Computation and Spacetime Structure

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.