## Tuesday, 21 February 2012

### Call for Papers for Inaugural Issue of International Journal of Machine Intelligence and Sensory Signal Processing

International Journal of Machine Intelligence and Sensory Signal Processing (IJMISSP)

Call for Papers for Inaugural Issue

The International Journal of Machine Intelligence and Sensory Signal Processing (IJMISSP) provides a forum for the publication of papers on the most recent developments in the area of devices, circuits and systems applied to machine intelligence and sensory signal processing problems. IJMISSP is a fully refereed international journal that is targeted at professionals, engineers, academics and researchers working in the field of machine intelligence.

IJMISSP publishes the following five types of articles: (1) original regular papers (max. 7000 words), (2) short papers (max. 2000 words), (3) technical reports (max. 5000 words), (4) research reviews (max. 7000 words), and (5) case studies (max. 3000 words). Original regular papers are full innovative findings that make significant theoretical and/or experimental contributions to knowledge in the field by using various theoretical and methodological approaches.  Short papers present interesting and significant results that warrant rapid publication. A technical report is a forum to raise and discuss open research problems; often such reports introduce new methods and techniques applied to unsolved research problems.  Research reviews are perceptive and prudently constructed articles that conceptualise research areas, blend previous innovative findings, advance the understanding of the field, and identify and develop future research directions. Case studies present practical implementation of theoretical ideas in real-time applications. Authors are welcome to submit manuscripts that qualify as any of the five categories.

Topics of interest include, but are not limited to, the following areas:

 Pattern recognition theory, fuzzy systems and implementations

 Neuron-inspired circuit models and memories

 Intelligent semiconductor devices and circuits

 Bio-inspired visual and speech processing circuits and systems

 Bionic eye/ear, electronic arm- and leg-based hardware systems

 Haptics, emotion and perception systems

 Neuron-inspired cellular evolutionary hardware

 Gesture and tracking in multimedia hardware

 Interactive AI gaming hardware

 Multisensory information fusion hardware architectures

 Neuron-inspired real-time pattern recognition hardware

 Bio-inspired informatics and systems hardware

 AI hardware in health informatics and knowledge discovery

 Applications in medical imaging and informatics

 Intelligent sensory signal processing circuits and applications

 AI hardware in engineering and science applications

Important Dates

Review to decision time: 1-3 months

We are now accepting submissions.  Papers will be reviewed on a rolling basis; early submission is therefore encouraged.

Notes for Prospective Authors

Submitted papers should not have been previously published nor be currently under consideration for publication elsewhere. (N.B. Conference papers may only be submitted if the paper was not originally  copyrighted and if it has been completely re-written). All papers are refereed through a peer review process. A guide for authors, sample copies and other relevant information for submitting papers are available on the Author Guidelines page. All papers must be submitted online. To submit a paper, please go to Online Submissions of Papers. If you experience any problems submitting your paper online, please contact  submissions@inderscience.com, describing the exact problem you experience. (Please include in  your email the title of the Journal).

A.P. James

Editor in Chief (IJMISSP)

Email: apj@ieee.org

## Sunday, 19 February 2012

### Nash's letter to NSA

At the following link -and at the attached file-  you may find a recently declassified correspondence between John Nash and the NSA from January 1955, where Nash makes the distinction between polynomial time and exponential time,conjectures that there are problems that cannot be solved  and proposes a cryptosystem of his own design...

http://agtb.wordpress.com/2012/02/17/john-nashs-letter-to-the-nsa

....simply awesome,

--
V/R,

Nikos Petrakos

MSc in Computer Science  & MSc in Applied Mathematics, 2009,NPS,USA,

npetrako@gmail.com
npetrako@snd.edu.gr

## Saturday, 18 February 2012

### Aida Bulucea (WSEAS)

Dear WSEAS

THANK YOU VERY VERY MUCH!!!!!!!!!!!

