neuron-request@HPLABS.HP.COM ("Neuron-Digest Moderator Peter Marvit") (12/05/89)
Neuron Digest Monday, 4 Dec 1989
Volume 5 : Issue 51
Today's Topics:
Data Complexity
Re: Data Complexity
Re: Data Complexity
Data Complexity
Data Complexity
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Subject: Data Complexity
From: aam9n@uvaee.ee.virginia.EDU (Ali Minai)
Organization: EE Dept, U of Virginia, Charlottesville
Date: 06 Oct 89 02:26:16 +0000
I have a question which might or might not make sense. Still, since it is
of great interest to me, I shall be grateful if people share their views,
suggestions and references on this topic with me. Of course, I shall be
particularly grateful if someone pointed out the absurdity of the whole
issue.
The question is this:
Given deterministic data {(X,Y)} where X is in a finite interval of n-d
real space and Y is in a finite interval of m-d real space, what
*structural* measures (if any) have people suggested for the "complexity"
of the data?
This immediately raises several other issues:
1) What possible notions of "complexity" might one admit in this regard?
2) What might be the *order* of this complexity? Should it, for example, be
defined over the set of all points, all pairs, all triplets etc.?
3) Given that the data is not uniformly distributed over the domain (i.e.
it is "lumpy"), what assumptions must be made regarding the blank areas?
Should these be statistical? (probably yes).
4) How can such a "complexity" measure be made scale-invariant?
etc.
Also, what about such complexity measures for continuous functions? I mean
measures defined structurally, not according to the type of the function
(e.g. degree for polynomials).
My gut feeling is that some sort of information-based measure is
appropriate, but whatever is used must be efficiently computable. Have
other people tried to deal with this problem? Since I am sure they have,
where can I find the material? What would be the appropriate area of
mathematics to look for such references? I am currently searching
approximation theory, statistical modelling, maximum-entropy theory,
measure theory, information theory, complexity theory (both system and
computational) and non-linear dynamics literature. Obviously, I will miss
more than I will find, hence this request. A particularly helpful thing
would be names of journals which regularly carry material relevant to this
issue. Also, is the International Journal of General Systems still
published? If not, what did it turn into? Or rather, who carries articles
about "the theory of things", "the calculus of indications" etc? I am
extremely interested.
Thank you.
Ali Minai
Dept. of Electrical Engg.
Thornton Hall,
University of Virginia,
Charlottesville, VA 22903.
aam9n@uvaee.ee.virginia.edu
------------------------------
Subject: Re: Data Complexity
From: pa1159@sdcc13.ucsd.EDU (Matt Kennel)
Organization: Univ. of California, San Diego
Date: 06 Oct 89 22:06:31 +0000
I don't know what other people have suggested, but I'll spout out a
suggestion of my own: crib ideas from the nonlinear dynamics people. For a
rough-and-ready measure of the complexity of the input space, why not use
fractal dimension? And for the output space, what about the sum of the
positive Lyapunov exponents of the nonlinear mapping?
>4) How can such a "complexity" measure be made scale-invariant?
That's exactly what the fractal dimension does?
>Also, what about such complexity measures for continuous functions?
>I mean measures defined structurally, not according to the type
>of the function (e.g. degree for polynomials).
I don't understand what you mean by "defined structurally."
Matt Kennel
UCSD physics
pa1159@sdcc13.ucsd.edu
------------------------------
Subject: Re: Data Complexity
From: cybsys@bingvaxu.cc.binghamton.edu (CYBSYS-L Moderator)
Organization: SUNY Binghamton, NY
Date: 09 Oct 89 17:06:14 +0000
[ Cross-posted from CYBSYS-L@BINGVMB ]
Really-From: heirich%cs@ucsd.edu (Alan Heirich)
I am also interested in the question of complexity measures raised by Ali
Minai. I think it is an important issue for anyone concerned with self
organizing systems -- i.e. how can you measure the extent to which a system
is organized?
Some candidates I've thought of are Shannon entropy, Kolmogorov entropy,
"integrality" (a measure defined by I-don't-remember-who, discussed by
Brooks in the recent book "Information, entropy and evolution", which I
assume readers of this list are familiar with, or should be). I would like
to hear about other possibilties.
