  We investigate the relationship between computation and spacetime structure,
focussing on the role of closed timelike curves (CTCs) in promoting
computational speedup. We note first that CTC traversal can be interpreted in
two distinct ways, depending on ones understanding of spacetime. Focussing on
one interpretation leads us to develop a toy universe in which no CTC can be
traversed more than once, whence no computational speedup is possible.
Focussing on the second (and more standard) interpretation leads to the
surprising conclusion that CTCs act as perfect information repositories: just
as black holes have entropy, so do CTCs. If we also assume that P is not equal
to NP, we find that all observers agree that, even if unbounded time travel
existed in their youth, this capability eventually vanishes as they grow older.
Thus the computational assumption "P is not NP" is also an assumption
concerning cosmological structure.
