Bostrom's 'Simulation Hypothesis" proposes that one of the three following possibilities is true:
1) The human race will go extinct before it ever reaches a "post-human" stage of evolution that possesses a technological level capable of running complex computer simulations of our universe (what he calls "ancestor simulations"),
2) The human race evolves to a "post-human" level and is capable of running ancestor simulations, but does not do so — possibly due to the energy requirements needed, or perhaps due to ethical concerns about simulating self-aware beings,
3) We (in the here and now) are most certainly living in a computer simulation.
Furthermore (and this is where it gets a bit 'trippy'), unless we are currently living in a computer simulation, it is highly unlikely that the human race will evolve to a post-human level that runs ancestor simulations.
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By "post-human", Bostrom meant that we would evolve to a point where we would be unrecognizable from what we currently define as 'human'. Either we evolve, biologically, to be completely different, or we transcend physical form into creatures of pure intellect, or we achieve the 'singularity' and merge ourselves with machines. Beings at this post-human level would, presumably, still be curious about the universe around them, and quite possibly about where they came from. With the fantastic levels of technology available to them, they could harness vast energies — such as tapping directly into the fusion furnace of a star — to power incredibly advanced computers which would run simulations of the universe to show them how they came to evolve from single-celled organisms all the way to their current state.
Bostrom's argument continues that if these post-humans ran ancestor simulations, they would be capable of running a very high number of them, and unless they constrained themselves (as in possibility 2), they indeed would run a very high number of them. Thus, because there would be thousands — possibly millions — of these simulated universes and only one real universe, just by looking at the probabilities, it is far more likely that we are currently living in a simulated universe, rather than the real universe.
Professor Martin Savage, from the physics department at the University of Washington, is already running some of these kinds of simulations. Savage works in a field of physics that deals with the motions and behaviours of the tiniest of sub-atomic particles — quarks and gluons — called 'lattice quantum chromodynamics' (QCD).
Lattice QCD is basically using very powerful supercomputers to simulate the universe from 'first principles' - starting with just the basic laws of physics and building up from there. With current computers, the biggest thing they can simulate is about the size of an atomic nucleus, but that will expand as computing power increases in the future.
"If you make the simulations big enough, something like our universe should emerge," Savage said, according to Science Daily.
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Savage is well-aware of the resource restrictions of his own simulations, which use certain assumptions and physical constraints in order to conserve computer power, such as limiting the size of the simulation and laying down the framework of the simulation on a cube-shaped grid or lattice. He believes that these same kinds of assumptions and constraints would show up in future simulations, including the ones run by Bostrom's post-humans. Proving whether we are living in a computer simulation or not would just be a matter of finding evidence of those constraints.
The constraint that would stand out the most, according to the research paper, is the spacing of the lattice. The more powerful your computer, the smaller the spacing of the lattice could be, essentially giving you a better and better 'resolution' for your simulated universe.