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%% Observer Localization in Multiverse Theories %%
%% Marcus Hutter (2008-2010) %%
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\title{
\vskip 2mm\bf\Large\hrule height5pt \vskip 4mm
Observer Localization in Multiverse Theories
\vskip 4mm \hrule height2pt}
\author{{\bf Marcus Hutter}\\[3mm]
\normalsize RSISE$\,$@$\,$ANU and SML$\,$@$\,$NICTA \\
\normalsize Canberra, ACT, 0200, Australia \\
\normalsize \texttt{marcus@hutter1.net \ \ www.hutter1.net}
}
\date{November 2010}
\maketitle
\begin{abstract}
The progression of theories suggested for our world, from ego- to
geo- to helio-centric models to universe and multiverse theories and
beyond, shows one tendency: The size of the described worlds
increases, with humans being expelled from their center to ever more
remote and random locations. If pushed too far, a potential theory
of everything (TOE) is actually more a theories of nothing (TON).
Indeed such theories have already been developed. I show that
including observer localization into such theories is necessary and
sufficient to avoid this problem. I develop a quantitative recipe to
identify TOEs and distinguish them from TONs and theories
in-between. This precisely shows what the problem is with some
recently suggested universal TOEs.
\\
\def\contentsname{\centering\normalsize Contents}
{\parskip=-2.7ex\tableofcontents}
\end{abstract}
\begin{keywords}
world models;
observer localization;
predictive power;
Ockham's razor;
universal theories;
computability.
\end{keywords}
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\label{secIntro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
A number of models have been suggested for our world. They range
from generally accepted to increasingly speculative to apparently
bogus. For the purpose of this work it doesn't matter where you
personally draw the line.
%
Many now generally accepted theories have once been regarded as
insane, so using the scientific community or general public as a
judge is problematic and can lead to endless discussions: for
instance, the historic geo$\leftrightarrow$heliocentric battle; and
the ongoing discussion of whether string theory is a theory of
everything or more a theory of nothing.
%
In a sense this paper is about a formal rational criterion to
determine whether a model makes sense or not.
%
In order to make the main point of this paper clear, below I will
briefly traverse a number of models
\cite{Harrison:00,Barrow:04,Hutter:10ctoex}. The presented bogus
models help to make clear the necessity of observer localization and
hence the relevance of this work.
%-------------------------------%
\paradot{Egocentric to Geocentric model}
%-------------------------------%
A young child believes it is the center of the world.
Localization is trivial. It is always at ``coordinate'' (0,0,0).
Later it learns that it is just one among a few billion other people
and as little or much special as any other person thinks of
themself. In a sense we replace our egocentric coordinate system by one with origin (0,0,0) in the center of Earth. The move away from
an egocentric world view has many social advantages, but dis-answers
one question: Why am I this particular person and not any other?
%-------------------------------%
\paradot{Geocentric to Heliocentric model}
%-------------------------------%
While being expelled from the center of the world as an individual,
in the geocentric model, at least the human race as a whole remains
in the center of the world, with the remaining (dead?)
universe revolving around {\em us}.
%
The heliocentric model puts Sun at (0,0,0) and degrades Earth to
planet number 3 out of 8. The astronomic advantages are clear, but
dis-answers one question: Why this planet and not one of the others?
Typically we are muzzled by semi-convincing anthropic
arguments \cite{Bostrom:02,Smolin:04}.
%-------------------------------%
\paradot{Heliocentric to cosmological model}
%-------------------------------%
The next coup of astronomers was to degrade our Sun to one star
among billions of stars in our milky way, and our milky way to one
galaxy out of billions of others, according to current textbooks.
Again, it is generally accepted that the question why we are in this
particular galaxy in this particular solar system is essentially
unanswerable.
%-------------------------------%
\paradot{Multiverses}
%-------------------------------%
Many modern more speculative cosmological models (can be argued to)
imply a multitude of essentially disconnected universes (in the
conventional sense), often each with its own (quite different)
characteristic: Examples are Wheeler's oscillating universe,
Smolin's baby universe theory, Everett's many-worlds interpretation
of quantum mechanics, and the different compactifications of string
theory \cite{Tegmark:04}. They ``explain'' why a universe with our
properties exist, since the multiverse includes universes with all
kinds of properties, but they cannot {\em predict} these properties.
