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SYSTEMICS OF EMERGENCE: RESEARCH AND DEVELOPMENT
SYSTEMICS OF EMERGENCE: RESEARCH AND DEVELOPMENT
SYSTEMICS OF EMERGENCE: RESEARCH AND DEVELOPMENT
Edited by
Gianfranco Minati\ Eliano Pessa^ and Mario Abram^ Italian Systems Society, Milano, Italy
^University of Pavia, Pavia, Italy
Spri inger
Gianfranco Minati Eliano Pessa Mario Abram Italian Systems Society University of Pavia Italian Systems Society Milan, Italy Pavia, Italy Milan, Italy
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Contents
Program Committee xi
Contributing Authors xiii
Preface xv
Acknowledgments xix
OPENING LECTURE 1
Uncertainty and Information: Emergence of Vast New Territories 3 G. J. KLIR
APPLICATIONS 29
Complexity in Universe Dynamic Evolution. Part 1 - Present state and future evolution 31 U. Di CAPRIO
Complexity in Universe Dynamic Evolution. Part 2 - Preceding history 51 U. Di CAPRIO
Mistake Making Machines 67 G. MINATI AND G. VlTIELLO
vi Systemics of Emergence: Research and Development
Explicit Velocity for Modelling Surface Complex Flows with Cellular Automata and Applications 79 M. V. AVOLIO, G. M. CRISCI, D . D'AMBROSIO,
S. Di GREGORIO, G. IOVINE, V . LUPIANO, R. RONGO,
W. SPATARO AND G. A . TRUNFIO
Analysis of Fingerprints Through a Reactive Agent 93 A. MONTESANTO, G. TASCINI, P. BALDASSARRI
AND L. SANTINELLI
User Centered Portal Design: A Case Study in Web Usability 105
M. P. PENNA, V. STARA AND D . COSTENARO
BIOLOGY AND HUMAN CARE 115
Logic and Context in Schizophrenia 117 P. L. BANDINELLI, C. PALMA, M . P. PENNA AND E. PESSA
The "Hope Capacity" in the Care Process and the Patient-Physician Relationship 133 A, RlCCIUTI
Puntonet 2003. A multidisciplinary and Systemic Approach in Training Disabled People Within the Experience of Villa S. Ignazio 147 D. FORTIN, V. DURINI AND M. NARDON
Intelligence and Complexity Management: From Physiology to Pathology. Experimental Evidences and Theoretical Models 155 P. L. MARCONI
Disablement, Assistive Technologies and Computer Accessibility: Hints of Analysis Through a Clinical Approach Based on the ICF Model 169 C. MASALA AND D . R. PETRETTO
Chaos and Cultural Fashions 179
S. BENVENUTO
COGNITIVE SCIENCE 191
Personality and Complex Systems. An Expanded View 193 M. MELEDDU AND L. F. SCALAS
Systemics of Emergence: Research and Development vii
Complexity and Paternalism 207 P. RAMAZZOTTI
A Computational Model of Face Perception 223 M. P. PENNA, V. STARA, M . BOI AND P. PULITI
The Neon Color Spreading and the Watercolor Illusion: Phenomenal Links and Neural Mechanisms 235 B. PINNA
Usability and Man-Machine Interaction 255 M. P. PENNA AND R. RANI
Old Maps and the Watercolor Illusion: Cartography, Vision Science and Figure-Ground Segregation Principles 261 B. PINNA AND G. MARIOTTI
EMERGENCE 279
Autopoiesis and Emergence 281 L. BiCH
Typical Emergencies in Electric Power Systems 293 U. Dl CAPRIO
Strategies of Adaptation of Man to his Environment: Projection Outide the Human Body of Social Institutions 311 E. A. NUNEZ
Emergence of the Cooperation-Competition Between Two Robots 317 G. TASCINI AND A. MONTESANTO
Overcoming Computationalism in Cognitive Science 341 M. P. PENNA
Physical and Biological Emergence: Are They Different? 355
E. PESSA
GENERAL SYSTEMS 375
Interactions Between Systems 377 M. R. ABRAM
viii Systemics of Emergence: Research and Development
Towards a Systemic Approach to Architecture 391 V. Di BATTISTA
Music, Emergence and Pedagogical Process 399 E. PlETROCINI
Intrinsic Uncertainty in the Study of Complex Systems: The Case of Choice of Academic Career 417 M. S. FERRETTI AND E. PESSA
A Model of Hypertextual Structure and Organization 427
M. P. PENNA, V. STARA, D. COSTENARO AND P . PULITI
LEARNING 435
Teachers in the Technological Age: A Comparison Between Traditional and Hypertextual Instructional Strategies 437 M. P. PENNA, V. STARA AND D. COSTENARO
The Emergence of E.Leaming 447 M. P. PENNA, V. STARA AND P. PULITI
Spatial Learning in Children 453
B. LAI, M . P . PENNA AND V. STARA
MANAGEMENT 461
Dynamics of Strategy: a Feedback Approach to Corporate Strategy-Making 463 V. CODA AND E. MOLLONA
