I was wondering if anyone had seen any proposals for (unrealistic) models of computation along these lines.
One needs to take some care to make sure that the state of the system remains well-defined even at limit-times. For instance, if one switches a light switch on and off at times 1/2, 3/4, 7/8, ..., the state of the light at time 1 is undefined.
Still, it seems that even if one imposes constraints to avoid the aforementioned "light switch problem", one can get idealized computers of this kind to do all kinds of things ordinary computers can't do, such as solve the halting problem.
If the computer is operating in discreet time, one way to model this is instead of having time take a value in the positive integers, have it take values in some higher ordinal. Non-well-ordered time seems to inevitably produce paradoxes like the light-switch paradox. If after taking steps at 1/2, 3/4, 7/8, 15/16... you can say that anything that gets to state and stays in that state has a well-defined state at time 1. But if the computer now takes steps at times ..., 17/16, 9/8, 5/4, 3/2, 2, It seems hard to figure out how to define the result of the computation at time 2, even given a well-defined transition function and defined state at time 1. Andy
There must be existing literature on computers that can be made to run faster and faster.
Jim Propp
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