A multiverse theory {\em plus} a theory predicting in which universe
we happen to live would determine the value of the inter-universe
variables for our universe, and hence have much more predictive
power. Again, anthropic arguments are sometimes evoked but are
usually vague and unconvincing.
%-------------------------------%
\paradot{Universal TOE (UTOE)}
%-------------------------------%
Taking the multiverse theory to the extreme, Schmidhuber
\cite{Schmidhuber:00toe} postulates a universal
multiverse, which consists of {\em every} computable universe.
Clearly, if our universe is computable (and there is no proof of the
opposite \cite{Schmidhuber:00toe}), the multiverse generated by UTOE
contains and hence perfectly describes our own universe, so we have a
Theory of Everything (TOE) already in our hands. Unfortunately it is
of little use, since we can't use UTOE for prediction. If we knew
our ``position'' in this multiverse, we would know in which
(sub)universe we are.
%
This is equivalent to knowing the program that generates {\em our}
universe. This program may be close to any of the conventional
cosmological models, which indeed have a lot of predictive power.
Since locating ourselves in UTOE is equivalent and hence as hard as
finding a conventional TOE of our universe, we have not gained much.
%-------------------------------%
\paradot{All-a-Carte models}
%-------------------------------%
Champernowne's normal number glues the natural numbers, for our purpose
written in binary format, 1,10,11,100,101,110,111,1000,1001,... to
one long string.
\beqn
1 10 11 100 101 110 111 1000 1001 ...
\eeqn
Obviously it contains every finite substring by construction. The
digits of many irrational numbers like $\sqrt{2}$, $\pi$, and $e$
are conjectured to also contain every finite substring. If our
space-time universe is finite, we can capture a snapshot of it in
a truly gargantuan string $u$. Since Champernowne's number contains
every finite string, it also contains $u$ and hence perfectly
describes our universe. Probably even $\sqrt{2}$ is a perfect TOE.
%
Unfortunately, if and only if we can localize ourselves, we can
actually use it for predictions. (For instance, if we knew we were
in the center of universe 001011011 we could predict that we will
`see' 0010 when `looking' to the left and 1011 when looking to the
right.)
%
Locating ourselves means to (at least) locate $u$ in the multiverse.
We know that $u$ is the $u$'s number in Champernowne's sequence
(interpreting $u$ as a binary number), hence locating $u$ is
equivalent to specifying $u$. So a TOE based on normal numbers is
only useful if accompanied by the gargantuan snapshot $u$ of our
universe. In light of this, such an ``All-a-Carte'' TOE (without
knowing $u$) is rather a theory of nothing than a theory of
everything.
%-------------------------------%
\paradot{Localization within our universe}
%-------------------------------%
The loss of predictive power when enlarging a universe to a
multiverse model has nothing to do with multiverses per se. Indeed,
the distinction between a universe and a multiverse is not absolute.
%
For instance, Champernowne's number could also be interpreted as a
single universe, rather than a multiverse. It could be regarded as
an extreme form of the infinite fantasia land from the NeverEnding
Story, where everything happens somewhere. Champernowne's number
constitutes a perfect map of the All-a-Carte universe, but the map is
useless unless you know where you are.
%
Similarly but less extreme, cosmological inflation models produce a
universe that is vastly larger than its visible part, and different
regions may have different properties.
%-------------------------------%
\paradot{Predictive power}
%-------------------------------%
The exemplary discussion above has hopefully convinced the reader
that we indeed lose something (some predictive power) when
progressing to too large universe and multiverse models.
%
Historically, the higher predictive power of the large-universe models
(in which we are seemingly randomly placed) overshadowed the few
extra questions they raised compared to the smaller
ego/geo/helio-centric models.