I am convinced that our WSEAS Forum will help a lot the scholars all across the world! Furthermore, this beautiful initiative shows doubtless the good intention and open vision of WSEAS Team!!!

My Faithful Regards,
Aida Bulucea

## Friday, 17 February 2012

### My new book: STOCHASTIC AND FUZZY MODELS

TITLE OF THE BOOK: Stochastic and Fuzzy Models
in Mathematics Education, Artificial Intellgence and Management

by Michael Gr. Voskoglou

http://www.lap-publishing.com , E-mail: info@lap-publishing.com
Printed also in U.S.A. and U.K.
ISBN:978-3-8465-2821-1

BOOK REVIEW: By Prof. I. Subbotin, LA-USA
( taken from http://azn.com/3846528218 )

A SOLID AND INFORMATIVE BOOK!
As the author notes mathematics does not explain the natural behaviour of
an object, it simply describes it. This description however is so much
effective, so that an elementary mathematical equation can describe simply
and clearly a relation, that in order to be expressed with words could
need entire pages. Inspired from this Michael Gr. Voskoglou, Professor of
Mathematical Sciences at the Graduate Technological Educational Institute
of Patras, Greece, worked during the last two decades (1991-2011) on
creating mathematical (stochastic and fuzzy) models for the better
description and understanding of several processes appearing in the areas
of Mathematical Education, Artificial Intelligence (Case-Based Reasoning)
and Management. His book Stochastic and fuzzy models, edited by Lambert
Publishing on October 2011, contains the main topics of this research.
Stochastic and fuzzy models consist of five chapters. In the first chapter
the main mathematical tools being necessary for the construction of the
mathematical models developed in the book, are presented including topics
from theories of Finite Markov Chains and of Fuzzy Sets. These topics are
discussed in a practical way accompanied with applications to Management
and Economics Among these applications a Markov chain model is developed
for the description of the decision making process and decision making in
a fuzzy environment is also studied. A method of evaluating the fuzzy data
obtained from a real situation, which is also presented in this
introductory chapter, plays a central role in constructing the fuzzy
models developed in the book.
The learning activity is a principal component of human cognition. In the
second chapter of Stochastic and fuzzy models three mathematical models
are developed for the description of the learning process in classroom
having as a common point of reference the general problem-solving
framework for learning introduced by Voss (International Journal of
Educational Research, 11, 607-622, 1987). The first of these models,
through which the teachers can get useful quantitative information of
their students leaning capacities, is based on the concept of conditional
probability. In the second model a finite absorbing Markov chain is
introduced on the steps of the learning process appeared in Vosse's
framework. In the third model the main steps of the learning process are
represented as fuzzy subsets of a finite set of linguistic labels
characterizing the students' performance to each step and the total
possibilistic uncertainty (Klir G. J, Fuzzy Sets and Systems, 74, p.28,
1995) is used as a measure of students' learning capacities. In fact, the
amount of information obtained by an action can be measured by the
reduction of uncertainty resulting from this action. Accordingly students'
uncertainty during the leaning process is connected to students' capacity
in obtaining relevant information. Classroom applications of the above
three models in the area of learning mathematics are also presented
illustrating their use and usefulness in practice.
Problem solving (PS) is a principal component of Mathematics Education
(and not only). In the first part of the third chapter of Stochastic and
fuzzy models the author presents a review of the PS process from the time
that Polya (How to solve it, Princeton Univ. Press, Princeton, 1945)
presented his first ideas on the subject until today and investigates the
role of the problem in learning mathematics and the future perspectives of
PS in Mathematics Education. A particular emphasis is given to analogical
PS , where the major focus has been on the reuse of a problem solved in
the past, what is called the mapping problem: Finding a way to transfer,
or map, the solution of an identified analogue (called source, or base
problem), to the present problem (called target problem).
In the second part of this chapter two mathematical models are developed
for the representation of the PS process and applications of these models
in classroom are presented. The first is a stochastic model constructed by
introducing a Markov chain on the stages of Schoenfeld's expert
performance model for PS (American Mathematical Monthly, 87, 794-805, 1980).
While early work on PS focused on describing the PS process, more recent
investigations have focused on identifying attributes of the problem
solver that contribute to successful PS. Carlson and Bloom (Educational
studies in Mathematics, 58,45-75, 2005) drawing from the large amount of
literature related to PS developed a broad taxonomy to characterize major
PS attributes that have been identifying as relevant to PS success. This
taxonomy gave genesis to their Multidimensional Problem-Solving Framework
(MPSF) that includes four phases: Orientation, Planning, Executing and
Checking.. Prof. Voskoglou introduces a fuzzy model for the PS process by
representing the phases of MPSF as fuzzy subsets of a set of linguistic
labels and by using the total possibilistic uncertainty as a measure of
students PS capacities in a way similar to that described above for the
process of learning. This model has the additional advantage of giving to
the user the possibility to study the combined results of performance of
two or more student groups during the solution of the same problems, or
alternatively of the same group during the solution of different problems.

