(article publié par dans les Nouvelles d'APL, revue trimestrielle d'AFAPL, numéro 26 de mars 1998, pp 7-22)
Crisp and Soft Computing with HyperCubical Calculus
New approaches to Modeling in Cognitive Science and Technology with Parity Logic, Fuzzy Logic, and Evolutionary Computing
Michael Zaus
DYNARRAY CORPORATION
1280 SPACEPARK WAY MOUNTAIN VIEW
CALIFORNIA 94043, USA
318 pp. To appear in Spring 1998 in press by Physica-Verlag
/ Springer Verlag Heidelberg
Series "Studies in Fuzziness and Soft computing ", Editor-in-chief
: Prof. Janussz Kacprszyk
| Dr. Michael Zaus DynArray Corporation Mountain View 1280 Spacepark Way e-mail dynarray@ibm.net Silicon Valley California 94043, USA |
Arbeitsgemeinschaft Fuzzy Logik und Soft Computing
Norddeutschland (AFN) http://fuzzy.cs.uni-magdeburg.de/afn |
Dr. Michael Zaus Hatter Landstr. 12 D-26209 Oldenburg-Hatten Tel/Fax 49 4481 98711 Germany |
¹\
In Memory of
Gérard A. Langlet
CEA. Laboratoire d'Informatique Théorique
C.E. Saclay, Gif sur Yvette, France
¹\
This text grew out of a research project on the foundations of parity logic, fuzzy logic, and evolutionary computing at the Institute for Cognitive Science of the University of Oldenburg in Germany, and out of a series of seminars on evolutionary and fuzzy computing in cognitive science. What these seemingly diverse fields have in common from a conceptual and computational Point of view is that all of them are based on hypercubical calculus. That was not apparent at the beginning, but arose gradually along intensive computational work. To provide an idea of what it implies, we list seven representative hypercubes together with their specific field of application.
| 1 |
Boolean Hypercube Parity Logic & Evolutionary Computing Bn = {0,1}n = {x = (x1,x2,...xn) Î Rn : xi Î {0,1} for all 1£i£n} |
|---|---|
| 2 |
Trivalent Hypercube: Fuzzy Logic & Fuzzy Cognitive Maps {-1,0,1}n = {x = (x1,x2,...xn) Î Rn : xi Î {-1,0,1} for all 1£i£n} |
| 3 |
Bipolar Hypercube: Fuzzy Logic & Fuzzy Cognitive Maps [-1,1]n = {x = (x1,x2,...xn) Î Rn : xi Î [-1,1] for all 1£i£n} |
| 4 | Unit Hypercube: Fuzzy Logic & Fuzzy Cognitive Maps In = [0,1]n = {x = (x1,x2,...xn) Î Rn : xi Î [0,1] for all 1£i£n} |
| 5 | Discrete Bipolar Hypercube: Fuzzy Logic & Fuzzy Cognitive Maps {-1,1}n = {x = (x1,x2,...xn) Î Rn : xi Î {-1,1} for all 1£i£n} |
| 6 | Schemata- viz. Hyperplane Cube: Evolutionary Computing {-1,0,*}n = {s = (s1,x2,...sn) Î {{0,1}È{*}}n : si Î {-1,0,*} for all 1£i£n} |
| 7 | GRAY-Coded Hypercube: Parity Logic & Evolutionary Computing Gn = {0,1}n = {x = (x1,x2,...xn) Î Rn : xi Î {0,1} for all 1£i£n} where ånixiÅyi = 1 and Gn is pathwise Hamiltonian. |
This is not the place to unpack their details, but it
helps to guide the reader around the central topics of this book. The research
project centered on formal models of connectionist information processing
regarding the emergence of meaning in the brain By investigating the formal
foundations of emergent computing, an by being persistently faced with
the target question of how meaning emerges in the brains of living systems,
it turned out soon that this was not only one of the last questions science
may try to answer, but also one that we are far from being able to treat
in terms of traditional mathematics, at least with respect to explanatory
models with high epistemic value. Rather than trying to model the formally
intractable emergence of meaning phenomenon, the author decided to search
for more fundamental information generating algorithms.