%
But the discussion of the (physical, universal, and all-a-carte)
multiverse theories has shown that pushing this progression too far
will at some point harm predictive power. We saw that this has to do
with the increasing difficulty to localize the observer.
%-------------------------------%
\paradot{Contents}
%-------------------------------%
Classical models in physics are essentially differential equations
describing the time-evolution of some aspects of the world. A Theory
of Everything (TOE) models the whole universe or multiverse, which
should include initial conditions. As argued above, it can be crucial to
also localize the observer. I call a TOE with observer
localization, a {\em Complete TOE} (CTOE).
%
Section \ref{secCTOEs} gives an informal introduction to the
necessary ingredients for CTOEs, and how to evaluate and compare
them using a quantified instantiation of Ockham's razor.
%
Section \ref{secCTOE} gives a formal definition of what accounts for
a CTOE, introduces more realistic observers with limited perception
ability, and formalizes the CTOE selection principle.
%
The Universal TOE is a sanity critical point in the development of
TOEs, and will be investigated in more detail in Section \ref{secUTOE}.
%
Important extensions listed in Section \ref{secDisc} are
detailed in \cite{Hutter:10ctoex}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Complete TOEs (CTOEs)}\label{secCTOEs}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
A TOE by definition is a perfect model of the universe. It should
allow to predict all phenomena. Most TOEs require a specification of
some initial conditions, e.g.\ the state at the big bang, and how
the state evolves in time (the equations of motion). In general, a
TOE is a program that in principle can ``simulate'' the whole
universe.
%
An All-a-Carte universe perfectly satisfies this condition but
apparently is rather a theory of nothing than a theory of
everything. So meeting the simulation condition is not sufficient
for qualifying as a Complete TOE.
%
We have seen that (objective) TOEs can be completed by specifying
the location of the observer. This allows us to make useful
predictions from our (subjective) viewpoint. We call a TOE
plus observer localization a Complete TOE.
%
If we allow for stochastic (quantum) universes we also need to
include the noise. If we consider (human) observers with limited
perception ability we need to take that into account too. So
%-------------------------------%
\paranodot{A complete TOE needs specification of}
%-------------------------------%
\begin{itemize}\parskip=0ex\parsep=0ex\itemsep=0ex
\item[(i)] initial conditions
\item[(e)] state evolution
\item[(l)] localization of observer
\item[(n)] random noise
\item[(o)] perception ability of observer
\end{itemize}
We deal with limited perception ability (o) in Section
\ref{secCTOE}. Space prevents discussing stochastic theories (n);
they are dealt with in \cite{Hutter:10ctoex}. Next we need a way to
compare TOEs.
%-------------------------------%
\paradot{Predictive power and elegance}
%-------------------------------%
Clearly we can never be sure whether a given TOE makes correct
predictions in the future. After all we cannot rule out that the
world suddenly changes tomorrow in a totally unexpected way. We have
to compare theories based on their predictive success in the past.
It is also clear that this is not enough: For every model we can
construct an alternative model that behaves identically in the past
but makes different predictions from, say, year 3000 on. Popper's
falsifiability dogma is little helpful. Beyond postdictive success,
the guiding principle in designing and selecting theories,
especially in physics, is elegance (and mathematical consistency).
The predictive power of the first heliocentric model was not
superior to the geocentric one, but it was much simpler. In more
profane terms, it has significantly less parameters that need to be
specified.
%-------------------------------%
\paranodot{Ockham's razor}
%-------------------------------%
suitably interpreted tells us to choose the simpler among two or more
otherwise equally good theories. For justifications of Ockham's
razor, see \cite{Li:08}. Some even argue that by definition, science
is about applying Ockham's razor, see \cite{Hutter:04uaibook}.
For a discussion in the context of theories in physics, see \cite{Gellmann:94}.
It is beyond the scope of this paper to repeat these considerations. In
\cite{Hutter:10ctoex} I prove that Ockham's razor is
suitable for finding TOEs.
