Riffing on Wolfram

From: Eric Hawthorne <egh.domain.name.hidden>
Date: Sun, 10 Nov 2002 22:40:05 -0800

Any comments? Can anyone point me to similar speculations?

Thanks, Eric


A collection of thoughts (very much a work in early progress)
provoked by chapters 9 and 12 of "A New Kind of Science"
by Stephen Wolfram.

Caveat: The following was written hastily and in somewhat sloppy,
informal terms, with casual or vague use of some arguably
pseudo-scientific terms, like "de-quantized" or "classicized" by which
I mean something like "the process whereby a single state or average
of quantum probabilities seems to take on importance so as be considered
the "actual" state of some particle etc. after it is observed."

Wolfram postulates that space-time is a network (of nodes and connections),
manipulated by simple programs which have the
characteristics that:

1. the only thing they do is make local adjustments
to the configuration of the network (e.g. replace a node by 3 nodes joined
by connections, erase a connection etc.)
2. They are order-invariant (causally-invariant he calls it)
in their global effect. It doesn't matter which time-order the local
replacement rules "fire" in.

and he goes on to begin to prove how relativity, gravity, matter etc.
work out nicely in such a model.
But he doesn't say what the substrate of the "universe" network is,
and he cannot yet fit quantum theory into his model,
which got me to thinking:

Quantum Computational Cosmology?? - E.H. 2002

--- The universe is information. More specifically, it is emergent              
--- order within an infinite-bandwidth signal, or in other words, is just
--- a particular, priveleged view of "all-possible information, all at once."
On reading Wolfram's book, and in particular the part about physics as CAs operating on
a network to produce space-time, matter, energy, I was prompted to have the following
ideas. Please excuse the lack of rigour. I'm just trying to convey intuitions here
and get some feedback on whether anyone thinks there's promise in this direction
or if there are other references people can point me to.
These questions arise: 
1. What would the network of nodes and arcs between nodes, in Wolfram's 
   "spacetime-as-network" be made of? i.e. what is the substrate of Wolfram's 
   universe network?
2. How do we define the "time arrow" and what makes the universe 
   as it appears to be?  
My essential concepts are these:
Principle 1
The substrate is simply (all possible arrangements of "differences")
or perhaps put another way, the substrate of the universe is
"the capacity for all possible information", 
The fundament is the binary difference. Each "direct difference" is an arc,
and network nodes are created simply by virtue of being the things at either end of
a "direct difference".
Let's posit that there is a multiverse, which we can think of as
all possible states of all possible universes, or as the information substrate
of the universe.
An information-theoretic interpretation of the multiverse might say that it is
defined as:
a universe with just one "thing" and no differences (boring) +
a universe with one difference (ergo, two things) +
all possible configurations of two differences +
all possible configurations of three differences + etc.
       -------        <-- A binary difference - the fundamental unit of information
     A ------- B   <-- two "things", A and B, created just by virtue of being defined
                       to be at the opposite poles of the binary difference.
To define a particular configuration of the universe, that is, a network
of binary direct-difference relationships between a certain number of
postulated individuals, you can use binary bits, as follows: 
The individual "things" are denoted A,B,C...
A "1" in the matrix (below left) denotes that a direct difference exists between the
column-labeling individual and the row-labeling individual. 
  E D C B A                        B - C     
E   1 0 1 1                       / \          
D     0 0 1       equivalent to  A - E
C       1 0                       \ /
B         1                        D                      
Every fundamental-level "thing" that exists is either at the end of a 
direct difference from another "thing", or is reachable by some chain of
direct differences from the other thing. "things" which are not reachable
by a chain of direct differences from some other "thing" do not exist.
So why don't we posit that the "Wolfram network" that describes the form
of spacetime at its smallest-grained (i.e. plank-length) level is in fact 
comprised of nodes and arcs which have no other reality (no other material 
that they are made of) other than binary differences. i.e. it is a network 
formed of pure information; of chains of direct and indirect differences, 
and of nothing else.
Space Network 1
A1 --- A2 --- A3 --- A4 --- A5 --- A6 --- A7 
|      |      |      |      |      |      |
|      |      |      |      |      |      |
B1 --- B2 --- B3 --- B4 --- B5 --- B6 --- B7
|      |      |      |      |      |      |
|      |      |      |      |      |      |
C1 --- C2 --- C3 --- C4 --- C5 --- C6 --- C7
A "Wolfram" space-filling (space-defining) network (2D space version)
wherein the topology of the arrangement of arcs (direct differences) and nodes
fits neatly into a 2 dimensional space (and so this particular network topology
can be viewed as DEFINING what 2D space is.) Notice that traversing a "straight
line" in space from A1 to C7 is best approximated by taking a 
"minimum-number-of-arcs" traversal from A1 to C7 over the network. This kind of
elegant property of the mapping of the network to properties of space of a certain 
dimensionality is what makes the notion that it defines 2D space
so compelling.
Space Network 2
Space curved by gravity = a network with a lower-dimensional topology
in some local part of itself.
A1 --- A2 --- A3 --- A4 --- A5 --- A6 -- 
|      |      |      |      |      |     \ A7 
|      |      |      |      |      |     / |   
B1 --- B2 --- B3 --- B4 --- B5 --- B6 --   |
|      |      |      |      |    /    \    |  
|      |      |      |      |  /        \  /   
C1 --- C2 --- C3 --- C4 --- C5 --------- C7
|      |      |      |      |  \       /  \      
|      |      |      |      |    \   /     |      
D1 --- D2 --- D3 --- D4 --- D5 --- D6 --- D7 
|      |      |      |      |    /   \    |       <-- A reduction in the number
|      |      |      |      |  /       \  |           of nodes (more precisely,     
E1 --- E2 --- E3 --- E4 --- E5  -------  E7           in the dimensionality of
|      |      |      |      |  \        / |           the topology of node connections)
|      |      |      |      |    \    /   |           occurs in space in the vicinity  
F1 --- F2 --- F3 --- F4 --- F5 --- F6 --- F7          of a mass, thus curving space
|      |      |      |      |    /  \     |           in on itself, and resulting
|      |      |      |      |  /      \  /            in gravity. -- Wolfram
G1 --- G2 --- G3 --- G4 --- G5         G7  
|      |      |      |      |  \       /              Perhaps the nodes have been "captured"
|      |      |      |      |    \   /  \             by the mass itself. The mass may be
H1 --- H2 --- H3 --- H4 --- H5 --- I6     \           modelled as a local surplus (or tangle)
|      |      |      |      |      |      J7          of overly connected nodes. -- Wolfram
|      |      |      |      |      |     /  
I1 --- I2 --- I3 --- I4 --- I5 --- J6 -- 
The speed of light, Wolfram says, can be defined as the maximum speed of
propagation of information through such a network. And that maximum speed
is one arc-traversal per "clock cycle" of the universe. This is equivalent
to saying that each change to the configuration of the network can only 
be propagated to other nodes by means of the execution of local-rule
programs, and each single execution of a local rule can only examine and 
affect adjacent nodes and arcs. 
Here is a binary, bitstring representation of network 1:
(Each "1" represents that there is a direct binary difference between
 the node indicated by the column label and the node indicated by
 the row label.)
   C7 C6 C5 C4 C3 C2 C1 B7 B6 B5 B4 B3 B2 B1 A7 A6 A5 A4 A3 A2 A1
C7    1  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0
C6       1  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0
C5          1  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0
C4             1  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  
C3                1  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  
C2                   1  0  0  0  0  0  1  0  0  0  0  0  0  0  0  
C1                      0  0  0  0  0  0  1  0  0  0  0  0  0  0  
B7                         1  0  0  0  0  0  1  0  0  0  0  0  0  
B6                            1  0  0  0  0  0  1  0  0  0  0  0  
B5                               1  0  0  0  0  0  1  0  0  0  0  
B4                                  1  0  0  0  0  0  1  0  0  0  
B3                                     1  0  0  0  0  0  1  0  0  
B2                                        1  0  0  0  0  0  1  0
B1                                           0  0  0  0  0  0  1
A7                                              1  0  0  0  0  0
A6                                                 1  0  0  0  0
A5                                                    1  0  0  0
A4                                                       1  0  0
A3                                                          1  0
A2                                                             1