The search for generic and most elementary information
processing mechanisms led the author into the field of algorithmic compression,
and by hitting upon Gérard Langlet's St. Petersburg paper on a Theory
of Everything in terms of the programming language APL. the ice was broken
[LAN92]). However, neither Langlet's T.o.E. nor the incredible scope it
covered was of primary interest to the author, but the mere fact that the
cumulative n-bit parity function
p : {0,1}n ® {0,1}n
constituted a
minimal and irreducible algorithm for generating increasingly complex structures
out of almost nothing by starting from the elementary bit. It provided
not only a unique tool for scientific modeling from scratch, but also the
birth of parity logic, as the reader will learn from part I of the book.
What followed from 1994 to 1996 was intensive computational research, partially
rewarding due to successful progress with magnificient support and encouraging
from Gérard Langlet, and partially frustrating because of an ignorant
and denigrating academic environment. In 1996, the frustration turned into
reward when the author introduced parity logic as an invited speaker to
the audience of the APL96 conference held at Lancaster University, England.
Then the big shock came by the end of 1996, when the author learned about
the death of Gérard Langlet. My greatest debt is therefore acknowledged
in the dedication. Whatever merits part I of this book possesses may truthfully
be credited to his influence, support, and encouragement. I wished we had
the chance for a bigger research project on the subject matter, since so
many things remain to be done, in particular with respect to multidimensional
Langlet transforms, parallel data compression, and binary dynamically systems.
Efforts in this direction are now subject to a joint venture project in
Silicon Valley, arranged by Amir Bukhari. chairman of DynArray Corporation,
Mountain View, and the author.
The second approach pursued in part II on fuzzy logic
is at the very heart of hypercubical calculus. In that respect the work
of Bart Kosko ([KOS92], [KOS97]) is greatfully acknowledged. It is argued
that fuzzy logic offers a paradigm shift in social and behavioral science
by virtue of providing a sound framework for soft computing, for constructing
non-linear dynamical predictor systems, and for making knowledge engineering
a prospering business. The current hands-off attitude of psychologists
towards fuzzy logic is quite puzzling, for they exclude a logic in their
practical and scientific activities which is common to any human individual,
namely approximate reasoning. What comes next door to approximate reasoning
is causal reasoning. Causal cognition abounds in problem solving, decision
making, and in trying to predict future events. Causal modeling is a domain
of fuzzy logic, and it is best pursued with fuzzy cognitive maps, that
is, knowledge projections par excellence. No knowledge, no map. No map,
no cognitive guidance. No cognitive guidance, no intelligible behavior.
Causal knowledge is a benchmark for competence in almost
any field of human activity, and the more complex the knowledge domain,
the harder it is to achieve. Our structural modeling approach to fuzzy
cognitive maps is precisely tailored to this task. Hypercubical calculus
and fuzzy cognitive maps ease a gentle entrance into non linear modeling,
because of their generic character that allows us to start small and grow
in fuzzy knowledge engineering. It emphasizes the state-space approach,
whereby state-space dynamics and the dynamics of interactions become apparent
and tractable. It subscribes to the Ganzheitsproblem, i.e. the way
how the whole is contained as a part in one of its own parts, thereby revealing
the real nature of fuzzy mutual subsethood. And it admits a highly desired
feature in research and practice, namely the growth of knowledge through
cooperative learning processes in terms of aggregating individual fuzzy
cognitive maps into an expanded reliable knowledge framework. The price
psychologists have to pay is to accept the nature of natural thinking,
that is, fuzzy thinking. That's presumably not too hard. A little harder
is the acceptance of multi-valued logic by incorporating fuzzy logic into
the curriculum of psychology. That precisely is the price, for otherwise
it wouldn't be a paradigm shift. Readers on the sceptical side should notice
that cognitive science gets currently more advanced outside psychology,
particularly in biology, physics, engineering, and computer science, so
something is going astray in psychology. It risks to loose another professional
domain. as so many before. and that in view of the fact that fuzzy logic
has become an international multimillion dollar business. If nothing happens,
the real loosers will be the students, for there is no excuse when faced
with reality by saying "Fuzzy logic? I thought that's only for machine
learning!". It certainly is not, as shown in part 11 of this book.
