Transcript Arithmetic - Brown University

Arithmetic Done by Brains and
Machines:
The Ersatz Brain Project
James A. Anderson
[email protected]
Department of Cognitive and Linguistic
Sciences
Brown University, Providence, RI 02912
Our Goal:
We want to build a first-rate, secondrate brain.
Ersatz Participants
Faculty:
Jim Anderson, Cognitive Science.
Gerry Guralnik, Physics.
David Sheinberg, Neuroscience.
Students:
Socrates Dimitriadis, Cognitive Science.
Brian Merritt, Cognitive Science.
Private Industry:
Paul Allopenna, Aptima, Inc.
Andrew Duchon, Aptima, Inc.
John Santini, Alion, Inc.
Acknowledgements
This work was supported by:
A seed money grant from the Office of the
Vice President for Research, Brown
University.
Phase I and Phase II SBIRs, “The Ersatz
Brain Project,” to Aptima, Inc. (Woburn
MA), Dr. Paul Allopenna, Project
Manager. Funding from the Air Force
Research Laboratory, Rome, NY
Comparison of Silicon Computers
and Carbon Computer
Digital computers are
• Made from silicon
• Accurate (essentially no errors)
• Fast (nanoseconds)
• Execute long chains of logical
operations (billions)
• Often irritating (because they
don’t think like us).
Comparison of Silicon Computers
and Carbon Computer
Brains are
• Made from carbon
• Inaccurate (low precision, noisy)
• Slow (milliseconds, 106 times
slower)
• Execute short chains of parallel
alogical associative operations
(perhaps 10 operations/second)
• Yet largely understandable
(because they think like us).
Comparison of Silicon Computers
and Carbon Computer
• Huge disadvantage for carbon: more
than 1012 in the product of speed and
power.
• But we still do better than them in
many perceptual skills: speech
recognition, object recognition, face
recognition, information integration,
motor control.
• One implication: Cognitive “software”
uses only a few but very powerful
elementary operations.
Major Point
Brains and computers are very different in their
underlying hardware, leading to major
differences in software.
Computers, as the result of 60 years of
evolution, are great at modeling physics.
They are not great (after 50 years trying and
largely failing) at modeling human cognition.
One possible reason: inappropriate hardware leads
to inappropriate software.
Maybe we need something completely different: new
software, new hardware, new basic operations,
even new ideas about computation.
So Why Build a Brain-Like Computer?
1. Engineering.
Computers are all special purpose devices.
Many of the most important practical computer applications
of the next few decades will be cognitive in nature:





Natural language processing.
Internet search.
Cognitive data mining.
Decent human-computer interfaces.
Text understanding.
We claim it will be necessary to have a cortex-like
architecture (either software or hardware) to run these
applications efficiently.
2. Science:
Such a system, even in simulation, becomes a
powerful research tool.
It leads to designing software with a particular
structure to match the brain-like computer.
If we capture any of the essence of the cortex,
writing good programs will give insight into
biology and cognitive science.
If we can write good software for a vaguely brain
like computer we may show we really understand
something important about the brain.
3. Personal:
It would be the ultimate cool gadget.
A technological vision:
In 2057 the personal computer you buy in Wal-Mart
will have two CPU’s with very different
architectures:
First, a traditional von Neumann machine that
runs spreadsheets, does word processing, keeps
your calendar straight, etc. etc. What they do
now.
Second, a brain-like chip

To handle the interface with the von Neumann
machine,

Give you the data that you need from the Web
or your files (but didn’t think to ask for).

Be your silicon friend, guide, and confidant
(Because you understand each other.)
Ersatz Basic
Assumptions
The Ersatz Brain Approximation:
The Network of Networks.
Conventional wisdom says neurons are the basic
computational units of the brain.
The Ersatz Brain Project is based on a different
approximation.
The Network of Networks model was developed in
collaboration with Jeff Sutton then at Harvard
Medical School, now at NSBRI.
