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From: 93funkst@wave.scar.utoronto.ca (FUNK STEVEN LESLIE)
Subject: Why NN's are limited to small domain
Message-ID: <D7pq0D.L4K@wave.scar.utoronto.ca>
Organization: /wu/news/news/organization
References: <PETER.95Apr23123122@swamp.indigo.co.il> <D7KBwA.9Ju@freenet.carleton.ca> <1995Apr26.170213.21833@janix.mfr.dec.com>
Date: Thu, 27 Apr 1995 21:21:48 GMT
Lines: 389

Hi Folks,

	I've included an essay that I wrote for a seminar on
consciousness, in response to penroses 'shadows fo the mind'.  It
includes a reasoned arguement explaining that both symbolic logic, and
connectionism are unable to model consciousness because they
incompatible with it.  I also present a 'new' system which does not
suffer from the same limitations as artificial neural networks.  Its the
end of the year, and so I thought it might be fun to see your collective
reaction.  Anywho, I'm sure I'll hear what you think.

Steve Funk
93funkst@wave.scar.utoronto.ca



		Modelling Consciousness:
	Consistency and 'Paradigmatic Solipsism'

		   By Steven Funk
		   For Dr. Seager
		     PHL C95S
	      Seminar on Consciousness

	
	Penrose argues about the limitations of the computational
modelling of cognition and consciousness, suggesting that they
may be incapable of capturing either.  The algorithmic approach,
which has been used in most of the research to date, is
described as a system which is based on the foundations of logic
as they are applied to Turing machines.  Logic is described as a
system, extracted from the platonic world, where a set of
symbols are manipulated in some formal way.  The goal is to
produce a consistent formal system which fully describes how the
modelled phenomenon works.  It is important to understand the
precise definition of logic, as this will become a central point
in Penrose's argument later on.  By formal, I mean a system
which follows an explicit set of rules, and never deviates from
them.  By consistent, I mean that a system could never
contradict itself.  So, P and ~P (i.e. NOT P) could never
coexist.  The Turing machine is a hypothetical mechanism that is
able to apply the rules of a formal system in such a way that
any computable function may be processed.

  

	For some decades the emphasis in modelling research has been to
develop a logical system which is able to describe human
behaviour.  A Turing machine of the mind would initially be
limited to simple, dedicated tasks like phoneme recognition. 
However, the goal was to produce an intelligent machine. 
Furthermore, this 'robot' would, if all went well, be conscious.
 The research, so far, has not proved to be successful for
several reasons.  While the effort has provided a fair amount of
information about how humans think, the intelligent machine
never materialized.  Some aspects of even  the simplest, most
dedicated tasks seemed unattainable.  The level of intelligence
that has been displayed has been barely enough to master limited
domains like chess, but even these have problems.  For example,
chess programs which seem unable to accommodate the concept of
short term sacrifice for long term gain.  For many the failure
of strictly logical systems was predicted by Godels
Incompleteness Theorem. Godels Theorem states that no formal
system may be proven consistent.  It seemed as though logic was
fundamentally flawed.  But, Godels theorem does not state that a
logical system is inconsistent.  It simply shows that even if
the system is consistent, this can never be proven.  So, while
computationalists and logicians were not proven wrong, they were
still compromised.  It seems to be an obvious problem for a
system to be unable to prove that it is abiding by its own
fundamental rule of consistency.  



	The issue of consistency within logic maybe, for the time
being, set aside as it can never proven.  However, the issue of
consistency within consciousness, seems to provide an even
greater problem.  If it is the goal of logic to produce a
consistent formal system, then it might be safe to say that
logic can only model consistent formal systems.  If the mind
were shown to be inconsistent, or informal, then logic would be
for the most part inapplicable.  While the issue of formalism
might be difficult to resolve, the issue of consistency is clear
cut.  One can easily demonstrate the inconsistency of the mind
by asking: "are all of your beliefs true?"  Most people will
respond no, recognizing that there is a difference between
objective truth and subjective truth, were objective truth takes
precedence.  If you then ask people: "what are beliefs?"  Most
will be forced to respond that a belief is something that is
held to be true.  So then the word 'belief' takes on two
contradictory definitions.  The mind is inconsistent.  If the
issues of belief, knowledge, and truth seem to controversial to
build a solid argument upon, consider perceptual psychology.  In
particular the case of the Necker cube.  The Necker cube is an
illusion which consists of a projection from a wire frame cube
onto a picture plane such that the orientation of the cube is
ambiguous.  The viewer is able to choose how they will view the
cube, how the cube will appear to them.  In order to do this the
mind must entertain two, separate and inconsistent notions of
the cubes orientation.  Yet most people are not only able to see
the cube clearly, but are also able to shift the orientation of
the cube from one interpretation to another.  Again, the mind is
able to entertain inconsistent notions.