A Cognitive Approach to Organizational Complexity 495 G. FlORETTI AND B. ViSSER
Normative Commitment to the Organization, Support and Self Competence 515 A. BATTISTELLI, M . MARIANI AND B . BELLO
A Multivariate Contribution to the Study of Mobbing, Using the QAM 1.5 Questionnaire 527 P. ARGENTERO ANDN. S. BONFIGLIO
Systemics of Emergence: Research and Development ix
Representation in Psychometrics: Confirmatory Factor Models of Job Satisfaction in a Group of Professional Staff 535 M. S. FERRETTI AND P. ARGENTERO
SOCIAL SYSTEMS 549
The Impact of Email on System Identity and Autonomy: A Case Study in Self-Observation 551 L. BIGGIERO
Some Comments on Democracy and Manipulating Consent in Western Post-Democratic Societies 569 G. MiNATi
Metasystem Transitions and Sustainability in Human Organizations. Part 1 - Towards Organizational Synergetics 585 G. TERENZI
Metasystem Transitions and Sustainability in Human Organizations. Part 2 - A Heuristics for Global Sustainability 601 G. TERENZI
SYSTEMIC APPROACH AND INFORMATION SCIENCE 613
Scale Free Graphs in Dynamic Knowledge Acquisition 615 I. LICATA, G. TASCINI, L . LELLA, A. MONTESANTO AND W. GIORDANO
Recent Results on Random Boolean Networks 625 R. SERRA AND M . VILLANI
Color-Oriented Content Based Image Retrieval 635
G. TASCINI, A. MONTESANTO AND P. PULITI
THEORETICAL ISSUES IN SYSTEMICS 651
Uncertainty and the Role of the Observer 653 G. BRUNO, G. MINATI AND A. TROTTA
Towards a Second Systemics 667 G. MINATI
X Systemics of Emergence: Research and Development
Is Being Computational an Intrinsic Property of a Dynamical System? 683 M. GlUNTi
The Origin of Analogies in Physics 695 E. TONTI
Prisoner Dilemma: A Model Taking into Account Expectancies 707 N. S. BONFIGLIO AND E. PESSA
The Theory of Levels of Reality and the Difference Between Simple and Tangled Hierarchies 715 R. POLI
General System Theory, Like-Quantum Semantics and Fuzzy Sets 723 L LiCATA
About the Possibility of a Cartesian Theory Upon Systems, Information and Control 735 P. ROCCHI
Program Committee
G. Minati (chairman) Italian Systems Society
E. Pessa (co-chairman) University of Pavia
G. Bruno University "La Sapienza", Rome
S. Di Gregorio University of Calabria
M, P, Penna University of Cagliari
R. Serra University of Modena and Reggio Emilia
G. Tascini University of Ancona
Contributing Authors
Abram M. R. Argentero P. Avolio M. V. Baldassarri P. Bandinelli P. L. Battistelli A. Bello B. Benvenuto S. Bich L. Biggiero L. BoiM. Bonfiglio N. S. Bruno G. Coda V. Costenaro D. Crisci G. M. D'Ambrosio D. Di Battista V. Di Caprio U. Di Gregorio S. Durini V. Ferretti M. S. Fioretti G. Fortin D. Giordano W. Giunti M.
AIRS, Milano, Italy Universita degli Studi di Pavia, Italy Universita degli Studi di Calabria, Rende (CS), Italy Universita Politecnica delle Marche, Ancona, Italy ASL Roma "E", Roma, Italy Universita degli Studi di Verona, Italy Universita degli Studi di Verona, Italy CNR, Roma, Italy Universita degli Studi di Pavia, Italy Universita dell'Aquila, Roio Poggio (AQ), Italy Universita degli Studi di Cagliari, Italy Universita degli Studi di Pavia, Italy Universita "La Sapienza", Roma, Italy Universita Commerciale "L. Bocconi", Milano, Italy Universita degli Studi di Cagliari, Italy Universita degli Studi di Calabria, Rende (CS), Italy Universita degli Studi di Calabria, Rende (CS), Italy Politecnico di Milano, Italy Stability Analysis s.r.l., Milano, Italy Universita degli Studi di Calabria, Rende (CS), Italy Villa S. Ignazio, Trento, Italy Universita degli Studi di Pavia, Italy Universita degli Studi di Bologna, Italy Villa S. Ignazio, Trento, Italy Universita Politecnica delle Marche, Ancona, Italy Universita degli Studi di Cagliari, Italy
XIV Systemics of Emergence: Research and Development
lovine G. Kiir G. J. LaiB. Leila L. Licata I. Lupiano V. Marconi P. L. Mariani M. Mariotti G. Masala C. Meleddu M. Minati G. Mollona E. Montesanto A. Nardon M. Nunez E. A. Palma C. Penna M. P. Pessa E. Petretto D. R. Pietrocini E. Pinna B. Poll R. PulitiP. Ramazzotti P. Rani R. Ricciuti A. Rocchi P. Rongo R. Santinelli L. Scalas L. F. Serra R. Spataro W. Stara V. Tascini G. Terenzi G. Tonti E. Trotta A. Trunfio G. A. Villani M. Visser B. Vitiello G.
