Infinity is a wonderful tool if you want to make stuff, a cornucopia: shake it a little, and things fall out of it without limit, but it is also infinitely hard to make by itself. No finite process can create an infinity, but once you have it, you can easily create universes.
The definition of reversible computation is that every state can have only exactly one possible preceding state. This way, you can construct an inverse rule, and derive the past from the present (if you have full information).
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Yes, and in case of periodicity this is given, even (and this was what I learned from your tweet) if this reversibility is not a property of the step-function alone, but of the step-function and the (subset of) states that are possible in this universe.
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You can also have a deterministic or indeterministic infinite memory universe that is reversible and not periodic, and an indeterministic finite memory universe that is not periodic.
End of conversation
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