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We humans usually think in words; to represent our opinion about, e.g., the size of an object, it is sufficient to pick one of the few (say, five) words used to describe size (‘‘tiny,’’ ‘‘small,’’ ‘‘medium,’’ etc.). Indicating which of 5 words we have chosen takes 3 bits. However, in the modern computer representations of uncertainty, real numbers are used to represent this ‘‘fuzziness.’’ A real number takes 10 times more memory to store, and therefore, processing a real number takes 10 times longer than it should. Therefore, for the computers to reach the ability of a human brain, Zadeh proposed to represent and process uncertainty in the computer by storing and processing the very words that humans use, without translating them into real numbers (he called this idea granularity). If we try to define operations with words, we run into the following problem: e.g., if we define ‘‘tiny’’ 1 ‘‘tiny’’ as ‘‘tiny,’’ then we will have to make a counter-intuitive conclusion that the sum of any number of tiny objects is also tiny. If we define ‘‘tiny’’ 1 ‘‘tiny’’ as ‘‘small,’’ we may be overestimating the size. To overcome this problem, we suggest to use nondeterministic (probabilistic) operations with words. For example, in the above case, ‘‘tiny’’ 1 ‘‘tiny’’ is, with some probability, equal to ‘‘tiny,’’ and with some other probability, equal to ‘‘small.’’ We also analyze the advantages and disadvantages of this approach: The main advantage is that we now have granularity and we can thus speed up processing uncertainty. The main disadvantage is that in some cases, when defining symmetric associative operations for the set of words, we must give up either symmetry, or associativity. Luckily, this necessity is not always happening: in some cases, we can define symmetric associative operations.  1997 John Wiley & Sons, Inc.

Granularity via nondeterministic computations: What we gain and what we lose