Broadly construed Case-Based Reasoning (CBR) is the process of solving new
problems based on the solution of past problems. The CBR systems expertise
is embodied in a collection (library) of past cases rather, than being
encoded in classical rules. Each case typically contains a description of
the problem plus a solution and/or the outcomes. A case-library can be a
powerful corporate resource allowing everyone in an organization to tap in
the corporate library, when handling a new problem. CBR allows the
case-library to be developed incrementally, while its maintenance is
relatively easy and can be carried out by domain experts. As an
intelligent-systems' method CBR enables information managers to increase
efficiency and reduce cost by substantially automating processes such as
diagnosis, scheduling and design. In the first part of the fourth chapter
of Stochastic and fuzzy models the author gives a detailed account of the
history and methodology, of the tools and applications and of the
development trends of CBR. In the second part a mathematization of the CBR
process is developed. First a Markov chain is introduced on the steps of
the CBR process Through this a measure is obtained for the effectiveness
of a CBR system Second a fuzzy model is constructed for the
representation of a CBR system following the general lines of the
development of the fuzzy models for learning and PS that we have already
described. Examples are also presented illustrating the application of
these models to real situations.
Mathematical modelling appears today as a dynamic tool for teaching
mathematics, because it connects mathematics with our everyday life and
gives to students the possibility to understand its usefulness in
practice; it has also the potential to enhance students' performance in
mathematics in general. In the last (5th) chapter of Stochastic and fuzzy
models Prof. M. Voskoglou gives an account of the history of mathematical
modelling as a teaching method of mathematics and presents the models used
by mathematical educators and researcher through the years for this
purpose. Next two models are developed for the process of mathematical
modelling in classroom. The first is a stochastic model obtained by
introducing a finite Markov chain having as states the stages of the
mathematical modelling process. In the second model the main stages of the
mathematical modelling process ((mathematization, solution and validation
of the model) are represented as fuzzy subsets of the set of linguistic
labels characterizing students' performance in classroom that we have
already mentioned above. A generalized form of Shannon's entropy measuring
the uncertainty in probability distributions expressed in terms of
Dempster-Shafer mathematical theory of evidence ((Klir G. J, Fuzzy Sets
and Systems, 74, p.20, 1995) is used as a measure of students' model
building capacities. Classroom applications are also presented to
illustrate the use in practice of the models (stochastic and fuzzy)
developed in this chapter.
A list of 55 author's publications related to its topics appears at the
end of the book.
The book Stochastic and fuzzy models is a welcome addition to the
literature in this subject. The exposition is very clear and
comprehensive, and it does not assume much beyond a basic knowledge of
probability theory and linear algebra. The book is addressed to
scientists, researchers and mathematical educators. It should also be of
background knowledge needed to become successful researchers in this
vibrant area.

