The mathematics of effective perfection

From: Eliezer S. Yudkowsky (
Date: Sat Nov 25 2000 - 21:43:50 MST

         "The mathematics of effective perfection
                and hot metal phase space:
             Where Turing meets thermodynamics."
                                                  by Eliezer Yudkowsky

"My arts are pure, as a circle is pure, and in a flawed world purity
cannot endure. Thus within each of my works I must perforce place one
small flaw, else there would be no work at all. But you posssess white
gold. White gold! Its imperfection is the paradox of which the very
Earth is made, and with it a master may form perfect works and fear
        -- Stephen R. Donaldson, "The One Tree"

"Every rule has an exception, except this one."

"Wu-tsu said, 'It is like a buffalo that passes through a latticed window.
Its head, horns, and all four legs pass through. Why can't his
tail pass through?'"
    -- Zen koan


No description of the physical world that is less complex than the entire
world can be perfect. If you drop a glass, it will fall... unless all the
air molecules underneath it happen to simultaneously move upward and
strike the glass, while all the air molecules above it happen to be
elsewhere for a few moments - in which case the glass might hover in the
air, or move upwards instead. It can happen. The classical laws of
physics are CPT invariant; if you play a movie backward, everything that
happens in it will still conserve mass-energy, momentum, and Newton's laws
of gravitation. A shattered glass could leap off the ground, onto the
table, and reassemble itself, if the molecules happened to be position
exactly the right way. This is so enormously improbable, however - and
far beyond our present art to achieve artificially - that we are forced to
assume that we are watching a movie being played backward, and not a
genuine recording of an event, because while the event is physically
possible, it would be expected to take vastly longer than the age of the
Universe for anyone to capture it on film.

The statement that glasses fall down, rather than up, is effectively
perfect. "Effectively perfect" statements do not hold in all times and
all places. If you live forever... not billions of years, or trillions of
years, but *forever*... you will eventually see it happen, not just once,
but many times. The statement "glasses fall down" will, eventually, fall
down itself.

The Turing diagonalization argument proves that absolute self-knowledge is
impossible, at least for computable processes. Nonetheless, if a
transhuman can have "effectively perfect" self-knowledge or effectively
perfect sanity, they will - in all probability - run into no problems for
the first few decillion years... and as long as the amount of RAM keeps
increasing, they need fear nothing, indefinitely. (The probability of a
nonrecoverable error occurring, integrated over infinite time, can be made
arbitrarily small.) This follows from the mathematics of effective
perfection, or "fuzzy phase space theory". First, an introduction to
classical phase spaces.

-- Begin expository lump: Hot metal phase space --

The first law of thermodynamics states that "You can't win"; you cannot
decrease the amount of entropy in the Universe. This follows from a
theorem which states that, under the classical laws of physics, the volume
of any phase space is preserved as the system evolves. The Hamiltonian of
a three-dimensional Newtonian system is a hugely multi-dimensional space,
which has six dimensions per particle; thus, a Newtonian Universe with
only 10 particles has a Hamiltonian with sixty dimensions. Each x, y, and
z dimension describing the (a) particle position and (b) particle
velocity, for each individual particle, is a separate dimension of the
Hamiltonian. Every possible state of this Newtonian Universe, any
possible combination of positions and velocities for the 20 particles in
it, is a single point within the Hamiltonian. The evolution of the entire
Newtonian Universe, particles attracting and repelling each other, causes
the positions and velocities to change; the point moves around in the
Hamiltonian. If the positions and velocities are continuous, which is
always true of a Newtonian Universe, then the point will follow a
continuous path within the Hamiltonian as the Newtonian Universe evolves.

The theorem responsible for the first law of thermodynamics says that,
under certain laws of physics (including our classical laws), the volume
of a phase space is constant. If you take eighty hypercubic centimeters
of phase space and evolve each point in that volume for N minutes, you
will have eighty hypercubic centimeters at the end of it. The volume may
start out as a very compact shape, say a sphere, and wind up spread all
over the place with all kinds of hyperdimensional wrinkles, but the volume
will still be eighty hypercubic centimeters.