CNR-IRPI, Rende (CS), Italy State University of New York, Binghamton, NY Universita degli Studi di Cagliari, Italy Universita Politecnica delle Marche, Ancona, Italy ICNLSC, Marsala (TP), Italy CNR-IRPI, Rende (CS), Italy ARTEMIS Neuropsichiatrica, Roma, Italy Universita degli Studi di Bologna, Italy Universita degli Studi di Sassari, Italy Universita degli Studi di Cagliari, Italy Universita degli Studi di Cagliari, Italy AIRS, Milano, Italy Universita degli Studi di Bologna, Italy Universita Politecnica delle Marche, Ancona, Italy Villa S. Ignazio, Trento, Italy AFSCET, France Istituto d'Istruzione Superiore, Roma, Italy Universita degli Studi di Cagliari, Italy Universita degli Studi di Pavia, Italy Universita degli Studi di Cagliari, Italy Accademia Angelica Costantiniana, Roma, Italy Universita degli Studi di Sassari, Italy Universita degli Studi di Trento, Italy Universita Politecnica delle Marche, Ancona, Italy Universita degli Studi di Macerata, Italy Universita degli Studi di Cagliari, Italy Attivecomeprima Onlus, Milano, Italy IBM, Roma, Italy Universita degli Studi di Calabria, Rende (CS), Italy Universita Politecnica delle Marche, Ancona, Italy Universita degli Studi di Cagliari, Italy CRSA Fenice, Marina di Ravenna, Italy Universita degli Studi di Calabria, Rende (CS), Italy Universita Politecnica delle Marche, Ancona, Italy Universita Politecnica delle Marche, Ancona, Italy ATESS, Frosinone, Italy Universita degli Studi di Trieste, Italy ITC "Emanuela Loi", Nettuno (RM), Italy Universita degli Studi di Calabria, Rende (CS), Italy CRSA Fenice, Marina di Ravenna, Italy Erasmus University, Rotterdam, The Netherland Universita di Salerno, Baronissi (SA), Italy
Preface
The systems movement is facing some new important scientific and cultural processes tacking place in disciplinary research and effecting Systems Research.
Current Systems Research is mainly carried out disciplinarily, through a disciplinary usage of the concept of system with no or little processes of generalization.
Generalization takes place by assuming: • inter-disciplinary (when same systemic properties are considered in
different disciplines), and • trans-disciplinary (when considering systemic properties per se and
relationships between them) approaches.
Because of the nature of the problems, of the research organization, and for effectiveness, research is carried out by using local inter-disciplinarily approaches, i.e. between adjacent disciplines using very similar languages and models.
General Systems Research, i.e. the process of globally inter- and trans-disciplinarizing, is usually lacking. General systems scientists are expected to perform disciplinary and locally inter-disciplinary research by, moreover, carrying out generalizations. The establishing of a dichotomy between research and generalizing is, in our view, the key problem of systems research today.
Research without processes of generalization produces duplications inducing besides fragmentations often used for establishing markets of details, symptomatic remedies, based on the concept of system, but with no or poor understanding of the global picture, that's without generalizing.
xvi Systemics of Emergence: Research and Development
The novelty is that emergence is the context where generaHzation, that's globally inter- and trans-disciplinarizing is not a choice, but a necessity.
As it is well known emergence is, in short, the rising of coherence among interacting elements, detected by an observer equipped with a suitable cognitive model at a level of description different from the one used for elements. Well-known exempla are collective behaviors establishing phenomena such as laser effect, superconductivity, swarms, flocks and traffic.
It has been possible to reduce General Systems Theory (GST), by dissecting from it the dynamic process of establishment and holding of systems, that's by removing or ignoring the processes of emergence. By considering systems and sub-systems as elements it is still assumed the mechanistic view based on the Cartesian idea that the microscopic world is simpler than the macroscopic and that the macroscopic world may be explained through an infinite knowledge of the microscopic.
However, daily thinking is often still based on the idea that not only the macro level of reality may be explained through the micro level, but that the macro level may be effectively managed by acting on the micro level. Assumption of manageability of the emergent level through elements comes from linearizing the fact that it is possible to destroy the upper level by destroying the micro.
Studying systems in the context of emergence doesn't allow the dissection mentioned above because models relate to how new, emergent properties are established rather than properties themselves only.
In the GST approach it has been possible to focus on (emergent) systemic properties (such as open, adaptive, anticipatory and chaotic systems), by considering their specificity and not reducibility to the ones of components. GST allowed description and representation of systemic properties, and adoption of systemic methodologies. This reminds in some way initial (i.e. Aristotelian) approaches to physics when the problem was to describe characteristics, essences, more than evolution.
Emergence focuses on the processes of establishing of systemic properties. The study of processes of emergence implies the study of inter-and trans-disciplinarity. Research on emergence allows for modeling and simulating processes of emergence, by using the calculus of emergence. Examples are the studies of Self-Organization, Collective-Behaviors, and Artificial Life. In short. Emergence studies the engine of GST, while GST allowed focusing on results. Models of emergence relate, for instance to phase transitions, synergetic effects, dissipative structures, conceptually inducing inter- and trans- disciplinary research.
Because of its nature emergence is inter- and trans-disciplinary.
Systemics of Emergence: Research and Development xvii
Paradoxically, this kind of research is currently made not by established systems societies, but by "new" systems institutions, like, just to mention a couple, the Santa Fe Institute (SFI), the New England Complex Systems Institute (NECSI), the Institute for the Study of Coherence and Emergence (ISCE), and in many conferences organized world-wide
General Systems Research is now research on emergence. As it is well known emergence refers to the core theoretical problems of the processes from which systems are established, as implicitly introduced in Von Bertalanffy's General Systems Theory by considering the crucial role of the observer, together with its cognitive system and cognitive models.
