Dynamic Database by Inconsistency and Morphogenetic Computing

Abstract

Since Peter Chen published the article Entity–Relationship Modeling in 1976, Entity-Relationship database has become a hot spot for research. With the advent of the big data, it appears that Entity-Relationship database is substituted for attribute reduction map structure. In the big data we have no evidence of the relationship but only of attributes and maps. In this paper we give an attribute representation of the relationship. In fact we assume that any entity can be in two different attributes (states) with two different values. One is the attribute that sends a message that we denote as e₁ and the other is to receive the message that we denote as e₂. The values of the attributes are the names of the entities. A relationship is a superposition ae₁ + be₂ of the two states. When we change the values of the states we change the database. When we change the two states in the same way we have isomorphism among database, and when we change the two states in different way we have isomorphism with distortion (homotopic transformation). Given a set of independent data base we can generate (compute) all the other data base in a dynamical way. In this way we can reduce the database that we must memorize. Because we are interested in the generation of the form (morphology) of database we denote this new model of computation as morphogenetic computing.

Appendices

Mathematical Appendix

1.1 Morphogenetic as Isomorphis with Distortion or, Weak Equivalence or Homotopy in Database

To understand the previous morphogenetic computing, we show different examples of the particular transformations denoted as isomorphism with deformation or homology and so on.

In the geographic database, there are two principal data of the earth geography. The first is located on a sphere and the other is the projection of the geography information on a plane. Now we know that the two databases are related but we also know that they are not isomorphic because when we project one image from the sphere, the new image on the plane has a distortion. In fact, in Map Projection Distorts Reality (topological similarity) a sphere is not a developable solid, and transfering from 3D globe to 2D map must result in the loss of one or global characteristics. Figure 15 shows the earth grid and features projected from sphere to a plane surface. Figure 16 shows the distorted map of the earth by planar projection.

Fig. 15.

The projection of a sphere on the plane

Fig. 16.

Distorted map of the earth by planar projection

The distance is equidistance between sphere and plane projection. All the other elements are distorted. Given a path on the sphere, we project the initial point and the final point of the path into the plane. The path on the plane changes in a different way from one part of the sphere to another. The same path at the north pole has a little dimension but at the equator it is more or less the same on the sphere. We have no global transformation that changes paths from the sphere to the plane. We have only local transformation that changes point by point in the sphere. This is the gauge transformation due to the impossibility to have the curvature property of the sphere into the plane. When we move from the sphere to the plane, we lose properties because the plane has not curvature.

Formal Description of the Morphogenetic Computation

Morphogenetic computing is more than the abstract theory. In this part we make the formal morphogenetic computation on the dynamic relationships in database.

Given the relationship matrix R that reflects the from/to connection among entities.

$$ R = \left[ {\begin{array}{*{20}c} {r_{11} } & {r_{12} } & {r_{13} } & \ldots & {r_{1n} } \\ {r_{21} } & {r_{22} } & {r_{21} } & \ldots & {r_{2n} } \\ {r_{31} } & {r_{32} } & {r_{33} } & \ldots & {r_{3n} } \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ {r_{n1} } & {r_{n2} } & {r_{n3} } & \ldots & {r_{nn} } \\ \end{array} } \right] $$

Then general operators A and B are created. Now the isomorphism with deformation is given by the formal expression (A.1).

$$ \begin{aligned} & [\left[ {\begin{array}{*{20}l} 1 \hfill & 0 \hfill & 0 \hfill & \ldots \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & \ldots \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & \ldots \hfill & 0 \hfill \\ \ldots \hfill & \ldots \hfill & \ldots \hfill & \ldots \hfill & \ldots \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & \ldots \hfill & 1 \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {a_{11} } \hfill & {a_{12} } \hfill & {a_{13} } \hfill & \ldots \hfill & {a_{1n} } \hfill \\ {a_{21} } \hfill & {a_{22} } \hfill & {a_{23} } \hfill & \ldots \hfill & {a_{2n} } \hfill \\ {a_{31} } \hfill & {a_{32} } \hfill & {a_{33} } \hfill & \ldots \hfill & {a_{3n} } \hfill \\ \ldots \hfill & \ldots \hfill & \ldots \hfill & \ldots \hfill & \ldots \hfill \\ {a_{n1} } \hfill & {a_{n2} } \hfill & {a_{n3} } \hfill & \ldots \hfill & {a_{nn} } \hfill \\ \end{array} } \right]e_{1} \\ + & \left[ {\begin{array}{*{20}l} {b_{11} } \hfill & {b_{12} } \hfill & {b_{13} } \hfill & \ldots \hfill & {b_{1n} } \hfill \\ {b_{21} } \hfill & {b_{22} } \hfill & {b_{23} } \hfill & \ldots \hfill & {b_{2n} } \hfill \\ {b_{31} } \hfill & {b_{32} } \hfill & {b_{33} } \hfill & \ldots \hfill & {b_{3n} } \hfill \\ \ldots \hfill & \ldots \hfill & \ldots \hfill & \ldots \hfill & \ldots \hfill \\ {b_{n1} } \hfill & {b_{n2} } \hfill & {b_{n3} } \hfill & \ldots \hfill & {b_{nn} } \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {r_{11} } \hfill & {r_{12} } \hfill & {r_{13} } \hfill & \ldots \hfill & {r_{1n} } \hfill \\ {r_{21} } \hfill & {r_{22} } \hfill & {r_{23} } \hfill & \ldots \hfill & {r_{2n} } \hfill \\ {r_{31} } \hfill & {r_{32} } \hfill & {r_{33} } \hfill & \ldots \hfill & {r_{3n} } \hfill \\ \ldots \hfill & \ldots \hfill & \ldots \hfill & \ldots \hfill & \ldots \hfill \\ {r_{n1} } \hfill & {r_{n2} } \hfill & {r_{n3} } \hfill & \ldots \hfill & {r_{nn} } \hfill \\ \end{array} } \right]e_{2} ]\left[ {\begin{array}{*{20}c} {name_{1} } \\ {name_{2} } \\ {name_{3} } \\ \ldots \\ {name_{3} } \\ \end{array} } \right] \\ = & (Ae_{1} + BRe_{2} )D \\ \end{aligned} $$