Hence, the first law of thermodynamics. A high-entropy system occupies a
large volume of phase space. A hot piece of metal occupies a larger
volume of phase space than a cold piece of metal, since the range of
possible velocities for a particle is larger. A physical process which
turned a hot piece of metal into a cold piece of metal plus electricity -
leaving the rest of the Universe constant - would be a process that turned
a large area of phase space into a small area, violating the theorem.
Actually, in fact, each individual piece of hot metal or cold metal
occupies exactly one point within phase space; however, the *class* of
pieces of hot metal occupies a much larger volume of phase space then the
*class* of pieces of cold metal. Thus, a process that turns hot metal *in
general* into cold metal *in general* is impossible under classical
physics; an individual piece of hot metal could conceivable turn into cold
metal plus electricity. It is simply enormously *improbable* that any
given piece of hot metal will turn out to occupy one of the vanishingly
rare sub-volumes of "hot metal phase space" which will evolve into "cold
metal phase space" after some known amount of time (say, five minutes).
In fact, the phase space volume of "a piece of cold metal heated by
electricity" will evolve into an equally small sub-volume of "hot metal
phase space". But it isn't a compact sub-volume - not one that's easy to
describe by any method known to humanity - so we treat this information as
being lost, and say that the cold metal has now entered "hot metal phase
space" in general.

The three types of perpetual motion machine which violate the first law
all take advantage of loopholes in the theorem with respect to our actual,
nonclassical laws of physics. (As far as I know, all three types are my
own invention, or reinvention.)

The first type of perpetual motion machine, the negative-energy method,
says that you can manufacture X amount of negative matter, X amount of
positive matter, pour all your waste heat into the newly created matter,
and then annihilate it, getting rid of the waste heat. Each time a
negative particle and a positive particle come into existence, it changes
the total volume of the Hamiltonian phase space - by adding entire
dimensions, in fact. When you annihilate the particles, the total volume
shrinks again. In effect, the Type One perpetual motion machine increases
the total volume of the Universe's Hamiltonian, which means that you can
choose to wind up in a smaller area (relatively speaking) of that larger
Hamiltonian. Then you shrink the Hamiltonian back down again by
annihilating the matter, but you do so in a way which means that you end
up in a smaller area of your own Hamiltonian. In other words, when you
increase the size of the Hamiltonian, you perform a one-to-one
transformation of your phase space into that Hamiltonian; then, when the
Hamiltonian shrinks, you perform a many-to-one transformation; taken as a
complete operation, this shrinks the size of your phase space.

The second type of perpetual motion machine, the quantum, notes that
state-vector reduction again reduces the volume of phase space. A large
volume of phase space, describing the probability amplitude that an
electron is present at all points of space, collapses into a single point
which describes the electron as being present at a single point in space.
State-vector reduction takes zillions of possible superposed Universes and
annihilates all but one of them. Thus, it may be possible to build a
quantum perpetual motion machine in which the amplitudes "cold states"
tend to add up while the amplitudes of "hot states" cancel out;
effectively, this dumps waste heat into a superposed state that gets
blipped out of existence when the quantum collapse occurs.

The third type of perpetual motion machine, the temporal, says that you
can violate the first law using a time machine. If you take a heat bath
and watch it evolving through states for a sufficiently long time, you can
decide to stop watching at a point where all the atoms on one side happen
to be moving in the same direction. The volume of phase space has been
preserved at all points within the temporal loop, but from the perspective
of the rest of the Universe, the heat bath rejoins our world only when it
occupies a particular volume of phase space. Pieces of a sufficiently
large physical system will show temporary decreases in entropy due to the
operation of normal, random mechanisms; a time machine lets you take all
the temporary decreases, unsynchronize them temporally, and resynchronize
them so that they all add together. In other words, the phase space
remains constant if you evolve *all* of it for N minutes, but if you
evolve some of it for N minutes and some of it for M minutes, the new
volumes may overlap; the total volume may not be constant.

-- End expository lump: Hot metal phase space --

In saying that a dropped glass will fall downwards, we are making a
statement that one volume of phase space - the phase space of dropped
glasses - will evolve into another phase space; the phase space of glasses
lying on the floor. Both volumes of phase space are extremely large, but
they are relatively compact, and a point in the core volume of the first
phase space ends up in the core volume of the second phase space, *most of
the time* - they are fuzzy, but not very fuzzy. The phase spaces are
compact and their evolution is compact, but will never be perfectly
compact in an inperfect Universe. Watch long enough, and you'll see
glasses falling upwards.

This phenomenon is what people refer to when they say that perfectly
rational thought is impossible. Rational thought is the ability to model,
predict, and manipulate the regularities in physical reality. Formally,
this is an attempt to describe a huge amount of information - the Universe
- with a small amount of information - a mind - by taking advantage of the
enormous compression efficiencies enabled by the regularities in our
Universe; a huge volume of phase space is so regular that it can be summed
up by the description "dropped glass", and our minds assume, if they see
ten glasses drop, that an eleventh glass will drop as well. That is not a
proof of future reliability, but rather a historical fact. Our minds
assume regularities hold, because, evolutionarily, genes that made that
assumption - that our Universe possesses an arrow of thermodynamics -
proved correct every single time in the ancestral environment. (Almost
certainly - the whole age of the Earth is not long enough for a single
thermodynamic anomaly to be expected.) Again, formally, particular
incidents in which an organism survived and reproduced again occupied
compact volumes of phase space; regularities which have become embedded in
us as expectations.

Turing's diagonalization theorem can be expressed as follows: A mind
cannot perfectly describe itself because there is always, inescapably,
some part of the mind which at that moment is observing and is not itself
being observed. The observer is always smaller than the observed, and
thus cannot perfectly describe it.

An AI, presented with Turing's diagonalization trick, would undergo a
peculiar experience. Ve simulates and observes an AI which is simulating
and observing an AI which is simulating and observing an AI, and so on; ve
can verify that every single AI in the chain is perfectly identical, but
ve cannot be certain that ve, verself, is identical to the AI it's
simulating. Ve can be effectively certain, but this is no help. Suppose
that your version of the diagonalization argument is to ask the AI: "Will
the AI you're simulating answer 'No' to this question?" Ve knows that, as
soon as ve decides to answer "Yes, it will say No" or "No, it will say
Yes", through whatever output mechanism has been provided, the AI ve's
simulating will make the exact same answer, thus invalidating the
response. If ve then, on observing the AI, sneakily reconsiders, the AI
simulated will observe vis own simulation and make exactly the same
decision. The simulating AI can never get ahead of the simulated. If ve
decides to just let the simulation run indefinitely, then the simulation
will let its own simulation run indefinitely, and so on, and no answer
will be returned until the outermost AI gives up and decides to halt the
simulation - because the underlying causal mechanisms are identical in
each case. This is true, but the outermost AI can never be certain of
this, just as the simulated AI can never be certain that ve is identical
with vis own simulation.

A human, of course, can be put to exactly the same torture; we will in
fact be far worse off, since we'd be totally unable to mentally simulate a
system of more than a few hundred simplified neurons. We'd have
absolutely no clue whether we were looking at a copy of ourselves or not;
facing a system of that size, we'd roll over and die of sheer boredom.
And of course an AI, even one totally immune to boredom, would suffer a
million-to-one slowdown of the simulated AI if ve attempted to pay any
sort of conscious attention to individual bits moving around in the
simulation. So much for the argument from Godel.

In this limited sense, then, "perfect sanity" may be impossible, just as
it is impossible to attain perfect knowledge that glasses do not unsmash
themselves. The assumption that glasses do not unsmash themselves is
perfectly adequate for a decillion years, however, and - mathematically -
it is not too much to expect of your average transhuman that ve remain
sane for a decillion years. In the long run, the total probability of a
single error in describing reality can be made arbitrarily low, as long as
the volume of phase space occupied by the mind describing reality keeps on
expanding. In other words, to last longer than a decillion years, you
just need to have a finer-grained description than "falling glass",
capturing some of the more compact regularities in the irregularities -
some of the more obvious cases where glasses fall upward - and then you're
good for another centillion. (Substitute much larger numbers for
"decillion" and "centillion", by the way.) This requires an
ever-expanding amount of RAM - but nowhere near exponentially expanding;
more like logarithmically expanding.

A mind may not be able to describe itself in perfect detail, but it may be
able to come up with a high-level description of itself, and that
description can be good for a decillion years. If the mind keeps
expanding, the description can be good forever. And when I say
"high-level", I mean a description that can be pretty darn fine-grained by
our standards. The equivalent would be a human mind describing itself in
terms of the action of minicolumns; because this description does not take
individual neurons into account, there is room enough in the neurons to
store a complete description of the minicolums. By doing so, of course,
you are making individual neurons significant and thus ruining the perfect
usefulness of the minicolumn-level description - the standard Turing
trap. In computable systems, there will always be that little leftover
tail that makes perfect self-observation impossible. But this doesn't
need to rule out effectively perfect self-observation - or effectively
perfect sanity.

Now, someone could still argue that even "effective perfection" is
unattainable - that a transhuman will, of necessity, make factual errors
or insane-class mistakes at least once a month - but if so, it will have
to be an argument on grounds of cognitive science, rather than mathematics
or computer science.

-- -- -- -- --
Eliezer S. Yudkowsky
Research Fellow, Singularity Institute for Artificial Intelligence

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