Igor Ya. Subbotin, PhD,
Professor of Mathematics, Lead Faculty for mathematics programs,
Department of Mathematics and Natural Sciences, College of Letters and
Sciences,
National University, 5245 Pacific Concourse Drive,
Los Angeles, CA 90045-6904
isubboti@nu.edu<mailto:isubboti@nu.edu>

### Online Panel on Uncertainty

Following is a brief status report on the Online Panel on Uncertainty. Currently, the members of the Panel are: G. Coletti (coherent conditional probability approach), D. Dubois/H. Prade (an uncertainty theory), G. Klir (generalized information theory), H. Nguyen (random set approach to uncertainty), S. Li (nonlinear-mathematics-based theory of uncertainty), B. Liu (a theory of uncertainty), S. Liu (grey systems), N. Singpurwalla (Bayesian approach to uncertainty), A. Skowron (rough sets) and L. Zadeh (generalized theory of uncertainty). Members of the Panel are spokespersons for principal approaches to uncertainty. The Panel will have three rounds. In the first round, the panelists will post summaries of their approaches to uncertainty, aimed at explaining the principal concepts and ideas which underlie their theories and their relation to other approaches to uncertainty. In the second round, the summaries will be opened for comments and discussion by the panelists and members of the BISC Group. In the third round, the panelists will respond to comments and questions. It is planned to publish the proceedings of the Online Panel on Uncertainty as a special issue of a journal or as a book.

So far as I know, there is no precedent for an online panel like the Online Panel on Uncertainty. I believe that the Online Panel on Uncertainty will serve an important purpose--a purpose similar to that of the World Conference on Uncertainty which was advocated by Hans Kuijper. The first round is expected to be completed soon.

Regards to all.

Lotfi
--  Lotfi A. Zadeh  Professor in the Graduate School Director, Berkeley Initiative in Soft Computing (BISC)   Address:  729 Soda Hall #1776 Computer Science Division Department of Electrical Engineering and Computer Sciences University of California  Berkeley, CA 94720-1776  zadeh@eecs.berkeley.edu  Tel.(office): (510) 642-4959  Fax (office): (510) 642-1712  Tel.(home): (510) 526-2569  Fax (home): (510) 526-2433  URL: http://www.cs.berkeley.edu/~zadeh/

## Thursday, 16 February 2012

### A new e-journal on applications of fuzzy sets

International Journal of Applications of Fuzzy Sets (IJAFS)
ISSN 2241-1240
Graduate Technological Educational Intitute (T.E.I.), Patras, Greece

GENERAL INFORMATION
The International Journal of Applications of Fuzzy Sets (IJAFS) is a
peer-reviewed ELECTRONIC journal published through the site of the
Graduate T. E. I. of Patras, Greece. The journal is devoted to publication
of original research results on a wide specter of problems on Fuzzy Sets,
Fuzzy Systems and Fuzzy Logic. The journal will also publish critical
survey articles, comprehensive review articles giving details of research
progress made recently in a particular area, book reviews, dissertations'
abstracts etc. We would like to invite you to submit manuscripts of your
original papers (up to 25 pages) for possible publication in IJAFS.
Authors are requested to submit their papers electronically to
submission see "Template-Instructions to the authors" in the site of the
journal.
Aims and scope: The notion of fuzzy sets was introduced by Zadeh in 1965
(Information and Control, 8, 338-353) in response to have a mathematical
representation of situations in everyday life in which definitions have
not clear boundaries (e.g. high mountains, good players, tall people,
etc). Since then the relevant theory was expanded rapidly, to cover almost
all sectors of human activities. Today one can see fuzzy sets theory both
as a formal theory which embraced classical mathematical areas such as
algebra, graph theory, topology, etc and as a powerful modelling tool that
can cope with a large function of uncertainties in real life situations.
IJAFS is a new high quality international academic journal devoted to
publication of original research results on a wide specter of problems on
Fuzzy Sets, Fuzzy Systems and Fuzzy Logic with emphasis on applications to
all sectors of human activities including (but not limited to)
applications of Fuzzy Sets on Algebra, Graph Theory and Discrete
Mathematics in general, Geometry, Topology, Education, Natural, Life and
Social Sciences, Design Sciences, Management and Economics, Information
Theory, Decision Analysis, Engineering, Materials' Technology, Medicine
and other Health Sciences, Systems Science, Telecommunications, Traffic
and Aircraft Control, Robotics, Computer Science and Expert Systems,
Learning Theories, Problem Solving and Modeling, Artificial Intelligence,
Pattern Recognition and Clustering, etc. Theoretical contributions on
Fuzzy Sets formal theory with promising applicable character are also
welcome.
Frequency: 1 issue per year
Cost of Publication: There is NO cost for publication and the e-journal is
free to everybody (OPEN ACCESS).
Publication policy: After its acceptance for publication a paper is
published IMMEDIATELY. This means that each volume of the journal is
completed progressively by the articles submitted and accepted for
volume is added at the end of the year.