Cerebral cortex contains intermediate level
structure, between neurons and an entire
cortical region.
Intermediate level brain structures are hard to
study experimentally because they require
recording from many cells simultaneously.
Network of Networks Approximation
We use the Network of
Networks [NofN]
approximation to structure
the hardware and to reduce
the number of connections.
We assume the basic
computing units are not
neurons, but small (104
neurons) attractor
networks.
Basic Network of Networks
Hardware Architecture:
• 2 Dimensional array of
modules
• Locally connected to
neighbors
Cortical Columns: Minicolumns
“The basic unit of cortical operation is the
minicolumn … It contains of the order
of 80-100 neurons except in the
primate striate cortex, where the
number is more than doubled. The
minicolumn measures of the order of
40-50 m in transverse diameter,
separated from adjacent minicolumns
by vertical, cell-sparse zones … The
minicolumn is produced by the
iterative division of a small number of
progenitor cells in the
neuroepithelium.” (Mountcastle, p. 2)
VB Mountcastle (2003). Introduction [to a special
issue of Cerebral Cortex on columns]. Cerebral
Cortex, 13, 2-4.
Figure: Nissl stain of cortex in planum
temporale.
Columns: Functional
Groupings of minicolumns seem to form the
physiologically observed functional columns.
Best known example is orientation columns in
V1.
They are significantly bigger than minicolumns,
typically around 0.3-0.5 mm.
Mountcastle’s summation:
“Cortical columns are formed by the binding together of many
minicolumns by common input and short range horizontal connections.
… The number of minicolumns per column varies … between 50 and
80. Long range intracortical projections link columns with similar
functional properties.” (p. 3)
Cells in a column ~ (80)(100) = 8000
The activity of the nonlinear attractor
networks (modules) is
dominated by their
attractor states.
Attractor states may be
built in or acquired
through learning.
We approximate the
activity of a module
as a weighted sum of
attractor states.That
is: an adequate set of
basis functions.
Activity of Module:
x = Σ ciai
where the ai are the
attractor states.
Elementary Modules
The Single Module: BSB
The attractor
network we
use for the
individual
modules is
the BSB
network
(Anderson,
1993).
It can be
analyzed
using the
eigenvectors
and
eigenvalues
of its local
connections.
Interactions between Modules
Interactions between modules are described by state
interaction matrices, M.
The state interaction matrix elements give the
contribution of an attractor state in one module to the
amplitude of an attractor state in a connected module.
In the BSB linear region
x(t+1) = Σ Misi
+
f
weighted sum
input
from other modules
+
x(t)
ongoing
activity
The Linear-Nonlinear Transition
The first BSB processing stage is linear and sums
influences from other modules.
The second processing stage is nonlinear.
This linear to nonlinear transition is a powerful
computational tool for cognitive applications.
It describes the processing path taken by many
cognitive processes.
A generalization from cognitive science:
Sensory inputs  (categories, concepts, words)
Cognitive processing moves from continuous values
to discrete entities.
Sparse Connectivity
The brain is sparsely connected. (Unlike most neural
nets.)
A neuron in cortex may have on the order of 100,000
synapses. There are more than 1010 neurons in the
brain. Fractional connectivity is very low: 0.001%.
Implications:
• Connections are expensive biologically since they
take up space, use energy, and are hard to wire up
correctly.
• Connections are valuable.
• The pattern of connection is under tight control.
• Short local connections are cheaper than long ones.
Our approximation makes extensive use of local
connections for computation.
Biological Evidence
Biological Evidence:
Columnar Organization in Inferotemporal
Cortex
Tanaka (2003)
suggests a columnar
organization of
different response
classes in primate
inferotemporal
cortex.
There seems to be
some internal
structure in these
regions: for
example, spatial
representation of
orientation of the
image in the
column.
IT Response Clusters: Imaging
Tanaka (2003) used
intrinsic visual
imaging of cortex.
