Biljana Percinkova, Charles Berner, Marika Apostolova, Taxhedin Selimi: SELF-REFERENCED SYSTEMS IN ARTIFICIAL INTELLIGENCE AND QUANTUM PHYSICS; RADICAL THEORY AND APPLICATION
ABSTRACT:
Deep analysis of the barriers in Artificial
Intelligence and Robotics based on Goedel's Law of
Incompleteness will be given by introducing propositional
calculus, typographical number theory and building formal
logical systems. Mathematical proofs of consistence and
completeness of formal logical systems (or simply robots in
artificial intelligence) will be analyzed by pointing out the
limitations of the contemporary approach and a radical theory
offering resolution will be presented. Interpretations of
quantum physics will be analyzed step by step as in the diagram
presented and solution based on a radical theory will be
offered. Strong support in terms of comparison of the results
with the measurements of relevant parameters of particle physics
done in the Institute for particle Physics Research in CERN,
Geneva, will be clearly presented and mathematical proofs
offered.
Picture 1
The theory proposed is a 'non dualist' resolution (to the
consciousness/observer problem in quantum physics and
completeness problem in artificial intelligence) as defined by
Jonathan Shear (1997), which states that rather than conscious
experience being generated from the physical world, or the
physical world being a product of conscious experience - both
are considered to be aspects of a deeper fundamental reality.
Each individual's state of conscious experience is a real state
had by a real entity, and that which that conscious state is of
is also real - in that it is based on those same real entities.
The subjective state and its object arise inextricably linked
from that which underlies them which is a finite set of agents
and their acts of choices. The underlying mathematics is based
on random directed graphs and matrices.
After the limitations and pitfalls of contemporary approaches
known in artificial intelligence and quantum physics will be
given, the results obtained by the Radical theory will be
computed by computer programs and comparisons with measurements
offered. Monte Carlo simulation method will be applied for
simulating the behavior of the agents and their connections in
complex networks. Validation of the simulation model offered
will be provided.
INTRODUCTION
It is unlikely that the problem
of understanding what subjective conscious experience is can be
solved unless, as suggested by Roger Penrose (1996), there are
"important changes in our picture of physical reality". Even
more definitively Penrose says,
"I certainly do not expect to find any answers [about mind,
consciousness, space-time, etc.] in sub-atomic physics, for
example. What I am arguing for is a radical upheaval in the very
basis of physical theory."
That is to say, without a theory that upsets the standard
paradigm of science we may not be able to understand the
relation of first person conscious experience to the physical
world.
Resolution of Goedel's theorem of non completeness is also
provided by the new Paradigm.
The theory proposed here by Charles Berner and mathematically
supported by Biljana Percinkova is a 'non dualist' resolution to
the consciousness/observer problem as defined by Jonathan Shear
(1997) which states that rather than conscious experience being
generated from the physical world, or the physical world being a
product of conscious experience, both are considered to be
aspects of a deeper fundamental reality. Each individual's state
of conscious experience is a real state had by a real entity,
and that which that conscious state is of is also real, in that
it is based on those same real entities. The subjective state
and its object arise inextricably linked from that which
underlies them. The present form of the theory was inspired by
John Wheeler's (1990) suggestion that information is fundamental
to the physics of the universe and by David Chalmers' (1995)
speculation that information may be "truly fundamental" with
"two basic aspects, corresponding to the physical and the
phenomenal features of the world." Wheeler also states (1979)
that that on which the laws of physics are based cannot also be
physical, and Chalmers argues (1996) that conscious experience
must be nonphysical.
An Overview of the
Theory
We propose that the fundamental reality that underlies both the
physical world and conscious experience is a number
1 of nonphysical
agents.
2 each agent has three qualities:
(i)
intrinsic uniqueness, or 'whoness' (the agent is itself
and not any other agent);
(ii) individual initiative, or ability and,
(iii) existence.
The first two qualities are agent specific, and the last is
common to all agents.
Diagram 1
The common existence of unique agents can be thought of in at
least three other ways:
(1) All of the
agents make up a single realm of existence.
(2) Each agent is in
as many states as there are agents, each of those states
being a state of (nonphysical) relation to a different
agent, in other words, any one of an agent's states is a
first person relation with some particular agent.
(3) Each agent is
in as many states as there are agents, each of those states
being informed by a different agent; in other words, the
content of any one of an agent's states is information based
on some particular agent.
Hypothesis or research questions
The basic hypothesis is that the Information (or Lila Paradigm)
could give us the total number of non physical agents as well as
some basic parameters known in particle physics.
The derivation of the number of existing nonphysical
agents (N).
