The following are two emails of a continuing dialogue held on the UC Berkeley discussion group in Soft Computing, BISC, with this particular topic focussing on the differences and correspondences between standard Probability Theory - being the mathematical base for Artificial Intelligence and Neural Net research, and, Fuzzy Logic - which is the basis for Soft Computing and systems management. The commentary I sent in reply to Lotfi Zadeh's proposal is a rather concise reprise of my proposal of Stochastic Logic originally mentioned in "UIU"(1992), and so I've re-printed it for you here for your consideration.

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To: BISC Group                                                                                                           November 11, 1998
From: L. A. Zadeh (zadeh@cs.berkeley.edu)

Probability Theory Needs an Infusion of Fuzzy Logic
to Enhance its Ability to Deal with Real World Problems

Discussions and debates centering on the relationship between fuzzy set theory and fuzzy logic, on one side, and probability theory, on the other, have a long history starting shortly after the publication of my first paper on fuzzy sets (1965). There is an extensive literature on this subject, including several papers of mine, of which the latest [Zadeh 1995] appears in a special issue of Technometrics that focuses on the relationship in question.

The following is an updated expression of my views. Your comments would be appreciated. Please do not hesitate to disagree with me.

First, a point of clarification. By probability theory in the heading of this message is meant standard probability theory (PT) of the kind found in textbooks and taught in courses. By infusion is meant generalization. Thus, the crux of my argument is that PT is in need of generalization, first by f-generalization (fuzzification) and second by g-generalization (granulation). In combination, f-generalization and g-generalization give rise to f.g-generalization
(fuzzy granulation). F-generalization of PT leads to what might be denoted as PT+. Then, g-generalization of PT+ leads to PT++, which may be viewed as f.g-generalization of PT. Basically, fuzzy granulation reflects the finite cognitive ability of humans to resolve details and store information. In summary, my contention is that to enhance its effectiveness PT should be generalized to PT++. This has been done already to a significant extent by a number of contributors, but much more remains to be done.

What is not in dispute is that standard probability theory (PT) provides a vast array of concepts and techniques which are effective in dealing with a wide variety of problems in which the available information is lacking in certainty. But what may come as a surprise to some is that along with such problems stand many very simple problems for which PT offers no solutions. Here is an sample. It should be noted that the underlying difficulties in these problems are well-known to probability theory professionals.

DB 1. What is the probability that my tax return will be audited? DB 2. What is the probability that my car may be stolen? DB 3. How long does it take to get from the hotel to the airport by taxi? DB 4. What is the probability that Mary is telling the truth? DB 5. A and B played 10 times. A won 7 times. The last 3 times B won.           What is the probability that A will win?

Questions of this kind are routinely faced and answered by humans. The answers, however, are not numbers; they are linguistic descriptions of fuzzy perceptions of probabilities, e.g., not very high, quite unlikely, about 0.8, etc. Such answers cannot be arrived at through the use of PT. What is needed for this purpose is PT++.

I discussed Problems 1-5 with Professor David Blackwell (UC Berkeley), who is one of world's leading authorities on probability theory. The initials DB signify that he agrees with my assessments.

What are the sources of difficulty in using PT? In Problem 1, the difficulty comes from the basic property of con- ditional probabilities, namely, given p(X), all that can be said is that the value of p(X|Y) is between 0 and 1. Thus, if I start with the knowledge that 1% of tax returns are audited, it tells me nothing about the probability that my tax return will be audited. The same holds true when I add more detailed information about myself, e.g., my  profess- ion, income, age, place of residence, etc.. IRS may be able to tell me what fraction of returns in a particular cat- egory are audited, but all that can be said about the probability that my return will be audited that is between 0 and 1.

As I have alluded to earlier, when I prepare my tax return, I do have a fuzzy perception of the chance that I may be audited. To arrive at this fuzzy perception what is needed is fuzzy logic, or, more specifically, PT++. I will discuss in a later installment how this can be done through the use of PT++.

In Problems 3 and 5, the difficulty is that we are dealing with a time series drawn from a nonstationary process. In such cases, probabilities do not exist. Unfortunately, this is frequently the case when PT is used.

In Problem 3, when I pose the question to a hotel clerk, he may tell me that it would take approximately 20-25 minutes. The point is that the answer could not be deduced in a rigorous way by applying PT. Again, to arrive at clerk's answer what is needed is PT++.

In Problem 4, the difficulty is that truth is a matter of degree. For example, if I ask Mary how old she is, and she tells me that she is 30 but in fact is 31, the degree of truth might be 0.9. On the other hand, if she tells me that she is 25, the degree of truth might be 0.5. The point is that the event "telling the truth" is, in general, a fuzzy event, PT does not support fuzzy events.