%-------------------------------%
\paradot{Complexity of a TOE}
%-------------------------------%
In order to apply Ockham's razor in a non-heuristic way, we need to
quantify simplicity or complexity. Roughly, the complexity of a
theory can be defined as the number of symbols one needs to write
the theory down. More precisely, write down a program for the state
evolution together with the initial conditions, and define the
complexity of the theory as the size in bits of the file that contains
the program. This quantification is consistent with our intuition,
since an elegant theory will have a shorter program than an
inelegant one, and extra parameters need extra space to code, resulting in
longer programs.
%-------------------------------%
\paradot{Standard model versus string theory}
%-------------------------------%
To give an example, let us pretend that the standard model of
particle physics + gravity (P) and string theory (S) both qualify as
TOEs. P is a mixture of a few relatively elegant theories, but
contains about 20 unexplained parameters that need to be specified
(although some regularities can be explained
\cite{Hutter:97family}). String theory is truly elegant, but
ensuring that it reduces to P needs sophisticated extra assumptions
(e.g. the right compactification). It would require a major effort
to quantify which theory is the simpler one in the sense defined
above, but I think it would be worth the effort. It is a
quantitative objective way to decide between theories that are (so
far) predictively indistinguishable.
%-------------------------------%
\paradot{CTOE selection principle}
%-------------------------------%
It is trivial to write down a program for an All-a-Carte multiverse
A. It is also not too hard to write a program for the universal
multiverse U, see Section \ref{secUTOE}. Lengthwise A easily wins over
U, and U easily wins over P and S, but as discussed A and U
have serious defects. Given all of the above, it now nearly suggests
itself that we should include the description length of the observer
location in our TOE evaluation measure. That is,
\beqn
\mbox{among two CTOEs, select the one that has shorter overall length}
\vspace{-2ex}
\eeqn
\beq\label{orctoe}
\Length(i)+\Length(e)+\Length(l)
\eeq
For an All-a-Carte multiverse, the last term contains the gargantuan
string $u$, catapulting it from the shortest TOE to the longest
CTOE.
%-------------------------------%
\paradot{TOE versus UTOE}
%-------------------------------%
Consider any (C)TOE and its program $q$, e.g.\ P or S. Since U runs
all programs including $q$, specifying $q$ means localizing (C)TOE
$q$ in U. So U+$q$ is a CTOE whose length is just some constant
bits (the simulation part of U) more than that of (C)TOE $q$. So
whatever (C)TOE physicists come up with, U is nearly as good as
this theory. This essentially clarifies the paradoxical status of
U. Naked, U is a theory of nothing, but in combination with
another TOE it excels to a good CTOE, albeit slightly longer=worse
than the latter.
%-------------------------------%
\paradot{Localization within our universe}
%-------------------------------%
So far we have only localized our universe in the multiverse, but
not ourselves in the universe. Assume the about $10^{11}\times
10^{11}$ stars in our universe are somehow indexed. In order to
localize our Sun we only need its index, which can be coded in about
$\log_2(10^{11}\times 10^{11})\approx 73$ bits. To localize earth
among the 8 planets needs 3 bits. To localize yourself among 7
billion humans needs 33 bits. These localization penalties are tiny
compared to the difference in predictive power (to be quantified
later) of the various theories (ego/geo/helio/cosmo). This explains
and justifies theories of large universes in which we occupy a
random location.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Complete TOE - Formalization}\label{secCTOE}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-------------------------------%
\paradot{Objective TOE}
%-------------------------------%
Since we essentially identify a TOEs with a program generating a
universe, we need to fix some general purpose programming language
on a general purpose computer. In theoretical computer science, the
standard model is a so-called Universal Turing Machine ($\UTM$)
\cite{Li:08}. It takes a program coded as a finite binary string
$q\in\{0,1\}^*$, executes it and outputs a finite or infinite binary
string $u\in\{0,1\}^*\cup\{0,1\}^\infty$. The details do not matter
to us, since drawn conclusion are typically independent of them. In
this section we only consider $q$ with infinite
output
\beqn
\UTM(q)=u_1^q u_2^q u_3^q\,... =:u_{1:\infty}^q
\eeqn
In our case, $u_{1:\infty}^q$ will be the universe (or multiverse)
generated by TOE candidate $q$. So $q$ incorporates items (i) and
(e) of Section \ref{secCTOEs}. Surely our universe doesn't look like
a bit string, but can be coded as one as explained in
\cite{Hutter:10ctoex}. We have some simple coding in mind, e.g.\
$u_{1:N}^q$ being the (fictitious) binary data file of a
high-resolution 3D movie of the whole universe from big bang to big
crunch, augmented by $u_{N+1:\infty}^q\equiv 0$ if the universe is
finite. Again, the details do not matter.
