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CELLULAR AUTOMATA; ALGEBRAIC FRACTALS
The goal of this paper is the description of a theoretical instrument based on a mathematical procedure that is able to provide technical solutions required by the development of technological applications of the cellular automata.
Different scientists starting with computer age initiated cellular automata theory. This theory pretends that universe is information and that every structure in universe is formed by a minimal computer. From strings to living creatures these minimal computers that form cellular automata induce a universe of information. Although this theory is accepted there is no description yet concerning the structure of cellular automata information structures. The main characteristics of information are:
-Is a computer unit able to connect information in a predictable way.
-Is complex revealing different components in a different situation.
-May connects to other information contained in cellular automata.
-Is able to develop to different stages of complexity
-Can be described in unitary style
-Is universal
-Can characterize any phenomena including life.
Theoretical Approaches
An attempt to describe these cellular automata information units is made in this paper using the concept of feedback cycle, and algebraic fractals that are developed in a theoretical approach as follow:
A feed back cycle can be characterized mathematically in different ways. All are isomorphic. If we consider numbers if we have three numbers a, b, c, such as a*b*c=1 were* is multiplication, then a*b; a*c; and b*c form the second set of generators, their produce being also 1. For example 2/5; 5/3 and 3/2 have the produce equal to1. 2/5* 5/3=2/3; 2/5*3/2=3/5; 5/3*3/2=5/2 and2/3*3/5*5/2=1. More than that 2/3*3/5=2/5; 2/3*5/2=5/3; 3/5*5/2=3/2 obtaining the first set of numbers. So the first set generates the second set, and reciprocally.
The same property can be obtained with functions. In this case * means composition of functions, and 1F means the unitary function f(x)=x. The property will be in this case the following theorem:
If F1;F2; and F3 are functions such as F1*F2*F3=1F than F1*F2;F1*F3 and F2*F3 are functions with the same property, and each set generates the other one. Demonstration is trivial as long as F1*F2= [F3] were [F3] is the inverse function of F3.
Definition
A feedback cycle is a double set of mathematical objects connected with operators with the following properties:
-Any two objects of the first set generate one object of the second set, and any two objects of the second set generate one object of the first set. Generating objects is determined by the operation induced on the set of objects.
-Operators forms also a double set of generators with the same property.
Properties:
-Only six generators and six operators form a feedback cycle. Composing operators one after another in sets of three we obtain the neutral element for the operation.
This kind of relationship describes a closed feed back, an-noticed in mathematics because it disappears in the context due to its neutrality.
Universality Property
If we have a group (A+) with additive notation than the following set form a feedback cycle:
a-b; b-c; c-a generating by operating inside the group: a-b+b-c=a-c; b-c+c-a=b-a; c-a+a-b=c-b. We have obtained so, a feedback cycle inside the group. In a group we find numerous feedback cycles, cycles, or circuits. Any mathematical structure containing a group contains also feedback cycles. We find feedback cycles in rings, manifolds, algebra, vectorial spaces, etc.
We characterize as a feedback cycle a set of six elements such as three of them determines the other three and reciprocal. We say that a set of six elements is a cycle if operating the elements in order we obtain the neutral element of the group.
We say that a set of six elements is a circuit if composing them in order we obtain a different element from the neutral element of the group.
The relationship between these kinds of objects characterizes the grammar of internal connections. Due to universality property the relationship between feedback cycles, cycles, and circuits
Definition
-A cycle is a set of 6 elements that gives 1 when are operated with the operation of the group.
-A circuit is a set of 6 elements that gives on operating a different value than 1.
In this paper there is an example of such kinds of relationships. Universality property may assure another kind of mathematical approach able to detect informational connections in different kinds of structures that are described in a mathematical way. These kinds of approaches might be used for mathematical modeling in living structures, chemistry, physics, or different mathematical domains.