	This leaves Penrose with an argument which is common to those
who call for some special component to consciousness.  If the
brain follows the consistent laws of biology, chemistry, and
eventually physics, then how can the mind be inconsistent?  We
have already stated that a consistent formal system cannot model
a phenomenon which is either inconsistent or informal.  So, how
can a consistent formal system produce inconsistent or informal
behaviour.  For Penrose there is only one way out, quantum
mechanics.  It is only in quantum mechanics that he believes we
can find the strange and mystical component needed to produce
consciousness.  But, this may not be necessary.



	There are two central problems with what Penrose is saying. 
They both have to do with the way the problem is described.  The
algorithmic approach, as defined by Penrose, requires a
cognitive level description of behaviour, described by a
'classical' logical system.  By cognitive level description, I
mean that the behaviour is defined in terms of higher level
functions.  So that when dealing with object recognition, the
actions of the individual neurons never come into play.  By a
classical logical system I mean one in which there is no
consideration of space or time.  This is because the underlying
principles of logic make no reference to either space or time.



	As a demonstration, consider the real world, and a cognitive
level description of the things in it.  It is important to
remember that it is a fundamental goal of logic to produce a
system were P and ~P can not coexist.  The descriptions might
include among them:  "all people born on the same day are the
same age."  This would imply, quite correctly, that twins will
always be the same age.  However, if we take into consideration
relativity, and in particular "The Twin Paradox", then it
becomes apparent that this level of description causes problems.
 Special relativity requires that time slow down as one
approaches the speed of light.  While time appears to proceed
normally for every person regardless of their speed, there is a
relative difference.  So, if a person boarded a spaceship and
travelled near the speed of light for 1 year, when they returned
more than 1 year would have passed back on earth.  In fact,
depending on the speed, thousands of years could have passed. 
The human race could be extinct, and the sun burnt out.  So,
what happens to our cognitive level description of the world if
we send one twin on a short trip near the speed of light.  Lets
say that during a 1 year trip 2 years have passed on earth.  In
this case our model of the world would have to include the
statement: "not all people born on the same day are the same
age."  So, P and ~P have emerged.  And all of this has occurred
within the consistent laws of physics.  Quantum Mechanics was
not only not required, but  is specifically incompatible with
the theories of relativity that produce this behaviour.  Not
only is it not possible to prove logic's consistency, it is easy
to make logic inconsistent.  But, is this such a bad thing. 
Where does the flaw lie, in the global nature of logic?



	The answer is that there really is no flaw.  It is important to
understand that the Platonic world is a reference to an
imaginary, abstracted world where all possible formal systems
exist.  Within the Platonic world exists symbolic logic, integer
mathematics, complex number mathematics, etc.  Symbolic logic is
just one of the formal systems that exists in the Platonic
world.  So, while it cannot be wrong or flawed in that it
performs exactly as its principles require, it might be better
to say that the problem is one of inappropriate application.  