Emergence is not intended as a process taking place in the domain of any discipline, but as "trans-disciplinary modeling" meaningful for any discipline. We are now facing the process by which the General Systems Theory is more and more becoming a Theory of Emergence, seeking suitable models and formalizations of its fundamental bases. Correspondingly, we need to envisage and prepare for the establishment of a Second Systemics -a Systemics of Emergence- relating to new crucial issues such as, for instance: • Collective Phenomena; • Phase Transitions, such as in physics (e.g. transition from solid to liquid)
and in learning processes; • Dynamical Usage of Models (DYSAM); • Multiple systems, emerging from identical components but
simultaneously exhibiting different interactions among them; • Uncertainty Principles; • Modeling emergence; • Systemic meaning of new theorizations such as Quantum Field Theory
(QFT) and related applications (e.g. biology, brain, consciousness, dealing with long-range correlations).
We need to specify that in literature it is used, even if not rigorously defined, the term Systemics intended as a cultural generalization of the principles contained in the General Systems Theory. We may say that, in short. General Systems Theory refers to systemic properties considered in different disciplinary contexts (inter-disciplinarity) and per se in general (trans-disciplinarity); disciplinary applications; and theory of emergence.
More generally the term Systemic Approach refers to the general methodological aspects of GST. In base of that a problem is considered by identifying interactions, levels of description (micro, macro, and mesoscopic levels), processes of emergence and role of the observer (cognitive model).
At an even higher level of generalization, Systemics is intended as cultural extension, corpus of concepts, principles, applications and
xviii Systemics of Emergence: Research and Development
methodology based on using concepts of interaction, system, emergence, inter- and trans-disciplinarity.
Because of what introduced above, Systemics should refer to the principles, approaches and models of emergence and complexity, by generalizing them and not popularizing or introducing metaphorical usages.
The Systems Community urges to collectively know and use such principles and approaches, by accepting to found inter- and trans-disciplinary activity on disciplinary knowledge. Trans-disciplinarity doesn't mean to leave aside disciplinary research, but apply systemic, general (disciplinary-independent) principles and approaches realized in disciplinary research. Focusing on systems is not anymore so innovative. Focusing on emergence is not anymore so new.
What's peculiar, specific of our community? In our view, trans- and global inter-disciplinary research, implemented as cultural values.
The third national conference of the Italian Systems Society (AIRS) focused on emergence as the key point of any systemic processes. The conference dealt up with this problem by different disciplinary approaches, very well indicated by the organization in sessions: 1. Applications. 2. Biology and human care. 3. Cognitive Science. 4. Emergence. 5. General Systems. 6. Learning. 7. Management. 8. Social systems. 9. Systemic approach and Information Science. 10. Theoretical issues in Systemics.
We conclude hoping that the systemic research will continuously accept the general challenge previously introduced and contained in the paper presented. This acceptance is a duty for the systems movement when reminding the works of the founding fathers.
The Italian Systems Society is trying to play a significant role in this process.
Gianfranco Minati, AIRS president
Eliano Pessa, Co-Editor
Mario Abram, Co-Editor
Acknowledgments
The third Italian Conference on Systemics has been possible thanks to the contributions of many people that have accompanied and supported the growth and development of AIRS during all the years since its establishment in 1985 and to the contribution of "new" energies. The term "new" refers both to the involvement of students and to the involvement and contribution of researchers realizing the systemic aspect of their activity.
We have been honoured by the presence of Professor George Klir and by his opening lecture for this conference.
We thank the Castel Ivano Association for hosting this conference and we particularly thank Professor Staudacher, a continuous reference point for the high level cultural activities in the area enlightened by his beautiful castle.
We thank the Provincia Autonoma of Trento for supporting the conference and the University of Trento, the Italian Association for Artificial Intelligence for culturally sponsoring the conference.
We thank all the authors who submitted papers for this conference and in particular the members of the program committee and the referees who have guaranteed the quality of the event.
We thank explicitly all the people that have contributed and will contribute during the conference, bringing ideas and stimuli to the cultural project of Systemics.
G. Minati, E. Pessa, M Abram
OPENING LECTURE
UNCERTAINTY AND INFORMATION: EMERGENCE OF VAST NEW TERRITORIES
George J. Klir Department of Systems Science & Industrial Engineering, Thomas J. Watson School of Engineering and Applied Science, State University of New York, Binghamton, New York 13902-6000, U.S.A.