Another class of simple problems which PT cannot handle relate to common sense reasoning exemplified by:

6. Most young men are healthy; Robert is young. What can be said about Robert's health?
7. Most young men are healthy; it is likely that Robert is young. What can be said about Robert's health?
8. Slimness is attractive; Cindy is slim. What can be said about Cindy's attractiveness?

In what ways do PT+ and PT++ enhance the ability of probability theory to deal with real world problems? In relation to PT, PT+ has the capability to deal with:

1. fuzzy events, e.g., warm day
2. fuzzy numbers, quantifiers and probabilities, e.g., about 0.7,  most, not very likely
3. fuzzy relations, e.g., much larger than
4. fuzzy truths and fuzzy possibilities, e.g., very true and quite possible

In addition, PT+ has the potential -- as yet largely unrealized -- to fuzzify such basic concepts as independence, stationarity and normality.

PT++ adds to PT+ further capabilities which derive from the use of granulation. They are, mainly:

1. linguistic (granular) variables
2. fuzzy rule sets and fuzzy graphs
3. granular goals and constraints

At this juncture there exists an extensive and growing literature centering on the calculi of imprecise and fuzzy
probabilities. The concepts of PT+ and PT++ serve to clarify some of the basic issues in these calculi and suggest directions in which the ability of probability theory to deal with real worlds problems may be enhanced.

Reference:
L. A. Zadeh, "Probability Theory and Fuzzy Logic are Complementary Rather
Than Competitive," Technometrics, vol. 37, pp. 271-276, 1995.

Warm regards to all,          Lotfi           

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To: BISC Group                                                                                              November 16, 1998

Re: Probability Theory Needs an Infusion of Fuzzy Logic to Enhance its
Ability to Deal with Real World Problems [Zadeh 19981111]

In all deference to Prof. Zadeh and everyone else who perceives a need to reconcile conventional Probability Theory with Zadeh (nee, "fuzzy") Logic, the issue really goes to the entire architecture of mathematics. It calls into question the relationships among and accessibility between, functions and content, in whatever possible   configurations are defined, discerned, discovered, or proposed. Especially as we're required to establish viable connections between inductively reachable, but not necesarily immediately available, information domains.

Succinctly - we can't meld PT with Zadeh Logic (ZL) unless we discuss them in terms of Godel's theorems, and, find a reasonable way to surmount the Incompleteness conclusions.

For any bounded set (defined more by the finite content than by any explicit boundary ... such as the temporal moment "now", which is forever changing and inclusive of more "explicit information" as it advances), any particular fixing of a temporal locus along a timeline can formalize the corresponding "content" to an amount that is
normalizable as "one", the principal upper-bound of standard probability and Boolean Logic.

Essentially, it randomly fixes and defines the Godel boundary of a considered system. And yet, we recognize that
there is transboundary information, potential, factors, and relationships, which can't be willy nilly left unconsidered
merely because we don't have the mathematical language or functions to fully acknowledge or access what is
potentially there. And, we're prompted to establish just such valid working relationships with all of that open
potential, if only for the fact that ZL says it's viable useful/usable information.

That notion is inherent in all ZL functions, because the "net probability" can and more typically does, surpass the value "one". It explicitly uses the probability field outside the Boolean/Godelian accepted 'unit' boundary(ies). So, it becomes unavoidably necessary to examine and specify these factors and relationships which are seen as common to both systems.

I did this analysis in 1991, and included it in "Understanding the Integral Universe" (1992), which relevant text is
online at  ceptualinstitute.com/uiu_plus/UIUcomplete11-99.htm   Section 5. "Charting a Path through the Maze". My conclusion is that all of these relationships and functions can be canopied under the nominal
nomenclature of Stochastic Logic.  It seems that Probability Theory does not need "an infusion of Fuzzy Logic to enhance its ability to deal with real world problems", as Prof Zadeh has phrased it, because Zadeh Logic is already the more general form of probabilities-relations, of which standard PT is a restricted sub-set.  Comprehensively, Stochastic Logic also includes the companion logic system of Quantum Mechanics, and, even the post-quantum system of Bohm/Hiley (in which case, information coding and transduction takes center stage).

The fact of this has been masked, ever since Huntington (1905) reductionistically pared down Boole's extensive
1853 treatise which was originally titled "An Investigation of the Laws of Thought on which are founded the
Mathematical Theories of Logic and Probabilitites." Much of Boole's original discourse is very reminiscent of the
soft-bounded relationships and flexible causal interdependencies of Zadeh Logic and soft computing.