The value for N that produces computations of values for
our physical Universe that agree most closely with measured
values is (10e)π (10 to the e to
the π ); however, this is also the value that has the
weakest explanation for it. As shown below, e and π
are the result of two different progressions of embedded
bifurcated sub-states in a giant circuit system, and it may be
that there is a third major progression based on 10 (Arneodo
1992). To show how π and e are intertwined, both
π and e are described in terms of the information
paradigm. We begin by writing the standard formula for π:1
π = the sum of
the limit N
à ∞ [6(1/12 + 1/22
+ 1/32 + . . . + 1/N2)]½
where N is any
whole number. There are many agents out of the 1023 agents that
have arrows extending across the circuit to most, if not all, of
the agents in the giant circuit arrangement. Although in Diagram
4b only four arrows are shown extending from agent I, consider
agent I to be one of those agents. In the information paradigm,
the value of π varies from the early universe to the
present, and the value is proportional to the following formula
based on the standard formula.
a/b1+ a/b2
+ a/b3 +. . .+ a/bN
The square, the square root and the 6 are ignored here as they
are constant for every term and thus do not affect the
proportionality.
The numerator in all of the terms is the conscious sub-state
produced by the 'single' circuit arrow of common time,
represented by IàF . This
sub-state, being the irreducible basis of the agents'
consciousness of a common world, counts as one, 1. The
denominator, conscious sub-state b1, is produced by
the sub arrangement IàO and also
counts as 1 because it is also irreducible, in that it has no
sub-states. Because both conscious sub-states a and b1
are embedded in the bifurcated 1st term sub-state of
consciousness, produced from the arrangement:
they
are compared (placed in ratio relative to each other) as a/b1
which is 1/1, the first term of the standard formula.
The 2nd term in the proportional formulation has the denominator
b2, which has two cross-over arrows in it, IàO
and IàZ , each of which produces
an irreducible sub-state of consciousness and, therefore, counts
as one. Because IàO and IàZ
each combine to produce sub states of consciousness that are
embedded in sub-state b2, these sub-states of
consciousness are thereby in a single state of consciousness
where those sub-states are added and so count as 2, that is, as
1 + 1 together. Because both sub-states a and b2 are
in the bifurcated 2nd term sub-state, they are compared as the
ratio a/b2, which is, therefore, 1/2, the second term
in the standard formula.
The 3rd. term
denominator, b3: has three cross-over arrows in it
and for the same reasons that the 2nd term has a ratio of 1/2,
the 3rd term has a ratio of 1/3, which is the 3rd term of the
standard formula. And so on to N terms, which, in this
case, are finite in number. All of the terms are summed in the
consciousness of each agent in the circuit due to all the
sub-states of consciousness that are the different terms being
in the single, overall state of consciousness which is produced
by the entire arrangement symbolized in Diagram 4b.
The usual formula for
e is:
e = the sum
limit Nà∞
(1/1! + 1/1! + 1/2! + 1/3! +. . .+ 1/N!)
where N is any
whole number. The number e is found in a particular combination
of sub-states in the circuit arrangement. Refer to Diagram 4b in
the following. The proportional formula for e is:
a/a' + a/c1 +
a/c2 + a/c3 +. . .+ a/cN
The 1st term numerator, produced by the arrangement I
à F, which is an irreducible
sub-state of consciousness, and thus has a value of 1, is called
sub-state a. The denominator is sub-state a', which is
the sub-state produced by the entire circuit, including an
iteration of I à F. Such a linear
arrangement of a single state of consciousness has a value of 1 multiplied by
1, which is 1!. Since both a and a' are in the
overall state (the entirety of Diagram 4b) they are (linearly)
compared, as a/a', which is 1/1!, the first term of the standard
formula.
The 2nd term numerator is also I à
F, sub-state a, which is 1; however, the denominator is any one
of the possible bifurcations extending from agent I, which are:
Bifurcation No. 2. is used in the 2nd term example. The
sub-state of consciousness produced from it is called sub-state
c1, shown in the next diagram embedded in the 2nd
term, along with sub-state a:
2nd term sub-state a/c1
= 1/1!
The bifurcated sub-state cl has the numerical value of 1 1
because there are two single states of consciousness embedded in
it, those produced by I à F and by
I à O. Both sub-state a and
sub-state c1 are embedded in the bifurcated 2nd term
sub-state and are thereby compared in the ratio a/c1,
which is 1/1!, the 2nd term of the standard formula.
In the third term, the numerator is 1, for the same reasons as
above, as in all the terms. The denominator, the sub-state c2,
however, has two bifurcated arrangements in it, cla and
clb each producing a separate state of consciousness:
3rd term sub-state a/c2
= 1/1 2 = 1/2!
The two sub-states cla and clb are embedded in
sub-state c2 and, since c2 is a single
state of consciousness, those two sub-states are thereby added
and count as 2; and in sub-state c2 the two
sub-states, each of which is 1, combine as 1 x 1 and also
combine with sub-state c2 as 1 x 1 x 2, which is the
factorized term 2!. Also, c2 is embedded along with
sub-state a in the 3rd term sub-state, comparing them in the
ratio a/c2, which is the ratio 1/2!, the 3rd term of
the standard formula.
For the same reasons as
above, the fourth term is a/c3:
4th term sub-state a/c3
= 1/3!
There are three c2 consciousness sub-states each of
which is made up of two cl sub-states embedded in a c3
sub-state, which, therefore, has the factorization of 1 x 2 x 3
= 3!, the 4th term of the standard formula; and, so on through
the rest of the finite number (N) of terms. Again, these are
summed due to all the terms being part of the single state of
consciousness of the entire arrangement in Diagram 4b.
The number π is based on the ratio of the number of
cross-over arrows in a sub-state to the arrow in the circuit
from a single agent in the circuit, while e is based on the
ratio of the number of bifurcations in a sub state to the arrow
from a single agent in the circuit. As can be seen, the same
arrows embedded in different groupings of sub-states produce
both π and e. The numeric consequence of this for
a finite N is proportional to e π .
Using the values of e and π as computed from the standard
formulas, where N is equal
to infinity, a close approximation to
the number of agents in our
Universe can be computed from N = (10e)π
. Using base 10 values for e and π in the
Mathematica program, the result is:
138,258,752,126,535,254,782,246.6528....
This should probably be rounded down to 6 in the last place
before the decimal because the actual N is finite and a
whole number, whereas the Mathematica computations of e and π
and were made assuming that N is infinite. In the
computations throughout this paper, we use N as equal to
1.382587521 x 1023.
It is obvious that π should be involved with spatial and,
therefore, all energy phenomena, because the very bifurcations
that produce space are the same as those which produce the π
ratio. This also applies to e, though the effect is less
widely spread.
Three less precise derivations of N follow to provide a check on
the assumption that (10e)π is the
relationship that produces the real value of N.
The point of inflection of the inflationary curve was estimated
by the New Inflationary Theory (Guth 1984) at about 6 x
10-33sec. A later (Guth 1997) estimate is about 10-34sec. Due to
many parameters being guessed at in the theory, the time of
inflection could vary from about 10-35sec to about 10-31sec.
The information model formula for the point of inflection is
given by π /2 N tq (see Section VII), so tq.
According to this, N is about 3 x 1022, using the 1984
estimate; however, this could vary from 6 x 1019 for the
10-35sec estimate, to about 5.5 x 1023 for the 10-31sec
estimate.
Using the rest mass formulas for two leptons pairs, each of
which contains N in the formula, and using the best
measured values for the rest masses of equivalent leptons from
those pairs, a value for N can be found. 2M(5)/M(3)
is the ratio of rest mass numbers of the tau-anti-tau pair and
the electron-positron pair, where
M(5) is
(120 N4)1/5 and M(3) is (6 N2)½
The best measured value for the tau particle/electron ratio is
3491ฑ6 (Perl 1990). Using this value to solve the equation,
3491 = 2(120 N4)/(6 N3) for N; then, N =
(3/219/2 53/2) 3491, which is 1.3836 x 1023 (ฑ~0.002).
This compares to the N = (10e)π value of
1.3826...X 1023, a difference of about +0.07%.
The formula for the Compton wavelength of the electron (λce) is
eฑKn. Using the best measurement of the elementary charge, eฑ,
expressed as length (Misner 1973), 1.38114132 x 10-34 cm; and,
the formula for n = NN/ eK. Then λce = eฑ K (N-N/eK), and
solving for N gives
= 1.382587517 x 1023,
where λce is 2.42531058 x 10-14cm (Cohen 1990).
This is different from
(10e)π by about -3 x 10-7%.
Given the closeness of the correlations, the assumption in a.
above, while not carefully supported by logic, nevertheless
seems likely to be correct and, therefore, the value for N
derived there is likely to be the number of nonphysical agents
that exist.
Universal Physical
Constants c, h, and G
Using the values derived in appendices A and B,
new standard values of the universal physical constants, c,
h, and G can be easily calculated to much greater
accuracies. The calculated value of G constitutes a
prediction that is likely to be tested by measurements in the
coming decade.
The speed of light, c is which is equal to 2.99792458 x
1010 cm/sec, which compares to the currently measured
value of 2.92792458 x 1010cm/sec.
The
value of Planck's reduced constant (ħ), expressed as length
squared, is 1p2, which is
2.61403186 x10-66cm2
G, the Newtonian
constant of gravitation, is G, therefore, is equal to
6.67876984 x 10-11m3Kg-1sec-2. This compares to the currently
measured value of 6.67(+0.1,-0.01) x 10-11m3Kg-1sec-2. 1
STUDY DESCRIPTION
In the paradigm the agents' states are called states of
information. This information paradigm is the most useful in
formulating a theory of consciousness. The conditions under
which an agent is conscious of information from one or more of
its states of information are given below.
Each agent is in one of two states with regard to each of its
states of information (relation), either a state of denial of
that information state or a state of non-denial of that
information state, which of the two being determined (made to be
so) by the agent itself. That the agent determines its own
states of denial and non denial of its own information states is
what is meant by the agent's individual initiative or ability.
Each of an agent's information states is based on the entire
domain of a particular agent, which consists of the qualities of
that particular agent, its information states, and its
self-determined states of denial or non denial with regard to
its information states. In Diagram 2, agent A has three
information states because in this example three agents exist:
A, W, and I. Agent A is in a state of non-denial of its
information state based on agent W, a state of denial of its
information state based on itself, and a state of denial of its
information state based on agent I (the denied information
states are crossed out). The qualities of each of the agents are
summarized as 'an existence who (the particular agent: A, W, or
I) determines' (See the Diagram below.)
On the above Diagram 2 the domain
of agent A is presented.
In order for an agent to be in a conscious state, the agent must
be in a state of non-denial of at least one of its information
states. An agent in a self-determined state of non-denial of a
state of information based on a particular agent is conscious of
any information in that information state that is based on a
quality of the particular agent that is the same as any of the
original agent's own qualities. Consciousness is a nonphysical
agent's state of the sameness of not-denied information with
some aspect of itself. For example, if agent A is in a state of
non-denial of its information state based on itself, agent A is
conscious of the information in that state that is based on all
three of agent A's qualities: intrinsic uniqueness (whoness),
individual initiative (ability), and existence. In other words,
agent A is conscious of its own existence as the particular
nonphysical agent that it is. If, alternatively, agent A is in a
state of non-denial of its state of information based on some
other agent, say agent W, agent A is conscious of the
information in that state that is based on agent W's existence
quality, which is the same as agent A's own existence quality,
but not of the information based on agent W's qualities of
whoness and ability, because who agent W is is not the same as
who agent A is, and agent W's ability is not the same ability as
agent A's ability (see Diagram 3).5 Consciousness of existence
without whoness and ability is consciousness of physical
existence; agent A is conscious of a physical entity. A
nonphysical agent's state of the sameness of not-denied
information based on the existence quality of an agent other
than itself with its own existence quality is consciousness of a
physical entity. We call this simplest type of physical entity a
proto-fermion.6 (Diagram 3 and future diagrams do not show any
denied information states in an agent's domain because they do
not contribute to the agent's consciousness.)
THE DOMAIN OF AGENT
A
Diagram 3
Due to the fact that an agent's information state of the domain
of a particular agent contains information based on that
particular agent's denied and not-denied information states, an
agent in a state of non-denial of an information state may be
conscious of information based on qualities of agents other than
the particular agent on whom the information state is based. For
example, if agent A is in a state of non-denial of its
information state based on agent W, and agent W is in a state of
non-denial of its information state based on agent I (in labeled
directed graph notation, AàWàI),
agent A is conscious of its information based on agent I's
existence quality in addition to its information based on agent
W's existence quality. Agent A is conscious of two
proto-fermions. How two or more proto-fermions are physically
(temporally, spatially, energetically) related in an agent's
state of consciousness depends on the configuration of the
system of agents in not-denied information states on which that
conscious state is based.
In AàWàI,
agent W, like agent A, is conscious of a proto-fermion due to
the sameness of agent I's existence quality with its own
existence quality. This is an example of a 'baby' common
universe. If the average number of not-denied information states
per agent is more than two, directed graph theory predicts the
existence of a giant system of agents in not-denied information
states that includes most of the agents.
Such a giant system is the basis of the individual conscious
state of each of the agents in it. shows (a) a toy model (or
baby-universe) of a giant system patterned on the giant system
on which our universe is based and (b) the same system re-drawn
to emphasize the largest circuit-type arrangement in that
system. The absence of an arrow from one agent to another means
that the corresponding information state is denied.
Diagram 4. a
Diagram 4. b
Each agent in or connecting into the circuit has an individual
state of consciousness of a physical universe that is the same
as every other agent's, except that each agent's viewpoint
differs according to that agent's position in the giant system.
Thus the agents seem to themselves to be conscious of an
independently existing physical universe.
In this information paradigm, we, the reader, the writers, and
all other individuals that can be conscious, are the nonphysical
agents, rather than being bodies, conscious minds, or even
physically located view point souls. A nonphysical agent is not
ordinarily conscious of itself as nonphysical because it is
ordinarily in a state of denial of its information state of
itself, and is therefore only conscious of it self by means of a
circuit arrangement. If this is the case, the nonphysical agent
is conscious of itself as an intrinsically unique entity with
individual initiative that is physically related to physical
entities
Click on the
picture for bigger view, then again click to zoom it
or right click to download it.