The third and final approach pursued in part III on evolutionary
computing is based equally well on hypercubical calculus inasmuch as the
Boolean hypercube Bn serves as the search space in function optimization
and multivariate feature analysis, while the schemata hypercube
Sn = {0,1,*}n
provides an analytical frame of reference for hyperplane analysis
in genetic and autogenetic algorithms. The purpose of part III is two-fold:
First, to examine where and in what respects parity logic affects the foundations
of evolutionary computing. Second, to outline a conceptually and computationally
coherent approach to a new type of genetic algorithms. called autogenetic
algorithms (AGAs). It is not the intention to develop another theory for
genetic computing. Instead, the last chapter on AGAs will explore the possibility
of reducing the algorithmic structure to a minimum of computational complexity.
This is foraally motivated by algorithmic compression.
As to the methodological aspect of evolutionary computing
it is important to note that we are mostly interested in multivariate search
in complex feature spaces. To make the respective fundamentals of AGAs
as selfcontained as possible, we present at first their conceptual framework,
secondly their theoretical foundations by examining the hypercube
Sn = {0,1,*}n,
thirdly by elaborating on their computational foundations, and
finally by discussing their applications to uni- and multivariate search
tasks in cognitive science and technology.
This book adopts the transdisciplinary view of modeling
in science wholeheartedly and is therefore aimed at a large audience in
hard and soft computing. Part I on parity logic is certainly more on the
side of hard computing and of special importance to readers interested
in scientific modeling from scratch in computer science, informatics, signal
and image processing, mathematics, physics, biology, and psychology. Part
11 on fuzzy logic is definitely on the side of soft computing and of particular
relevance to causal knowledge engineering in psychology, medicine, sociology,
political science, ecology, and economics. Part III on evolutionary computing
belongs as a model-free estimation approach both to hard and soft computing.
Search and optimization abound in any of the above fields, so part III
covers a truly multidisciplinary approach and is as such highly adaptable
to intradisciplinary target questions and research strategies.
Part I on parity logic owes its deepest debts to the late
Dr. Gérard A. Langlet for his advice, support and encouragements
in the years from 1994 to 1996. At that time G. Langlet was the head of
the Laboratoire d'Informatique Théorique of the Commissariat d'Energie
Atomique (CEA) at Saclay, France. Communicating and cooperating with him
on the subject matter was not only a lot of fun, but also an unforgettable
learning history. I am also greatly indebted to M. Sylvain Baron, president
of the "Association Francophone pour la promotion du langage APL"
(AFAPL), as both G. Langlet and S. Baron supported my work by publishing
parts of it in French in -Les Nouvelles d'APL". As to the connection
of parity logic and APL, I thank Dieter Latterniann from "APL-Cermany"
and Adrian Smith from the "British APL Association- for inviting me
as a speaker to the APL96 conference at Lancaster University in the summer
of 1996. The conference ignited the "Silicon Valley connection"
with Amir Bukhari, and the confidence that underground work in the rarefied
atmosphere of a hostile academic environment pays off by following rule
No. 1
"Never give up!" .
Regarding part II on fuzzy logic I thank particularly
Prof. Dr. Bart Kosko from the University of Southern California for sending
me a number of valuable papers on fuzzy logic and fuzzy cognitive maps.
His work has influenced my views of fuzzy logic eminently and formed the
idea of synthesizing parity logic, fuzzy logic. and evolutionary computing
into hypercubical calculus. I am not satisfied with the results obtained
so far, but the vision to make it a computational power tool is all the
more motivating, since the "State-Space as Hypercube" paradigm
is present in so many diverse fields that it cries for unification. Many
thanks are also due to my students who went patiently through the methodology
of concept mapping, mind mapping, fuzzy cognitive mapping, and a dozen
of fuzzy cognitive maps in the seminars of 1996/97 and 1997. They furnished
the sunny side of this research, thanks to all of them.