(A.1)

When A = B, we have (A.2).

$$ (Ae_{1} + ARe_{2} )D = A(e_{1} + Re_{2} )D{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} $$

(A.2)

In this case, the input graph and output graph are the same but names of the entities D change. So the morphogenetic result is an isomorphic database where the structure is the same but the names change.

When A ≠ B and B = A C we have (A.3).

$$ (Ae_{1} + BRe_{2} )D = (Ae_{1} + ACRe_{2} )D = A(e_{1} + CRe_{2} )D{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} $$

(A.3)

This appears to be an isomorphism but with correction by C.

Example 1.

For relationship R1 shown Fig. 17, we have the representation (A.4).

Fig. 17.

Relation R₁

$$ R_{1} D = (\left[ {\begin{array}{*{20}l} 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right]e_{1} + \left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]e_{2} )\left[ {\begin{array}{*{20}c} {class} \\ {classroom} \\ {enrollment} \\ {teacher} \\ {student} \\ \end{array} } \right]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} $$

(A.4)

Given the permutations

$$ A = \left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right],\;B = \left[ {\begin{array}{*{20}l} 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ \end{array} } \right] $$

$$ \begin{aligned} & R_{2} = Ae_{1} + BR_{1} e_{2} = (A\left[ {\begin{array}{*{20}l} 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right]e_{1} + B\left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]e_{2} ) \\ = & (\left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right]e_{1} + \left[ {\begin{array}{*{20}l} 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]e_{2} ) \\ = & (\left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right]e_{1} + \left[ {\begin{array}{*{20}l} 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]e_{2} ) \\ = & \left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right]e_{1} + \left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]e_{2} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \\ \\ \end{aligned} $$

(A.5)

The inconsistent Fig. 18 shows the mixed names in one entity.

Fig. 18.

The inconsistence after permutation

Particular case for the diagram

$$ \begin{aligned} & 1)\;\;A = B = A \\ & (\left[ {\begin{array}{*{20}l} 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right]Ae_{1} + A\left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]e_{2} ) = A(\left[ {\begin{array}{*{20}l} 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right]e_{1} + \left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]e_{2} ) \\ & {\text{For}}\;A = \left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right] \\ \end{aligned} $$

We have the coherent graph Fig. 19.

Fig. 19.

The coherent graph

Multi-dimensional transformation is shown in Fig. 20.

Fig. 20.

Multi-dimension space reference (e ₁₁ , e ₁₂ , e ₂₁ , e ₂₂ )

The two dimensional reference space (e ₁₁ , e ₂₁ ) can be expanded in four dimensional space reference (e ₁₁ , e ₁₂ , e ₂₁ , e ₂₂ ). From the same entities D we have the first relationship structure, form or topology denoted R₁₁.

$$ D = \left[ {\begin{array}{*{20}c} {class} \\ {classroom} \\ {enrolment} \\ {teacher} \\ {student} \\ \end{array} } \right]{\text{ and R}}_{11} = \left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right] $$

Now with A and B we change the “out” state and “in” state and generate the new relationship R₁₂. In Fig. 6, we have the principal relation R₁₁ as the father of the all the other relations. The father generates two children, one is R₁₂ and the other is R₂₁. The children R₁₂ and R₂₁ join to generate R₂₂. This is the morphogenetic process. Figure 6 is an aggregation of different databases with different states not only two of “out” and “in”. The building of the new meta database gives us the instrument to move dynamically from one database to another or to separate one big database into more simple database one connected with the other by homotopic transformation or isomorphic transformation with deformation.