The matrix formalism as and addition to the graph formalism,
introduced and developed by Biljana Percinkova, is presented on
the Diagram above. Percinkova has also developed different
algorithms for finding specific sub-graphs and structures of
interest in particle physics.
PERCINKOVAS ALGORITHM FOR FINDING PROTO-PHENOMENA IN ONE UNIT
1-D SPACE CONTINUUM
ARRANGEMENT
The following algorithm has been proposed by Prof. Biljana
Percinkova:
The matrix
could be used for finding many (if not all) combinations of
proto-phenomena since it is just another view of the graphs,
only spread out in plane.
Ive found an algorithm for discovering all the structures of
Linear plus Bifurcated type of combination forming
space-continuum between units of matter. The "lists" in this
mini-spanning trees are one unit of 1-D space from the root. The
discovering of the structures is done in systematic way, so that
no sub-structures are missed. This is like Mendeleev System of
elements.
First the ones (1-s) in the row are found. Then a(i,j) cell,
being one from the central circuit (the Hamiltonian) is marked
as reference cell; in this case it is the cell ((I,F). Then its
origin is being determined from the (i-1) row, in this case the
W row; that is the cell (W,I). Finally the (j-1) column is being
checked in this case this the I column for finding other
possible roots for the sub-structures, denoted with ones in the
column.
(Beautiful
synchronicity has happened. Namely, when starting the research
I've commenced with the L row having E P R as its members so
nicely pointing to Einstein-Podolsky-Rosen experiment; this
experiment is my favorite and mentioned in Henry Stapp's
writings.)
Hence we have:
W
↓
I
↓ ↓ ↓ ↓
F O
Z P
I N
↓
↓
F F
↓ ↓ ↓ ↓
X Y X Y
F
↓
Y
↓ ↓ ↓
E M T
-
From the
row E proto-fermions Q, L are one unit 1-D space from
Y through E. Checking (j-1) column, in this case E column,
two other sub-structures (1-s in that column) are being
discovered. Here we have L being 1-D space from itself L (L à
E à L):
Y L K
↓ ↓ ↓
E E E
↓ ↓ ↓ ↓ ↓ ↓
Q L Q L Q L
E
↓
Q
↓ ↓
M H
-
From the
row M proto-fermions O, R are one unit 1-D space from
Q through M. (O and R are 1-s in the M row; the cell (M,O)
is referential (a(i,j)). The origin a(i-1, j-1) is the cell
((Q,M) showing the space distance from Q through M. Finally
the (j-1) column, the M column is being checked for other
roots: these are Y and D.
Q Y D
↓ ↓ ↓
M M M
↓ ↓ ↓ ↓ ↓ ↓
O R O R O R
B
↓
N
↓ ↓ ↓
I O Z
Z F
↓ ↓
X X
↓ ↓ ↓ ↓
L V L V
-
From the
row L proto-fermions E, P, R are one unit 1-D space
from X through L. The column L, being (j-1), gives
additional sub-structure. Here we have E being one unit 1-D
space from itself ! (E à
L à E).
X E
↓ ↓
L L
↓ ↓ ↓ ↓ ↓ ↓
E P R E P R
P
↓
C
↓ ↓
T U
-
From the
row D proto-fermions W, M, R are one
unit 1-D space from T through D, cell (T,D).
T
↓
D
↓ ↓ ↓
W M R
R
↓
G
↓ ↓
B V
Connectedness of a random directed graph, digraph, as seen by
Baker and Szekeres:
It is as if we are
dealing with the following situation:
There are N labeled individuals (i1 to iN
). In an arbitrary sequence one individual chooses to be in a
state of knowledge of another individual. We may say that at
knowledge choice number q, individual i chooses to
be in a state of knowledge of individual j (i à
j, i ≠ j). This situation is a random labeled digraph, where
the number q of directed edges increases with each knowledge
choice made.
We can also write i àà
j (i precedes j, or i is connected to j) if there is a set of
knowledge choices (directed edges) (i1 = i à
i2 ), (i2à
i3) ,...., (ip1 à
ip = j) linking i to j. We can say that the
individuals are (the graph is) totally connected at knowledge
choice q if i àà j
for all ordered pairs of individuals i, j (i ≠ j).
i ~àà
j indicates that i is not connected to j,
and i ~à
j indicates that i does not choose to be in
a state of knowledge of j.
At knowledge choice q let c(q) be the total number of
ordered pairs of individuals i, j such that i àà
j. The individuals will be totally connected at knowledge choice
q if c(q) = N (N 1). That is every individual is
connected to every other individual. We can describe the
connectedness of the individuals (digraph) to be given by a
parameter:
x(q) = 0 ≤ x(q) ≤ 1
The total number of connections c(q) can be broken down into two
parts. Let cd(q) be the number of ordered pairs of
individuals i, j such that the knowledge choice i à
j is the first knowledge choice for which i àà
j. We call cd(q) the number of direct connections.