The work of part III on evolutionary computing has been
influenced mostly by Prof. Dr. John Holland, Prof. Dr. Ingo Rechenberg,
and Prof. Dr. Hans-Paul Schwefel, but in view of hundreds of different
models I have developed my own ideas, guided by parity logic and the canons
of scientific modeling from scratch.
Finally, I would like to thank Prof. Janusz Kacprzyk for
endorsing the book's publication by Physica-Verlag, Prof. Dr. Alf Zimmer
for his commitment of time and energy to reviewing the initial draft of
the book. and Dr. Martina Bihn from Springer Verlag for arranging its final
publication.
Acknowledgements
This research was supported in parts by the German Science
Foundation (DFG), Grant Sche 298/5-2 to the late Prof. Dr. Eckart Scheerer
from the "Interdisciplinary Research Group on Cognitive Science"
of the Universities Bremen and Oldenburg, Germany. Moreover, by Prof. Dr.
Hans Colonius, director of the Institute for Cognitive Science, University
of Oldenburg, for providing office room and facilities to complete the
book, and by Dr. Amir Bukhari, chairman of DynArray Corporation, Mountain
View, California, for establishing a joint venture on developing a hypercubical
calculus technology with the author at Silicon Valley in 1998.
Contents
This chapter previews the underlying approaches comprising parity logic, fuzzy logic, and evolutionary computing. Parity logic centers on efficient binary computing through automatic Boolean differentiation and integration, fast and entropy preserving transforms of binary arrays for reversible or nondissipative computing, and a broad scope of Boolean feedback machines called parity logic engines (PLEs). Fuzzy logic, in turn, centers on the design of nonlinear dynamical predictor systems in terms of fuzzy cognitive maps (FCMs) which endorse and enhance causal reasoning in decision processes, cross-impact analysis, and complex expert systems. Parity logic engines include evolutionary genetic search and optimization, thereby extending the computational framework of genetic algorithms (GAs) with autogenetic algorithms (AGAs). Altogether, these approaches provide new ways for fast computing, intelligent information processing, and highly adaptive system design, i.e. one type of system fits a wide range of diverse but formally intimately related problems. This holds for parity logic engines, fuzzy cognitive maps, and autogenetic algorithms.
Definition 1.1 The parity function p : Bn ® {0,1} computes whether an n-dimensional bit vector x has an even or odd number of 1s:
| p(x) = { | |
| 0 if x Î Bn contains an even number of 1s |
The parity function p may take on at least three
forms. First, the 2-bit parity function
: B2 ® {0,1},
known as the eXclusive-OR operation XOR. Second, the
n-bit parity function
: Bn ® {0,1},
which determines the binary scalar integral of its argument
x Î Bn.
Third, the cumulative n-bit parity function :
: Bn ® Bn,
which determines the binary vector integral of its argument
x Î Bn,
and henceforth the parity integral
z = (z1, z2,...,zn) Î Bn.
What we are being faced with is essentially XOR, its generalization
to monadic operators. and their fusion to parity logic systems, defined as follows.
Definition 1.2
The quadruple
PLS = {Bn,,ni=1 xiÎx,ni=1 xi Î x}
is called a finite Parity Logic
System if, and only if, Bn is a finite Boolean space such that
(1) elements x,y,z,w Î Bn = {0,1}n are binary vectors
of length n,
(2) x y is the eXclusive-OR operation, defined on elements
x, y Î Bn,
where is
symmetric: x y º y x,
associative: (x y) z º x (y z), and
bisymmetric: (x y) (z w) º (x z) (y w)
|
(3) (ni=1 xiÎx)
=
(x1 x2 ... xn)
=
{
|
|
| 1 if number of 1s in x is even |
(4) (ni=1 xiÎx)
=
(x1, (x1 x2),
...,
(x1 x2 xn)) = z Î Bn
where
z = (z1,z2,...zn) is the parity integral of
x Î Bn.