In his attempt to break away from Aristotelian Logic and that of Clarke and Spinoza, Boole in fact enunciated the
notion of 'conditional existence' (my labeling). Where existence - and non-existence - occur in relation to other
events, dynamics and things - being present or not, in diverse combinations. So, while he tried to oppose the logic
systems of 2.5 millenia, he in fact placed them in context of a larger group of relations and boundaries.

And this is what Zadeh Logic does also. Let me explain how.

                I'll use the notations "a", and "<a>" (for bar-a), and (dot) for the dot function.

                The basic Boolean identities are :

                                                            (1)   a (dot) <a> = 0
                                                            (2)   a (plus) <a> = 1

and are an entrée to replacing the Aristotelian, A=A, identity function which is inherent in equation (2) ... the binary and equal probabilities that "a" either exists or it doesn't.

               If we translate (1,2) in terms of probabilities, the relationships become clearer.

                      (1) The probability that "a" and "<a>" can co-exist simultaneously, is, zero.
                      (2) The probability of "a"'s existence, plus, the probability of "<a>"s existence, sums to, one.

Under Boolean Logic, the probabilities of (2) are equal by assumption and convention.   Under ZL, they need not be. Nor are the number of parameters limited to two. Nor is the function product fixed as, or limited to, "one".

In the Quantum Mechanical application of function (1), the function product is allowed - in fact expected - to be
"not-zero", where the formal enunciation of QM fixes the value at "one"; and the EPR-Bell-Aspect formulations
inherently specify "not-zero, not-one". Bohm/Hiley, too, allow different product values - which is why parameter
specifications of their general relations perforce 'reduce to' standard QM equation forms.

Now, in essense, we are describing the most general form of mathematical architecture possible, allowing and
expecting "open unboundedness" as the most accessible apriori form. This, in contra-position to the strict
interpretation of Godel Incompleteness. The full house can be sub-partitioned in many alternative ways, (including the Godel Cut) but it is still one-house, where all aspects and qualities are presumed compatible and accessible,
regardless of cut, and regardless of local access restrictions. Each of the partitionings illuminates important
relationships unique to the chosen partitions and definitions, so it is not a case of "which one is better or more
accurate". Each is valid under the chosen or applied constraints.

Any model will always display a 'variance from observation' since any and all events must be subject to the full
spectrum of relations and dynamics of all possible partitionings rather than merely any local focus-group of
parameters used by a particular partitioning method (PT, ZL, etc.).

This basically sums up why I concur with Prof Zadeh's essential message, that some form of melding and
compatibility among the two immdediate approaches (partitionings) of concern - PT and ZL - needs to be secured, to the benefit of both perspectives. And specifically, Zadeh Logic is the more expansive of the two approaches, being able to embrace PT, even as they have their individual information-processing capabilities, which the other method may not be able to handle or produce.


...There are 2 other threads which grow out of the above thoughts.

The above has a bearing on the Fractal approach to chaos & complexity. It can be considered a specific partitioning also, which is modular within a larger frame of reference. In this case, fractal exponents ought to be perceived as dimension-states, albeit non-integer ones. In general, all exponents are therefore referential to the dimensionality of a function or partial function. With this notion, the landscape of Complexity broadens, and expands to include the notion that non-zero probabilities of information/energy exchange .... exponents greater than zero ... are requisite for the enactment of complex organizations which are behaviorally interconnected. There may be specific recursive exponent relationships ... Mandelbrot sets, etc...., but the primary dynamic is implicit ... the greater than zero probability of information transferred from one configuration set to some next one ... which in turn illuminates emergent patterns that viewing the simple general base/exponent definition might not show too readily.

...The last thing I'll talk about here is a specific example (DB 6, if you will) of another probability problem
experienced everyday, but having no methodology of solution at the present. It is the "substitution problem" :

If a basketball team is defined as 5 players actively participating in ball handling on the court, in competition with
another team, what single formula can process the full team roster of non-active as well as active players? The
"team" is the "complexity", as defined above. It "exists" as long as certain parameters of interaction/participation are met. But at any given time the players on the bench have a participation probability of zero. And, there are
conditions which enable the complexity called "team" to exist in potentia when no games are going on.

In other words, if the team is bounded by the definition of "5 players actively engaged against another 'team'", then it is unavoidable that transboundary trans-Godel Limit information/energy must be mathematically accessible in order to specify the alternative conditions under which a "team" remains a viable entity and complexity. We can't limit the probabilities to only 5 specific players and whether they are active in a game.

My regards to all. I thank you for reading and considering these ideas.

Ceptual Institute

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