Here is the binary matrix above laid out as a single long bitstring,
by placing each row of bits after the preceding row of bits:
To represent the connectivity configuration of a network with
n nodes, in this way, you need a bitstring of length n x (n-1) / 2 .
In the case of network 1 above, it has 21 nodes, so we need a bitstring
of length (21 x 20 / 2) = 210 bits.
But what do we get if we consider all of the different binary values
that a bitstring of length 210 can represent? Well we get all of the
different possible "directly-connected" or "not directly connected"
arrangements of a network with 21 nodes in it.
How many possible arrangements of a network of 21 nodes are there,
anyway? Well it is 2 to the power (n x (n-1) / 2) = 2 to the power 210.
Of course, a bitstring of size 210 can also represent all of the
configurations of a network of 20, or 19, or 18, ... or 2 nodes as well,
simply by having a lot of leading-zero bits in it.
So in summary, a bitstring of length 2 to the power (n x (n-1) / 2)
can represent all possible connectivity (direct-difference) arrangements
of all networks of up to n nodes.
Coming back to physics, we can think about spacetime as a "wolfram network" 
of nodes (defined solely as the poles of binary differences) and arcs 
(the direct differences themselves). 
In this physical model, now imagine that some very, very large n is the 
maximum number of different entities (or different spacetime locations, at a
planck-length scale) postulated to be realizable in the universe. 
In this situation, the number of possible different configurations of
such a "discrete" spacetime network (with n nodes i.e. n "different"
spacetime locations) is 2 to the power (n (n-1)/2) for the very very large
n. In total this very large set of possible configurations of spacetime nodes
can be viewed as all possible configurations of the universe. Or equivalently,
this number, 2 to the power (n(n-1)/2), can be viewed as the "information
potential" of the multiverse-substrate of the universe.
Philosophical aside: 
With this information-theoretical model of the universe, an expansion of the 
universe might best be viewed as an expansion of the number of nodes in the
network, or equivalently an expansion of the number of bits in the 
"difference matrix" bitstring. So "expanding universe" is best understood as
expanding "information-potential" of the network in which time and space can 
be defined, rather than simply an expansion of space and time themselves.
An expansion of the universe is an expansion of the information 
canvas in which space and time can evolve. It is a memory upgrade.
If an ordinary computer has an 8-bit register in its CPU chip, the CPU can 
represent, or "hold" exactly 1 of the 256 possible states of an 8-bit
bitstring in its register memory at one time.
But research is presently underway into Quantum Computers. What is a quantum
computer? It is something which can have an 8-bit "quantum register", which can
hold simultaneously, as quantum states of atoms, all 256 possible states of
the 8-bit bitstring. If the quantum computer is asked to add two 8-qubit
registers together and put the result in a third 16-qubit register, the result,
in a single quantum computation operation, is a register containing simultaneously
the additions of all possible pairs of 8-bit values. In simplistic terms, the
quantum computer "does all possible computations at once."
What if the multiverse substrate of our universe is like a quantum computer,
doing "all possible computations at once" on a bitstring of length 
2 to the power (n (n-1)/2), where n is the maximum number of distinct discrete
spacetime locations posited to exist.
Principle 2
If the multiverse is "all possible states "simultaneously" of a 
length (n (n-1)/2) bitstring, 
then the "time-arrow" and the "actual universe"is defined as an order-producing 
"selection" or "view" of a subset of the "potential states" of the multiverse.
If we imagine the multiverse as kind of holding (or being the potential for)
all possible states of the long bitstring, then you can make a selection
from all of those states. i.e. you can define 
U1 = a particular sequence of states of the bitstring.
The word "sequence" rather than "set" is chosen deliberately here, 
because my contention is that, of all possible sequences Ui, some
sequences will be "order-producing", when the sequence of states is
traversed in the direction (through state-space) which leads from the beginning 
to the end of the sequence of states.
So why don't we just make the bold claim that "the time arrow"
is the direction through state-space from the beginning to the end
of an "order-producing" sequence (U1) of states. And that U1,
the "order-producing" sequence of states, is the "observable 
The information-theoretic interpretation of the weak anthropic
Why is U1 the observable universe? Well because its evolution
of states was order-producing in just the right measure to 
produce just the right mixture of randomness and order to
produce matter and energy, the rules of physics, and 
"emergent behaviour" systems such as intelligent observers.
There need not be a single "order-producing" state-sequence, and indeed 
it would be surprising if there were. More likely, there is a highly 
constrained subset of all possible sequences of information-states which 
can produce order in the manner required to produce an observable universe. 
The observable state sequences are constrained by requirements which boil down to
spatio-temporal regularity and continuity, and a certain critical
mixture of just enough order accretion and coherence ability, and just enough
random variation of form to generate many trials of potential self-ordering
These factors are influenced by the values of the physical constants,
which may in turn correspond to particular highly constrained execution
sequences of very particular local-rule programs modifying the 
spacetime network.
How does this relate to Wolfram's CAs?
Well we can define programs as being simply the things which
specify the state transitions from "state i to state i+1" of a sequence Ui
of states of the multiverse.  
In (vaguely recollected) Hoare logic terms, my contention is that the
multiverse can be "viewed" as simultaneously (in an extra-time sense) 
executing every possible program Pij such that S(i) Pij -> S(j).
Most sequences of executions of programs will be like executing
"garbage" programs on "garbage data", but some sequences of
program executions will produce interesting evolutions of states exhibiting
complexity and order, and emergent behaviour.
A good guess is that some of the "interesting" program executions
may be understandable as, modellable as,
executions of particular universal CLASS 4 automata.
The details are left to the wonks ;-)
--- Summary --------
The multiverse (or substrate for our universe) is precisely
the potential for all information. That is, it is equivalent
to the "simultaneous" exhibition of or capacity for
all possible states of a long and possibly growing bitstring.
The time-arrow of an observed universe IS the ORDER OF VISITATION of the 
information-states of the multiverse which corresponds to 
the execution of "Wolfram-automata" which locally-evolve the configuration 
of the individuals and differences of the substrate in such a way as to 
generate just the right mix of randomness and order which produces 
stable, organized systems and ultimately observers.
With this formulation, we need not assume that there is some
magical, extra-universal "supercomputer" busily computing a 
"Fredkin-Wolfram-information" universe for us. Computation 
of ordered complexity just falls out, as just being a particular
path through a very large set of information-states all of
which co-exist in the multiverse substrate. The huge, but
unrealized set of information states in the substrate, in 
fact IS the substrate. It only becomes (or hosts) an observable
universe when viewed by observers existing within 
a set of its states that is consistent enough to be "real".
Received on Mon Nov 11 2002 - 01:39:51 PST

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