane (in fact, all of classical analysis falls in this category); the latter has not been achieved to date.

[ProfHewitt]

Digital arbiters can theoretically take an arbitrary amount of time to settle although statistically they tend to settle soon rather than later. Also, if an Actor sends itself a 'stop' message over the Internet via Timbuktu, it can take an arbitrary amount of time to be received back. Also, an Actor can take an arbitrary amount of time to process a message.

In the above models of computation, arbitrary means absolutely arbitrary, i.e., there is no a priori bound on the amount of time that it can take.

Your trouble may be with Plotkin's proof, which shows that state machine models of nondeterministic computation are inadequate.

[pron]

I have no problem with the proof. It's easy to come up with non-computable abstractions. All of calculus is one (in fact, Turing himself pointed it out in "On Computable Numbers", and he invented and used others when he found them useful), yet it's commonly used to model natural systems without anyone considering it a proof against Church-Turing. So the fact that an abstraction is non-computable is unsurprising and has nothing to do with the thesis. The Turing thesis is relevant when you're talking about a physical realization, and you have not provided any proof that you've found one that's not Turing-computable.

Every physical object does have an a priori bound on the amount of time it can take to do something, unless that time could possibly be infinite. The reason is that it needs some sort of a counter, so it needs some state, and there's only so much state storable in the universe to store a counter.