Implicit deterministic structures

Despite being notoriously difficult to analyze formally, the behavior of general CA rules is nonetheless often amenable to an almost complete mathematical characterization. In this section we look at a simple method that exploits the properties of certain implicit deterministic structures of elementary one-dimensional rules to help determine the existence of periodic temporal sequences, rule inverses and homogeneous states. Additional details appear in [jen86a] and[jen86b]. 

We recall that elementary CA are defined by nearest-neighbor rules of the form 

While the explicit deterministic structure of such rules - by which me mean the set of sites whose value is fixed by values of other sites - appears as [ai-i t),ai t), ai+i t) Gift + 1), there may also exist an implicit underlying deterministic structure not immediately obvious from the definition of the rule itself. It may be 

It is also easy to see what rules are of type specified by conditions given in equation 5.11. Since we must have 

---

Finally, we should refer to situations where both independent and response variables are subject to experimental error regardless of the structure of the model. In this case, the experimental data are described by the set (yf,x,), i=l,2,...N as opposed to (y,Xj), i=l,2,...,N. The deterministic part of the model is the same as before however, we now have to consider besides Equation 2.3, the error in Xj, i.e., x, = Xj + ex1. These situations in nonlinear regression can be handled very efficiently using an implicit formulation of the problem as shown later in Section 2.2.2... 

Engineers, physicists and, in general, scientists dealing with real phenomena nsnally have to deal with uncertainty which often raises serious theoretical and computational difficulties. With the aim of reducing these complications that frequently make many problems irresolvable, standard methods have been developed in structural analysis assuming implicitly that all involved parameters are deterministically known. This remains is an oversimplification of real conditions because parameters are only partially known. Due to economical and technical reasons, only few problem parameter measurements are available if not simply intrinsically uncertain. The uncertainty of structural problems may afflict many involved factors, such as dynamic loads intensity, material mechanical parameters and geometrical configurations, all commonly considered as deterministic in standard analysis. 

---