The evolution and understanding of hierarchical complexity in biology froman algebraic perspective

We develop the rigorous notion of a model for understanding state transitio n systems by hierarchical coordinate systems. Using this we motivate an alg ebraic definition of the complexity of biological systems, comparing it to other candidates such as genome size and number of cell types. We show that our complexity measure is the unique maximal complexity measure satisfying a natural set of axioms. This reveals a strong relationship between hierar chical complexity in biological systems and the area of algebra known as gl obal semigroup theory. We then study the rate at which hierarchical complex ity can evolve in biological systems assuming evolution is "as slow as poss ible" from the perspective of computational power of organisms. Explicit bo unds on the evolution of complexity are derived showing that. although the evolutionary changes in hierarchical complexity are bounded, in some circum stances complexity may more than double in certain "genius jumps" of evolut ion. In fact, examples show that our bounds are sharp. We sketch the struct ure where such complexity jumps are known to occur and note some similariti es to previously identified mechanisms in biological evolutionary transitio ns. We also address the question of, How fast can complexity evolve over lo nger periods of time? Although complexity may more than double in a single generation, we prove that in a smooth sequence of t "inclusion" steps, comp lexity may grow at most from N to (N + 1)t + N, a linear function of number of generations t, while for sequences of "mapping" steps it increases by a t most t. Thus, despite the fact that there are major transitions in which complexity jumps are possible, over longer periods of time, the growth of c omplexity may be broken into maximal intervals on which it is bounded above in the manner described.