Recent Results on Ordering Parameters in Cellular Automata and Boolean Networks

Jürgen Klüver

University of Duisburg

Ordering parameters are numerical values that characterize the rule systems of cellular automata (CA) and of Boolean networks (BN). Well-known ordering parameters are the P-parameter (Weissbuch and Derrida) and the logical equivalent parameter (Langton), which both measure the proportion of different cell states generated by the respective CA or BN rules. Other parameters are the Z-parameter (Wuensche and Lesser) that measures the probability of “computing backwards” and the proportion of canalyzing functions in a BN (Kauffman).

Besides the ordering parameters of transition rules there also exist topological ordering parameters, in particular the v-parameter (Klüver and Schmidt), which measures the proportion of influence the different cells have on each other. In this case the v-parameter is a characteristic of the adjacency matrix and not of the transition rules. In all cases particular values of the ordering parameters generate special forms of systems dynamics.

Because all known parameters measure in a certain dimension the respective degree of difference it is possible to derive a “theorem of inequality” (Klüver): the more equal a system is in the dimensions measured by the ordering parameters, the more simple the dynamics of the system will be and vice versa (simplicity is measured via the complexity classes of Wolfram). A corollary is the theorem of the high probability of order.

A serious problem is the case where if two or more ordering parameters are combined, in particular ordering parameters with “opposing” values. These cases are only seldom analyzed. We undertook many experiments with the combination of the P-parameter, the proportion of canalyzing functions, and the v-parameter for Boolean networks with six units. The results are that there exist different regions of parameter combinations that can be characterized as regions of order (simple dynamics) and regions of very complex dynamics that can be called “quasi chaotic regions”. Unfortunately these regions are not so simple as the early pioneers of ordering parameters believed. Yet the analysis of such ordering parameters seems to be the only way to obtain a theoretical understanding of the potential universal systems.