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DonGeddis comments on The Absent-Minded Driver - Less Wrong
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Rather than using a PRNG (which, as you say, requires memory), you could use a source of actual randomness (e.g. quantum decay). Then you don't really have extra memory with the randomized algorithm, do you?
I thought of this as well, but it does not really matter because it is the ability to produce the different output in each case event that gives part of the functionality of memory, that is, the ability to distinguish between events. Granted, this is not as effective as deterministically understood memory, where you know in advance which output you get at each event. Essentially, it is memory with the drawback that you don't understand how it works to the extent that you are uncertain how it correlates with what you wanted to remember.