Alan Heirich Comp. Sci. & Eng., Cognitive Science
C-014 University of California, San Diego 92093
heirich@cs.ucsd.edu
aheirich@ucsd.bitnet
------------------------------
Subject: Data Complexity
From: ib@apolling (Ivan N. Bach)
Date: 12 Oct 89 16:24:47 +0000
Many scientists believe that you cannot accept something as a science
unless you can measure it in some way. Scientists often prove or disprove
their theories by performing experiments in which they measure some key
quantities, and then decide whether the results of their measurements are
in accordance with the proposed theory. You must also be able to repeat
such experiments.
When it comes to expert systems and neural nets, many authors tend to make
all kinds of extravagant claims that are usually not supported by any
objective measurements. For example, they claim that a given expert system
has achieved the capability of a human expert. This would, presumably,
mean that the expert system is as complex as those components of the human
brain that are the material basis for the human expertise, unless the
expert system is designed more efficiently than the human brain.
When I worked as a consultant at AT&T Bell Laboratories, I did some
research on the measuring of the complexity of biological and artificial
information systems. I was surprised by how much work has already been
done in this area. I came to the conclusion that you can use entropy
calculations to determine the MAXIMUM amount of information that can be
stored in a particular information system, i.e., its maximum information
capacity.
For example, I calculated the maximum amount of information that can be
stored in a fused human egg cell because that egg cell contains a blueprint
for a human being. One way of measuring the complexity of any system is to
find out how much information is needed to completely describe a blueprint
for that system, and, for example, transmit it to a nearby star. The
instructions for building all the proteins in a human body are stored in
DNA molecules in the nucleus of the egg cell. The information is encoded
by using an alphabet that consists of just four characters A (adenine), C
(cytosine), G (guanine), and T (thymine). To calculate the maximum
information capacity of a DNA DNA molecule, you have to assume that the
distribution of bases is completely random, i.e., that there are no
restrictions imposed on the order in which bases can appear in the DNA
chain. The distribution of bases in real cells is not random, and,
therefore, the actual information capacities of such cells are less than
the potential maxima. The maximum capacity of a fused human egg cell is
several gigabits.
Once the genes of a human being are completely mapped, we will be able to
calculate exactly the information capacity of those genes. In the
meantime, we can calculate the maximum possible capacity of human genes.
That is better than not having an objective measurement, and it has a
number of precedents in mathematics. Mathematicians can sometimes
determine the limits within the values of a particular function must be,
but they may not be able to calculate the actual function values.
I also came to the conclusion that the information capacity of an
information system depends on the number of internal states that are
actually used to store information. For example, you can view a stone as a
system with just one internal state (0 bits of information) if you do not
use its atoms to store information. You can view an electrical relay that
can be closed or open as a system with just two internal states (1 bit of
information) if you use the open state to store a 0 and the closed state to
store a 1, etc. You can calculate the maximum information capacity of an
expert system by taking into account the total number of characters and the
number of characters in the alphabet.
Researchers have calculated the information capacities of viruses, insects,
mammals, etc. It is interesting that the ordering of biological systems by
their information capacity corresponds very closely with the usual ordering
of biological systems into lower (less complex) and higher (more complex)
species. After you calculate the information capacity of different
artificial and biological systems, you can present your results in the form
of a diagram.
relay expert system
| | information
V V capacity in bits
0 ++--...---+--------+---------------------------------...---+-------------->
^ ^ ^
| | |
stone virus human egg cell
I came to the conclusion that an artificial system could achieve the same
level of intelligence (information processing) as a human being with a
smaller number of internal states if its design was more optimal than the
design of the human being. Unfortunately, over several billion years
nature has used genetic algorithms to produce biological information
systems of incredible complexity and optimization. The level of
miniaturization used in DNA is about four orders of magnitude greater than
the level of miniaturization used in integrated circuits with a 1-micron
geometry.
British researchers could not figure out how the information for producing
all the proteins needed to construct a virus could be encoded in just a
couple of thousand bases. The information stored in DNA is read in
triplets. They discovered that the code for one protein started at a
certain location on the DNA chain. The code for another protein started on
the next base, i.e., the codes overlapped.
I came to the conclusion that an artificial intelligence system that would
be at the same intellectual level as a human being would have to be
extremely complex even if its design is highly optimized. We will need a
very large number of internal states to implement an artificial subsystem
that would imitate the capabilities of, for example, a human eye simply
because of the large amount of information that must be processed in real
time. I do not believe that such complex systems can be produced by
conventional coding of computer systems. I think that we will have to use
automatic, self-organizing methods based on genetic algorithms to
automatically and gradually develop more and more complex artificial
systems until we eventually achieve and surpass the complexity and
intellectual capability of human beings.
