Novikovian computation (1/2)

... or, "why time loop logic will almost surely not be developed". (This is not quite correct: see the end result.)

The Novikov self-consistency principle states that the probability of a paradoxical event is zero. This is generally only relevant when closed timelike curves come into play.

For a concrete example, let's suppose I have a time machine, and that I intend to go back in time to shoot my own grandfather. Or Hitler, if you prefer. Now obviously, even in the absence of the NSP, I have a certain chance of failing to do so, for any of several reasons:

• my time-machine may fail to operate correctly;
• my gun may fail to operate correctly;
• I may have a sudden attack of conscience or sanity or heart;
• I could be caught by the police (in either time period);
• I could be intercepted by the Hounds of Tindalos;
  
• I might successfully shoot my grandfather but fail to kill him, explaining the shortness of breath that had always plagued him;
• I might shoot and kill someone that happened to look like my grandfather, and return to the future incorrectly believing myself successful; or
  
• something else.

In a universe in which the Novikov self-consistency principle holds, the chances of the above are effectively increased significantly: I cannot succeed in the attempt any more than I can fly by flapping my arms. If all predictable sources of failure have been ruled out, an unpredicted one will emerge.

So let's consider that briefly in the context of time loop logic:

Note that if N is itself prime, i.e., there is no such prime F ≠ N, then some event will prevent the execution of step 3 that receives the value F from the future.

Assume for the sake of concreteness that the endochronic communications channel c mentioned in the article carries 32 bits of information, and that the base probability of failure due to an external event is `epsilon`. Note also that N has `omega(N)``-2` prime factors. Then, in the absence of the NSP, the probability of receiving a valid `F` is `{:(omega(N)-2)*(1-epsilon):} / 2^32`; the probability of receiving an invalid `F` is `{:(2^32 - (omega(N)-2))*(1-epsilon):} / 2^32`; and the probability of external failure is of course `epsilon`.

(continued in next post)

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Edit 2009-02-12: corrected count of prime factors.