Deterministic rules

Fig. 3.16 Space-time pattern of fc — 2, r — 1 rule R18 kink-nites (i.e.. neighboring a = 1 site.s) are indicated in solid black. Notice the stochastic-like kink trajectories, despite the strictly deterministic rule.

Figure 3.31 shows sample evolutions for p = 0, 1/4 and 3/4. The space-time pattern for p = 0 rapidly settles into an ordered state consisting of checkerboard-pattern domains, separated by two-site kinks once formed, the kinks remain locked in place. As p is slowly increased, these kinks begin to undergo annihilating random walks, much like the ones we saw earlier in the evolution of (the deterministic) rule R18. Their density decreases like pkink Grassberger, et.al, observed... 

In the case of the threshold rules defined in this section, we must consider sequential iterations of deterministic rules. Also, the choice of spins that may change state is not random but is fixed by some random permutation of the sites on the lattice. Such rules may be shown to correspond to spin glasses in the zero-temperature limit. 

As before, we set po = 0 to have o = 0 as the single absorbing state.Table 7.2 shows the eight deterministic-rule corners of the (pi,p2,p3)-cube. Figure 7.7 shows three slices of the phase-diagram of this system. We see that there always exists... 

The qualitative nature of the phase boundaries, however, is not simply related to the behavioral class of the deterministic rule corners. For example, while figure 7.7-c shows a phase transition to a class-3 rule, figures 7.7-a and 7.7-b show that the boundaries end at the class-3 rule. Similarly, while in figure 7.7-a the phase transition ends at a class-2 rule, there is only the absorbing stationary state close to the class-2 rule in figure 7.7-c. 

For small enough values of p so that pf p) < p for all 0 < p < 1, p = 0 will be the only fixed point. As p increases, there will eventually be some density p for which pf p ) > p in this case, we can expect there to be nonzero fixed point densities as well. Qualitatively, the mean-field-predicted behaviors will depend on the shape of the iterative map. If / has a concave downward profile, for example (i.e. if/" < 0 everywhere), then, as p decreases, Poo decreases continuously to zero at some critical value of p = Pc- Note also that the iterative map /jet for the deterministic rule associated with its minimally diluted probabilistic counterpart is given by /jet = //p-... 

We begin by recalling that Conway s original deterministic rule is an outer-totalistic (code OT224) k = 2 rule defined on the two-dimensional Moore neighborhood ... 

To be more precise, P is the probability that a site having exactly k living (i.e. (7=1) neighbors either survives a = S and the given site had previously been alive) or is born (a = P and the given site had previously been dead). Conway s deterministic rule, of course, is reproduced by setting Pg = Sk,3 and P = Sk,2 + 6k,3-... 

Forget for a moment that you know that figure 12.11 shows the space-time pattern due to a well defined local deterministic rule, and that the underlying universe really consists of nothing but bits. Suppose you are told only that this figure represents some sort of alien physics, and that you may see as many different samples of this alien world s behavior as you wish. How are you to make any sense of what is really going on ... 

Cellular automata are simple mathematical idealizations of natural systems. They consist of a lattice of discrete identical sites, each site taking on a finite set of say integer values. The values of the sites evolve in discrete time steps according to deterministic rules that specijy the value of each site in terms of the values of neighboring sites. Cellular automata may thus be considered as... 

Wolfram has elaborated on this description elsewhere [11,12], As we shall see, the restriction to deterministic rules is unnecessary, and we shall in fact make extensive use of probabilistic rules in our studies of real physical and chemical systems. 

When such a transition rule is applied, the state of each cell and, therefore, of the entire system varies completely unpredictably from one cycle to the next (Figure 6.9), which is unlikely to be of much scientific interest. No information is stored in the model about the values of the random numbers used to determine the next state of a cell, thus once a new pattern has been created using this rule there is no turning back All knowledge of what has gone before has been destroyed. This irreversibility, when it is impossible to determine what the states of the CA were in the last cycle by inspecting the current state of all cells, is a common feature if the transition rules are partly stochastic. It also arises when deterministic rules are used if two different starting patterns can create the same pattern in the next cycle. 

Deterministic rules, or a combination of deterministic and random rules, are of more value in science than rules that rely completely on chance. From a particular starting arrangement of cells and states, purely deterministic rules, such as those used by the Game of Life, will always result in exactly the same behavior. Although evolution in the forward direction always takes the same course, the CA is not necessarily reversible because there may be some patterns of cells that could be created by the transition rules from two different precursor patterns. 

The value of each site evolves according to the same deterministic rules. 

As we shall see, the fourth characteristic can be modified to include probabilistic rules as well as deterministic rules. An important feature sometimes observed in the evolution of these computational systems was the development of unanticipated patterns of ordered dynamical behavior, or emergent properties . As Kauffman has expressed it,22 Studies of large, randomly assembled cellular automata. .. have now demonstrated that such systems can spontaneously crystallize enormously ordered dynamical behavior. This crystallization hints that hitherto unexpected principles of order may be found, [and] that the order observed may have significant explanatory import in [biology and physics]. This proposal has borne considerable fruit, not only in biology and physics, but also in chemistry. Readers are referred to reviews for applications in physics23-26 and biology 27 selected physical and chemical applications have been reviewed by Chopard and co-workers.28... 

Machine learning may produce an entire spectrum of decision rules in terms of their predictive power. At one end of spectrum we have strong, or quasi-deterministic, rules with error rates close to zero, which very rarely fail and are almost always valid. At the other end, we have very weak, or probabilistic, decision rules with error rates close to one, which usually fail and are very rarely valid, and then only for very few special situations. In the area of machine learning, such rules are sometimes called heuristics. Unfortunately, there is no clear division between decision rules and heuristics, and our judgment must be used to distinguish between them. 

One of the most widely set of deterministic rules was developed by Dennis Klatt [253], [10]. Although designed for use with die MlTalk formant synlliesiser, these rules can be used with any synthesis technique. 

Let us consider the case of deterministic fractals first, i.e. self-similar substrates which can be constructed according to deterministic rules. Prominent examples are Sierpinski triangular or square lattices, also called gasket or carpet (in d = 2) and sponge (in d = 3), respectively, Mandelbrot-Given fractals, which are models for the backbone of the incipient percolation cluster, and hierachical lattices (see for instance the overview in Ref. [21]). In this chapter, however, we restrict the discussion to the Sierpinski triangular and square lattice for brevity. 

A one-dimensional cellular automation consists of a line of sites, with each site carrying a value (0 and 1 in the simplest case). The value of each position is updated in discrete time steps by an identical deterministic rule depending on the neighbourhood of sites around it. 

GA is a new parallel optimization search method which is different from the traditional optimization methods in the field of application. Goldberg [114] summarized the differences between GA and traditional optimization method as follows GA operates the code of the solution set not the solution set itself GA searches from one population, not a single solution GA uses the compensation information (fitness function), not derivatives or other complementation knowledge GA uses methods of probability, not the deterministic rule of state transition. 

The probabilistic information is an additional information to the deterministic information. So a difference in the evaluation is not possible. The probabilistic information helps to decide whether a deviation from a deterministic rule is acceptable. 

The purpose of the analysis is to evaluate the failure probability of a component when it is subject to a severe situation. The failure occurs when the loading L applied to the component exceed the strength S of its composing materials. L and S are estimated by a chain of complex thermal-hydraulics and mechanics calculations. In the deterministic assessment, a margin factor ME), which is equal to the ratio S L, must exceed a safety factor equal io 1.2. The deterministic rule to be complied with is thus ME >1.2. The failure probability is therefore defined as ... 

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