The pair
áBn, ni=1 xi Î xñ
had been overlooked by Knuth,
Minsky, Grossberg, Kosk-0, Holland. Roth, Hamming, Conway, Wolfram, Walsh,
Hadamard, Chaitin, and many others involved with binary corriputing. Kenneth
Iverson QIVE781) introduced it into the programming language APL as the
operator
¹\x Î Bn,
called "unequal-scan", and Gérard Langlet
([LAN92]) discovered, investigated, and exploited its ubiquitious role
in a great number of seminal papers on binary computing in APL. Since the
mathematical trilogy
á, , Åñ
occurs nowhere in the literature on binary
modeling, the author has christened it parity logic
in view of chapters
2 to 6, and by virtue of the fact that all contravalent phenomena are expressible
and analyzable by sequential, parallel. iterative, and recursive parity
integration.
The essence of the approach is as follows. In the same way as physics builds on an elementary, indivisible entity that depends on the act of observation, namely Max Planck's quantum, so does subormation theory build on the binary unit, the bit. However, the bit is as such not merely an abstract unit, but also a computational and simultaneously a representational unit for a manifold of contravalent phenomena. Examples are either one or zero, true or false, pro or con, head or tail, positive or negative.. yes or no, left or right, on or off, male or female, spin-up or spin-down, anion or cation. attractor or repellor, passive or active, dominant or recessive, activated or inhibited, and so forth. Parity integration mimics and models elementary mechanisms, whereby we understand kinds of reaction topologies. A biochemical mechanism, for instance. "explains" what the molecules do in forming a reaction chain or a reaction topology. A reaction chain is a sequence of elementary states, say, of either passive or active states, whereas a reaction topology is an organized field induced by a propagated reaction chain. The underlying elementary differences are formally representable by the binary operation XOR, whereas the propagation of differences - the selforganizing wavefronts of differences - are representable only by generalized XOR, the operator of parity integration. It acts as a force-the-force mechanism, and depending on the nature of its argument x Î Bn, it generates, propagates. accumulates, transforms, organizes, stores, decodes. structures, restructures. combines and recombines it in a unique way.
No other operator than parity integration reveals this
broad scope of implicit parallelism, whereby it gains so much computational
power. It is the binary counterpart of the Riemann-Liouville-Weyl integro-differential
operator for generating simultaneously binary vector integrals and binary
vector differentials. It evolves reversible transforms, symmetrical, periodical,
self-similar, auto-organized and iso-entropical transformation operators
and three-fold symmetry operators, whose vertical reflection generates
nonsymmetrical and self- inverse transformation operators, hence two-fold
symmetry operators. Working with
áBn, ni=1 xi Î xñ
revealed more than
100 specific properties which are, however, only the tip of an iceberg.
Parity integration works at the binary level. thus at the computer's operation
level. and precisely that makes computing with parity logic extremely fast
and efficient, for it bypasses any gigantic number crunching mod 2. Whoever
is involved with binary computing, and interested in scientific modeling
from scratch, will have to reconsider his or her allocation of foundations
in view of the generic triple
á, , ñ.
Parity logic shares presently the same status as fuzzy
logic did in the late sixties. It is a rather young and still largely unexploited
field of hard computing in soft computing, but it contrasts and complements
fuzzy logic in several attractive directions. The same holds for genetic
and neural computing, because the pair
áBn, ni=1 xi Î xñ
is in itself a generic
quantum algorithm for binary problem solvers, difference machines, parallel
parity feedback engines, reversible Turing machines, fast involutive transforms
without matrix operators, and iso-entropic trigonal transforms for research
regarding excitable media. That may suffice, for otherwise we are getting
too far ahead of what the next five chapters have to offer.
Part I on parity logic is an attempt to organize the subject
matter into a coherent framework useful to students, researchers, and practitioners
alike. Chapter 2 is the main chapter from a theoretical point of view and
provides the fundamentals of XOR and the foundations of generalized XOR
operators. Chapter :3 shows how this theoretical framework is applied to
binary signal analysis, why parity logic constitutes the basis towards
the binary counterpart of Fourier analysis, and how t lie most important
Langlet- and Shegalkin transforms are used for binary computing Chapter
4 is both an adventure and a tour-de-force through modeling perception
and action in parity logic. Chapter 5 is the main chapter regarding applied
parity logic, since it centers on parity logic engines with a detailed
exposition of genitons, paritons, and fanions, the three main structures
of parity logic. All of this is embedded into the context of excitable
media, thereby showing its potential to simulate the evolution of structure
formation without any numerical ad-hoc assumptions as used in conventional
fractal modeling approaches. Chapter 6 closes part I by discussing about
20 further t ransdisci pli nary perspectives of parity logic, and how we
may couple it with other research fields.
