I don't believe either the main 'dust' premiss of Greg Egan's Permutation City, or the way it was supposed to have been invented.
Paul Durham, as a Copy, invents the dust theory when, during an experiment, he is computed 'backwards', yet experiences time running 'forwards'.
Reverse order. After an initial leap into the future, he was now travelling backwards through real time. [...] In real time, the first thing to be computed would be his model-time-final brain state, complete with memories of everything that 'had happened' in the 'preceding' ten seconds.
The problem is, I just don't believe it is possible to calculate such a final state 'complete with memories' without calculating the intermediate states that give those memories -- that's what computational irreducibility is all about.
This would be a minor quibble, however, if it were my only objection, since the dust theory is independent of what process lead to its invention. But I don't believe the dust theory either.
But when his subjective time was scrambled, the flow of cause and effect within the computer bore no relationship whatsoever to the flow of his experience -- so how could it be an essential part of it? When the program spelt out his life DBCEA, but it still felt exactly like ABCDE ... then surely the pattern was all, and cause and effect were irrelevant. The whole experience might just as well have arisen by chance. Was that fatuous? ... As ludicrous as claiming that ... every warehouse full of paint and canvas contained the complete works of Rembrandt and Picasso ... solely by virtue of the potential redefinition of the coordinates of space-time?
The idea, as I understand it, is that the various states that make up a Copy, or even the whole VR world, do not need to be computed sequentially, or even computed at all. If the required state exists anywhere in time and space, then it can be linked into the next state, somewhere completely different in time and space, then to the following state, again somewhere completely different, and the Copy's consciousness will experience a smooth series of state change running forwards in its own subjective timeframe.
The key is hiding in the phrase potential redefinition of the coordinates of space-time. In the limit, it would seem that the dust theory needs only two bits of state, a zero and a one, to support any and all Copies. A suitable coordinate transform on that zero and one defines just the required sequence of states. But where is this 'suitable coordinate transform' to be found? That transform now has all the information that was previously in the sequence of states. Egan has some kind of internal magic finding its way from state to state, finding the transform. But the state space is just so Vast, it would have to be pretty clever magic, or, in reality, just as sophisticated a computation as the one the dust theory is trying to avoid.
In "Oceanic", the protagonist discovers the premises the author has chosen to be true about his fictional world, but what else can he discover? Of course he treats these discoveries as empirical evidence that sheds light on the nature of reality, but this hardly constitutes an attempt by the author to prove those premises true about the real world.
-- Greg Egan, rec.arts.sf.written, 1999
In the sequel to Oceanic the protagonist realises that he is merely one of an infinite number of conceivable characters written by an infinite number of conceivable Egans. He then realises that not only is it not meaningful to ask which is the "real" character, but that his own identity might be shared by an infinite subset of characters who share identical plots except for the placement of a comma, or an arbitrary misspelling.
Greg Egan then realises that it is not necessary to actually write the third book of the trilogy, as all possible stories exist within a list of allowable English words - it is the act of reading the words which creates the story in the mind of the reader, and the reader is therefore the one who selects the story out of an almost-infinite universe of possible stories by reading different words in an arbitrary sequence. He therefore supplies readers with a copy of the Macquarie Australian Dictionary.
Greg Egan's final novel, and the last novel that can ever be written, is a sheet of white cardboard with a spot of black ink. All possible letters, characters and hieroglyphs exist in virtuality within the card/spot boundary and create potential stories of haunting although evanescent beauty within the minds of not only existing readers, but all readers who come to hear about the novel. All publishing houses close because, really, what's the point? The world's economy collapses and we are left with nothing but empty streets haunted by the occasional literary figure in rags stumbling along, holding a sheet of spotted cardboard.
-- Joe Slater, rec.arts.sf.written, 30 May 1999