%-------------------------------%
\paradot{Observational process and subjective complete TOE}
%-------------------------------%
As we have demonstrated it is also important to localize the
observer. In order to avoid potential qualms with modeling human
observers, consider as a surrogate a (normal not extra cosmic) video
camera filming=observing parts of the world. The camera may be fixed
on Earth or installed on an autonomous robot. It records part of the
universe $u$ denoted by $o=o_{1:\infty}$. (If the lifetime of the
observer is finite, we append zeros to the finite observation
$o_{1:N}$).
In a computable universe, the observational process within it, is
obviously also computable, i.e.\ there exists a program
$s\in\{0,1\}^*$ that extracts observations $o$ from universe $u$.
Formally
\beq
\UTM(s,u_{1:\infty}^q) = o_{1:\infty}^{sq}
\eeq
where we have extended the definition of $\UTM$ to allow access to
an extra infinite input stream $u_{1:\infty}^q$. So
$o_{1:\infty}^{sq}$ is the sequence observed by subject $s$
in universe $u_{1:\infty}^q$ generated by $q$.
Program $s$ contains
all information about the location and orientation and perception
abilities of the observer/camera, hence specifies not only item (l)
but also item (o) of Section \ref{secCTOEs}.
A Complete TOE (CTOE) consists of a specification of a (TOE,subject)
pair $(q,s)$. Since it includes $s$ it is a Subjective TOE.
%-------------------------------%
\paradot{CTOE selection principle}
%-------------------------------%
So far, $s$ and $q$ were fictitious subjects and universe programs.
Let $o_{1:t}^{true}$ be the past observations of some concrete
observer in our universe, e.g.\ your own personal experience of the
world from birth till today. The future observations
$o_{t+1:\infty}^{true}$ are of course unknown. By definition,
$o_{1:t}$ contains {\em all} available experience of the observer,
including e.g.\ outcomes of scientific experiments, school
education, read books, etc.
The observation sequence $o_{1:\infty}^{sq}$ generated by a
correct CTOE must be consistent with the true observations
$o_{1:t}^{true}$. If $o_{1:t}^{sq}$ would differ from $o_{1:t}^{true}$
(in a single bit) the subject would have `experimental' evidence
that $(q,s)$ is not a perfect CTOE.
We can now formalize the CTOE selection principle as follows
\beqn
\mbox{Among a given set of perfect ($o_{1:t}^{sq}=o_{1:t}^{true}$) CTOEs $\{(q,s)\}$}
\vspace{-2ex}
\eeqn
\beq\label{Lqs}
\mbox{select the one of smallest length } \Length(q)+\Length(s)
\eeq
Minimizing length is motivated by Ockham's razor.
Inclusion of $s$ is necessary to avoid degenerate TOEs like U and A.
%
The selected CTOE $(q^*,s^*)$ can and should then be used for
forecasting future observations via
$...o_{t+1:\infty}^{forecast}=\UTM(s^*,u_{1:\infty}^{q^*})$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Universal TOE - Formalization}\label{secUTOE}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-------------------------------%
\paradot{Definition of Universal TOE}
%-------------------------------%
The Universal TOE generates all computable universes.
The generated multiverse can be depicted as an infinite
matrix in which each row corresponds to one universe.