Figure 3: A Feedback Cycle in Modeling Vision
These structures will characterize more complex structures connected in feedback cycles. Feedback cycles are seen from different points of view. Functions describing feedback cycles are connected with vectors in a feedback circuit. Structures are connected to vertexes in the circuit. Both these kinds of structures can give a self-determination, or self-generating system, characterizing a well-balanced feedback circuit. They can also give opened circuits. Applying vectors characterized by functions to structures we will obtain a characterization of structure’s functioning. This kind of modeling characterizes the function of a global structure from the same perspective and from the same level of perception.
To understand how a complex system is functioning regarded from different perspective or levels of perception we need another approach. This will be done using structural algebraic properties of the space that may create different levels of complexity on the same system of generation.
Feedback Substructures
There are two sets of generators:f1,f2, f3 and g1, g2, g3 such as f1.f2=g3, f1.f3=g2, f2.f3= g1 and g1.g2=f3, g1.g3=f2, g2.g3=f1.
These properties may develop a characteristic metric, in our case anarmonic rapport. We have here again a feedback cycle. If we have four points A, B, C, D on the same line, than we gave the following list of anarmonic rapports:
AB/AD-CB/CD; DC/DB-AC/AB; AD/AC-BD/BC
We can notice that the product of the left side members is 1 the same for the right side members. In this paper is studied the example of a group of automorphisms describing the feedback given by sets of four points. Each automorphism include such a feedback described by four points but expressed in analytic way.
So if these rules have universality in their construction it will be normal to find them in different other structures like biological or social structures.
We have the following connections that might be used in determination of steps of demonstration for fractal varieties:
A-Establish a double set of generators able to generate each-other
B-Describing the interior structure of symmetries
C-Describing the metric induced by symmetries
D-Finding the transformations that preserve the metric
E-Find the invariant structures to transformations
F-Find the geodesics produced by these invariant structures
The last objects might produce a new system of generators
Network Of Information
What is the connection between mathematical feedback cycle and feedback cycles used in the practical models (data entrance, processing data; using data, exit data; processing exit data and using exit data)? The answer is different from the two points of view. If we take the vectors connected to the points of the feedback cycle we will obtain a system in which the informational values of the points are enriched by composing the vectors functions of the cycle from the first one to the point characterizing one characteristic of the feed back. Composing the vector between data entrance and processing entered data, the information entered in processing entered data contains the image of the function between these two points. The information entered in using entered data contains the composed information of two functions F1 and F2. F1 connecting entered data to processing entered data, and F2 connecting processing entered data to using entered data. In order to apply on a practical case we need to consider both; the cycle of generators given by points and the cycle given by vectors (functions). This mathematical modeling part uses different graphs structures that respect algebraic rules that will be developed in this chapter in algebraic fractal’s description.
We need to consider the points of the cubic network in algebraic or graphs language considering that this level of structuring is minimal for a brief description of complexity. Any derived and more complex description will respect the results obtained at categorical level. Rotations in the cube will destroy informational feedback cycles on different components, preserving some of them. If the balance of feedback cycles will be destroyed by induced information, rotations might rebalance some of the cycles. So changes become necessary in informational metabolism. All these characteristics describe only the complexity of one stage working with similar phenomena. Across stages description needs a different approach. To go from individual characteristics towards network characteristics needs also a between stages approach. This means to develop opened cycles instead of closed cycles, and to determine mathematical properties as a consequence of different generators obtained using the same patterns, and characterizing different fractal varieties stages. Most of the demonstrations describing cubic structures don’t need too much effort, but are difficult to be represented because of the multidimensional image. A great problem is to find an example of functions able to describe the model. Most functions are not commutative to composition so all the theory must transfer to non-commutativity.
Non-commutativity
If we take the subgroup of automorphisms of one- dimensional projective space, we discover the following group of functions: f1=x; f2=1-x; f3=1/x; f4=1-1/x; f5=1/1-x; f6=x/x-1
The table of composition of these functions follows
Table 3
Table of Composition for the Subgroup of Automorphisms of the Projective Space
Function / F1 / F2 / F3 / F4 / F5 / F6F1 / F1 / F2 / F3 / F4 / F5 / F6
F2 / F2 / F1 / F4 / F3 / F6 / F5
F3 / F3 / F5 / F1 / F6 / F2 / F4
F4 / F4 / F6 / F2 / F5 / F1 / F3
F5 / F5 / F3 / F6 / F1 / F4 / F2
F6 / F6 / F4 / F5 / F2 / F3 / F1
The following list of functions form a feedback cycle: f5; f6; f2; f4; f3; f2
We can notice that f2*f6=f5; f2*f4=f3; f6*f4=f2 and f2*f3=f4; f2*f5=f6; f3*f5=f2. More than this the following table will give an f1result on composition for any row or column.
Table 3
Any Line or Column is Formed by Feedback Cycles Obtained by Circular Permutations
F2 / F3 / F4 / F2 / F6 / F5F3 / F4 / F2 / F6 / F5 / F2
F4 / F2 / F6 / F5 / F2 / F3
F2 / F6 / F5 / F2 / F3 / F4
F6 / F5 / F2 / F3 / F4 / F2
F5 / F2 / F3 / F4 / F2 / F6
The lines are obtained by circular permutations of the first line. This is an implicit argument for rotations in cubic structures.
There are other feedback cycles contained in this group,: f2,f3,f4,f2,f6,f5 ; f2,f3,f4,f2,f6,f4 ; f2,f4,f6,f2,f4,f6 ; f2,f3,f4,f3,f6,f5 ; f2,f5,f6,f3,f4,f6 ; f2,f4,f6,f3,f4,f6 ; f2,f6,f5,f3,f6,f5 ; f2,f5,f6,f2,f5,f6 ; f2,f6,f5,f2,f3,f5 ; f2,f6,f5,f3,f6,f4 ; f2,f3,f5,f3,f6,f4 ; f2,f6,f4,f2,f3,f5 ; f2,f3,f4f6,f3,f4 ; f2,f4,f3,f2,f4,f3 ; f2,f6,f4,f6,f3,f5 ; f2,f3,f4,f6,f3,f5 ; f2,f4,f3,f6,f5,f6 ; f2,f4,f3,f6,f5,f3 ; f2,f3,f5,f2,f3,f5 ; f2,f5,f3,f6,f5,f3 ; f3,f4,f6,f3,f4,f6 ; f2,f4,f2,f3,f5,f6 ; f2,f5,f2,f6,f4,f3 ; f3,f5,f6,f3,f5,f6 ; and their circular permutations. We can find other kind of structures which don’t form feed back cycles, but which are involved in generative structures: f2;f6;f3 and f5;f4 and f1.In this case the first set generates the second set and the second set generates f1.We can find also sets of three functions generating each other by a specific composition (f2;f3;f4: f2*f3=f4; f2*f4=f3; f4*f3=f2) . Other triplets are: f2,f3,f5; f6,f3,f4; f5,f3,f6; f6,f5,f2; f2,f4,f6. There are component parts of any feedback cycle formed by automorphisms. All these structures have informational values and can describe and model different phenomena.
The Next Step, Structure of Information
Looking more attentively to feedback cycles we may notice that there are four categories of cycles with symmetric proprieties.
In each set of cycles we may describe compositions and generators. The following examples describe the vectors between objects. Both objects and vectors form feedback cycles (See Figure 6).
Figure 6
Automorphisms Have Four Kinds of Structural Behavior
Sets are counted in the same order and noted A, B, C, D obtaining four composed feedback cycles (See Figure 14)
Using the two kinds of symmetries A, and B being generated in a different way than C and D; and the symmetry between A and D; B and C we are now on the same position as on the beginning. There are two sets of generators able to determine a feedback cycle. Each set (one of each A, B, C, D category arranged in symmetric way), has two kinds of symmetries, and may develop substructures with relevance in the new space metric. E and F were obtained composing the other feedback cycles. E was obtained composing A and D. F was obtained composing D and C. The composition between A and C is irrelevant, obtaining one part of B. Each one of these complex cycles might be connected to a geometrical transformation. (Fig 13, A,B,C,D).
Figure 7
The Structure of Elements From the New Stage of Complexity Develops Transformations as Internal structural Components