Abstract: A research program whose objective is to study uncertainty and uncertainty-based information in all their manifestations was introduced in the early 1990's under the name "generalized information theory" (GIT). This research program, motivated primarily by some fundamental methodological issues emerging from the study of complex systems, is based on a two-dimensional expansion of classical, probability-based information theory. In one dimension, additive probability measures, which are inherent in classical information theory, are expanded to various types of nonadditive measures. In the other dimension, the formalized language of classical set theory, within which probability measures are formalized, is expanded to more expressive formalized languages that are based on fuzzy sets of various types. As in classical information theory, uncertainty is the primary concept in GIT and information is defined in terms of uncertainty reduction. This restricted interpretation of the concept of information is described in GIT by the qualified term "uncertainty-based information". Each uncertainty theory that is recognizable within the expanded framework is characterized by: (i) a particular formalized language (a theory of fiizzy sets of some particular type); and (ii) a generalized measure of some particular type (additive or nonadditive). The number of possible uncertainty theories is thus equal to the product of the number of recognized types of fuzzy sets and the number of recognized types of generalized measures. This number has been growing quite rapidly with the recent developments in both fuzzy set theory and the theory of generalized measures. In order to fully develop any of these theories of uncertainty requires that issues at each of the following four levels be adequately addressed: (i) the theory must be formalized in terms of appropriate axioms; (ii) a calculus of the theory must be developed by which the formalized uncertainty is manipulated within the theory; (iii) a justifiable way of measuring the amount of relevant uncertainty (predictive, diagnostic, etc.) in any situation formalizable in the theory must be found; and (iv) various methodological aspects of the theory must be developed. Among the many
4 George J. Klir
uncertainty theories that are possible within the expanded conceptual framework, only a few theories have been sufficiently developed so far. By and large, these are theories based on various types of generalized measures, which are formalized in the language of classical set theory. Fuzzification of these theories, which can be done in different ways, has been explored only to some degree and only for standard fuzzy sets. One important result of research in the area of GIT is that the tremendous diversity of uncertainty theories made possible by the expanded framework is made tractable due to some key properties of these theories that are invariant across the whole spectrum or, at least, within broad classes of uncertainty theories. One important class of uncertainty theories consists of theories that are viewed as theories of imprecise probabilities. Some of these theories are based on Choquet capacities of various orders, especially capacities of order infinity (the well known theory of evidence), interval-valued probability distributions, and Sugeno /l-measures. While these theories are distinct in many respects, they share several common representations, such as representation by lower and upper probabilities, convex sets of probability distributions, and so-called Mobius representation. These representations are uniquely convertible to one another, and each may be used as needed. Another unifying feature of the various theories of imprecise probabilities is that two types of uncertainty coexist in each of them. These are usually referred to as nonspecificity and conflict. It is significant that well-justified measures of these two types of uncertainty are expressed by functionals of the same form in all the investigated theories of imprecise probabilities, even though these functionals are subject to different calculi in different theories. Moreover, equafions that express relationship between marginal, joint, and conditional measures of uncertainty are invariant across the whole spectrum of theories of imprecise probabilities. The tremendous diversity of possible uncertainty theories is thus compensated by their many commonalities.
Key words: uncertainty theories; fuzzy sets; information theories; generalized measures; imprecise probabilities.
1. GENERALIZED INFORMATION THEORY
A research program whose objective is to study uncertainty and uncertainty-based information in all their manifestations was introduced in the early 1990's under the name "generalized information theory" (GIT) (Klir, 1991). This research program, motivated primarily by some fundamental methodological issues emerging from the study of complex systems, is based on a two-dimensional expansion of classical, probability-based information theory. In one dimension, additive probability measures, which are inherent in classical information theory, are expanded to various types of nonadditive measures. In the other dimension, the formalized language of classical set theory, within which probability measures are
Uncertainty and Information: Emergence of.., 5
formalized, is expanded to more expressive formalized languages that are based on fuzzy sets of various types. As in classical information theory, uncertainty is the primary concept in GIT and information is defined in terms of uncertainty reduction. This restricted interpretation of the concept of information is described in GIT by the qualified term "uncertainty-based information."
Each uncertainty theory that is recognizable within the expanded framework is characterized by: (i) a peirticnlar formalized language (a theory of fuzzy sets of some particular type); and (ii) generalized measures of some particular type (additive or nonadditive). The number of possible uncertainty theories is thus equal to the product of the number of recognized types of fuzzy sets and the number of recognized types of generalized measures. This number has been growing quite rapidly with the recent developments in both fuzzy set theory and the theory of generalized measures. In order to fully develop any of these theories of uncertainty requires that issues at each of the following four levels be adequately addressed: (a) the theory must be formalized in terms of appropriate axioms; (b) a calculus of the theory must be developed by which the formalized uncertainty is manipulated within the theory; (iii) a justifiable way of measuring the amount of relevant uncertainty (predictive, diagnostic, etc.) in any situation formalizable in the theory must be found; and (iv) various methodological aspects of the theory must be developed.
Among the many uncertainty theories that are possible within the expanded conceptual framework, only a few theories have been sufficiently developed so far. By and large, these are theories based on various types of generalized measures, which are formalized in the language of classical set theory. Fuzzification of these theories, which can be done in different ways, has been explored only to some degree and only for standard fuzzy sets. One important result of research in the area of GIT is that the tremendous diversity of uncertainty theories emerging from the expanded framework is made tractable due to some key properties of these theories that are invariant across the whole spectrum or, at least, within broad classes of uncertainty theories.
One important class of uncertainty theories consists of theories that are viewed as theories of imprecise probabilities. Some of these theories are based on Choquet capacities of various orders (Choquet, 1953-54), especially capacities of order infinity (the well known theory of evidence) (Shafer, 1976), interval-valued probability distributions (Pan and Klir, 1997), and Sugeno y^-measures (Wang and Klir, 1992). While these theories are distinct in many respects, they share several common representations, such as representations by lower and upper probabilities, convex sets of probability distributions, and so-called Mobius representation. All
6 George J. Klir
representations in this class are uniquely convertible to one another, and each may be used as needed.