--
Michael Gr. Voskoglou (B.Sc., M.Sc., M.Phil., Ph.D)
Professor of Mathematical Sciences
School of Technological Applications
26334 Patras- Greece
E-mail: <voskoglou@teipat.gr> ; <mvosk@hol.gr>
URL: http://eclass.teipat.gr/eclass/courses/523102
Tel.-Fax
: 00302610328631, Mobile: 00306978600391,

The concept of truth has a position of centrality in logic. In
discussing the concept of truth, Tarski said "Snow is white if snow is
white." At first, this appears to be a tautology. What Tarski meant is that
the proposition "snow is white," is true if in fact snow is white. Here is
a more realistic version of Tarski's assertion, in the form of a question.
Given that usually snow is white, what is the truth-value of the
proposition "snow is white?" How would you answer this question?

Regards to all.

Lotfi

--
Director, Berkeley Initiative in Soft Computing (BISC)

729 Soda Hall #1776
Computer Science Division
Department of Electrical Engineering and Computer Sciences
University of California
Tel.(office): (510) 642-4959
Fax (office): (510) 642-1712
Tel.(home): (510) 526-2569
Fax (home): (510) 526-2433

### From Daniel Schwartz

From: Daniel Schwartz

Dear

I first learned about fuzzy sets theory from my doctoral advisor, George
Lendaris, at Portland State University.  George had been a student of Lotfi
at UC Berekely.  I wrote a dissertation in fuzzy logic and Lotfi was the
external examiner.  That was 1981.

Subsequently I continued to work in fuzzy logic and approximate reasoning as
an Assistant Professor in Systems Science at what is now Binghamton
University in New York, and then starting in 1984 at the Computer Science
Department of Florida State University.  I am still employed at FSU, now as
an Associate Professor, and, while my research has taken me down numerous
time to time.

My contributions in this area include:

1. Axioms for a theory of semantic equivalence.  {\it Fuzzy Sets and
Systems}, 21 (1987) 319--349.

2. Dynamic reasoning with qualified syllogisms, {\it Artificial
Intelligence}, 93, 1-2 (1997) 103--167.

3. Layman's probability theory: a calculus for reasoning with linguistic
likelihood, {\it Information Sciences}, 126, 1-4 (2000) 71--82.

4. Agent-oriented epistemic reasoning: subjective conditions of knowledge
and belief, {\it Artificial Intelligence}, 148, 1-2 (2003) 177-195.

I am deeply grateful to Lotfi for the advice an assistance he has given me
at various crucial stages in my career.