Part I was written from the standpoint of a theoretical, computational, and cognitive psychologist with a personal choice of main themes. As such it is a modest tribute to G6rard A. Langlet's life work. Representatives from science and engineering will treat it differently. and hopefully more comprehensively, but for parity logic coming into existence, it was and will be well worth its efforts.
A number of reasons motivated the author to enter into the foundations of fuzzy logic. First, fuzzy logic complements parity logic to the extent that the former resides within the n-dimensional unit hypercube [0,1]n = In, whereas the latter operates on the vertices of In. So they coexist in harmony. Second, fuzzy logic offers currently the most powerful approach to causal reasoning in cognitive science and technology by providing a mathematically sound framework for qualitative and quantitative nonlinear dynamical predictor systems in terms of fuzzy cognitive maps. Their mathematical structure has been derived from hypercubical state space and stability analysis, and the operational concept of causality is grounded on a firm set-theoretical and differential calculus base, in particular, John Stuart Mill's law of concomitant variation in terms of differential Hebbian learning laws ([KOS92]).
Third, the Boolean space Bn is the discrete envelope of the unit hypercube In. It is used in current VLSI chip technology for fuzzy set representations on the one hand, and as the search space of optimal fuzzy unit viz. fit vectors in evolutionary computing on the other hand. So, no bits, no fits, no flips. Moreover, the power of XOR in Bn should be inheritable to In. In other words, if fuzzy equal exists, and it does exist, then there is no escape for fuzzy unequal, hence fuzzy XOR, and thus fuzzy differences. Strangely enough, this leads to the conclusion that fuzzy equal in fuzzy unequal is the degree to which the whole is a part of its own part. See chapter 7 for more on fuzzy XOR. Fourth. fuzzy and parity logic are decisively concerned about algorithmic compression. Parity logic by virtue of its compact and fast transforms, fuzzy logic by virtue of data compression and granulation. Fifth, parity logic may unite fuzzy logic and evolutionary computing into the cohesive framework of an adaptive hypercubical calculus.
Chapter 7 on the mathematical foundations provides the respective background for these aspects by introducing the space In, then a series of conceptual and computational foundations including a powerful algorithm for fuzzy Hamming and Euclidean distance measures, the quest for emergent meaning in fuzzy linguistic terms including fuzzy entropy, a detailed discussion of fuzzy mutual subsethood, a comparison of the YinYang equation A = Ac, i.e. A equals not-A, in parity and fuzzy logic, an in-depth discussion of the Yin-Yang equation regarding modeling in social and behavioral science with critical comments on probability theory, knowledge space concepts, and nonlinear systems. A short survey on generalized fuzzy inner- and outer products completes chapter 7 and supports both the sets-as-points and sets-as-arrays views in fuzzy logic.
Chapter 8, the main contribution of part 11 on fuzzy logic,
invites the reader to an extensive discussion of fuzzy cognitive maps.
It is treated throughout from the viewpoint of knowledge engineering with
emphasis on the construction of individual fuzzy cognitive maps and their
interpersonal aggregation. To capture their importance to empirical research
we define them briefly as follows.
Definition 1.3
A fuzzy cognitive map (FCAI)
is a nonlinear dynamical causal web network in terms of a fuzzy digraph
with feedback. It consists of nodes and directed edges such that
(1) the nodes Ci represent concepts in terms of crisp
or fuzzy sets (or even fuzzy systems) associated with domain knowledge
from single experts, distributed expert pooling, cooperative learning,
documents, transscripts, or empirical data bases, and
Fuzzy cognitive maps are dynamical systems based on causal
knowledge. They are most useful as nonlinear dynamical predictor systems,
where the predictions are limit cycles. i.e. sequences of events like
{Sell A, Buy B, Invest C}.
