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**Recent Results on Ordering Parameters in Cellular Automata and Boolean Networks**

Jürgen Klüver Information Technologies and Educational Processes University of Duisburg-Essen (Germany)

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Ordering Parameters Ordering Parameters are numerical values that characterize the rule systems of cellular automata (CA) and of Boolean networks (BN); "rules" are on the one hand rules of transition and on the other hand "topological rules" – who interacts with whom

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Ordering Parameters Examples of ordering parameters: The P-parameter (Weissbuch and Derrida; already Ashby in the Sixties) and the logical equivalent -parameter (Langton), which both measure the proportion of different cell states generated by the respective CA- or BN-rules. A B: P(Imp) = 0.75 The proportion of canalyzing functions in a BN (Kauffman): Logical Implication is a canalyzing function. K, i.e. the number of variables of a BF is not an ordering parameter.

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Ordering Parameters Another ordering parameters is the Z-parameter (Wuensche and Lesser) that measures the probability of "computing backwards". 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.

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Ordering Parameters In this case the V-parameter is a characteristic of the adjacency matrix and not of the transition rules. V = (OD – ODmin)/(ODmax – ODmin) OD is the factual out degree of a graph or network respectively, ODmin the minimal possible out degree, ODmax the maximal possible one.

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Ordering Parameters In all cases particular values of the ordering parameters generate special forms of systems dynamics.

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Because all known parameters measure in a certain dimension a respective degree of difference it is possible to derive a "theorem of inequality": The more unequal 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).

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**A corollary is the theorem of the high probability of order:**

The probability of simple dynamics and even point attractors is much greater than the probability of complex dynamics. From this we obtain the paradox of democratic reforms.

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**A specific BN is characterized by BN = (C, V, P)**

A serious problem is the case if two or more ordering parameters are combined, in particular ordering parameters with "opposing" values. Experiments with the combination of the P-parameter, the proportion of canalyzing functions and the V-parameter for Boolean networks with six units and three Boolean functions (BF). In all cases K = 2. A specific BN is characterized by BN = (C, V, P)

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Hypothesis: There is a domain in the (three-dimensional) "parameter space" of BN, defined by 0.5 P , 0 v 0.3, and 0 C 0..3. In this region the occurrence of complex dynamics is very probable; outside of this region no complex dynamics will occur or only with a very low probability. Complex dynamics: periods p of attractors with p 2n/2.

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Restriction to and analysis of the 1499 non-isomorphic BN (isomorphic in the graph theoretical sense). Each combination of three BF obtains ca. 106 different BN. Restriction to K = 2: in a strict mathematical sense there is no K 2.

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**Some examples from the "hyperspace" of parameter combinations:**

A class of trajectories of a specific BN is a point in this hyperspace, generated by the region of C, V, and P.

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**Most important results:**

The probability of complex dynamics inside the region is about p = 0.43; outside p = 0.09.; In this sense the theorem of inequality is confirmed and so is the theorem of the probability of order: only 8% of all BN are characterized by complex dynamics. From a sociological point of view rather satisfactory.

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**But: This statistical measure (P, C. V) is very coarse.**

Most of BN with complex dynamics lie outside the region, although the probability of finding them is rather small. Further investigations of this hyperspace obviously are necessary. Enlarging the dimensions of the hyperspace by adding more ordering parameters, e.g. the Z-parameter and/or other topological characteristics of the adjacency matrix.

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**Perhaps even the theorem of inequality is not deep enough, i. e**

Perhaps even the theorem of inequality is not deep enough, i.e. too near at the surface of the observable behavior of BN and CA. Yet the analysis of such ordering parameters seems to be the only way to obtain a theoretical understanding of these potential universal systems.

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This could give us a Kantian Stance (Kauffman): Measuring mathematical characteristics all real systems must have, regardless of their empirical contents - a mathematical a priori.

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Thank you

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