Fuzzy Sets and Artificial Intelligence

Many patterns of Nature are so irregular and fragmented, that, compared with Euclid — a term used in this work to denote all of standard geometry — Nature exhibits not simply a higher degree but an altogether different level of complexity … The existence of these patterns challenges us to study these forms that Euclid leaves aside as being "formless," to investigate the morphology of the "amorphous." - Benoit Mandelbrot, as quoted in a review of The Fractal Geometry of Nature by J. W. Cannon in The American Mathematical Monthly, Vol. 91, No. 9 (November 1984), p. 594

Artificial intelligence requires a new way of thinking about both Nature and computation. Alpha Zero has demonstrated a fundamentally new form of chess playing that did not exist before. Its style of play has been described as alien, resembling neither the style of human play nor the style of classical machine play.

Fuzzy set theory (or fuzzy logic) is an alternative approach to standard set theory or "crisp" set theory. With fuzzy sets, every element in the universe of discussion has a "degree of set membership" in one or more sets. The degree of membership is a value between 0 and 1 that can, under certain conditions, be interpreted as a probability (it is a mistake to treat degree of set membership as identical with probability, however).

Let's consider the category of image-recognition - not merely machine image-recognition but the category, in general (including human or other image-recognition). Let us say we have two large sets of images T and L. T contains images of tigers shot from a wide variety of distances, angles and visual conditions. L has a similarly wide selection of images of lions.

Using T and L as our ground truth, let us randomly select an image from one or the other set and submit this image to a test subject for identification. The test subject can answer "lion," "tiger" or "unknown". When the subject answers "lion" for an image drawn from T - or vice-versa - we can say that this is an error as measured against the ground truth. But sometimes the subject will not be able to make any positive identification, no matter how carefully they attempt to do so - perhaps because the image is too fuzzy or the animal is too distant or the particular angle or image conditions cause the animal's appearance to be equivocal with the appearance of the other animal. From the perspective of the ground truth, "unknown" is always an erroneous response. But this contradicts our intuition that "unknown" is a perfectly reasonable response for cases where the image information required to distinguish an element of one set from the elements of the other (on the basis of the image alone) is lacking. In such cases, "unknown" is the correct answer and an answer of "tiger" or "lion" would be flatly incorrect or, at best, a mere guess.

Instead of categorizing answers according to the ground truth, we can allow the test subject to associate some degree of confidence with the answer - [L,1.0] is "more of an element" of L than [L,0.9]. As we submit the images to the test subject for review, two new sets - L' and T' - will be formed, describing that subject's classification of the images, along with an associated confidence parameter. This approach allows us to directly express equivocation, since the test subject may answer [L,0.5],[T,0.5] in order to classify an image as being equally a member of either set - this would have been an "unknown" in our previous arrangement. But the test subject can now express degrees of equivocation, so that [L,0.6],[T,0.4] classifies an image as slightly more a member of L than of T. Of course, any item in the universe of discourse can be fully included in more than one set, so membership is not normalized to 1.0; an answer of [L,0.1],[T,0.1] may express the test subject's doubt that either animal is in the picture at all.

But now let us introduce the liger.

Ligers are a fact of physical reality (they are a real, existing hybrid). But ligers break our L/T dichotomy. Even our fuzzy sets don't help - [L,1.0],[T,1.0] should indicate the situation where a lion and (separately) a tiger are present in the image. That is, a liger - being its own hybrid - requires its own classification, let's call it G. What I am asserting is that, at the macroscopic level of observation, reality is inherently continuous and, thus, the category of categorization itself is always liable to breakage. This is the black swan theory - no matter how complete we feel our theory is, the possibility of a black swan always exists. No matter how much we rationalize having missed the possibility of a black swan (after the fact), the fact remains that we overlooked this possibility because reality is fundamentally continuous - if there are lions (L) and there are tigers (T), there is always the possibility that there are ligers (G), a fundamentally new category that does not belong to any already-known category.