\beqn\arraycolsep0ex
\begin{array}{c|cccccc}
q\; & \multicolumn{3}{c}{\UTM(q)} & \\ \hline
\epstr & u_1^\epstr & u_2^\epstr & u_3^\epstr & u_4^\epstr & u_5^\epstr & \cdots \\
0 & u_1^0 & u_2^0 & u_3^0 & u_4^0 & \cdots & \cdots \\
1 & u_1^1 & u_2^1 & u_3^1 & \cdots & \cdots & \\
00\; & u_1^{00} & u_2^{00} & \cdots & \cdots & & \\
\vdots & \vdots & \vdots & \vdots & \\
\end{array}
\eeqn
To fit this into our framework we need to define a single program
$\breve q$ that generates a single string corresponding to this matrix.
The
standard way to linearize an infinite matrix is to dovetail in
diagonal serpentines though the matrix:
\beqn
\breve u_{1:\infty} := u_1^\epstr u_1^0 u_2^\epstr u_3^\epstr
u_2^0 u_1^1 u_1^{00} u_2^1 u_3^0 u_4^\epstr u_5^\epstr
u_4^0 u_3^1 u_2^{00} ...
\eeqn
Formally, define a bijection
$i=\langle q,k\rangle$ between a (program, location) pair $(q,k)$
and the natural numbers $\SetN\ni i$, and define $\breve u_i:=u_k^q$.
It is not hard to construct an explicit program $\breve q$ for
$\UTM$ that computes $\breve u_{1:\infty}=u_{1:\infty}^{\breve
q}=\UTM(\breve q)$.
%-------------------------------%
\paradot{Remarks}
%-------------------------------%
Cutting the universes in bits and interweaving them into one string
might appear messy, but is unproblematic for two reasons: First, the
bijection $i=\langle q,k\rangle$ is very simple, so any particular
universe string $u^q$ can easily be recovered from $\breve u$.
Second, such an extraction will be included in the localization/
observational process $s$, i.e.\ $s$ will contain a specification of
the relevant universe $q$ and which bits $k$ are to be observed.
%-------------------------------%
\paradot{TOE versus UTOE}
%-------------------------------%
We can formalize the argument in the last section of simulating TOE
by UTOE as follows: If $(q,s)$ is a CTOE, then $(\breve q,\tilde s)$
based on UTOE $\breve q$ and observer $\tilde s:=rqs$, where program
$r$ extracts $u^q$ from $\breve u$ and then $o^{sq}$ from $u^q$, is
an equivalent but slightly larger CTOE, since $\UTM(\tilde s,\breve
u)=o^{qs}=\UTM(s,u^q)$ by definition of $\tilde s$ and
$\Length(\breve q)+\Length(\tilde s) = \Length(q)+\Length(s)+O(1)$.
%-------------------------------%
\paradot{The best CTOE}
%-------------------------------%
Finally, one may define the best CTOE (of an observer with experience
$o_{1:t}^{true}$) as
\beqn
UCTOE := \arg\min_{q,s}\{\Length(q)+\Length(s) : o_{1:t}^{sq}=o_{1:t}^{true} \}
\eeqn
where $o_{1:\infty}^{sq} = \UTM(s,\UTM(q))$. This may be regarded as
a formalization of the holy grail in physics; of finding such a TOE.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}\label{secDisc}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Respectable researchers, including Nobel Laureates, have dismissed
and embraced each single model of the world mentioned in the
introduction, at different times in history and concurrently.
(Excluding All-a-Carte TOEs which I haven't seen discussed before.)
As I have shown, Universal TOE is the sanity critical point.
%
The most popular (pseudo) justifications of which theories are
(in)sane have been references to the dogmatic Bible and Popper's
limited falsifiability principle. This paper contained a more
serious treatment of world model selection. I introduced and
discussed the usefulness of a theory in terms of predictive power
based on model {\em and} observer localization complexity.
%
Extensions to more practical and realistic (partial, approximate,
probabilistic) theories (rather than TOEs), more motivation
and examples, and a proof that Ockham's razor is suitable for
finding TOEs can be found in \cite{Hutter:10ctoex}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Bibliography %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\addcontentsline{toc}{section}{\refname}
\begin{small}
\begin{thebibliography}{ABCD}\parskip=0ex
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\end{small}
\end{document}
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