	Instead there is another issue which may be the problem.  It
may be considered inappropriate to describe consciousness at the
'cognitive level'.  By this I mean a description of human
behaviour in terms of cognitive functions.  In this respect the
study of cognition has been heavily influenced by developments
in computation as well as the functionalist camp in philosophy. 
A cognitive function might be considered something like the
rehearsal of a word so that it might be encoded in long term
memory.  While this level of description does not need to
account for the neural mechanisms required for memory, it does
not become limited by behaviouralist constraints.  In essence
the cognitive level of description attempts to describe human
performance in terms of a black-box (or set of black-boxes).  It
would provide an explanation as to why the level of description
would be so important, if describing things at different levels
provided the modelling system with more, less, or different
information.  After all, if you discuss such a system solely in
terms of higher level behavioural performance, understanding the
nature of the underlying complexity becomes impossible. 
Consider a painting.  Examining the individual brush strokes
from centimetres away tells very little about the image. 
Examining the entire image from meters away, tells little about
the brush strokes.  Now, in either case how would you describe
the way in which the image is created?  In one case the response
might be: "what image?" In another, the response might involve
no reference to brush strokes at all.  And yet there must be a
clear line of reasoning from one to the other.  A change in the
brush strokes would produce a qualitative change in the image. 
One promising possibility which would fit this notion is the
idea of emergent phenomena.  In emergent phenomena the behaviour
of small scale components produces an unexpected behaviour that
seems to emerge at some level of complexity.  But, if we believe
that consciousness emerges at some level of complexity, we must
ask why.  This has lead to a number of involved issues including
panpsychism and the notion of critical mass.  In this case I
raise the issue as not just being troublesome for logic, but
potentially for any descriptive framework.    



	In discussion of  the global nature of logic we will be
examining a limitation that is for the most part specific to
logic.  It may apply to several components of the platonic
world, however, there may be other systems to which it does not
apply.  There are two examples of the global nature of logic and
the limitations that they entail.  First, there is the case of
the twin paradox.  Logic fails to account for this world, partly
because of the level of description.  However, this cognitive
level could be accommodated in a system which allows for a kind
of modular consistency.  The two statements P and ~P may not
coexist, at the same place and time, however they could coexist
in the greater world.  So that one might say that either a set
of twins are the same age or they are not, but that they may not
be at the same time the same and different ages.  Put another
way, each incidence of twins is assigned either P or ~P, but not
both simultaneously.  In this way a set of twins may obtain
consistency within its own world.  But, the greater world of all
sets of twins would be inconsistent.  The second example is
given by Wittgenstein.  The issue is raised in relation to a
number series.  While the series: 1,2,3,4,5.. might be defined
as x=(x-1)+1, a number series such as: 2,4.. causes problems. 
This is because the series might be defined as x=(x-1)*2,
x=(x-1)2, or as x=(x-1)+2.  In this example all descriptions of
the series must be entertained, yet the three definitions are
inconsistent.  If the third number in the sequence is 8 the
first definition is appropriate, if the number is 16 then the
second is, if the number is 6 then the third is.  But, until the
third number is provided three logically inconsistent notions of
the series must be maintained.  Local consistency allows global
inconsistency to exist, by isolating the separate descriptions. 
As long as the third number is absent, each definition could be
interchanged with the other.  Each is consistent onto their own,
it is only when they are all used to describe the series that
they become inconsistent.  But, this inconsistency only occurs
if they are used at the same time and in the same place.  This
principle of locality could be formalized in an alternative
version of logic.  However, the current system of logic is
incapable of making the distinction.  The important thing here
is that everything has its own nature, and that this nature has
consequences.  In the case of logic, it is its nature to deal
solely with global consistency.  In the case of the real world,
it is the nature of the universe to have locality (barring
quantum mechanical voodoo).  It is this incompatibility that
poses problems for the notion of a formal descriptive system. 
In order to be successful, the modelling medium and the
phenomenon being modelled must share certain characteristics
that are a consequence of their individual qualities.



	The issues of level of description and form of description are
intimately intertwined.  It may be possible to apply symbolic
logic to a neural description of the mind.  However, this
method, known as connectionism, has its own limitations. 
Connectionism by its nature is a system that utilizes
prototypical neural activity to perform tasks.  The primary
difference between this scheme and that of logic is the ideas of
space and time.  These become important only in the simplest
way.  In fact they are defined in terms of strictly ordinal
relationships between the neural units.  With this exception
connectionism can be equated with more classical logical
systems, all be it inductive ones.  The units are taken to
represent symbols, and the neural activity known as activation
is representative of a truth value.  In addition there is a
connection weight which links one unit to another, this may be
considered a rating of the importance of one symbols truth value
to that of the other.  An initial pattern is put into the
collective system and the truth values of the individual symbols
interact in an evolutionary way until the system settles into a
stable state.  When this is done the resulting output is the
computational consequence of organization and topology of the
network, and the underlying rules that govern connectionism. 
These underlying rules may be considered to be another system
extracted from the platonic world.        