The implications go down to the foundations of math itself. Modern math is based on standard set theory. This kind of set theory is ideal for symbolic reasoning, the kind that mathematicians use almost exclusively. But note that not all reasoning must be symbolic. Here is the proof of Pythagoras's theorem. I will explain it using symbols but it is not comprehended symbolically:

The image on the left shows two gray, square regions. The long sides of the triangles are labeled a (all equal), the short sides are labeled b (also all equal). To transform the left image to the right image:

• The red triangle stays where it is
• The blue triangle slides all the way down
• The green triangle slides all the way to the left
• The yellow triangle slides to the upper-right

The hypotenuses of the triangles are, obviously, equal, and labeled c. Since these are right-triangles, the angle formed by placing the long-side and short-side of the triangles abutting on the same line must be 90 degrees, since 180-90=90. Thus, the gray region in the center of the right image is a square and its area must be c². This is a proof of Pythagoras's theorem.

There is nothing about this proof that requires the use of symbols. You could even build a physical model of this proof, if you wanted. It is even possible to perform numerical calculation without the use of symbols. Techniques involving only a straight-edge and compass easily allow numerical calculations to be performed to a handful of significant digits.

Standard set theory - and the mathematics built on it - naturally assumes noiselessness in the symbols themselves. This means that we always recognize the symbol 3 as the number it represents and never confuse it with another number, such as the number four. It also means that there are no categories that break the syntax of our formal system - there are no ligers among the symbols of mathematics (imagine a 3 and 4 superimposed, for example).

In the real world, noiselessness is never absolute, it is always a matter of degree, based on the redundancy and other error-correcting features of the chosen encoding. In the limit, we must admit that this is even true of human mathematics. Human knowledge and human memory - even with all its external, material aids - is not perfectly noiseless.

But noisy symbols are like fuzzy sets, or ligers. We live in a Universe where absolute noiselessness is simply not in the attainable set of conditions, but where our most effective theories of reasoning and material causality are built on symbols that are supposed to be made out of noiselessness. Ligers break noiseless theories.

So, what we want is a system that ligers can't break. A liger doesn't break reality, it just makes it different when we discover one. There is no reason we cannot build formal systems that act like the material world - systems that are noise-tolerant. It is still possible to reason with a quasi-consistent system of symbols - fuzzy symbols. In fact, the world just is fuzzy symbols being interpreted by the mind-body system. In short, the mathematics of reality is fuzzy mathematics.

Unlike classical computation systems, AI systems are inherently fuzzy. They are good at fuzziness, unlike their brittle forebears. But AI systems are going to face increasing headwinds as they improve at fuzziness. The human brain can easily distinguish a lion from a tiger, even at a very young age after seeing only a tiny number of examples - and perhaps entirely schematic! Yet our brain has great difficulty performing long-division on numbers more than a handful of digits in size. This is a result of the brain's fuzzy orientation - it does not expend precious mental resources on noiselessly encoding decimal digits which would enable us to perform rapid long-division in our heads. The more fuzzy AI becomes, the more it is going to face the same obstacle - when it encounters a formal problem with high logical depth, it will need to utilize an external, classical computational process to handle this problem in the same way that the human utilizes a calculator to handle such problems.

Fuzzy sets are not exactly identical with quantum mathematics but it is tempting to wonder if it is possible to naturally represent a fuzzy set theory as a quantum system. This could even establish a correspondence between the limits of algorithmic complexity - which we have explored in previous posts - and the a priori limits of physical observation that quantum theory predicts (the Planck limits).

In this view, the reason that quantum systems act more like fuzzy sets than like crisp sets would be a consequence of the limitations of the observer (us). An observer of limited complexity can only distinguish two distinct objects up to the limit of complexity. If two objects are different but this difference can only be perceived at a level of complexity beyond that possessed by the observer, they will appear to that observer to be the same. By the same token, if two objects are the same (have the same properties) but this identity can only be perceived at a level of complexity beyond that possessed by the observer, they will appear to that observer to be different. Today, this is purely pedantic speculation. But a world in which Artificial Super Intelligence exists, this will no longer be a pedantic matter. We may end up in a world where AI is able to distinguish between things that look the same to us - no matter how much scientific instrumentation we apply; and vice-versa. Between here and there, we're going to need a robust language in which to discuss fuzziness. Such a language may look as different from traditional mathematics as the above proof of the Pythagorean theorem looks different from an algebraic proof of that theorem.