Ninjaed, probably
It'd work if you could get something up past 2c, [...]
I still don't get it. (Either me or thee, but right now I'll go with it being me.)
If you can send your signals on a nominally 8 year round trip at twice the speed they should, you get your echoed signal back in four years. You send "Hello!" and you wait for years to hear... "Hello!" Can't see it rivalling Cleverbot, it's just an echo chamber.
Are you confusing this with the speed increase point beyond which communication is instantaneous? Even then, getting the echo before you sent it can only tell you what you are about to shout, and is only useful if you can hear what you were about to shout[1], modify what you're about to do and then shout out something different[2], thus doing an effectively parallel calculation. Assuming you can do that, which may be considered a similar magnitude of speculation to the whole negative time-delay itself (if not completely linked in the first place).
[1] e.g. "I tried this particular travelling salesman route, and it was the shortest so far, at 5 hours travelling!"
[2] e.g. "I have now tried the first fifty possible routes, and the 14th was the shortest at 4 hours, so, you now need to have checked the 51st and tell us if this changes things in the next iteration"
This is exactly what I'm saying, and it's the reason why I included the clauses I did - you'd have to have a very large gap between each transmitter. Even at 2c, you'd still have to have a significant gap, but we're also talking on the timescale of computers, remember, so you could work pretty close to the absolute boundry it would require, which helps some.
Say you had the equation 2(256/(8 * 16)). Solving it step by step inside a computer, you have to first pass 8 and 16 to an operator, and the operator has to process it and send it back. But to begin with, with conventional computers, in your program memory you know that it's 8 and 16 that you have to pass for the first step, and you know where in the CPU you're passing it.
Now say you have an input stream (the transmitters) that operate on your memory and instructions directly. This is done with things like Cheat Engine all the time. You come to the step in your program that you're going to solve (time A), you solve it (time B), and then you send the next step in the program through your transmitter to the other side (time C). The other side receives the answer to the step at a speed not greater than half the time period it takes from A to C, and sends it back to be queued by the home computer at or before time A occurred, ready to inject as the answer so that the next step (4/128) can begin immediately. Rinse, repeat.
With this solution you're limited by memory access speed. I don't know if there's something faster, but it still jacks your processing capability waaaay the hell above normal physical limitations, if I understand those correctly. Of course, it's blatant violation of causality, which may or may not be an issue