I'd like to compute the identity function. Let me denote by 1 the presence of a rock, and 0 the absence of a rock. I'll set up the problem (the input) by placing a rock appropriately, and after ten seconds, I'll check to see the if a rock is present. If so, the output is 1, otherwise it is 0.
This is a theory, a valid interpretation, highly reliable, scales well, and it commutes. Why isn't it a computation by their definition?
Now it is a computation because you are the computing entity that encoded a problem into the system and later decoded the answer. They paper essentially establishes the distinction of a system just evolving under the laws of physics and using this evolution to compute something by assigning an interpretation to the state of the system. I could use the same stone to compute something different, for example the boring trajectory of a stone placed on a surface in a gravitational field, but without specifying what you are computing the stone is not computing anything.
This is a theory, a valid interpretation, highly reliable, scales well, and it commutes. Why isn't it a computation by their definition?