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The Temporal Singularity: Time-Accelerated Simulated Civilizations and Their Implications



Provided significant future progress in artificial intelligence and computing, it may ultimately be possible to create multiple Artificial General Intelligences (AGIs), and possibly entire societies living within simulated environments. In that case, it should be possible to improve the problem solving capabilities of the system by increasing the speed of the simulation. If a minimal simulation with sufficient capabilities is created , it might manage to increase its own speed by accelerating progress in science and technology, in a way similar to the Technological Singular-ity. This may ultimately lead to large simulated civilizations unfolding at extreme temporal speedups, achieving what from the outside would look like a Temporal Singularity. Here we discuss the feasibility of the minimal simulation and the potential advantages, dangers, and connection to the Fermi paradox of the Temporal Singularity. The medium-term importance of the topic derives from the amount of computational power required to start the process, which could be available within the next decades, making the Temporal Singularity theoretically possible before the end of the century.
The Temporal Singularity: time-accelerated
simulated civilizations and their implications
Giacomo Spigler1
The Biorobotics Institute, Scuola Superiore Sant’Anna, Pisa, Italy,,
Abstract. Provided significant future progress in artificial intelligence
and computing, it may ultimately be possible to create multiple Artificial
General Intelligences (AGIs), and possibly entire societies living within
simulated environments. In that case, it should be possible to improve
the problem solving capabilities of the system by increasing the speed of
the simulation. If a minimal simulation with sufficient capabilities is cre-
ated, it might manage to increase its own speed by accelerating progress
in science and technology, in a way similar to the Technological Singular-
ity. This may ultimately lead to large simulated civilizations unfolding at
extreme temporal speedups, achieving what from the outside would look
like a Temporal Singularity. Here we discuss the feasibility of the mini-
mal simulation and the potential advantages, dangers, and connection to
the Fermi paradox of the Temporal Singularity. The medium-term im-
portance of the topic derives from the amount of computational power
required to start the process, which could be available within the next
decades, making the Temporal Singularity theoretically possible before
the end of the century.
Keywords: temporal singularity; simulated civilization; multi-agent systems;
simulated society; Fermi paradox; artificial life; technological singularity; arti-
ficial general intelligence; deep reinforcement learning; simulation hypothesis;
post-biological civilization
1 The Temporal Singularity
It seems possible, if not likely, that artificial agents with general intelligence
(AGI) will be built in the future [21, 25]. It also seems likely that such agents
could be further improved to achieve super-human degrees of intelligence (ASI).
A simple way to increase the capabilities of an agent is to execute the same
algorithms on a faster (super-)computer, so to provide it with more time to
think and solve problems, thus resulting in a shorter solving time in the external
world. In practice, a simulated environment may be required for the agent to
work in, as the ‘slow’ external world would be a limitation to the performance
the agent even at moderate speedups. It is interesting to note that this approach
is already regularly used, for example in training deep reinforcement learning
(DRL) agents, that is usually performed in simulated environments whose ex-
ecution speed is limited only by the available computing power [8,4, 19]. For
example, DRL agents learning to play Atari games can experience thousands
of game frames per second even on a regular desktop computer, compared to
human players that play them at 15-60 frames per second.
Another approach to improve the effective capabilities of the system without
any modification to its algorithms is to simulate multiple agents each with its
specific differences, so that they can come up with different ways of solving
the problem individually or cooperatively by exploiting dynamics of collective
intelligence. An interesting outcome of simulations of this type is the potential
to simulate the unfolding of entire “civilizations”, possibly pursuing complex
sets of goals like general progress in science and technology. The potential of the
approach relies not only on the possibly advanced intelligence level of the agents
(ASI), but also on the temporal speedups that could be achieved by increasing
the computing resources available for the simulation. Throughout this paper
intelligent agents and civilizations will be referred to as ‘simulated’ only to mean
that they experience a simulated environment in contrast to the ‘real’ external
world, but there is no reason not to consider them as real as any intelligent agent
outside the simulation.
Here we suggest that if it will be possible to create at least a limited group
of AGIs in a simulation unfolding faster than the external time, then such sim-
ulation may be able to accelerate the rate of progress in science and technology,
possibly by continually self-improving its core technologies such as its intelligence
algorithms and its computing systems in a manner similar to the Technological
Singularity [14, 30, 21, 18, 9], and thus potentially achieve a runaway increase in
its capabilities. Specifically, the rate of progress may be so high that in a very
short time the simulations could progress to producing entire civilizations span-
ning thousands or millions of years or even more in an arbitrarily short time
interval elapsed in the external world, achieving what from the outside would
be a Temporal Singularity. In particular the Temporal Singularity is defined
as the moment in time where a minimal simulation capable of beginning the
runaway exponential self-improvement is started. We will discuss the feasibility
of the minimal simulation in Section 2.
It is difficult to imagine what such a quick progress would look like, as even
a single century of progress at the present rate is challenging to forecast. Even
more, we can only wonder what the world would become after the Temporal
Singularity has allowed the unfolding of millions or billions of years of an ad-
vanced civilization [18], during which potentially any questions our species may
ever ask could have been answered.
This result is compatible with the idea of the Technological Singularity, of
which the Temporal Singularity can represent a component or a way to achieve
it. Contrary to the main definitions of the Technological Singularity, however,
the Temporal Singularity would not necessarily require a runaway increase in the
cognitive capabilities of the artificial agents, but rather only a runaway increase
in the temporal speedups of the simulations. We should note that speeding up
the execution of AGIs has been already suggested in this context, for example
by Vernor Vinge, who discusses an AI whose ‘mind clock’ is significantly faster
than its creator and the problem of AI boxing [30], or by Solomonoff in the
context of an exponential increase in the number of simulated agents [28]. Most
notably Marcus Hutter [18] explored what the Technological Singularity would
look like for both the outside and the inside of a virtual software society under-
going it, also discussing the difference between speeding up the simulation time
and increasing the intelligence of the agents. However, the focus of the discus-
sion was put on the extreme progress and changes achieved in the traditional
Technological Singularity, rather than on the implications of a drastic increase
in the temporal speedups of the simulations and its potential implications on
the Fermi Paradox.
The idea of simulated civilizations is also not novel, although it has been
generally applied to us being in the simulation ourselves, rather than focusing
directly on the benefits, limits and implications of us producing it, and in partic-
ular on the possibility to speed up the elapsing of the simulated time. Philoso-
phers have always wondered about the nature of reality and the possibility of it
being an illusion. In recent times, the argument has been especially developed
by Hans Moravec [23] and Nick Bostrom [5] in the explicit context of computer
simulations. A more closely related investigation was proposed by Vidal, who
explored the possibility that scientific simulations will improve significantly in
the future and finally result in simulating an entire universe, in order to better
probe and understand our own universe and the processes of physical, biological
and cultural evolution [29]. However, most of the discussions such as Vidal’s and
Bostrom’s only focus on a very special type of simulations restricted to detailed
versions of our physical universe and our same society and life as we know it,
which although intriguing from a scientific point of view, constitute only a tiny
fraction of the potential uses of time-accelerated simulations, and possibly an
inefficient use of the computing resources. For example, as we discuss in section
2, it may be that fooling the simulated agents to prevent them from discov-
ering that they belong to a simulation may not be necessary, which would in
turn lower the computational requirements for the simulated environment. In
any case, whether our own world is itself simulated or not does not reduce the
potential advantages of running our own time-accelerated simulations.
Section 2 will next overview the feasibility and broad computational require-
ments for simulations capable of achieving and sustaining the Temporal Sin-
gularity, while Section 3 will explore some of the advantages and risks of such
simulations, and the implications of the Temporal Singularity for the Fermi
2 Feasibility
The minimal simulation. It is difficult to estimate what are the minimal
requirements for a simulation capable of starting the runaway exponential pro-
cess of self-improvement and time-acceleration that characterizes the Tempo-
ral Singularity. In general, we should expect the minimal simulation to pro-
vide a problem-solving capability sufficient to compete with teams of human ex-
perts, either by providing significant temporal speedups, by using more capable
AGIs/ASIs or by creating a larger number of individuals. Even small advan-
tages, compared to traditional research and development, may be sufficient to
start the process by exploiting the compound nature of progress [21, 28]. The
minimal requirements could thus be reasonably low (see the discussion on the
computational requirements below), especially after achieving human-level AGI,
which itself however may not be required, as a super-human narrow intelligence
in specific fields like improving the computing technology may be sufficient.
