To analyse the evolutionary emergence of structural complexity in physical processes, we introduce a general, but tractable, model of objects that interact to produce new objects. Since the objects--epsilon-machines--have well-defined structural properties, we demonstrate that complexity in the resulting population dynamical system emerges on several distinct organizational scales during evolution--from individuals to nested levels of mutually self-sustaining interaction. The evolution to increased organization is dominated by the spontaneous creation of structural hierarchies and this, in turn, is facilitated by the innovation and maintenance of relatively low-complexity, but general individuals.

Complex systems may often be characterized by their hierarchical dynamics. In this paper do we present a method and an operational algorithm that automatically infer this property in a broad range of systems; discrete stochastic processes. The main idea is to systematically explore the set of projections from the state space of a process to smaller state spaces, and to determine which of the projections that impose Markovian dynamics on the coarser level. These projections, which we call Markov projections, then constitute the hierarchical dynamics of the system. The algorithm operates on time series or other statistics, so a priori knowledge of the intrinsic workings of a system is not required in order to determine its hierarchical dynamics. We illustrate the method by applying it to two simple processes; a finite state automaton and an iterated map. a Strictly speaking, Lie's work is not limited to classical mechanics and can be applied to any differential equation.

Current analyses of genomes from numerous species show that the diversity of their functional and behavioral characters is not proportional to the number of genes that encode the organism. We investigate the hypothesis that the diversity of organismal character is due to hierarchical organization. We do this with the recently introduced model of the finitary process soup, which allows for a detailed mathematical and quantitative analysis of the population dynamics of structural complexity. Here we show that global complexity in the finitary process soup is due to the emergence of successively higher levels of organization, that the hierarchical structure appears spontaneously, and that the process of structural innovation is facilitated by the discovery and maintenance of relatively noncomplex, but general, individuals in a population.