David Barrett-Lennard writes: > Why is it assumed that a multiple "runs" makes any difference to the > measure?

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One reason I like this assumption is that it provides a natural reason for simpler universes to have greater measure than more complex ones. Imagine a Turing machine with an infinite program tape. But suppose the actual program we are running is finite size, say 100 bits. The program head will move back and forth over the tape but never go beyond the first 100 bits. Now consider all possible program tapes being run at the same time, perhaps on an infinite ensemble of (virtual? abstract?) machines. Of those, a fraction of 1 in 2^100 of those tapes will start with that 100 bit sequence for the program in question. And since the TM never goes beyond those 100 bits, all such tapes will run the same program. Therefore, 1/2^100 of all the executions of all possible program tapes will be of that program. Now consider another program that is larger, 120 bits. By the same reasoning, 1 in 2^120 of all possible program tapes will start with that particular 120-bit sequence. And so 1/2^120 of all the executions will be of that program. Therefore runs of the first program will be 2^20 times more numerous than runs of the second. If we use the assumption that each of these multiple executions or runs contributes to the measure, we therefore can conclude that the measure of the universe generated by the first program is 2^20 times greater than the measure of the universe generated by the second. And more generally, the measure of a universe is inversely related to the size of the program which creates it. Therefore, QED, universes with simple programs have a higher measure than universes with more complex programs. This conclusion then allows us to further conclude that observers are likely to evolve in lawful universes, that is, universes without "flying rabbits", i.e. rare, magical exceptions to otherwise universal laws. And we can conclude that the physical laws are likely to be stable or at least predictable over time. All of these are very properties of the universe which are otherwise difficult or impossible to explain. The fact that the multiverse hypothesis can provide some grounds for explaining them is one of the main sources of its attractiveness, at least for me. However, all this is predicated on the assumption that multiple runs of the same program all contribute to the measure. If that is not true, then it would be harder to explain why simple programs are of higher measure than more complex ones. > If the computation is reversible we could run the simulation backwards - > even though the initial state make seem contrived because it leads to a > low entropy at the end of the computation. Given that the simulated > beings don't know the difference (their subjective time runs in the > direction of increasing entropy) the fact that the simulation is done in > reverse is irrelevant to them. > > Would a simulation done in reverse contribute to the measure? When I think of the abstract notion of a universal TM that runs all possible programs at once, I don't necessarily picture an explict time element being present. I think of it more as a mapping: TM + program ==> universe. The more programs which create a given universe, the higher the measure of that universe. However, I don't think I can escape from your question so easily. We could alternately imagine an actual, physical computer, sitting in our universe somewhere, simulating another universe. And that should contribute to that other universe's measure. In that case we should have some rule that would answer questions about how much reversible and reversed simulations contribute. I would consider applying Wei Dai's heuristic, which I discussed the other day. It says that the measure of an object is larger if the object is easier to find in the universe that holds it. I gave some rough justifications for this, such as the fact that a simple counting program eventually outputs every million bit number, but no one would say that this means that the complexity of a given million bit number is as small as the size of that program. In this context, the heuristic would say that the contribution of a physical computer simulating another universe to the measure of that simulated universe should be based on how easy it is to find the computation occuring in our own universe. Computations which occur multiple times would be easier to find, so by Wei's heuristic would have higher measure. This is another path to justify the assumption that multiple simulations should contribute more to measure. I'd say that a computation running backwards contributes as well, by making it easier to locate. Now take a complex case, where a computation ran forwards for a while, then backwards, then forwards. I'd say that this heuristic suggests that the portion of the simulated universe which was repeated 3 times (forwards, backwards, forwards) would have higher measure than the rest of it. > Once we say that all possible computations exist in the Platonic sense, > it seems to me that running them is irrelevant. Of course it is agreed > that the existence hypothesis tells us nothing about their relative > measure. Does anyone have some principles to go by? See above. > I presume a theory of measure along the lines described by Jesse would > need to account for the measure of mappings between computations. > Presumably a simple correspondence would have higher weighting than some > complicated mapping between two computations. Yes, this gets into the difficult issue of telling whether two computations are running the same program, i.e. whether a mapping exists between the two. I'm hoping to write more about this soon. Hal