Let ci(q) be the number of ordered pairs of
individuals i,j such that i ~à
j until after the first knowledge choice for which i àà
j. We call ci(q) the number of indirect connections.
c(q) = cd(q) +
ci(q)
We can describe the direct connectedness and indirect
connectedness of the individuals (graph) to be given by xd(q)
and xi(q) respectively where:
xd(q) = ; xi(q)=
; x(q) = xd(q) + xi(q)
How do x(q), xd(q), and xi(q) vary with
the number of knowledge choices? Clearly monotonically. We can
derive differential equations that describe the average growth
of c(q), cd(q) and ci(q) as follows:
For each i let:
d(i,q) be the number of
individuals k such that k àà
i, which is the number who are connected to i.
For each j let:
e(j, q) be the number of
individuals l such that j àà
l, which is the number to whom j is connected.
At knowledge choice q
suppose that i à j. Then
how many new connections of the form k àà
l are established?
Clearly if i àà
j before knowledge choice q, the probability of which is
x(q) , then no new connections are established. That is:
c(q + 1 ) = c(q) ; cd(q
+ 1 ) = cd(q); ci(q + 1 ) = ci(q)
However if i ~àà j
at knowledge choice q, the probability of which is 1 x(q),
then not only is the new direct connection i à
j established but also the indirect connections k àà
l, such that k àà i,
j àà l and k ~àà
l. [D. Perhaps this should read,
not k~àà l, but kààj
and iààl.]
Let the number that are
already connected at knowledge choice q be f (i,j,q).
There will also be extra connections due to those that j
is connected to that i is not connected to, and due to
those that are connected to i that are not connected to
j. Let their numbers be:
g(i,j,q) = number of l
such that j àà l & i
~àà l
and
h(i,j,q) = number of k
such that k àà i & k
~àà j
So in this case:
cd(q + 1) =
cd(q) + 1
ci(q + 1) = ci(q)
+ d(i,q) e(j,q) f(i,j,q)+ g(i,j,q) + h(i,j,q)
c(q + 1) = c(q) + 1 +
d(i,q) e(j,q) f(i,j,q) + g(i,j,q) + h(i,j,q)
Given the relative
probabilities of the two cases:
cd (q + 1) =
x(q) cd(q) + {1 x(q)} {cd(q) + 1}
= cd(q) + {1
x (q)} ...................................(1)
ci(q + 1) =
x(q) ci(q) + {1 x(q)} {ci(q) + d(i,q)
e(j,q) f(i,j,q) + q(i,j,q) + h(i,j,q)}
= ci(q) + {1
x(q)} {d(i,q) e(j,q) f(i,j,q) + g(i,j,q) + h(i,j,q)}
c(q + 1) = c(q) +{1 x(q)}
{ 1 + d(i,q) e(j,q) f(i,j,q) + g(i,j,q) + h(i,j,q) }
The probability at
knowledge choice q of i àà
j is x(q). This applies equally to all pairs so we may estimate:
d(i,q) = e(j,q) = x(q)
(N 1)
f(i,j,q) = d(i,q) e(j,q)
x(q) = x(q)3 (N 1)2
g(i,j,q) = e(j,q) x(q) =
x(q)2 (N 1)
h(i,j,q) = d(i,q) x(q) =
x(q)2 (N 1)
So we can estimate:
ci(q + 1) ci(q)
+ {1 x(q) } { x(q)2 (N 1)2 x(q)3
(N 1)2 + 2 x(q)2 (N 1)}
= ci(q) +
x(q)2 {1 x(q)} (N 1) [N { 1 x(q)}+ {1+ x (q) }
]
xi (q + 1 )
= xi(q) + x(q)2 { 1 x(q) } [ { 1 x (q)
} + { 1 + x(q) } / N ]
From equation (1) we can
see:
xd (q + 1) =
xd (q) + { 1 x(q) } / { N (N 1)
For N large we may
approximate:
xd(q + 1 ) =
xd(q) + {1 x(q) } / N2
xi(q + 1 ) =
xi(q) + x(q)2 { 1 x(q)}2
Replacing the difference
equations by differential equations:
dxd/dq = (1
x) / N2 .....................................(2)
dxi /dq =
x2(1 x)2 .......................................(3)
dx/dq = (1 x) / N2+
x2 (1 x)2 ...............(4)
Equation (4) can be replaced by two
differential equations, the first for x à
1 and the second for x2 (1 x) > 1 / N2 .