An artificial information system will have a different internal model of
the outside world than a human being, and a totally alien set of values,
unless such a system has the shape of a human baby and goes through the
experiences of adolescence, adulthood, and old age, or we control the
information it receives from the outside world in such a way that it gets
the impression that it is a human baby and that it is going through the
experiences of a human being. I am not saying that we should try to create
such an artificial system. I am just saying that an immobile artificial
system that does not have the shape of a human body, and that receives
information from the outside world through some sensors that have no
relationship to human senses, cannot possibly develop an intellect similar
to ours. We are more likely to develop an alien race of artifical beings
than to develop artificial beings which cannot be distinguished from human
beings as described in Isaac Asimov's novels on robots.
If you are interested in this topic, I can post detailed entropy
calculations and references to relevant books and articles.
------------------------------
Subject: Data Complexity
From: ib@apolling (Ivan N. Bach)
Date: 15 Oct 89 22:08:08 +0000
How can we measure the capacity to store and transmit information? Claude
Shannon, founder of the theory of information, suggested that the capacity
of a system to store and transmit information can be determined by
calculating the degree of randomness or disorganization in the system [1].
He decided to call this measure "entropy," because he realized that his
formula:
H = -K x sum of (pi x log pi) for all i=1 to i=a
(where "H" is the entropy of a system with "a" possible states or events,
"K" is a constant, and "pi" is the probability of occurrence of the i-th
state or event) was essentially the same as the formula that is used in
statistical thermodynamics to calculate the entropy or disorganization of a
thermodynamic system.
It can be proved mathematically [2] that the entropy of a system will be at
its maximum when all system states or events are equally probable and
independent, i.e., when the probability of each event in a system with "a"
states or events is equal to 1/a, and the probability of an event is
independent from the probabilities of all other events. This maximum
entropy is equal to log a.
When we set the constant "K" to 1, and use base 2 to calculate logarithms,
the unit of entropy is called a "bit."
One of the best books on the use of Shannon's entropy formula to calculate
the information capacity of biological information systems is the book
"Information Theory and The Living System" written by Lila Gatlin [3].
If a system has only one state, the probability of occurrence of this state
is equal to 1. The entropy of such a system is equal to zero.
The entropy of an information system with 2 states will be maximum when the
probability of each event is equal to 1/2. This follows from the basic
requirement that the sum of all probabilities of all possible events must
always be equal to 1. The entropy of such a system is:
Hmax = -(p1 x log2 of p1 + p2 x log2 of p2) bits
= -(1/2 x log2 of 1/2 + 1/2 x log2 of 1/2) bits
= -(log2 of 1/2) bits
= log2 of 2 bits (because -log2 of 1/a = log2 of a)
= 1 bit
Computer programmers usually think of a single-bit variable as a variable
which can have either value 0 or value 1. The maximum entropy or
information capacity of a single-bit variable is 1 bit.
Lila Gatlin showed that Claude Shannon's formula could be used to calculate
the entropy of a randomly chosen base in the DNA chain of bases in the
nucleus of a living cell. If all four types of bases in a DNA chain are
equiprobable, the entropy of a randomly chosen base in the DNA chain is
equal to:
Hmax = -log2 of 1/4 bits
= log2 of 4 bits
= 2 bits
Computer programmers know that they have to use at least 2 bits to encode 4
different values. The maximum entropy for a randomly chosen symbol from a
sequence of symbols selected from an alphabet of "a" symbols (a=4 for DNA)
is equal to log2 of "a" bits.
If all bases in a DNA chain are not equiprobable, the probability of a
certain type of base occurring at a random position in a DNA chain is
proportional to the number of bases of its type in the whole chain, i.e.,
it depends on the composition of DNA. All bases in the DNA of "Escherichia
coli" bacteria are equiprobable. Bases in the DNA of "Micrococcus
lysodeikticus" appear with the following probabilities:
p(A)=p(T)=.145 and p(C)=p(G)=.355
because adenine (A) bonds only to thymine (T), and cytosine (C) bonds only
to guaning (G). Therefore, the entropy of a randomly chosen base in the
DNA chain of "Micrococcus lysodeikticus" is equal to:
H = -(.145 x log2 of .145 + .145 x log2 of .145 + .355 x log2 of .355 +
.355 x log2 of .355) bits
= 1.87 bits
Notice that this value is smaller than the maximum value of 2 bits for
equiprobable bases.
