Nick Bostrom Published in Philosophical Quarterly (2003) Vol. 53, No. 211, pp. 243‐255. (First version: 2001)

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The simulation argument states that unless we are now living in a simulation, we shall either go extinct or our descendants will almost certainly never run an ancestral-simulation.

Assume that a simulated civilisation is indeed possible. And a posthuman civilisation may perform such ancestor simulations (to some utility). Such an assumption denominates two factors – (a) consciousness & (b) computability.

Consciousness

An implicit assumption of the argument is that consciousness is substrate independent, i.e., our biological wetware is not a necessary property of consciousness. Bostrom doesn’t delineate sufficient conditions to attain consciousness. Rather, the simulation should generate subjective experiences such that the computational processes map to human brains to a sufficiently fine-grained detail, such as on the level of individual synapses.

Computability

Another assumption is that future humans would wield the required computational capability to perform such ancestor simulations.

<aside> <img src="/icons/calculator_gray.svg" alt="/icons/calculator_gray.svg" width="40px" /> Fermi estimating computational requirements

Human brain FLOPS

\begin{align*} \text{neurons} &= 10^{11} \\ \text{synapses/neuron} &= 10^3 \\ \text{synapse freq} &\approx \text{500 HZ} \\ \therefore \text{no.\ of\ brain\ FLOPS} &\approx 10^{16} \end{align*}

Total computer operations to simulate all brain FLOPS

It is difficult to estimate the population & duration of the simulation, as it is contextual. But we can assume a relative upper-bound to gauge pessimistic estimates.

\begin{align*} &\bm {simulation\ parameters} \\ & \text{population} = 10^{10} \ \text{people}\\ & \text{duration} = 10^{10}\text{s (}\sim \text{300 years)} \\ \end{align*}

$$\bold {total\ computational\ operations \approx 10^{36}} \\ OR\\ \sim \bold{10^{26}\ operations/s}$$

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The fastest supercomputer (as of 2022) can perform $\sim 10^{18}$ FLOPS. Of course, the rate of technological progress will bump this magnitude, notwithstanding any fundamental breakthroughs in Physics or Computer Science. Bostrom cites computational capabilities in the order of $10^{42}$ operations/s that only presume known nanotech designs.

One such computer can run the entire ancestral simulation using a millionth of its processing power. Thus we can safely conclude that a posthuman civilisation would possess enough computational prowess to run simulations.

Formal interpretation

Let’s assume $f_p$ is the fraction of all posthuman civilisations. $n$ is the number of avg ancestral simulations. $h$ is the avg number of humans before civilisations reach a posthuman era.

\begin{align*} f_p &\rightarrow \text{prob of posthuman civilisation} \\ n &\rightarrow \text{avg no. of simulations }\\ h &\rightarrow \text{total humans before posthuman era} \end{align*}

The fraction of humans living inside a simulation ($f_{sim}$):

\begin{align*} f_{sim} &= \frac{\text{total humans in a simulation}}{\text{total humans before posthuman era}} \\ &= \frac{(f_p \times n \times h)}{(f_p \times n \times h) + h} \\ f_{sim}&= \frac{(f_p \times n)}{(f_p \times n) + 1} \end{align*}

Although, not all posthuman civilisations would run ancestral simulations; $f_{actual} \le f_p$. And actual simulations ($n_{actual}$) would likewise be $n_{actual} \le n$.

\begin{align*} f_{actual} &\rightarrow \text{fraction of } f_p \text{ who actually run simulations} \\ n_{actual} &\rightarrow \text{avg. simulations run by } f_{actual} \\ \therefore n &= n_{actual} \times f_{actual} \end{align*}

Substituting,

\begin{align*} f_{sim}&= \frac{(f_p \times n)}{(f_p \times n) + 1} \\ \therefore \bold { f_{sim}} &= \bold{\frac{(f_p \times n_{actual} \times f_{actual})}{(f_p \times n_{actual} \times f_{actual}) + 1}} \end{align*}