For x << 1 we may
replace (1 x ) by 1 and get:
dx/dq = 1 / N2
+ x2 = (1 + N2 x2) / N2
.....(5)
This differential
equation is readily solved:
q = N arctan(N x) +
constant
At q = 0, x = 0, so
constant = 0, and:
q = N arctan(N x)
............................................(6)
∴
x = (1 / N) tan(q / N) ....................(7)
For x << 1 we also have from equation (2):
dxd/dq = 1 /
N2.
xd = q / N2
+ constant
At q = 0, xd
= 0, so constant = 0, and:
xd = q / N2
........................................................(8)
xi = (1 /
N) { tan(q / N) (q / N) } ........(9)
For x2 (1
x) > 1 / N2 we can ignore (1 x) / N2 and
get from equation (4):
dx/dq = x2 (1
x)2 .......................................(10)
The errors involved in
making the approximations for the differential equations (5) and
(10), e1 and e2 respectively, are:
e1 = (x / N2+
x4)
e2 = (1 x )
N2
The differential
equations (5) and (10) are equally in error when:
| e1| = | e2 |
x / N2 + x = (
1 x ) / N2
Comments on connectedness of a random digraph
(1.....)
The expected proportion of individuals who have not made a
knowledge choice to be in a state of knowledge of another
individual by knowledge choice q is exp( q/N). The
proportion of individuals who have not become known by any
individual (i.e. no individual has made a choice to be in a
state of knowledge of them) by knowledge choice q is
also exp( q/N). Consequently, by knowledge choice q,
1 exp( q/N) individuals have made a knowledge choice and
1 exp( q/N) individuals have become known. The maximum
number of connections c(q) will occur when all individuals
who have made a knowledge choice are connected to all
individuals who have become known. So the maximum
connectivity that is possible by knowledge choice q is
x ≤ {1 exp( q/N)}2.
At q = (1 + π/2) N there will be about 8% of individuals who
have not made a knowledge choice. This means that x can be
no more than 0.85 at this point, which is at variance with x
= 1 near (1 + π/2) N. It can also be shown that the last
individual is likely to make its first knowledge choice or
the last individual will become known when q = log(4 N). In
other words the connections cannot be complete until about q
= log(4 N) .
It seems likely that the reason for this discrepancy is that
no account was taken above of the numbers of individuals who
have not made a knowledge choice and have not become known
as a function of q. The problem arises due to using
x as an average connectivity when the underlying
distribution is skewed due to the Poisson distribution of
knowledge choices.
The Origin of Time - Charles Berner:
Agents in not-denied information states are the basis of an
agent's conscious experience of the passage of time. There is no
background of time (or space) against which the agents exist.
There is not a blankness or nothing, or any other state, that
'precedes' or 'follows' their existence. The agents do not exist
at a certain time, remain the same over time, or change. All of
this is also true of the agents' self-determined states of
denial and non-denial. Each agent determines which denial and
non-denial states it is in, and the determinations of all of the
agents decide the form of each agent's conscious experience,
including each agent's consciousness of the passage of time.
Diagram 5 shows the states of consciousness produced by the
arrangement of agents in not-denied information states A à
W à I, and Diagram 6
concentrates on agent A's state of consciousness, which is
consciousness of the passage of time. In Diagrams 5 and 6, and
in the explanatory text following them, the non-directed graph
symbol called a vertex (a dot) placed next to a letter
representing an agent indicates that some agent is conscious of
that agent as a physical particle. For example, in the
arrangement A à W à
I, agent A is conscious of two proto-fermions, W and I ;
agent W is conscious of one proto-fermion, I; and agent I is
not conscious of any proto-fermions. Labeling proto-fermions
with the letter representing the agent on whom they are based is
for ease of explanation only, because 'who determines' is not
included in an agent's consciousness of a proto-fermion. An
agent only distinguishes proto-fermions by their physical
characteristics, which are determined by the arrangement that is
the basis of the agent's conscious state.
If an agent is conscious, that consciousness is always a single
state of consciousness, because the agent is a single
individual, although there may be many sub-states of
consciousness embedded within that single state of
consciousness. In Diagrams 5 and 6, agent A is in a single state
of consciousness (based on the entire arrangement A à
W à I) in which there is
an embedded state of consciousness (based on the sub arrangement
A à W). The embedded state
is a state of the sameness of agent A's information based on
agent W's existence quality with agent A's own existence
quality, i.e., a state of consciousness of a proto fermion based
on agent W (labeled W) . The single state consists of that
embedded consciousness of W plus a state of the sameness of
agent A's information based on agent I's existence quality with
agent A's existence quality, i.e., consciousness of I.
However, the consciousness of I is contingent on the existence
of the sub-arrangement (A à
W) that is the basis of the embedded consciousness of W ;
therefore, in agent A's single state of consciousness, I's
existence is contingent on W ' s existence. This contingency is
experienced by agent A as temporal: W existed before I. Thus
agent A's single state of consciousness is agent A's present
moment in which I exists, and the embedded state of
consciousness is agent A's memory, also in that present moment,
of a 'past' moment in which W existed. Because the contingency
is asymmetric, agent A is conscious of time as passing in one
direction only, and because agent A's conscious experience is a
single state, the passage of time of which agent A is conscious
is a time continuum. The continuum of time in this example is
bounded by the non-durational past instant of time in which W
existed, and by the non-durational present instant of time in
which I exists.