Another unifying feature of the various theories of imprecise probabilities is that two types of uncertainty coexist in each of them. These are usually referred to as nonspecificity and conflict. It is significant that well-justified measures of these two types of uncertainty are expressed by functionals of the same form in all the investigated theories of imprecise probabilities, even though these functionals are subject to different calculi in different theories. Moreover, equations that express relationship between marginal, joint, and conditional measures of uncertainty are invariant across the whole spectrum of theories of imprecise probabilities. The tremendous diversity of possible uncertainty theories is thus compensated by their many commonalities.
Uncertainty-based information does not capture the rich notion of information in human communication and cognition, but it is very useful in dealing with systems. Given a particular system, it is useful, for example, to measure the amount of information contained in the answer given by the system to a relevant question (concerning various predictions, retrodictions, diagnoses etc.). This can be done by taking the difference between the amount of uncertainty in the requested answer obtained within the experimental frame of the system (Klir, 2001a) in the face of total ignorance and the amount of uncertainty in the answer obtained by the system. This can be written concisely as
Information {A^ \S,Q) = Uncertainty {Aj^j,^ \ EF^.Q)
- Uncertainty ( 4 J S, 0
where • S denotes a given system • EFs denoted the experimental frame of system S • Q denotes a given question • Aj^^j, denotes the answer to question Q obtained solely within the
experimental frame EFs • As denotes the answer to question Q obtained by system S. This allows us to compare information contents of systems constructed within the same experimental frame with respect to questions of our interest.
The purpose of this paper is to present a brief overview of GIT. A comprehensive presentation of GIT is given in a forthcoming book by Klir (2005).
Uncertainty and Information: Emergence of.,, 7
2. CLASSICAL ROOTS OF GIT
There are two classical theories of uncertainty-based information, both formalized in terms of classical set theory. The older one, which is also simpler and more fundamental, is based on the notion oi possibility. The newer one, which has been considerably more visible, is based on the notion oi probability.
2.1 Classical Possibility-Based Uncertainty Theory
To describe the possibility-based uncertainty theory, let X denote a finite set of mutually exclusive alternatives that are of our concern (diagnoses, predictions, etc). This means that in any given situation only one of the alternatives is true. To identify the true alternative, we need to obtain relevant information (e.g. by conducting relevant diagnostic tests). The most elementary and, at the same time, the most fundamental kind of information is a demonstration (based, for example, on outcomes of the conducted diagnostic tests) that some of the alternatives in X are not possible. After excluding these alternatives from X, we obtain a subset E oi X. This subset contains only alternatives that, according to the obtained information are possible. We may say that alternatives in E are supported by evidence.
To formalize evidence expressed in this form, the characteristic function of set E, rji, is viewed as a basic possibility function. Clearly, for each x G X, r/Xx) = 1 when x is possible and r^ix) = 0 when x is not possible. Possibility function applicable to all subsets of X, Post<, is then defined by the formula
Pas J, (A) = max r , (x) (1) xeA
for all A ^ X. It is indeed correct to say that it is possible that the true alternative is in set A when A contains at least one alternative that is also contained in set E.
Given a possibility function Pos^ on the power set of X, it is useful to define another function, Necj.;, to describe for each A c X the necessity that the true alternative is in ^ . Clearly, the true alternative is necessarily in A if and only if it is not possible that it is in A , the complement of A. Hence,
Nee J, (A) = l- Posj, (A) (2)
f o r a l l ^ e X . The question of how to measure the amount of uncertainty associated
with a finite set E of possible alternatives was addressed by Hartley (1928).
8 George J. Klir
He showed that the only meaningful way to measure this amount is to use a functional of the form
or, alternatively, c log/, | E \ where | E \ denotes the cardinality of E, and b and c are positive constants. Each choice of values b and c determines the unit in which the uncertainty is measured. Requiring, for example, that
clog^2 = l ,
which is the most common choice, uncertainty would be measured in bits, One bit of uncertainty is equivalent to uncertainty regarding the truth or falsity of one elementary proposition. Choosing conveniently Z) == 2 and c = 1 to satisfy the above equation, we obtain a unique functional, //, defined for any possibility function, Posj.:, by the formula
H(Pos,) = \og,\E\. (3)
This functional is usually called a Hartley measure of uncertainty. Its uniqueness was later proven on axiomatic grounds by Renyi (1970).
Observe that the Hartley measure satisfies the inequalities
0<H{PoSj,;)<\og^\X\
for any E ^ X and that the amount of information, I{Posi?), in evidence expressed by function Posu is given by the formula
I{Pos,:) = \og,\X\-\og,\E\. (4)
It follows from the Hartley measure that uncertainty associated with sets of possible alternatives results from the lack of specificity. Large sets result in less specific predictions, diagnoses, etc., than their smaller counterparts. Full specificity is obtained when only one alternative is possible. This type of uncertainty is thus well characterized by the term nonspecificity.