Lila Gatlin came to the conclusion that the entropy of a complex system is
equal to the sum of entropies of its components.
The number of base pairs per sex or germ cell ranges from 10,000 for
bacteria to over 1,000,000,000 for mammals. If we assume equiprobable
bases, the maximum total entropy of the DNA chain in a human sex cell is
about 5.8 gigabits (2.9 billion nucleotide pairs x 2 bits per nucleotide
pair). When a sperm cell fuses with an egg cell, the maximum entropy of
the DNA chain in the nucleus of the fertilized cell is equal to 11.6
gigabits.
Notice that the DNA in a fertilized egg cell contains just a "blueprint"
for a human body. Dancoff and Quastler [4] [5] tried to estimate very
roughly the entropy of an adult human body by calculating the amount of
information required to specify the type, position, and orientation of
atoms in the body. There are probably about 60 different types of atoms in
the human body. Assuming 64 different atoms which appear with equal
frequency, the maximum entropy of a randomly selected atom is equal to 6
bits. However, not all types of atoms occur with the same frequency. Only
three types of atoms, carbon, hydrogen and oxygen, are present in
significant amounts. Their approximate relative frequencies in the human
body are 0.66, 0.22 and 0.07, respectively. The frequency of all other
types of atoms is about 0.05. Therefore, the actual entropy of a human
body is about 1.4 bits per atom. About 23 more bits are needed to specify
proper orientation of each atom. That brings the total entropy per atom to
about 24.4 bits. It is not necessary to specify that many bits per each
atom, because more than 90% of body's weight consists of water and other
comparatively featureless substances. Therefore, the total entropy of a
human body is:
H = 24.4 bits x 10% of 7x10 to the power of 27 (the total number of
atoms in the body)
= 2x10 to the power of 28 bits (or 10 to the power of 24 pages of
English text)
Similar astronomical figures were calculated in different ways by Linschitz
[6] and Morowitz [7].
What I find fascinating is that the information needed to specify the
blueprint for a human body is so efficiently compressed that it can be
placed into the nucleus of each of about 50 trillion cells in that body.
Each sex cell contains only half a blueprint. Each non-sex cell contains
all the information needed to build every type of protein in a human body.
Which types of proteins are actually built depends on the position of
biological switches.
Several valuable lessons relevant to the design of extremely complex neural
networks can be deduced from the studies of biological systems.
1. The storage of a neural network is usually measured in the number of
interconnects. If you can calculate the number of different states of a
neural network, you can calculate its maximum information capacity by
assuming that all states are equiprobable. If the probability of
occurrence of each state is either known or can be calculated, you can
calculate the actual information capacity of your network.
Once you can calculate the capacity of different networks, you can
objectively compare them, and determine which network represents the
most efficient way of storing information.
In Fig. 2.13 on page 33 of the DARPA Neural Network Study [27], you can
see a comparison of the storage and speed of a leech, worm, fly, etc.
It is estimated that a human brain has the storage capacity of 10 to the
power of 14 interconnects, and the speed of 10 to the power of 16
interconnects per second.
2. Many researchers are trying to find analytical methods for designing
optimal neural networks. I suggest that we use genetic algorithms to
design extremely complex and optimized neural networks automatically.
Genetic algorithms require an objective function or a goal. We can
specify the minimum number of neural nodes or the minimum number of
layers in a neural network as our goal. We can then use a random
process to generate generations of neural nets. We can use the
objective function to evaluate members of each generation, and discard
those members which do not achieve a good enough result with respect to
the objective function. We can then cross the most successful members
in each generation until we get an optimal network.
Research has shown that application of a priori knowledge of which areas
of the domain of a complex objective function should be explored usually
results in suboptimal results, i.e., in the finding of a local optimum.
Genetic algorithms based on random processes, thoroughly but not
exhaustively explore all areas of the objetive function's domain, and
usually result in the finding of an overall or global optimum [7][8].
I have come to the conclusion that very intelligent systems must be very
complex. We cannot use conventional computer programming methods to
specify each detail of an extremely complex information system. If we
restrict ourselves to only analytical methods of constructing neural
networks, we will not be able to construct networks that will be complex
enough to have the information capacity equal to or greater than the
capacity of a human brain.