	There are essentially two problems with connectionism, each
relating to consistency in a different way.  The first problem
has to do with the way in which the symbols are represented. 
The binding of each symbol to a unit limits the systems
flexibility.  The processing of such a symbol is limited to its
interactions with other symbols.  These interactions are limited
to the transmission of a single truth value.  Those symbols
which are not directly connected to a symbol do not have access
to it.  So, once the system is trained its relations of symbols
become constant, none may be dismissed, nor new ones added. 
This produces consistent behaviour, however, it does not allow
for an on going dynamic adaptation to changing conditions.  The
second problem has to do with the consistency of the underlying
rules which govern the systems performance.  In particular the
summation of incoming activation.  Because all of the units obey
the same principle of addition, there is no adaptation.  In real
neural systems the same inputs produce different outputs in
different neurons.  However, in connectionism the same inputs
will always produce the same outputs.  Similarly, there may be
more then one input, within the framework of connectionism,
which will produce the same output.  While this does occur in
neurons, it is a result of adaptation, not the platonic system
which governs neuronal behaviour.  So there is a consistent
application of the principles of mathematics which is
inappropriate.



	As an alternative approach I propose a system which carries the
underlying principles further.  'Temporal Computation', for lack
of a better name, changes the computational medium which seems
to cause problems for connectionist systems.  As discussed
previously, in conventional connectionism, the underlying
mechanics are handled by straight forward math.  However, the
nature of mathematics carries over to the behaviour of
connectionism.  As an example consider the case were two units
are connected to a third.  Connectionsim uses simple summation
to evaluate the incoming activation.  So, inputs of 0 & 9, 1 &
8, 2 & 7, 3 & 6, and 4 & 5, all produce the same net input of 9.
 This is not the consequence of learning or adaptation, but of
mathematics itself.  In temporal computation the medium becomes
time.  The actual spiking rates of, say 4 & 5 are integrated to
produce a unique output pattern that is distinct from that
produced by inputs of 3 & 6.  The individual spikes are summed
together, and then a decay rate is applied.  This allows two
spikes which are transmitted at relatively similar times, to be
integrated into a single spike.  This summed value may then
trigger a threshold, which would transmit a different pattern
for each different possible value. 

Figure 1: The dendritic integration of spiking patterns with
frequency 4 & 5, with a Combined signal decay rate of 0.5
units/time step.  If the firing threshold were set at 1.1 units,
then an irregular output pattern would result.

Figure 2: The dendritic integration of spiking patterns with
frequency 3 & 6, with a Combined signal decay rate of 0.5
units/time step.  If the firing threshold were set at 1.1 units,
then an regular output pattern would result.  Note that this
produces a different pattern from inputs of 4 & 5.



	This gains two things.  First, as already shown some of the
problematic assumptions of mathematics no longer apply, allowing
greater latitude in the behaviour of the system.  Figures 1 and
2 show a comparison of two members of the set already described
as they would be represented in the new system.  Bear in mind
that the variation in outputs results from two regular and
persistent inputs.  Second, the symbol is now independent of the
unit.  Allow the pattern generated in Figure 1 to represent
'car', and the pattern in Figure 2  to represent 'tree'. 
Because the firing patterns have become unique and meaningful,
they have in essence become symbols.  These new symbols may be
transmitted from unit to unit so that a unit may represent more
than one symbol and a symbol may be instantiated in more than
one unit.  



	What has been done here might be considered a kind of
'Paradigmatic Solipsism', in that we are isolating the
representational paradigm and analysing it independently of any
particular model.  It is this method that has revealed some
fundamental problems with symbolic logic, and connectionism, as
well as leading to the presentation of a third alternative. 
While I believe that this approach would provide better results
in attempting to describe consciousness, there is one important
thing to remember.  Each of the three representations described
here are drawn from the platonic world.  None is correct or
incorrect, it is merely a matter of finding one that is
appropriate.  However, I feel that progress is being made in
moving towards a paradigm with a greater descriptive power.