Best wishes to all,

--Dan Schwartz

************************************************************************
Dr. Daniel G. Schwartz                            Office    850-644-5875
Dept. of Computer Science, MC 4530                CS Dept   850-644-4029
Florida State University                          Fax       850-644-0058
Tallahassee, FL 32306-4530                        schwartz@cs.fsu.edu
U.S.A.                                   http://www.cs.fsu.edu/~schwartz
************************************************************************

## Wednesday, 15 February 2012

### From Professor Aida Bulucea

This is one of the marvellous moments of my life, your message is a Heaven dream
I love WSEAS, and your gentleness and sensitive generosity make me the most happy human in this world. I almost cannot write because of my warm emotion.
My mind cannot find the words for expressing the feelings of my soul. I hope do never disappoint you
Thank you so much!!!!
My Faithful Regards,
Aida

*********************************************************************
Berkeley Initiative in Soft Computing (BISC)
*********************************************************************
Dear Members of the BISC Group:

The concept of uncertainty is of fundamental importance. Of particular importance is the role of uncertainty in the realm of decision analysis. To improve our ability to make rational decisions in an environment of uncertainty and imprecision we need a better understanding of existing approaches to management of uncertainty. To this end, Hans Kuijper and others have suggested holding a World Conference on Uncertainty. This suggestion has merit but its implementation will be a formidable task.

A much less ambitious suggestion is put forward in the following. Basically, my suggestion is to hold an Online Panel on Uncertainty. Such a panel and the World Conference would not be mutually exclusive.

The Panel would consist of spokesmen for principal approaches to management of uncertainty, among them probability theory, fuzzy logic, random set theory, rough sets, grey systems, etc. Each panelist would submit a 2,000 or less word statement summarizing his/her approach. These statements will be posted and opened for discussion by members of the BISC Group, including the panelists. The first round will be followed by a second round. The statements together with a selection of discussions will be published as a special issue of an appropriate journal or as a book. You are invited to offer suggestions regarding panelists, including yourself. To get the ball rolling, following is my statement. Comments will be posted after all statements are in. Your comments regarding the panel are welcome.

Regards.

Preamble
There is a deep-seated tradition in science and engineering of turning to probability theory when one is faced with a problem in which uncertainty plays a significant role. This tradition was justified when there were no tools for dealing with uncertainty other than those provided by probability theory. Today, this is no longer the case. And yet, the tradition persists. What is widely unrecognized is that uncertainty has many facetsthat cannot be dealt with adequately through the use of standard probability theory. The prevailing view, let us call it a sufficienist view, is that probability theory is all that is needed to deal with any kind of uncertainty. In the words of Dennis Lindley, a prominent Bayesian, the credo of sufficienists may be stated as:

The only satisfactory description of uncertainty is probability. By this I mean that every uncertainty statement must be in the form of a probability; that several uncertainties must be combined using the rules of probability; and that the calculus of probabilities is adequate to handle all situations involving uncertainty...probability is the only sensible description of uncertainty and is adequate for all problems involving uncertainty. All other methods are inadequate...anything that can be done with fuzzy logic, belief functions, upper and lower probabilities, or any other alternative to probability can better be done with probability. (Lindley 1987)
There is another school of thought, call it insufficienism. In relation to probability theory, I became an insufficienist even before I wrote my first paper on fuzzy sets in 1965. This is what I had to say in a paper published in 1962 entitled "From circuit theory to system theory."

There are some who feel that this gap reflects a fundamental inadequacy of conventional mathematics--the mathematics of precisely-defined points, functions, sets, probability measures, etc.--for coping with the analysis of biological systems, and that to deal effectively with such systems, which are generally orders of magnitude more complex than man-made systems, we need a radically different kind of mathematics, the mathematics of fuzzy or cloudy quantities which are not describable in terms of probability distributions. Indeed, the need for such mathematics is becoming increasingly apparent even in the realm of inanimate systems, for in most practical cases the a priori data as well as the criteria by which the performance of a man-made system is judged are far from being precisely specified or having accurately known probability distributions. (Zadeh 857)

