This Small Grant for Exploratory Research will support an investigation that will pursue the underpinnings and implications of the coevolution to the edge of chaos, based on entirely novel models of molecules or goods as symbol strings which act on one another to produce new symbol strings according to a "random grammar". The principal investigator seeks new insights into how complex adaptive systems collaborate, compete, and coevolve. In particular, the investigator expects to find new models of economic take-off based on achieving a sufficient diversity of goods and services in an economy. Similar concepts will apply to ecosystem structure. Most broadly, it is hoped that investigation of random grammars will yield insight into the self organized behaviors of far from equilibrium systems. Among the fundamental phenomena in biology and beyond, in economic and cultural systems, is the coevolution of complex adaptive systems. In the past few years, work carried out on diverse fronts by a small number of physicists, biologists, mathematicians, and economists, has begun to suggest a broad new conceptual framework and candidate universal features of such coevolution: Complex coevolving systems may evolve the the edge of chaos. The new theory is built upon the recognition that such coevolving systems, whether symbionts or interlocked firms, mutually build the world they jointly inhabit. Such coevolutionary systems can behave in three regimes. The first is a chaotic regime, where each adaptive agent struggles to improve on multipeaked "fitness landscapes" which deform so rapidly due to moves of other agents that no agent comes to rest. This regime is called the Arms Race in evolutionary biology. The ordered regime occurs when all agents climb to local fitness peaks that are mutually consistent. Adaptive chance ceases. The third regime is the boundary between order and chaos. Two new facts have emerged: 1) Mean fitness, or "payoff" is low in both the chaotic regime and in the ordered regime and is optimal at the edge of chaos. 2) A kind of "invisible hand" can act such that each agent tunes the structure of its own landscape and couplings to other landscapes to its own advantage, and by doing so the entire system coevolves to the phase transition between order and chaos. If generally true, the implications are broad. We must consider that adaptive agents mutually alter the games we play with one another, in evolution, in economies, and perhaps elsewhere, to a characteristic poised state between order and disorder. Signatures of this poised state include a power law distribution of avalanches of change initiated by small perturbations. Just such a distribution of sizes of extinction events in the evolutionary Record are known. Similar avalanches may occur in technological evolution.