3. The methods used to construct human neural networks are totally
different from the methods which are now used to construct artificial
neural networks and design complex VLSI circuits. The technology of
biological systems can be viewed as an extremely advanced and alien
technology. To not take advantage of this technology would be like
coming across a UFO or a planet with very advanced technology, and
passing up the opportunity to study and apply that technology.
The Turing machine is an abstract automaton with a head that is used to
read instructions from an abstract tape. Most computer scientists think
that the Turing machine is used to prove theorems in the theory of
automata, and that such a mechanism should not be implemented because
there is no practical use for it. If you compare the diagram of a
Turing machine in a textbook on the theory of automata with the diagram
of a ribosome in a textbook on genetics or molecular biology, you will
notice remarkable similarities.
A ribosome in the cytoplam of a cell has a head which is used to read
the instructions for the building of proteins which are stored on a
"tape" called messenger RNA. Since ribosomes (the work stations for the
manufacturing of proteins) are located outside the nucleus of a cell,
and the DNA molecules are safely stored in the nucleus, messenger RNA
molecules are needed to copy the instructions for building proteins from
DNA molecules, and carry them to ribosomes.
I would not be surprised if our colleagues who are reading the
"sci.nanotech" newsgroup expressed an interest in such a mechanism.
There must be a number of other ingenious mechanisms in our cells that
are just waiting to be discovered.
4. Instead of constructing a full-blown neural network, we could specify a
"blueprint" for contructing that network. I think that the amount of
information in the blueprint can always be smaller than the amount of
information in the network if there is any redundancy or repetition in
the structure of the network. If a very complex network is used to
store large amounts of similar types of information, whole sections of
that network should have identical or very similar structure. There is
no reason why we should use a different structure each time we want to
store a data item of a certain type. If whole areas of the network are
going to have the same structure, we could specify in the blueprint that
a certain number of subnetworks of a given type is needed in a
particular location, and we may not care how each detail of each
subnetwork is going to be constructed. I suspect that Mother Nature had
coded recursive subroutine calls into our DNA molecules long time before
computer programmers rediscovered such calls.
5. If you use a certain "alphabet" of components to construct your neural
net, you should use each type of component with the same frequency if
you want achieve the maximum information capacity.
Main references for the artificial evolution and genetic algorithms seem to be:
A. Turing (1936-1952) [8]
Turing machines, the chemical basis of morphogenesis.
J. von Neumann (1952-53) [9]
Reproduction and evolution of automata.
L. Fogel, A. Owens, and M. Walsh (1966) [10]
Artificial evolution through simulated evolution.
J. Bagley (1967) [11]
Behavior of adaptive systems.
R. Rosenberg (1967) [12]
Relationship between genotype and phenotype.
R. Weinberger (1970) [13]
Computer simulation of a living cell.
E. Goodman (1972) [14]
Adaptive behavior of simulated bacterial cells.
N. Foo and J. Bosworth (1972) [15]
Various aspects of genetic operators.
D. Frantz (1972) [16]
Non-linearities in genetic adaptive research.
J. Holland (1962, 1975 and 1976) [17][18][19]
Theory of adaptive systems and spontaneous emergence of self-reproducing
automata.
K. DeJong, A. Brindle, and A. Bethke (1975-81) [20]
Genetic algorithms and function optimization.
S. Smith (1980) [21]
Learning systems based on genetic algorithms.
J. Grefenstette (1983) [22]
Optimization of genetic search algorithms.
M. Mauldin (1984) [23]
Maintaining diversity in genetic search.
A. Goldberg and I. Pohl (1984) [24]
Complexity theory and AI research.
D. Goldberg (1985, 1989) [25][26]
Application of genetic algorithms.
REFERENCES
1. Shannon, C. E., "A Mathematical Theory of Communication," "Bell System
Technical Journal," Vol. 27, pp. 379-423 and 623-656. (Reprinted in C.
E. Shannon and W. Weaver, "The Mathematical Theory of Communication,"
pp. 3-91, U. of Illinois Press, Urbana, 1949.)
2. Khinchin, A. I., "Mathematical Foundations of Information Theory,"
Dover, New York, 1957.
3. Gatlin, Lila, "Information Theory and The Living System," Columbia
University Press, New York, 1972.
4. Dancoff, S. M. and H. Quastler, "The Information Content and Error Rate
of Living Things," in "Information Theory in Biology" edited by H.
Quastler, U. of Illinois Press,Urbana, 1953.
5. Calow, Peter, "Biological Machines: A Cybernetic Approach to Life,"
Edward Arnold Publishing Co., London, England, 1976.
6. Linschitz, H., "Information Content of a Bacterial Cell" in "Information
Theory in Biology," edited by H. Quastler, U. of Illinois Press, Urbana,
1953.
7. Morowitz, H. J., "Bull. Maths. Biophys.," Vol. 17, 1955, pp. 81-86.
8. Turing, A. M., "The Chemical Basis of Morphogenesis," "Phil. Trans. Roy.
Soc.," ser. B, No. 237, London, 1952.
9. Von Neumann, John, a lecture at the U. of Illinois published in: Arthur
W. Burks, ed., "Theory of Self-Reproducing Automata," U. of Illinois
Press, Urbana, 1966.
10. Fogel, L. Owens et al., "Artificial Intelligence Through Simulated
Evolution," John Wiley and Sons, Inc., New York, 1966.
11. Bagley, J. D., "The Behavior of Adaptive Systems Which Employ Genetic and
Correlation Algorithms," Ph.D. thesis, U. of Michigan, Ann Arbor, 1967.
12. Rosenberg, R. S., "Simulation of Genetic Populations with Biochemical
Properties," Ph.D. thesis, U. of Michigan, Ann Arbor, 1967.
13. Weinberger, R., "Computer Simulation of a Living Cell," Ph.D. thesis,
U. of Michigan, Ann Arbor, 1970.
14. Goodman, E. D., "Adaptive Behavior of Simulated Bacterial Cells
Subjected to Nutritional Shifts," Ph.D. thesis, U. of Michigan, Ann
Arbor, 1972.
15. Foo, N. Y. and J. L. Bosworth, "Algebraic, geometric, and stochastic aspects of
genetic operators," Tech. Rep. 003120-2-T, U. of Michigan, Ann Arbor, 1972.
16. Frantz, D. R., "Non-linearities in Genetic Adaptive Search," Ph.D.
thesis, U. of Michigan, Ann Arbor, 1972.
l7. Holland, J. H. "Concerning Efficient Adaptive Systems," pp. 215-230 of
"Self- Organizing Systems," edited by M. C. Yovits, G. T. Jacobi, and
G. J. Goldstein, Spartan Books, Washington, D.C., 1962.
18. Holland, J. H., "Outline for a Logical Theory of Adaptive Systems,"
Journal of the Assoc. for Computing Machinery, Vol. 9, No. 3, July
1962, pp. 297-314.
19. Holland, J. H., "Studies of the Spontaneous Emergence of
Self-Replicating Systems Using Cellular Automata and Formal Grammars,"
pp. 385-404 from "Automata, Languages, Development," edited by Aristid
Lindenmayer and Grzegorz Rosenberg, North-Holland Publishing Company,
Amsterdam, 1976.
20. DeJong, K. A., "Analysis of the Behavior of a Class of Genetic Adaptive
Systems," Ph.D. thesis, Dept. of Computer and Communications Sciences,
U. of Michigan, Ann Arbor, 1975.
21. Smith, S., "A Learning System Based on Genetic Adaptive Algorithms," Ph.D.
thesis, U. of Pittsburgh, Pa., 1980.
22. Grefenstette, J. G., "Optimization of Genetic Search Algorithms," Tech.
Rep. CS-83-14, Vanderbilt U., 1983.
23. Mauldin, L. Michael, "Maintaining Diversity in Genetic Search," Proc.
of the 1984 National Conference on Artificial Intelligence, pp.
247-250.
24. Goldberg, Allen and Ira Pohl, "Is Complexity Theory of Use to AI?," pp.
43-55 from "Artificial Human Intelligence," edited by A. Elithorn and
R. Banerji, Elsevier Science Publishers, Amsterdam, 1984.
25. Goldberg, David, "Dynamic System Control Using Rule Learning and
Genetic Algorithms," Proc., of the 1985 National Conference on AI, pp.
588-592.
26. Goldberg, E. David, "Genetic Algorithms In Search, Optimization and
Machine Learning," Addison-Wesley Publishing Company, Inc., Reading,
1989.
27. "DARPA Neural Network Study," AFCEA International Press, Fairfax, 1988.
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