Comments to "Bell Curves and Monkey Languages", J. Casti, Complexity, 1, 12-15 1995.

Whether there are universal laws or principles in complex systems is a fascinating and important question. Prof. John Casti uses the case of Normal Distribution (\bell curves") to illustrate that such universal principle is perhaps out there waiting to be discovered [1]. He suggests Zipf's law as a candidate for such universal principle. But as the author of one of the three publications to prove that \monkey languages" exhibit Zipf's law [2] (B. Mandelbrot and J. Nicolis wrote the other two [3,4]), I tend to believe that Zipf's law is a \law of transformation" (if it can be called a \law") rather than a \law of complex systems". I restrict the name \Zipf's law" to rank-frequency distributions that are power-law function P (r) 1=r , 1 (r is the rank, and P (r) is the frequency of events corresponding to that rank). Besides the rank-frequency distribution of English words and \monkey typed words", other rank-frequency distributions that obey power-law functions with exponent close to 1 were also known, including the population of cities [5]. The non-linguistic examples given in Ref.[5] may not have an explanation yet, but the Zipf's law in \monkey languages" has already been explained and understood. This explanation also shed light on the Zipf's law in English language. To summarize in one sentence, the Zipf's law in \monkey languages" is caused by an exponential transformation of variables, and if the rst variable follows an exponential distribution, the second would follow a power-law distribution. To be more speci c, the rst variable is the word's length, and the second variable is the word's rank. The details of this proof can be found in Ref.[2], but I can sketch the proof here easily: suppose there is an exponential function of variable x: f(x) = Aexp[ Bx], and a transformation from x0 to x: x = Cexp[Dx0], the new function in variable x will be: