I am still confused about the mapping between TM output and a universe. A TM produces abstract output, a list of symbols. Do we need to also specify some kind of mapping between the output and the universe?
Suppose we have a physical universe with some characteristics, and a TM which is offered as an implementation for the universe. We want to understand whether the TM actually does implement the universe. Maybe this is in Wei's model where you input the coordinates of a region and the TM produces the state of the region, or perhaps the earlier suggestion that the TM produces a sequence of strings which represent time slices of the universe. To actually answer the question, we'd want to create a mapping between the TM's abstract, symbolic output, and the universe. Maybe we would have one abstract symbol corresponding to an empty location of the universe, while other symbols correspond to the presence of certain elementary particles at each location. Using this mapping, we could test the TM by seeing whether its output actually corresponded to the state of the universe. My question is whether this mapping is fundamentally important. It seems necessary for us to interpret the TM output, but is it an important ingredient of the TM as a model for the universe? One issue is that there could be more than one TM which produced output which mapped to the universe. As a trivial case, one TM could produce binary output, while another produced the exact same output expressed in hexadecimal. If we could map one output to a given universe, we could about as easily map the other one to the same universe. My real worry, although I haven't come up with a clear example, is that this mapping might be called upon to do more of the work than is appropriate. Is there a danger that you could shift some of the complexity from the TM program into the mapping, leaving it ambiguous whether a given TM program actually implements a given universe? It's not clear to me. Hal