CAS model

While there is an enormous variety of particular CA models-each carefully tailored to fit the requirements of a specific system-most CA models usually possesses these five generic characteristics ... 

Coupled-map Lattices. Another obvious generalization is to lift the restriction that sites can take on only one of a few discrete values. Coupled-map lattices are CA models in which continuity is restored to the state space. That is to say, the cell values are no longer constrained to take on only the values 0 and 1 as in the examples discussed above, but can now take on arbitrary real values. First introduced by Kaneko [kaneko83]-[kaneko93], such systems are simpler than partial differential equations but more complex than generic CA. Coupled-map lattices are discussed in chapter 8. 

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. 

Chapter 9 provides an introductory discussion of a research area that is rapidly growing in importance lattice gases. Lattice gases, which are discretized models of continuous fluids, represent an early success of CA modeling techniques. The chapter begins with a short primer on continuum fluid dynamics and proceeds with a discussion of CA lattice gas models. One of the most important results is the observation that, under certain constraints, the macroscopic behavior of CA models exactly reproduces that predicted by the Navier-Stokes equations. 

It turns out, rather fortuitously, that if the desire is to merely obtain an overview of the general types of possible two-dimensional behaviors, then focusing only on T and OT- type rules is not really a restriction, as the set of all possible behaviors is well represented. Having said that, we should be quick to point out that if the desire is instead to study either a class of CA systems with a special set of behavioral characteristics or to find an appropriate CA model for a real physical system, specific rules and/or lattice connectivities and neighborhoods will have to be invented. For our brief introductory look in this section at generic two-dimensional behavior, however, we will be content to restrict ourselves (for the most part) to commentary on T- and OT type rules. 

In this section we introduce several CA models of prototypical reaction-diffusion systems. Such systems, the first formal studies of which date back to Turing , often exhibit a variety of interesting spatial patterns that evolve in a self-organized fashion. 

There exist many different CA models exhibiting BZ-like spatial waves. One of the simplest, and earliest, described in the next section, is a model proposed by Greenberg and Hastings in 1978 [green78], and based on an earlier excitable media model by Weiner and Rosenbluth [weiner46]. One of the earliest and simplest mathematical models of the BZ reaction, called the Orcgonator, is due to Field and Noyes [field74]. 

A recent example of a CA model of the immune response in AIDS is Pandley s four-cell model using interactions among macrophages (= M) containing parts of the virus on their surface, helper T cells (= H), cytotoxic T cells (= C) and the virus (= V) ([pand89], [pandQl]) ... 

Boolean Network with connectivity k- or N, )-net - generalizes the basic binary k = 2) CA model by evolving each site variable Xi 0,1 of according to a randomly selected Boolean function of k inputs ... 

Wiesenfeld, et. al. [wiesen89] compare the simplicity of the independent relaxation time interpretation of l//-noise fluctuations in the saudpile CA model to other recent models yielding 1// noise ... 

Hierarchical Structures Huberman and Kerzberg [huber85c] show that 1// noise can result from certain hierarchical structures, the basic idea being that diffusion between different levels of the hierarchy yields a hierarchy of time scales. Since the hierarchical dynamics approach appears to be (on the surface, least) very different from the sandpile CA model, it is an intriguing challenge to see if the two approaches are related on a more fundamental level. 

Conventional CA models are defined on particular lattice-networks, the sites of which are populated with discrete-valued dynamic elements evolving under certain local transition functions. Such a network with N sites is simply a general (undirected) graph G of size N and is completely defined by the NxN) connectivity matrix... 

This chapter is organized into two main parts. To give the reader an appreciation of real fluids, and the kinds of behaviors that it is hoped can be captured by CA models, the first part provides a mostly physical discussion of continuum fluid dynamics. The basic equations of fluid dynamics, the so-called Navier-Stokes equations, are derived, the Reynolds Number is defined and the different routes to turbulence are described. Part I also includes an important discussion of the role that conservation laws play in the kinetic theory approach to fluid dynamics, a role that will be exploited by the CA models introduced in Part II. 

In this section we introduce a mobile CA model of land combat called EINSTein, developed at the Center for Naviil Analyses for the US Marine Corps. We include a discussion of this model here becau.se it is an interesting blend of CA-like local dynamics and agent-based modeling techniques. 

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