“Majesty,” he said kneeling in front of the king, “give me a grain of wheat to put on the first square of this chessboard, and two grains to put on the second square, and four grains to put on the third, and so on, doubling the number for each succeeding square, give me enough grains to cover all 64 squares of the board.”
   “You did not ask for much, oh my faithful servant,” exclaimed the king. “Your wish will certainly be granted.” And he ordered a bag of wheat to be brought to the throne. But when the counting began, it soon became clear that with all the crop of Indian the king could not fulfill his promise to Sissa.
   To do so would have required:

18,446,744,073,709,551,615 grains!
   Assuming that a bushel of wheat contains about 5 million grains, one would need some 4×10¹² bushels to satisfy the demand of Sissa. Since the world production of wheat averages about 2×10⁹ bushels a year, the amount requested by the minister was that of the world's wheat production for the period of some two thousand years! (http:// www.rrting.com/English/othertm/57031/)

This is a kind of index increasing, i.e. a catastrophe or an emergency. The reason of this emergency is positive feedback. The more the number on an square of chessboard, the more the quantity to increase, the more the quantity to increase, the more the number on next square. So powerful the positive feedback is, that professor Cramer's puzzle could be easily solved.

New complex structures of non-living things and original creatures come from interaction between relatively simple materials or between simple material and environment. These interaction are random with low probability. But the probability of new complex structure is able to increase when the number of simple materials increases.

Each kind of creature, different from non-living things, is able to set up an index increasing of their body by breeding, which is a kind of positive feedback (with some certain environment). The more the basic quantity of a kind of creature is, the more the number to increase. The more the number to increase, the more the quantity of next generation is, just like the wheat. What is more, such processes have not only repeated for 64 times, but many billions of years. E. coli reproduces itself once per 20 minutes, in which rate it costs almost 304 years to reach 10^2400000. Man's cell divides once per 8-48 hours. If it is assumed 2 days, to reach 10^2400000 needs 44000 years, which is only a short instant in the evolution process.

What is self-reproduction? It is a special repro- duction.

Definition: If the motivity of reproduction is the function of the model and the process does not need any help from outside complex systems, this kind of reproduction is self-reproduction.

It should be noticed that self-reproduction does need helps from outside systems, which are simpler than itself, but does not need help from outside complex systems.