I was brought up on probability theory. My first paper (1949) was entitled "Probability criterion for the design of servomechanisms." My second paper (1950) was entitled "An extension of Wiener's theory of prediction." Many other papers which dealt with probability theory followed. As I acquired more experience in applying probability theory to real-world problems, I became increasingly aware of its limitations. Based as it is on bivalent logic, probability theory is not a good fit to reality--a reality in which most classes have unsharp boundaries--a reality in which almost everything is a matter of degree. Gradually, I evolved from being a sufficienist to being an insufficienist, believing that standard probability theory provides no effective tools for dealing with information which is described in a natural language.
An issue which has a close bearing on the sufficiency of probability theory is the relationship between probability theory and fuzzy set theory. This issue has been, and to some degree still is, an object of many discussions and debates--debates going back to the publication of my first paper on fuzzy sets in 1965. There are many different views, centering principally on random sets, conditional probabilities and voting models. (Zadeh 1995) These views are rooted in a basic property of fuzzy sets which is described in my 1965 paper, namely, a convex combination of crisp sets is a fuzzy set. What does not follow is that fuzzy set theory is subsumed by probability theory or vice versa. The concept of a fuzzy set is distinct from the concept of probability, and fuzzy set theory is distinct from probability theory. Consider a simple example. I am in an art gallery with Fay, my wife. I point to a painting and ask Fay to indicate, on the scale from 0 to 10, how much she likes the painting. Her answer, say 7, may be interpreted as the grade of membership of the painting in question in the set of paintings which Fay likes. Note that no probabilities are involved. Another simple example. I am filling out a medical questionnaire. The question is: On the scale from 0 to 10, how severe is your pain? Again, no probabilities and no voting models are involved. What underlies these examples is the remarkable human capability to graduate perceptions.

A real test of the effectiveness of an approach to uncertainty is the capability to solve problems which involve different facets of uncertainty. Here are a few such problems which are stated in a natural language.

1. Most Swedes are tall. What is the average height of Swedes?
2. Probably John is tall. What is the probability that John is short? What is the probability that John is very short? What is the probability that John is not very tall?
3. Usually, most United flights from San Francisco leave on time. I am scheduled to take a United flight from San Francisco. What is the probability that my flight will be delayed?
4. X is a real-valued random variable. Usually X is much larger than approximately a. Usually X is much smaller than approximately b. What is the probability that X is approximately c, where c is a number between a and b?
5. A and B are boxes, each containing 20 balls of various sizes. Most of the balls in A are large, a few are medium and a few are small. Most of the balls in B are small, a few are medium and a few are large. The balls in A and B are put into a box, C. What is the number of balls in C which are neither large nor small?
6. Usually Robert leaves his office at about 5 pm. Usually it takes Robert about an hour to get home from work. At what time does Robert get home?
7. A box contains about 20 balls of various sizes. There are many more large balls than small balls. What is the probability that a ball drawn at random is small?
8. If X is small then usually Y is small
If X is medium then usually Y is large
If X is large then usually Y is small.

What is Y if X is neither small nor large?

In essence, my thesis is that standard probability theory is in need of generalization aimed at enhancing its ability to deal with real-world problems. Such generalization was described in my 2002 paper, "Toward a perception-based theory of probabilistic reasoning with imprecise probabilities." My 2002 paper was a stepping stone to my 2006 paper, "Generalized theory of uncertainty (GTU)--principal concepts and ideas." The basic ideas which are put forward in this paper are summarized in the following.

Generalized Theory of Uncertainty (GTU)

GTU is based on fuzzy logic. The point of departure in GTU is the concept of a restriction (generalized constraint). Let X be a variable which takes values in U. A restriction on X, R(X), is expressed as X isr R, where X is the restricted (constrained) variable, R is the restricting (constraining) relation and r is an indexical variable which defines the way in which R restricts X. A restriction is hard if it is of the form X
ÎµA, where A is a set in U. A restriction is soft if it is not hard. Inequality and equality constraints are hard restrictions. A restriction is singular if R is a singleton. A restriction is nonsingular if R is not a singleton.

There are many different kinds of soft restrictions. A basic soft restriction is a possibilistic restriction, expressed as: X is A, where A is a fuzzy set in U. This expression implies that Poss(X=u)=ÂµA(u), where ÂµA is the membership function of A. A possibilistic restriction defines the fuzzy set of possible values of X, with the understanding that possibility is a matter of degree. A simple example of a possibilistic restriction is: X is small, where small is a fuzzy set in the space of reals.

Another basic restriction is a probabilistic restriction, X isp R, where R is the probability distribution of X. An example of a probabilistic restriction is: X isp normal distribution with mean m and variance ÏƒÂ².

Restrictions may be combined. An important combination is a Z-restriction, expressed as: X isz R, where R is a Z-number (Zadeh2011). Briefly, a Z-number is an ordered pair of fuzzy numbers, (A,B), where A is a possibilistic restriction on the values of a variable, and B is a possibilistic restriction on its probability. Typically, A and B are described in a natural language. Examples: (close to 5, very likely), (steep decline, unlikely). A Z-restriction may be expressed as what is referred to as a Z-valuation. Examples: (budget deficit, approximately 2 million dollars, very likely), (travel time, about 1 hour, usually). A Z-valuation, (X,A,B), is defined as Prob(X is A) is B. What is important about Z-valuations is that many propositions drawn from a natural language may be represented as Z-valuations. Examples:

Probably John is tallâ†’Prob(Height(John) is tall) is probable

Usually it takes Robert about an hour to get home from workâ†’(travel time, about 1 hour, usually)

There is an important connection between the concept of uncertainty and the concept of a restriction. More specifically, uncertainty is a concomitant of nonsingularity of a restriction. Concretely,

uncertainty=nonsingularity

This equality carries an important consequence. Since there are many kinds of restrictions, there are many different kinds of uncertainty. The principal uncertainties are possibilistic, probabilistic and their combinations. The principal combination is a Z-restriction. Viewed in this perspective, probability theory is concerned, in the main, with probabilistic restrictions while possibility theory is concerned, in the main, with possibilistic restrictions. Importantly, the generalized theory of uncertainty (GTU) opens the door to consideration of any kind of uncertainty.

In GTU, there is an important connection between the concept of information and the concept of a restriction. More specifically,

information=restriction

A proposition in a natural language, p, is a carrier of information. In GTU, a proposition is equated to a restriction

proposition=restriction

This basic equality--referred to as the meaning postulate--is the point of departure in what I call restriction-based semantics of natural languages (RS). More concretely, p is represented as X isr R, referred to as the canonical form of p. Example:

Most Swedes are tallâ†’Prop(tall Swedes/Swedes) is most,

where the fuzzy quantifier, most, plays the role of a possibilistic restriction on the variable Prop(tall Swedes/Swedes).

In GTU, computation with information which is described in a natural language is carried out in two phases. In Phase 1, the information carried by propositions is converted into a collection of restrictions through the use of restriction-based semantics. In Phase 2, restrictions serve as objects of computation and are computed with principally through the use of the extension principle of fuzzy logic (Zadeh 1965, 1975a,b & c). In essence, the extension principle is a rule which governs propagation of restrictions on the arguments of a function to a restriction on the values of the function.

Concluding Remark

The generalized theory of uncertainty (GTU) has a much higher problem-solving capability than standard probability theory.
Most importantly, GTU opens the door to construction of mathematical solutions of computational problems which are stated in a natural language. Standard probability theory does not have this capability. Lack of this capability is one of its principal limitations.

History of science is replete with debates between sufficienists and insufficienists. In most cases, the views put forward by insufficienists eventually become conventional wisdom. In large measure, scientific progress is driven by a quest for new ideas and new techniques.

--
Director, Berkeley Initiative in Soft Computing (BISC)

729 Soda Hall #1776
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