Greg Egan is the first fiction writer I've seen who takes the concept of AI somewhat seriously (see my detractions below). In virtually all science fiction, AI is either not present (Dune), artificially rare (Star Wars, William Gibson), somehow deficient relative to the "real" intelligence of humans (Star Trek), or easily tamed into servitude (Asimov). Egan thankfully avoids these lame cop-outs and provides a more realistic view of what might happen when our hardware can support human-mind-scale computation.
Some of the extrapolation is fairly straightforward, for example the idea that the first humans to have themselves "scanned" and instantiated within a computer as Copies will be the elderly and the fatally ill. Egan goes many orders beyond the straightforward, however, and hits on some big questions: If I get moved into a computer, is it still "me"? Should sentient software be considered legally human? If I am a program running in a computer and I edit my memories and my most basic desires, have I become a new person? If I halt a Copy's program and archive their data indefinitely, have I "killed" the Copy? What would it be like to be forced to live forever within a computer, with no ability to commit suicide ("bail out")? If these are interesting philosophical questions today, they will become much more tangible over the coming decades as (or if, depending on your view) AI develops.
Now, my caveats/complaints. A book that seriously considers AI must, I think, include the possibility of super-human AI as well. And Egan, like almost all other authors, conveniently leaves this possibility out. For example, in Permutation City there is an unexplained 17x slowdown of Copies relative to real time. In truth if the average Copy runs at a 17x slowdown, the millionaires among us would cobble together enough supercomputing power to run at a rate equivalent to real time. And the billionaires would have enough hardware to run laps around flesh-and-blood humans. I could easily envision a scenario where every company that doesn't have a management team of hyperspeed Copies would be left in the dust. But Egan tends to stay away from these kinds of unpleasant they-will-become-our-masters scenarios. (In another book of his called Diaspora, Egan does allow for faster-than-human robots called gleisners, but again assumes they will treat is well -- basically a variant of Asimov's stunted-AI). I would love to see Egan put on the Bill Joy hat and deal with superhuman intelligences fairly.
The second half of the book relies very heavily on the author's intriguing "Dust Theory". While I don't necessarily find the idea very compelling as a physical theory, it does touch on some ideas that could very well have validity, such as the notion that a universe will exist if it has internal mathematical consistency (the Platonic view to its logical conclusion). Unfortunately at some points in the story the Dust Theory feels like a cheap trick, a bit of magic that can push the story in whatever arbitrary direction the author desires. In this respect the plot is like a French art film: locally rational, globally irrational.
Despite the detractions, I enjoyed the book immensely and found the ending surprisingly poignant. Read it especially if you are intrigued by the notion of strong AI.