AGI. Still, while we could imagine some limited type of “civilization” com-
posed by agents with narrow intelligence (ANI), the development of artificial
general intelligence (AGI) is likely to be a core requirement for enabling complex
artificial civilizations. It is not known whether AGI itself will ever be possible,
though there do not seem to be strong reasons for it to be not. Unfortunately,
the field is known to have a poor track record of predictions about when such
a system wil be developed. Current predictions also vary greatly depending on
the expected requirements for specific types of implementations, with average
agreement placed around 2040 [25, 6] and possibly as early as 2029 [21, 2], and a
high confidence in any case that it may happen before the end of the century. We
could also wonder whether the artificial agents could instantiate consciousness,
but it may not be a strict requirement in this context. On the other hand, it
may turn out that consciousness is required, for example for the establishment
and maintenance of societies and complex civilizations (e.g., for consciousness
and sociality [15]).
Fooling the agents. The requirements for the simulations discussed here
also change significantly depending on whether the simulated agents are allowed
to know they belong to a simulation or whether they need to be fooled. In par-
ticular, fooling the agents may be challenging especially if the aim of the simula-
tions is to produce progress in science and technology that apply to the external
world, as a large degree of knowledge of it would be required. In the limit, a
perfect simulation of our physical world may be required for perfect fooling,
which would however limit the simulation (for an analysis of the requirements,
see for example [3]). It is however possible that fooling is not necessary, or that
perfect fooling can be achieved with simpler simulations. If fooling is not used,
the potential problems that may arise and their solutions would fall within the
traditional problem of AI boxing and containment (e.g., [1,6]).
Computational requirements. The computational requirements for the
simulations described here can be assessed by separately estimating the resources
required for the agents and for the simulated environment. It is difficult to predict
the requirements for a single AGI agent, but estimates have been suggested
for the calculations per second required for a real-time functional simulation
of the human brain. Such estimates range wildly from tens of Teraflops [24]
(1013 FLOPS) to Exaflops and more (1018 to 1025 FLOPS [26]). However, the
highest estimates have been mostly suggested for detailed whole brain emulation
approaches, which are unlikely to be the most computationally efficient approach
to AGI, and may thus constitute an upper-bound on the actual requirements for
computer-optimized implementations of the algorithms. A common intermediate
estimate is for the required power to be of the order of tens of Petaflops (1016
FLOPS) [21], comparable to the performance of present day supercomputers.
As for the computational requirements for the simulated environment, mul-
tiple answers may be correct. Even today we are performing time-accelerated
simulations in limited conditions, for example to train deep reinforcement learn-
ing agents, so there seems to be no strict lower bound on the required speed
of the system. However, it is likely that more complex environments will be re-
quired in order to support AGI agents performing complex tasks, especially to
allow progress in science and technology. While a certain degree of physically-
detailed simulation of the real world may be required, a perfect simulation of the
real world may not. Indeed, even present-day engineering software allow for part
of the development in engineering to be performed in simulation (for example,
using the COMSOL Multiphysics simulation software [10]).
Still, a perfect simulation may be required in case fooling of the agents was
desired. For example, even if an imperfect simulation was sufficient to fool the
agents, knowledge of the external world will be required to achieve progress in
science and engineering, which could allow the agents to ultimately discover
the truth. Nonetheless, while a perfect simulation would be computationally
prohibitive with our current technology, we might be able to achieve it in the
future [3]. In any case, it seems unlikely that a perfect simulation will be required
for the minimal simulation and thus to start the Temporal Singularity.
When. If we assume that the agents require a computational power on
the order of the average current estimates for the computational power of the
human brain, and a linear scaling of the total requirements with the number
of agents and temporal speedup, with negligible environment overhead, then
the computational requirements for a minimal simulation of tens to hundreds
of agents at faster than real-time may be as low as 1018 to 1021 FLOPS (e.g.,
1016 ·100 100 agents in real-time or 10 agents at 10×faster than real-time).
If the Moore’s law continues to hold, the world’s most powerful supercomputer
could achieve the required speed between the years 2020-2040, or alternatively
individual home workstations between the years 2055-2075. Specialized hardware
may however be developed to provide faster increases in the computational power
in the future, as it has happened for example in the specific case of deep learning
with the development of specialized accelerators like the Tensor Processing Unit
(TPU) [20]. It is also interesting that these estimates are similar to current
estimates for the development of AGI, which could be an important requirement
for the simulations.
