Random mappings

J f = 2 entries of the rule table be filled in. Each configuration can be interpreted as being the binary representation of some whole number between 0 and A/ — 1. This suggests that the behavior of N, k = N)-neis is equivalent to the behavior of a random mapping of a finite set of A/ integers to itself. 

Random maps have been studied by a large number of different authors, including Kruskal [krus54], Rubin and Sitgreaves [rubin54], Pittel [pittel83] and Kolchin [kolc86]. 

Consider an order W system and a random function 4 which maps each of the H = 2 possible binary states Si to unique successor states Sj = / )(Si) with probability p[(j)] = 1/A/ . We can immediately compute the asymptotic N-dependence of the cyclic structure of the corresponding state transition graph 

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Non-Homogeneous CA a characteristic feature of all CA rules defined so far has been that of homogeneity - each cell of the system evolves according to the same rule 0. Hartman and Vichniac [hartSfi] were the first to systematically study a class of inhomogeneous CA (INCA), in which the state-transition rules are allowed to vary from cell to cell. The simplest such example is one where there are only two different 0 s, which are randomly distributed throughout the lattice. Kauffman has studied the other extreme in which the lattice is randomly populated with all 2 possible boolean functions of k inputs. The results of such studies, as well as the relationship with the dynamics of random, mappings, are covered in detail in chapter 8.3. 

A simple mapping enables tire RSA fonnalism to be applied to binding of a ligand L to receptors R embedded in a surface (tire RSA-random site (RSA-RS) model) [149,... 

A data set can be split into a training set and a test set randomly or according to a specific rule. The 1293 compounds were divided into a training set of 741 compounds and a test set ot 552 compounds, based on their distribution in a K.NN map. From each occupied neuron, one compound was selected and taken into the training set, and the other compounds were put into the test set. This selection ensured that both the training set and the test set contained as much information as possible, and covered the chemical space as widely as possible. 

Chapter 8 describes a number of generalized CA models, including reversible CA, coupled-map lattices, quantum CA, reaction-diffusion models, immunologically motivated CA models, random Boolean networks, sandpile models (in the context of self-organized criticality), structurally dynamic CA (in which the temporal evolution of the value of individual sites of a lattice are dynamically linked to an evolving lattice structure), and simple CA models of combat. 

Mean Field Estimate It is easy to predict the value of 7 from A. Consider a one-dimensional = 2, r = 1 CA evolving according to rule (j) for which (j>) = A. The probability that two randomly chosen (2r + 1) neighborhood blocks map to the same value is equal to Psame = A + (1 - A). The average left spreading rate of (j> is then given by... 

It would appear that the tradeoffs between these two requirements are optimized at the phase transition. Langton also cites a very similar relationship found by Crutchfield [crutch90] between a measure of machine complexity and the (per-symbol) entropy for the logistic map. The fact that the complexity/entropy relationship is so similar between two different classes of dynamical systems in turn suggests that what we are observing may be of fundamental importance complexity generically increases with randomness up until a phase transition is reached, beyond which further increases in randomness decrease complexity. We will have many occasions to return to this basic idea. 

Strange Attractors The motion on strange attractors exhibits many of the properties normally associated with completely random or chaotic behavior, despite being well-defined at all times and fully deterministic. More formally, a strange attractor S is an attractor (meaning that it satisfies properties (i)-(iii) above) that also displays sensitivity to initial conditions. In the case of a one-dimensional map, Xn+i = for example, this means that there exists a <5 > 0 such that for... 

The discussion of the randomness of the Bernoulli map iterates therefore applies equally well to the behavior of the a = 4 attractor set of the logistic equation. 

Some or all of the vertices in each fragment may be representative of a water molecule. The trace of each fragment may be mapped onto a two-dimensional grid (Figure 3.1c). This trace is equated with the mapping of a cellular automaton von Neumann neighborhood. The cellular automata transition rules operate randomly and asynchronously on the central cell, i, in each von... 

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