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"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

nothing."

-- 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 http://singinst.org/

Research Fellow, Singularity Institute for Artificial Intelligence

**Next message:**Zero Powers: "Re: 137: Sheer blank-minded stupidity"**Previous message:**Jason Joel Thompson: "Re: 137: Sheer blank-minded stupidity"**Next in thread:**hal@finney.org: "Re: The mathematics of effective perfection"**Maybe reply:**hal@finney.org: "Re: The mathematics of effective perfection"**Reply:**scerir: "Re: The mathematics of effective perfection"**Reply:**Ben Goertzel: "RE: The mathematics of effective perfection"**Reply:**John Clark: "Re: The mathematics of effective perfection"**Maybe reply:**hal@finney.org: "Re: The mathematics of effective perfection"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

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