Train video camera
on exposed cortex,
cell activity can
be picked up.
At least a factor of
ten higher
resolution than
fMRI.
Size of response is
around the size of
functional columns
seen elsewhere:
300-400 microns.
Columns: Inferotemporal Cortex
Responses of a region
of IT to complex
images involve
discrete columns.
The response to a
picture of a fire
extinguisher shows
how regions of
activity are
determined.
Boundaries are where
the activity falls
by a half.
Note: some spots are
roughly equally
spaced.
Active IT Regions for a Complex Stimulus
Note the large number of roughly equally distant
spots (2 mm) for a familiar complex image.
Back-of-the-Envelope
Engineering
Considerations
Engineering Hardware Considerations
We feel that there is a size, connectivity, and
computational power sweet spot at the level of the
parameters of the network of network model.
If an elementary attractor network has 104 actual
neurons, that network might have 50 attractor
states. Each elementary network might connect to 50
others through state connection matrices.
A brain-sized system might consist of 106 elementary
units with about 1011 (0.1-1 terabyte) numbers
specifying the connections.
If 100 to 1000 elementary units on a chip gives a
total of 1,000 to 10,000 chips in a cortex sized
system. Well within the upper bounds of current
technology.
Modules
(Ersatz Processing Units:EPUs)
Function of EPU Modules:
• Simulate local integration: Addition
of inputs from outside, from other
modules.
• Simulate local network dynamics.
• Communications Controller: Handle long
range (i.e. not neighboring)
interactions.
Simpler approximations are possible:
• “Cellular automaton”. (Ignore local
dynamics.)
• Approximations to local dynamics.
Physical (Hardware) Module
We assume only
local
connections
for the
physical
hardware.
Reason:
Flexible,
easy to
build, easy
to work
with.
Software Based Connectivity
Cortical data
suggests more
connections
than just
nearest
neighbors
exist.
Simulate these
with EPU
module
software, in
the the
Communications
Controller.
Implications
Interesting bonus from this structure:
• Information transmission both local
and long range can be slow.
• It will take multiple steps (a long
time) to move data to distant
modules.
• But: This is a feature, not a bug!
Implications
Forces us to pay attention to the
Temporal aspects of module behavior
• Communication times
• Module temporal dynamics
• Note: The details of spatial
arrangement of data affects
communication times.
Consistent with cortical neuroscience
Implication: We can “program” the array
by manipulating these “analog”
properties to control array behavior.
Ersatz Programming
Peculiarities
How do you make this “computer” compute?
Not with logic!
It is like a hybrid analog-digital computer.
Programming Techniques:
• Spatial arrangement of data on array
• Integration of data from multiple sources
• Abstraction and discrete concept formation
• Control of computation using (analog) dynamical
system parameters
• Assemblies of interacting modules.
Give one example: performance of arithmetic by a
simple Ersatz-like system.
Ersatz Arithmetic
Cognitive Computation: Example Arithmetic
• Brains and computers are very different
in the way they do things, largely
because the underlying hardware is so
different.
• Consider a computational task that
humans and computers do frequently, but
by different means:
– Learning simple arithmetic facts
Learning the “Right Thing”
Cognition is not memory for facts (like
computer data) but remembering the
“right things” even if the right things
are constructed from many experiences
and don’t actually exist!
Most (99.9%) sensory input data is
discarded. (The essential process of
“creative data destruction.”)
What is kept are useful abstractions and
transformation of the inputs.
Arithmetic
Digital computers compute the answers to problem
using well-known logic based algorithms.
Humans do it very differently.
The human algorithm for elementary multiplication
facts seems to look like:
1. Find a number that is the answer to some
multiplication problem and
2. A product number that is about the right size.
This is a process involving memory and
estimation, not computation as traditionally
understood.
Next, develop advantages and disadvantages of
doing it this way.
A Problem with Arithmetic
• We often congratulate ourselves on the
powers of the human mind.
• But why does this amazing structure
have such trouble learning elementary
arithmetic?
• Adults doing arithmetic are slow and
make many errors.
• Learning the times tables takes
children several years and they find it
hard.
Brain Software: John von Neumann
Von Neumann: 1958, The Computer and the Brain
The nervous system is a complex machine which manages to do its
exceedingly complex work on a rather low level of precision.
Von Neumann, as a numerical analyst, knew that
errors would rapidly grow and the result would
be meaningless if there were more than a few
steps in the computation.
Computational Strategy
Ways to avoid problem:
• Use a small number of steps
• Use discrete (“logic-like”) operations
rather than hard (“analog”) operations.
Engineering rule: Digital is easy, analog
is hard.
Von Neumann:
… Whatever language the central nervous system is using is
characterized by less logical and arithmetical depth than
we are normally used to.
A small number of powerful operations are
strung together to form a mental
computation.
Teaching of Mathematics
Collaborators: Prof. Kathryn Spoehr, Dr.
Susan Viscuso, and Dr. David Bennett
My own interest goes back to a joint
paper with Prof. Phil Davis of Brown
Applied Mathematics.
Point of the paper:
The “Theorem-Proof” method of teaching
mathematics has ruined mathematics in
the 20th Century.
Reason for Ruination
Real mathematicians do not think this
way.
Mathematicians use a complex blend of
intuition, perception, and memory to
understand complex systems.
Proving theorems is the last stage, to
convince others that you are correct.
Effects very hard on consumers of
mathematics: Engineers and scientists.
They say, “I don’t think like this.” and
lose confidence in their intuitions.
Why is Arithmetic so Hard?
People are much worse than they should be at
elementary arithmetic.
Elementary arithmetic fact learning involves
making the right associative links between
pairs of the 10 digits to give products, sums,
etc.
Only a few hundred facts to learn ...
Arithmetic rules are orders of magnitude less
complicated than syntax in language.
But: Takes years for children to learn
arithmetic.
The Problem with Arithmetic
At the same time children are having
trouble learning arithmetic they are
knowledge sponges learning
– Several new words a day.
– Social customs.
– Many facts in other areas.
Association
In structure, arithmetic facts are simple
associations.
Example: multiplication:
(Multiplicand)(Multiplicand)  Product
Simple association (S-R learning) was
popular idea in the 1920’s (Thorndyke).
Formation of arbitrary associations is
the basic rationale behind flash cards.
Can learn this way, but hard and not
really with “understanding.”
Multiplication
• Arithmetic facts are not arbitrary
associations.
• They have an ambiguous structure that
gives rise to associative interference.
4 x 3 = 12
4 x 4 = 16
4 x 5 = 20
• Initial ‘4’ has associations with many
possible products.
• Ambiguity causes difficulties for
simple associative systems.
Number Magnitude
• One way to cope with ambiguity is to
embed the fact in a larger context.
• Numbers are much more than arbitrary
abstract patterns.
• Experiment:
– Which is greater?
– Which is greater?
17 or 85
73 or 74
Response Time Data
Number Magnitude
It takes much longer to compare 74 and
73.
When a “distance” intrudes into what
should be an abstract relationship it
is called a symbolic distance effect.
A computer would be unlikely to show such
an effect. (Subtract numbers, look at
sign.)
Magnitude Coding
Key observation: We see a similar effect
when sensory magnitudes are being
compared.
Deciding which of
– two weights is heavier,
– two lights is brighter,
– two sounds is louder
– two numbers is bigger
displays the same reaction time pattern.
Magnitude Coding
This effect and many others suggest
that we have an internal
representation of number that acts
like a sensory magnitude.
Conclusion: Instead of number
being an abstract symbol, humans
use a much richer coding of number
containing powerful sensory and
perceptual components.
Magnitude Coding
Argue that this “perceptual” elaboration
of number is a good thing. It
– Connects abstract “number” to the
physical world.
– Provides the basis for mathematical
intuition.
– Is perhaps responsible for the
creative aspects of mathematics.
Mathematics by Adults
Mathematics is the most lawful and
abstract of the sciences.
Real mathematicians would not crudely
associate a number with a weight?
Would they?
In fact, they do.
Consider Jacques Hadamard’s book The
Psychology of Invention in the
Mathematical Field. (1946)
How Experts do Mathematics
Hadamard (a world class mathematician)
interviewed his peers in 1943-5.
Conclusion: Most of them did not reason
abstractly.
They used
• Visualization
• Auditory imagery
• Kinesthetic imagery with imagined
muscle movements for insights in to
“abstract” systems.
Language and formal abstract reasoning
were conspicuous by their rarity.
Quotes:
The mental pictures of the mathematicians whose answers I have
received are most frequently visual, but they may also be of
another kind – for example, kinetic. There can be auditive
ones.”
… practically all of (them) avoided not only the use of mental
words but also the mental use of any algebraic or any precise
signs … they use vague images. There are two or three
exceptional cases, the most important of which is the
mathematician George D Birkhoff, one of the greatest in the
world, who is accustomed to visualize algebraic symbols and
work with them mentally …
Hadamard
Einstein
One of Hadamard’s informants was Einstein.
The words or the language as they are written or spoken do not seem to
play any role in my mechanism of thought.
Albert Einstein
To Einstein, thinking involves transforming of
received sense images into a series of “memory
pictures.”
Thinking began when he found a certain picture
recurring in a number of series. “… such an
element becomes a concept.”
Einstein
These concepts are not words but can
become linked to words.
It is by no means necessary that a concept must be connected
with a sensorily cognizable and reproducible sign (a word)
but when this is the case thinking becomes by means of
that fact communicable. (Albert Einstein, Autobiographical
Notes.)
Therefore, the function of words and
concepts is to convince others, not
necessarily yourself who had understood
the system through other means.
Richard Feynman
Richard Feynman was a “kinesthetic” thinker:
Feynman said to Dyson … that Einstein’s great work had sprung from
physical intuition and when Einstein stopped creating it was because
‘he stopped thinking in concrete physical images and became a
manipulator of equations.’ Intuition was not just visual but also
auditory and kinesthetic. Those who watched Feynman in moments of
intense concentration came away with a strong, even disturbing sense
of the physicality of the process, as though his brain did not stop at the
gray matter but extended through every muscle in his body. A Cornell
dormitory neighbor opened Feynman’s door to find him rolling about
on the floor beside his bed as he worked on a problem.
James Gleick, Genius: The Life and Science of Richard Feynman
Non-Verbal Science
Among the virtuosos of intuitive (nonverbal) science are physicists with
their “gedanken experiments.”
At the age of 16 Einstein performed a
powerful visual thought experiment.
He assumed an observer was moving along
side an electromagnetic wave.
Think of a boat moving in the same speed
and direction as an ocean wave.
Waves: Water
and ElectroMagnetic
See a stationary
hill of water.
See a stationary
electro-magnetic
field?
Waves
Water wave: See a stationary hill of
water.
If you traveled with the same speed and
direction as an electromagnetic wave,
you would see a motionless spatially
varying electric and magnetic field.
Einstein knew this had been looked for
and never found.
Relativity
Perhaps we did not see this because it
was impossible for an observer to
travel at the same velocity as an
electromagnetic wave.
Results of this insight:
… a paradox upon which I had already hit at the age of 16: if
I purse a beam of light with the velocity c … I should
observe such a beam as a spatially oscillatory
electromagnetic field at rest. However there seems to be
no such thing.
… One sees that in this paradox, the germ of the special
relativity theory is already contained.
Albert Einstein, Autobiographical Notes.
Visual Image of a Proof
Hadamard gives his own visual images of a
proof. The proof is by contradiction.
Theorem: There is no largest prime number.
• Suppose someone claims that P is the largest prime.
• Form the product of all the prime numbers up to P,
forming a large number, N.
• Add one to N, giving N+1.
• Given this construction, all the primes up to P must give a
remainder of 1 when they divide N+1.
Previously Shown: All integers are primes or the product of
primes.
Therefore, either (1) the number N+1 itself is prime or (2) It is
the product of two or more primes, each larger than any in
the sequence of known primes that formed N+1.
I consider all primes from 2 to
11, say 2,3,5,7,11. I see a
confused mass.
I form their product,
2x3x5x7x11. N being a
rather large number I
imagine a point far from the
confused mass.
I increase that product by 1, say
N+1. I see a second point a
little beyond the first.
That number, if not a prime,
must admit of a prime
divisor. … I see a place
somewhere between the
confused mass and the first
point.
Problems
These images are supposed to be
universal.
In fact: Hadamard’s image is wrong for
the number 11.
For N=11, N+1 is 2,311 which is itself
prime so the “place” in the last image
is identical to the “second point.”
If the number used is N=13, N+1 = 30,031
which is the product of 59 and 509.
N=13 agrees with Hadamard’s image.
A visual image can be misleading!
need formal proofs to check our
intuitions.
We
Model Makes Small Mistakes,
Not Big Ones
Model used a neural network based
associative system.
Buzz words: non-linear, associative,
dynamical system, attractor network.
The magnitude representation is built
into the system by assuming there is a
topographic map of magnitude somewhere
in the brain.
First Observation about
Arithmetic Errors
Arithmetic fact errors are not random.
• Errors tend to be close in size to the
correct answer.
• In the simulations, this effect is due
to the presence of the magnitude code.
Second Observation: Error
Values
• Values of incorrect answers are not
random.
• They are product numbers, that is, the
answer to some multiplication problem.
• Only 8% of errors are not the answer to
a multiplication problem.
Human Algorithm for
Multiplication
The answer to a multiplication problem
is:
1. Familiar (a product)
2. About the right size.
Human Algorithm for
Multiplication
• Arithmetic fact learning is a
memory and estimation process.
• It is not really a computation!
Flexible and programmable
Learning facts alone doesn’t get you far.
The world never looks exactly like what
you learned.
Heraclitus (500 BC):
• It is not possible to step twice
into the same river.
A major goal of learning is to apply past
learning to new situations.
Getting Correct What you
Never Learned: Comparisons
Consider number comparisons:
Is 7 bigger than 9?
We can be sure that children do not learn
number comparisons individually.
There are too many of them.
– About 100 single digit comparisons
– About 10,000 two-digit comparisons
– And so on.
Building a System to Perform
Simple Arithmetic Operations
•
•
We have a model for arithmetic learning.
Can we now make a system capable of performing some
simple mathematical operations on numbers?
• Techniques we can use include attractor networks,
differential weighting of portions of an array of
units, and a specialized data representation for
number.
• Examples of simple operations are
1. increment, decrement,
2. greater than, less than,
3. round off.
• The current version is restricted to the digits
from 1 to 10.
Bar
Codes
A bar code represents magnitude by position on a map.
There are ten patterns for the digits from 1 to 10.
The patterns for each digit overlap slightly.
Number Representation
Bar coding for number resembles the
number line:
1 2 3 4 5 6 7 8 9 0 (10)
Programming
Patterns
The number map is
weighted by
programming
patterns.
One pattern is used
for each
operation.
The pattern(s) for
number and for
operation
multiply.
System dynamics
gives the final
answer.
Basic Arithmetic Operations
•Count up
(starting number + 1)
•Count down (starting number – 1)
•Greater than: Given two digits, output the
larger.
•Lesser than:
smaller.
Given two digits, output the
•Round off: Given activity at a location
on the array, output the nearest integer.
Programming Pattern:
Count Up/Down
Count up
(starting
number + 1)
Count down
(starting
number – 1)
(mirror
image of
count up).
Programming Pattern:
Greater Than/Less Than
Greater than:
Given two
digits, output
the larger.
Lesser than:
Given two
digits, output
the smaller.
(mirror image of
“Greater than”
pattern.
Programming Pattern:
Round-Off
Round off:
Given
activity
at a
location
on the
array,
output the
nearest
integer.
Manipulating
Starting
Point
•We are manipulating the
starting point in the
attractor structure.
•Once the attractor
structure is formed many
operations can be
performed without further
learning.
•Operations are not
“logical” but based on
continuous mathematics.
•This might be considered
a very simple kind of
mathematical intuition.
Experimental Data: Single Digit
Number Comparisons
Assume something
like experimental
reaction time is
related to the time
taken to get the
answer.
The Greater-Than
operation shows a
“symbolic distance”
effect just like
humans do.
Physiological Evidence
Bars overlap.
Integers close in
magnitude show a
degree of similarity
in their
representations.
A 2002 paper in
Science showed this
effect in single
unit recordings in
primate prefrontal
cortex.
Note the similarity
to the symbolic
distance curves.
A Nieder, DJ Friedman, EK Miller
(2002). Representation of the quantity
of visual items in the primate
prefrontal cortex. Science 297, 1708
Numerosity
Numerosity:
A problem
joining
‘abstract’
quantities
with pattern
recognition.
Given a set of
identical
items
presented in a
field, report
how many items
there are.
Subitzing
For humans, number from one to about four works in
what is called the subitizing region.
Subjects “know” quickly how many objects are present.
Each additional item (up to 4) adds about 40 msec to
the response time.
In the counting region (more than 4 objects) each
additional item adds around 300 msec per item.
This figure is consistent with explicit counting..
Evidence that there is a strong “total activity”
component to subitizing.
Number Estimation with Lateral
Information Flow
The network
of networks
model
propagates
pattern
information
laterally.
Total
maximum
activity
gives
numerosity.
Which Plate has the Most
Segment the
Cookies?
field by using
boundary
modules in
attractor
states. (No
lateral
transmission.)
(Lateral
interactions
can be halted
by interposing
lines or
regions.)
Metacontrast
Counting Cookies
Program Counting Cookies:
1. The image is segmented.
2. The numerosity of objects in each segment is
computed using activity based lateral spread.
3. Activity measure is converted into an integer by
the round-off operation.
4. Integers are compared using the greater-than
operator with the largest integer is the output.
This very simple program is based on topographic
and dynamic representational assumptions.
Not just a toy problem: Can let you estimate
number of similar or identical objects with a
largely parallel and selective algorithm.
Magnitude
• We now see the usefulness of the
“sensory” magnitude number
representation.
• We can use magnitude to do
computations like number
comparisons without having to
learn special cases.
Implications
We have constructed a system that acts like
like logic or symbol processing in a
limited domain.
It does so by using its connection to
perception to do much of the computation.
These “abstract” or “symbolic” operations
display their underlying perceptual nature
in effects like symbolic distance and error
patterns in arithmetic.
Connect perception to abstraction
and gain the power of each approach
• Humans are a hybrid computer.
• We have a recently evolved, rather
buggy ability to handle abstract
quantities and symbols.
• (only 100,000 years old. We have the
alpha release of the intelligence
software.)
Connect perception to abstraction
and gain the power of each approach
• We combine symbol processing with
highly evolved, extremely effective
sensory and perceptual systems.
• Realized in a mammalian neocortex.
• (over 500 million years old. We have a
late release, high version number of
the perceptual software.)
• The two systems cooperate and work
together effectively.
Conclusions
A hybrid strategy is biological:
– Let a new system complement an old
one. Never throw anything away.
– Even a little abstract processing
goes a long way. Perhaps that is one
reason why our species has been so
successful so fast.