FCMs differ critically from Bayesian networks or
probabilistic predictor systems by being superior in flexibility and problem
adaptability. They are designable as decision support systems for complex
and nonlinear problem domains in medicine, economics, business, politics.
psychology, education, and many other branches in practice or science wherever
causal reasoning plays a central role. Chapter 8 discusses their history,
the nature of causal reasoning, causal algebra, a carefully designed scheme
for constructing and aggregating FCMs, and seven examples of real and virtual
world FCMs. The basic maxime is that no one can invent causality, but everyone
can learn causal reasoning with fuzzy cognitive maps. As model-free estimator,
approximator and predictor systems they admit qualitative analysis, field
experimental procedures, expert pooling via internet, interpersonal aggregation,
quantitative evaluation, computer based simulations, complex networks through
nested FCMs, interactive modeling, and accessibility to VLSI-implementation.
A discussion of several advanced topics concerning continuous FCMs and
methodological issues regarding evaluation, limitations, and implementation
completes chapter 8 and part II on fuzzy logic.
Chapter 9 on the foundations of EC deals primarily with
parity logic and parity integration as far as it concerns particular issues
in EC. It includes scientific modeling from scratch for evolving mathematical
structures and operators, the role of parity integration in EC, for instance
GRAY-code and its reversal to BCD viz. binary code, a survey of Langlet
transforms in terms of a mind map, symmetry operators and autogenetic growth,
architectures derivable from n x n-parity matrices called paritons, and
a compact survey of parity logic engines. All of this is presented in an
informative and less formal way in order to spread ideas evenly over potential
fields of application.
A more systematic approach is then adopted in chapter
10 by developing the fundamentals of AGAS. it is well known from EC-research
that genetic algorithms (GAs) may encode complicated structures through
simple binary representations, and that simple transformations have the
power of improving such structures through selective pressure. AGAs resemble
GAs by structure, that is, by the reproductive cycle comprising evaluation,
selection, mutation, and recombination. But AGAs differ from GAs with respect
to the choice of genetic operators that change, exchange, and transform
these structures for progressive improvements. In pseudo-code, the procedure
amounts to
(2) the edges
eij : Ci ® Cj
represent the degree of positive or negative causal strength
between any pair of concept nodes
Ci and Cj.
1.3 Evolutionary Computing
Part III on evolutionary computing (EC) contains a more
general survey on the foundations of EC through parity logic, and a comprehensive
chapter on fundamentals of autogenetic algorithms (AGAs). Over the past
40 years, from Box's ([BOX571) "Evolutionary Operation" to Jacob's
([JAC971) "Principia Evolvica", EC has become a vast and steadily
expanding field centering on the emulation or simulation of natural evolution
processes in order to build artificial evolution systems for solving complex
and analytically intractable problems. It is meanwhile hard to characterize
EC in a single statement, because it includes an extremely wide range of
approaches such as genetic algorithms (HOL92]), evolution strategies ([REC94],
[SCHW95]), evolution programs [MIC92]), and cellular automata ([WOL97]),
to mention only four major research lines in this respect.
| Autogenetic Algorithm, (AGA) begin
initialize population Pop(t) from search space evaluate Pop(t) in real space R while (stoprule not satisfied) do
select elitist x by compressing Pop(t-1) mutate structure x by Bernoullian bit-flip mask integrate and iterate ni=1 xiÎx
to form Pop(t) evaluate Pop(t) in real space R |
The basic idea consists in using the operator of parity
integration as the main operator in AGAs for exploring and exploiting the
underlying search space such that the pair
áBn, ni=1 xi Î xñ
becomes the backbone of AGAs. This is a definitely new approach and appears
so far nowhere in EC-research.
To provide a proper background we start out with a conceptual framework of EC, followed by a concise review of the theoretical foundations of GAs and AGAs. A good deal of formal details is then devoted to their computational foundations, including representation and coding, evaluation and scaling, selection and sampling, mutation and recombination. Each computational. phase will be assisted with specific computer programs in APL, and two sections on uni- and multivariate search with AGAs.