Diagram 5
An individual state of consciousness is produced for an agent
for each sub-arrangement beginning with that agent, and these
individual states are all embedded in the agent's single state
of consciousness based on the entire arrangement. An arrangement
is defined as a directionally connected series of agents in
not-denied information states. If one or more of the agents in
the arrangement are in more than one not-denied information
state, the arrangement is a furcated, or branched
Diagram 6
One or more conscious states being embedded in another conscious
state is a conscious comparison of the states, or an observation
of a physical relationship between proto-fermions, or an
observation of a protoboson.
For example, the consciousness that agent A has of a time
continuum relating the two proto-fermions W and I is an
observation of a proto-time boson, a 'message carrier' of the
temporal relationship between the two proto-fermions. In the
next section, in which space is modeled, space is shown to be
derivative on time, therefore a spatial relationship is also a
type of boson. Then, finally, an energy boson, the photon, is
modeled and shown to be derivative on space-time. In the above
examples there is no spatial or energy relationship between any
of the proto-fermions.
A comparison of one or more observations is a measurement. For
example, if there is the arrangement A à
W à I à
F, the sub-arrangement A à
W à I, producing an
observation for agent A of a time continuum, is embedded in and
thereby compared to agent A's single state of observation of the
entire time continuum based on the entire arrangement. This
comparison is a measurement of the entire time continuum using
the embedded time continuum as a measuring rod. If the duration
of the embedded time continuum is called one time quantum (1 tq),
the entire time continuum is of 2 tq duration. In agent A's
consciousness of that bounded time continuum there are no breaks
or parts; therefore, it is a true continuum in the Aristotelian
sense. A time continuum is not composed of time quanta, a time
continuum is only measured in time quanta. A future does not
appear in the consciousness of agents and thus has no physical
reality. The future is only a possibility.
Space
The linear embedding of one or more conscious states in one
conscious state is conscious observation of a time continuum.
Conscious observation of a space continuum, which is based on a
furcated arrangement, is the non-linear embedding of two or more
linear embeddings (or observations of time continua) in one
conscious state.
Agent W's conscious
observation of a space continuum is diagrammed below.
Diagram 7
In Diagram 7, agent W's conscious state based on sub-arrangement
W à I is linearly embedded
in agent W's conscious state based on sub-arrangement W à
I à F and it is also
linearly embedded in agent W's conscious state based on
sub-arrangement W à I à
Z. These two conscious observations of time continua are
non-linearly embedded in agent W's single state of consciousness
based on the entire arrangement. The non-linear embedding, or
comparison, of the two observations of time continua is a
measurement of the two time continua, each against the other. In
agent W's single state of consciousness, the two different time
continua have the same duration, therefore proto-fermions F and
Z are consciously experienced by agent W to be two different
proto-fermions that exist at the same time. This is observation
of a one-dimensional (1-D) space continuum bounded by F and Z.
Because each time continuum acts as the standard of measurement
for the other time continuum, the extent of the 1-D space
continuum is one quantum of length, 1 lq. The non-directed graph
symbol called an edge (a line) is drawn between F and Z in
Diagram 7, to illustrate the bounded 1-D space continuum. F and
Z are true point fermions; like electrons and quarks, they do
not extend in space.
In agent W's conscious memory of I, I is not located in space,
and agent W is not conscious of itself at all.
In the following arrangement, agent W is conscious of five
proto-fermions, four of which are located in bounded 3-D space.
If more arrows extended from agent I, agent W's consciousness
would include additional spatial dimensions up to N-l, where N
is the total number of agents that exist. In our current
universe, however, a circuit arrangement results in
consciousness of spatial dimensions being limited to three
because any spatial dimensions in excess of three are
consciously experienced by the agents in the circuit as quickly
decaying into energy.
In the Graphs A, B, C and D in the proceedings different aspects
of the Radical Theory are given:
Diagram A
Diagram B
Diagram C
Diagram D
CONCLUSION
The Information (or Lila )
Paradigm is a radical theory of the reality behind some of the
most important phenomena in physics and artificial intelligence
introduced by Charles Berner and mathematically supported by
Biljana Percinkova, Michael Baker and Peter Szekeres. It is
offering a new vision of physical reality enabling us to resolve
some of the main barriers known in contemporary Quantum
Mechanics and Artificial Intelligence, such as the
collapse/observer problem or Gödels Theorem of Incompleteness.
Strong scientific support as comparison of the parameters
obtained with the values measured in CERN institute for Particle
Physics is provided, as in the example given below:
