Possible Laws for Artificial Life Evolution
  Motivated by a recent article on open problems in artificial life, here I postulate three laws which form a mathematical framework to describe artificial life evolutionary dynamics. They are based on a continuous approximation of population dynamics. Four dynamical elements are required in this formulation: ascendant matrix, transverse matrix, fitness function, and the stochastic drive. The first law states that in the absence of stochastic drive the artificial life always seeks for a local fitness attractor and stay there. It gives the reference point to discuss the general evolutionary dynamics. The second law is explicitly expressed in a unique form of stochastic differential equation with all four dynamical elements. The third law defines the relationship between the focused level of description to its lower and higher ones, and also defines the dichotomy of deterministic and stochastic drives. These laws provide a coherence framework to discuss several current problems, such as emergency and stability. In particular, two quantities are emphasized: the fitness function as the standard for selection and the stochasticity as the source of creativity. Those three laws may appear almost self-evident from a statistical physics point of view. However, their equivalent to a most conventional approach for evolutionary dynamics is shown for the first time by the present author, to the best of his knowledge. The computational advantage of the present formulation in the study of artificial life evolution is also discussed. 
P. Ao