Allocation of the resources. We may further wonder how the available
computing resources could be allocated between different processes to achieve
the highest problem-solving capabilities of the system. For example, increased
computation could be traded off between creating a larger number of agents, in-
creasing the speed of the simulation and thus its temporal speedup compared to
the external world, increasing the cognitive capabilities of the individual agents
or simulating more complex environments. It may thus be required for the re-
sources to be re-allocated dynamically depending on the state of technology.
Potential limitations. Even if the minimal simulation would be possible,
there may be other limitations that could prevent or limit the Temporal Singu-
larity. For example, it may be that temporal acceleration will not be the most
efficient allocation of the computing resources, so that creating a larger number
of agents or stronger ASIs will produce the best results. However, the fact that
artificial agents working at faster than real-time are already being used and the
potential advantages of simulating societies and civilizations suggest that this is
unlikely to be the case. Temporal acceleration may also be helpful to speed up
the solution of time-critical problems given a current level of intelligence of the
available agents, in case improving the cognitive capabilities of the agent would
prove difficult and more time-consuming. Finally, increasing the number of simu-
lated agents may ultimately be limited by the intrinsic problem solving speed of
each agent, which could be then trivially improved with temporal acceleration.
Another potential limit is that perfect simulations may be required to enable
practical progress in science and technology. However, even present day research
involves significant portions of time for theoretical work and simulations, so there
seems to be a margin of speedup that can be achieved. Moreover, even if the
first environments limited the potential to advance science and technology, it
would be possible to iterate between time-accelerated work performed inside the
simulation and prototyping, testing, and conducting experimental work in the
external world, whose results and data could be fed back into the simulation to
start the next cycle. Also, some type of theoretical work like in mathematics,
computer science, philosophy and others may not need frequent access to data
from the external world, suggesting that it should still be possible to benefit
greatly from the temporal speedups of these simulations. In any case, interaction
with the external world will always be required for maintenance and upgrades, to
manufacture the newly developed technologies, and to acquire experimental data
[18]. This dependency may ultimately limit the maximum speedups that can be
achieved or their rate of growth. Still, even relatively low effective speedups could
be highly beneficial. Further, the processes performed in the external world may
be optimized inside the simulations to avoid wasting external time, for example
by providing efficient instructions distributed among a large number of external
world agents, although this may involve risks in the context of AI boxing [1,6].
3 Implications
Advantages. Similar to the Technological Singularity, the Temporal Singu-
larity would produce a runaway increase the rate of growth of scientific and
technological progress. In addition, however, it would also allow the study of the
potential future of advanced intelligent civilizations and societal structures that
will be required to be stable for extremely long intervals of time, which could
be useful for scientific purposes and may provide invaluable information, thus
impacting our society and guiding the future of our own civilization in a safe and
beneficial way. In the extreme, we might be able to simulate civilizations with
characteristics similar to our own, experimenting new societal designs and con-
ditions. Finally, due to the potential for significant technological development,
there is a clear competitive advantage for the first entity that will achieve a
minimal simulation, even at moderate temporal speedups, whether it would be
governments or private companies.
Dangers. In general, the Temporal Singularity shares all the potential dan-
gers related to AGI/ASI and the Technological Singularity (see for example [6]),
and in particular to the problem AI boxing [1]. However, the problems may be
worse in this context, as even moderate temporal speedups would make it diffi-
cult to track the events inside the simulation. Finally, the same extreme progress
in technology in a short span of time also constitutes a potential danger, as our
society may not be capable of metabolizing it in the available time. For example,
we can try to imagine what could have happened if we abruptly produced not
just the technology, but a full stockpile of thermonuclear weapons during the
Middle Ages.
Fermi Paradox. The Fermi Paradox is the contradiction between the ap-
parent high likelihood of the existence of other intelligent civilizations in our
galaxy or in the universe and the current lack of evidence of any. The Temporal
Singularity leads to interesting implications in this context. First, if intelligent
civilizations would achieve a degree of technology similar to our present one, and
in particular develop computing systems, it may turn out to be almost inevitable
that at some point they would produce a Temporal Singularity. Time-accelerated
simulations could thus be part of some or all the possible intelligent civilizations,
providing advantages like achieving a practical ‘subjective immortality’ within
the simulated environments, either for the individual agents or for their civiliza-
tion as a whole, and subjectively delaying its demise due to the heat death of
the universe or earlier extinction events. This can apply to either the external
agents ‘moving into’ the simulation, or for the simulated agents themselves as
‘mind children’ progeny, as put by Hans Moravec [22], which could then possi-
bly imply an abundance of post-biological civilizations in the universe [11, 12].
An interesting possible outcome of this process is that time in the real world
would be an important resource, and the speed of space colonization and com-
munication, that is already considered slow, would become unbearable. Future
civilizations may then prefer to avoid large-scale galactic colonization.
It is interesting to note that the Temporal Singularity shares features with
the transcension hypothesis [27] in the inevitable search for more energy and
computing power, but ultimately produces opposite predictions, as in the tran-
scension hypothesis advanced civilizations would try to slow down their subjec-
tive time by approaching black holes, rather than to accelerate it, in order to
forward time travel to a time where all civilizations may ultimately meet and
merge, and to optimize the acquisition of information.
On a negative side, the potential dangers that arise from this technology may
constitute a ‘Great Filter’ [17] that very few civilizations survive, thus explaining
the Fermi paradox. However, time-accelerated simulations may also be used to
escape traditional Great Filters by quickly providing us with solutions in time-
critical situations, including for example impacts of asteroids detected with short
notice or the Berserker scenario, in which an advanced intelligent civilization may
attack any newly emerging civilization.
Finally, a prediction of the Temporal Singularity in the context of the Fermi
paradox can be made in the rapid increase in the power used by a civilization,
tracking the super-exponential progress in technology, which could progress from
using the resources of its host planet to those of its entire solar system within
decades rather than millennia. Further, depending on the physical limits of tech-
nology, it may be possible that at least partial Dyson spheres [13] or Matrioska
brains [7] would be constructed in a relatively short time. The idea is particu-
larly interesting as present day technology should be capable of detecting even
partial neighboring Dyson spheres by changes in the infrared radiation of their
host star [31, 32, 16]. A prediction of the Temporal Singularity in the context of
the Fermi Paradox is then on the speed of construction of such mega-structures.
4 Conclusion
We have explored the idea that progress in computing and artificial intelligence
can lead to time-accelerated simulated civilizations unfolding in short time in-
tervals in the external world, due to a runaway increase in the rate of growth
of scientific and technological progress they could produce, that would quickly
increase the temporal speedups of the simulations themselves, ultimately result-
ing in a ‘Temporal Singularity’. The potential advantages and dangers of such
simulations have been briefly explored together with some implications of the
Temporal Singularity on the Fermi paradox.
The medium-term relevance of the topic comes from the potentially relatively
low computational power required to start the process, which could be as low as
1018 1021 FLOPS and thus be available within the next decades, making the
Temporal Singularity theoretically possible before the end of the century, and
possibly in its first half.
As a final remark, it is interesting to note that given the great competitive
advantages of running a simulation of the type described here, it is virtually in-
evitable that if it will ever be technically possible to create it, it will be created.
It should be noted, however, that this is unlikely to happen in a discontinuous
way, but rather we should expect an incremental progress, for example, starting
from the simple advantage of temporal speedups in simulated environments for
training artificial narrow intelligences (ANIs), as is already being done, to per-
haps accelerating simulated ‘childhood’ development and training of AGIs, to
actual simulated multi-agent systems, building towards complete societies and
civilizations following the increase in the available computing power.
Acknowledgments. I would like to thank Ivana Kolorici and Renato Spigler for
the helpful discussions and comments and the anonymous reviewers for the useful
suggestions and references, from which this manuscript benefited significantly.
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We describe the framework and strategy of the \^G infrared search for extraterrestrial civilizations with large energy supplies, which will use the wide-field infrared surveys of WISE and Spitzer to search for these civilizations' waste heat. We develop a formalism for translating mid-infrared photometry into quantitative upper limits on extraterrestrial energy supplies. We discuss the likely sources of false positives, how dust can and will contaminate our search, and prospects for distinguishing dust from alien waste heat. We argue that galaxy-spanning civilizations may be easier to distinguish from natural sources than circumstellar civilizations (i.e., Dyson spheres), although Gaia will significantly improve our capability to identify the latter. We present a "zeroth order" null result of our search based on the WISE all-sky catalog: we show, for the first time, that Kardashev Type III civilizations (as Kardashev originally defined them) are very rare in the local universe. More sophisticated searches can extend our methodology to smaller waste heat luminosities, and potentially entirely rule out (or detect) both Kardashev Type III civilizations and new physics that allows for unlimited "free" energy generation.