Consider now two universal sets, X and 7, and assume that a relation R Q, X X Y describes a set of possible alternatives in some situation of interest. Consider further the sets
Uncertainty and Information: Emergence of... 9
R^ ={x^ X\ {x^y) G R for some y ^Y]
Ry -{y ^Y\ {x^y) e R for some XG X}
which are usually referred to as projections of R on sets X, 7, respectively. Then three distinct Hartley measures are applicable, which are defined on the power sets of X, 7, and XxY. To identify clearly which universal set is involved in each case, it is useful (and a common practice) to write H(X), H{Y), H{X,Y) instead of
H{Pos,^\ H{Pos,^\ H(Pos,)
respectively. Functionals H(X)=\og2\R^^ and i/(y)^log2 |/?rl are called simple (or marginal) Hartley measures, while functional H(X,Y)=log2\R\ is called a joint Hartley measure.
Two additional functionals are defined,
H(X\Y) = log,l^ and H(Y \X) = l o g , ^ (5)
which are called conditional Hartley measures. Observe that the ratio |i?|/|/?y| in H(X\Y) represents the average number of elements of X that are possible alternatives under the condition that a possible element of Y is known. This means that H(X \Y) measures the average nonspecificity regarding possible choices from Xfor all possible choices from Y. Function H(X\Y) has clearly a similar meaning with the roles of sets X and Y exchanged. Observe also that the conditional Hartley measures can be expressed in terms of the joint measures and the two simple Hartley measures:
H(X I Y) = H(XJ) - H(Y) and H(Y \ X) = H(XJ) - H(X). (6)
If possible alternatives from X do not depend on selections from 7, and visa versa, then R = Xx Yand the sets Rx and Ry are called noninteractive. Then, clearly,
H{X\Y) = H(X) and H{Y\X) = H{Y), (7)
/ / ( X , Y) = H(X) + H(Y). (8)
In all other cases, when sets Rx and Ry are interactive, these equations become the inequalities
George J. Klir
H{X\Y)<H{X) and H{Y\X)<H{Y),
H(X,Y)<HiX) + H{Y). (9)
(10)
In addition, the functional
T„(X,Y) = H{X) + H{Y) - H(X,Y), (11)
which is usually called an information transmission, is a useful indicator of the strength of constraint between sets Rx and Ry.
The Hartley measure is applicable only to finite sets. Its counterpart for subsets of the ^-dimensional Euclidean space E" (« > 1) was not available in the classical possibility-based uncertainty theory. It was eventually suggested (as a byproduct of research on GIT) by Klir and Yuan (1995b) in terms of the functional
HL(PoSi;) = minclog^ 11(1 +ju(E,)) + juiE)-YljuiE,_) i=\
where E, T, Eu, and ju denote, respectively, a convex subset of E", the set of all isometric transformations from one orthogonal coordinate system to another, the /-th projection of E in coordinate system /, and the Lebesgue measure, and b and c are positive constants whose choice defines a measurement unit. This functional, which is usually referred to as Hartleylike measure, was proven to satisfy all mathematical properties that such a measure is expected to satisfy (Klir and Wierman, 1999; Ramer and Padet, 2001).
Let a measurement unit for the Hartley-like measure be defined by the requirement that HL{Posi)=\ when £" is a closed interval of real numbers of length 1 in some assumed unit of length. That is, we require that
clog, 2 1.
It is convenient to choose c = 1 and ^ = 2 to satisfy this equation. Then,
HL(PoSj,) = minclog^ YI(\ + JU(E,_)) + JU(E)-YIJU{E,^) i=]
(12)
Uncertainty and Information: Emergence of... 11
In these units, a unit square has uncertainty 2, a unit cube has uncertainty 3, etc. For any universal set X (a convex subset of E''), a normalized Hartleylike measure, NHL is defined for each convex subset E of Xby the formula
^ffl<Po.„) = M £ 2 i i . (,3) HL{Pos^)
Clearly, NHL is independent of the chosen unit and 0 < NHL(Posii) < 1.
2.2 Classical Probability-Based Uncertainty Theory
The second classical uncertainty theory is based on the notion of classical probability measure (Halmos, 1950). As is well known, the amount of uncertainty in evidence expressed by a probability distribution, /?, on a finite set X of considered alternatives is measured (in bits) by the functional
S(p(x) \xeX) = -J]p(x)log, p{x). (14) XEX
This functional was introduced by Shannon (1948) and it is usually referred to as Shannon entropy. Its uniqueness has been proven in numerous ways (Klir and Wierman, 1999).
For probabilities on XxY, three types of Shannon entropies are recognized: joint, marginal, and conditional. A simplified notation to distinguish them is commonly used in the literature: S(X) instead of S(p(x)\xeX), S(X,Y) instead of S(p(x,y) \xeX,yGY), etc. Conditional Shannon entropies are defined in terms of weighted averages of local conditional entropies as
S{x\Y) = -Y,Py(y)Zp(^\y)^''S2P(x\y), (15) yeY XGX
S(Y IX) = -Y^p,(x)Y^p(y I x)log2 p{y \ x). (16) xeX yeY
As is well known (Klir and Wierman, 1999), equations and inequalities (2) -(6) for the Hartley measure have their exact counterparts for the Shannon entropy. For example, the counterparts of (2) are the equations
S(X\Y) = S(XJ)-S(Y) and S(Y\X) = S(XJ)-S(X). (17)
12 George J. Klir
Moreover,
T,iX,Y) = S{X) + S{Y)-S{X,Y) (18)
is ii^Q probabilistic information transmission (the probabilistic counterpart of (7)).
It is obvious that the Shannon entropy is applicable only to finite sets of alternatives. At first sight, it seems suggestive to extend it to probability density functions, q, on E (or, more generally, on E", n> 1), by replacing in Eq.(14) p with q and the summation with integration. However, there are several reasons why the resulting functional does not qualify as a measure of uncertainty: (i) it may be negative; (ii) it may be infinitely large; (iii) it depends on the chosen coordinate system; and most importantly, (iv) the limit of the sequence of its increasingly more refined discrete approximations diverges (Klir and Wierman, 1999). These problems can be overcome by the modified functional
S\q{x\q\x)\x^E^=\q{x)\og,^dx^ (19) •'R q\x)
which involves two probability density functions, q and q\ Uncertainty is measured by 5" in relative rather than absolute terms.
When q in (19) is a joint probability density function on E^ and q^ is the product of the two marginals of ^, we obtain the information transmission
= L L?(^9>^)log2— —dxdy. ^^ (ix(^)'qY(y)
This means that (20) is a direct counterpart of (18).
2.3 Relationship Between the Classical Uncertainty Theories
It is fair to say that the probability-based uncertainty theory has been far more visible than the one based on possibility. Moreover, the distinction between the two classical theories was concealed in the vast literature on probability-based uncertainty theory, where the Hartley measure is almost routinely viewed as a special case of the Shannon entropy. This view, which is likely a result of the fact that the value of the Shannon entropy for the
Uncertainty and Information: Emergence of... 13
uniform probability distribution on some set is equal to the value of the Hartley measure for the same set, is ill-conceived. Indeed, the Hartley measure is totally independent of any probabilistic assumptions. Furthermore, given evidence expressed by a possibility function Posu, any probability measure Pro, not only the one representing the uniform distribution, is consistent with Posu when
Pro{a)<Pos,{A) (21)
for all A (^X. Function Posu captures thus a set of probability measures and we have no basis for choosing any one of them.
The fundamental distinction between the two classical theories, both conceptual and formal, was correctly recognized by Kolmogorov (1965), who refers to the possibility-based uncertainty theory as "the combinatorial approach to the quantitative definition of information" and makes a relevant remark: "Discussions of information theory do not usually go to this combinatorial approach at any length, but I consider it important to emphasize its logical independence of probabilistic assumptions."
In spite of Kolmogorov's clear discussion of the fundamental distinctions between the two classical uncertainty theories, many information theorists still continue in their writings to subsume the Hartley measure under the Shannon entropy or they dismiss it altogether. One rare exception is Renyi (1970), who developed an axiomatic characterization of the Hartley measure and proved its uniqueness.
It is shown in Sec. 2.1 that the type of uncertainty measured by the Hartley functional is well described by the term nonspecificity. One way of getting insight into the type of uncertainty measured by the Shannon entropy is to rewrite Eq. (14) as
S{p{x) I X e X ) = - ^ j9(x)log2 xeX
1 - Z M J ^ ) y^x
(22)
The term
Con{x) = Y^p{y) y^X
in Eq. (22) represents clearly the total evidential claim pertaining to elements that are distinct from x. That is, Con{x) expresses the sum of evidential
14 George J. Klir
claims that fully conflict with the one focusing on x. Clearly, Con{x) G [0,1] for each x e X. The function
CON{x) = -log2[l - Con(x)]
which is employed in Eq. (22), is monotonic increasing with Con(x) and extends its range from [0,1] to [0,oo]. Hence, it represents the same conflict in a different scale. The choice of the logarithmic function is a result of axiomatic requirements for the Shannon entropy.
It follows from these facts and from the form of Eq. (22) that the Shannon entropy is the mean (expected) value of conflict among evidential claims within a given probability distribution function. This is another demonstration that the two classical uncertainty theories deal with distinct types of uncertainty. It is significant that in every generalization of the classical theories both uncertainty types coexist.
3. JMATHEJMATICAL FRAJMEWORK FOR GIT
GIT is based on (i) generalizing classical measure theory (Halmos, 1950) by abandoning the requirement of additmty\ and (ii) generalizing classical set theory (Stoll, 1961) by abandoning the requirement that sets have sharp boundaries. The former generalization results in the theory of generalized measures, which are also commonly called fuzzy measures for some historical reasons of little significance (Denneberg, 1994; Pap, 1995; Wang and Klir, 1992). The latter generalization results in the theory of fuzzy sets (Klir and Yuan, 1995a, 1996; Yager et al., 1987). The following is an overview of these two generalizations.
3.1 Generalized Measure Theory
Classical measures are generalized by abandoning the requirement of additivity. However, for utilizing the generalized measures to represent uncertainty, it is essential to only replace additivity with a weaker requirement of monotonicity. The following is a formal definition of monotone measures, which represent one dimension of the mathematical framework for GIT.
Given a universal set X and a non-empty family C of subsets of X that contains 0 and X and has an appropriate algebraic structure (cr-algebra, ample field, power set, etc.), a regular monotone measure, g, on {X, C) is a function