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We motivate the \^G infrared search for extraterrestrial civilizations with large energy supplies. We discuss some philosophical difficulties of SETI, and how communication SETI circumvents them. We review "Dysonian SETI", the search for artifacts of alien civilizations, and find that it is highly complementary to traditional communication SETI; the two together might succeed where either one, alone, has not. We discuss the argument of Hart (1975) that spacefaring life in the Milky Way should be either galaxy-spanning or non-existent, and examine a portion of his argument that we dub the "monocultural fallacy". We discuss some rebuttals to Hart that invoke sustainability and predict long Galaxy colonization timescales. We find that the maximum Galaxy colonization timescale is actually much shorter than previous work has found ($< 10^9$ yr), and that many "sustainability" counter-arguments to Hart's thesis suffer from the monocultural fallacy. We extend Hart's argument to alien energy supplies, and argue that detectably large energy supplies can plausibly be expected to exist because life has potential for exponential growth until checked by resource or other limitations, and intelligence implies the ability to overcome such limitations. As such, if Hart's thesis is correct then searches for large alien civilizations in other galaxies may be fruitful; if it is incorrect, then searches for civilizations within the Milky Way are more likely to succeed than Hart argued. We review some past Dysonian SETI efforts, and discuss the promise of new mid-infrared surveys, such as that of WISE.
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There is no strong reason to believe that human-level intelligence represents an upper limit of the capacity of artificial intelligence, should it be realized. This poses serious safety issues, since a superintelligent system would have great power to direct the future according to its possibly flawed motivation system. Solving this issue in general has proven to be considerably harder than expected. This paper looks at one particular approach, Oracle AI. An Oracle AI is an AI that does not act in the world except by answering questions. Even this narrow approach presents considerable challenges. In this paper, we analyse and critique various methods of controlling the AI. In general an Oracle AI might be safer than unrestricted AI, but still remains potentially dangerous.
The human brain has some capabilities that the brains of other animals lack. It is to these distinctive capabilities that our species owes its dominant position. Other animals have stronger muscles or sharper claws, but we have cleverer brains. If machine brains one day come to surpass human brains in general intelligence, then this new superintelligence could become very powerful. As the fate of the gorillas now depends more on us humans than on the gorillas themselves, so the fate of our species then would come to depend on the actions of the machine superintelligence. But we have one advantage: we get to make the first move. Will it be possible to construct a seed AI or otherwise to engineer initial conditions so as to make an intelligence explosion survivable? How could one achieve a controlled detonation? To get closer to an answer to this question, we must make our way through a fascinating landscape of topics and considerations. Read the book and learn about oracles, genies, singletons; about boxing methods, tripwires, and mind crime; about humanity's cosmic endowment and differential technological development; indirect normativity, instrumental convergence, whole brain emulation and technology couplings; Malthusian economics and dystopian evolution; artificial intelligence, and biological cognitive enhancement, and collective intelligence.
The acceleration of technological progress has been the central feature of this century. We are on the edge of change comparable to the rise of human life on Earth. The precise cause of this change is the imminent creation by technology of entities with greater-than-human intelligence. The chapter presents some possible projects that take on special significance, given the intelligence amplification (IA) point of view. These examples illustrate research that can be done within the context of contemporary computer science departments. This discussion of IA would yield some clearly safer approaches to the Singularity. The problem is not simply that the technological Singularity represents the passing of humankind from center stage, but that it contradicts our most deeply held notions of being. A closer look at the notion of strong superhumanity can show why that is.
A possible resolution to the Fermi Paradox is that we are living in an artificial universe, perhaps a form of virtual- reality `planetarium', designed to give us the illusion that the universe is empty. Quantum-physical and thermo- dynamic considerations inform estimates of the energy required to generate such simulations of varying sizes and quality. The perfect simulation of a world containing our present civilisation is within the scope of a Type K3 extraterrestrial culture. However the containment of a coherent human culture spanning ~100 light years within a perfect simulation would exceed the capacities of any conceivable virtual-reality generator.