Dialectical Logic K-Model : A Mathematical Model for Machine

An axiom system for dialectical logic K-model which based on Kirchhoff energy-method is established by author in the paper. The author describes that subjective-laws is the mirror imagine reflected from objective-laws and defines that the three-step which named by sensation, abstraction and thinking in artificial intelligence. At same time, describes that axiom system for dialectical logic K-model, in which contains such as logic-variable energy conservation law, Mozi’s principle( mini-max principle) and forbidden law, etc. In the axiom system also contain such as a continuous true-value-function system valued on interval ] 1 , 1 [   and the K-graph for logic-variable. And describes the giving value method by matrix based on K-graph K G satisfied Kirchhoff laws to the logic variable. The author describes simply the linear and nonlinear logic variable system. And describes simply the logic variable involved three-dimension Euclidean space and topology networks space separately. Dialectical logic K-model would supply an computation algorithm idea for machine so that the machine is able to think by dialectical logic method, thus an important information-treated method maybe the dialectical logic.


Introduction
Objective world is consist of five factors which are MATTER, ENERGY, INFORMATION, SPACE and TIME.Among these factors, there are many inner or outer laws of causation can be called as objective-laws, also can be called as objective-logic.Corresponding to objective-laws, there is a mirror image within machine of artificial intelligence, which can be called as subjective-laws, also can be called as subjective-logic.The machine of artificial intelligence only holds objective-laws through subjective-laws, or with another word, only holds objective-logic through subjective-logic.Subjective-logic is built on infinite times circulation by the three-step which named Sensation, Abstraction and Thinking.Sensation is input-ability to machine from objective information with sensors.Abstraction is conception -classification-ability to sense information and to name it.Thinking is ability of machine researching from conceptionclassification information to get the subjective-logic.
The dialectical logic was established by Hegel, a great philosopher (Bencivenga, E., 2000;Kosok, M., 1966).By the opinion of dialectical logic, the objective laws are dynamical, varied and driven by the causality law.Its main opinion is: contradictions are the causes of object's variety.The establishment of mathematical model for dialectical logic is the necessary condition in order to make that the dialectical logic is compute-able in machine.
Dialectical logic K-model established in the paper would supply a computation-idea for machine so that the machine is able to think by dialectical logic method, so an important information-treated method would be the dialectical logic.The dialectical logic K-model combined with three main recognition technologies (image recognition technology, voice recognition technology, text recognition technology), data base technology and machine-self-programming technology will take a great progress to artificial intelligence.

Conception and the Conception-dimension
2.1.1.The Conception is the Name Which be Named the Object-thing's Main Property and neglected its secondary properties.Conception is the basis of artificial intelligence.conception; WHITE HORSE is a two-dimension conception; RUNNING WHITE HORSE is a three-dimension conception; ONE RUNNING WHITE HORSE is a four-dimension conception; etc.

Associate Database
The three main recognition technologies will setup three data bases, ( although there are other data base, for example physical data base and chemical data base, etc. ) These three data bases are associated by the conception-dimension.

Definitions
Denote objective-domain by B, and subjective-domain by A; define research operator  (research operator is an arithmetic to get subjective-logic from objective-logic) and inverse research operator ∇ −1 ( inverse research operator is an arithmetic which make subjective-logic to objective-logic in order to check its true or false) and thus

Research Error Alternating-convergence Law
To every correct logic-transformation F , always make 0 in the formulation as above C=+1, if j=2p+1; or C=-1, if j=2p, p=0,1,2,⋯ In fact, incorrect logic-transformation * F can bring the researching divergence.The faster the convergence of the operator j F is, the brighter the operator j F is. )) imagine-logic corresponding to emotional quotient.
The imagine-logic would be forbidden to put into machine, because the imagine-logic will make machine to produce emotional quotient so that the autonomous-mind will belong to the machine.The artificial intelligence with the autonomous-mind will not like to be "a willing tool" for human beings, so "a new creation" will be created, of cause this is harmful to human beings.

Logic Variable Infinite-separable-characteristic Law
The logic variable is infinite-separable, i.e.

Logical Inductive and Deductive Method Theorem
For cycle research operator arithmetic 2.6.3.(deductive) so that as above is true to all   j .Proof: through finite to prove infinite, there always exists error-function based on the 2.2.3.research error alternating-convergence law, if the 2.6.2.(hypothesis) is false, then producing the false to the 2.6.3.(deductive).Thus the truth to every step of j F would be carefully checked.
Proof is over.

Definitions of Logic Algorithm(Boolean operator)
The proof of formulation(1) is over. ) The proof of formulation(2) is over.
Remark1: As shown above , when time t is degenerated into a constant and destroy the contradictions in the ormulations, then the formulation (1) and (2) will be degenerated from the continuous true value function Boolean algebra valued on interval [-1,+1] into the mathematical model of formal logic, i.e. binary Boolean algebra.

Logic True Valued Function Composition Theorem
For logic true valued function , as below there is a composition theorem: as similar as above, also in which the possible augment root and complex root will be removed out.
in which select a maximum or minimum of pure function.
Proof:obviously based on definition of logic arithmetic.
Proof is over.

Linear Logical System And Non-Linear Logical System
According to the 2.2.1.as shown above, obviously we have the formulation as below Remark 2: The strong action influence by human beings to the object would produce non-linear logic; the weaker action influence or zero action influence by human beings to the object would produce linear logic.Because of, the error action influence of human beings would produce the non-linear properties.An example for linear logic is science, and an example for non-linear logic is engineering or technology.As similar as above , another example for linear logic is original ecology system, and another example for non-linear logic is a planted system in agriculture .

Expression by graph
In formulation (3), lower reaches node is left and upper reaches node is right, in the opposite direction would sign a negative in the front of ) ( e ij t .
In the graph K G , every cycle ij W is satisfied by

Work And Energy Law
In the graph G is a no-loop, no-multiple edge directed simple graph; 2.11.2.3.In the graph K G , n+1 positive nodes construct a perfect sub graph, another n+1 negative nodes construct 2.11.2.4.Node 1 V  connect only to node 1 -V , positive node  q V connect only to negative node - q V which right subscript is equal.2.11.2.5.The power-function of node 2.11.2.6.In the graph K G , other nodes and edges all be defined power-function ) ( v i t , power-function ) ( e ij t , flow-function ) ( f ij t and contradiction-function ) ( R ij t on them, these are all satisfied by Kirchhoff laws as above.
. The true value is given by the formulation as below then the positive factor and negative factor which formed the two sides of the contradiction are balance then the positive factor in the contradiction is superior factor; then the negative factor in the contradiction is superior factor.

Expression by power function matrix
at the first column is the transposed vector of the first row ; c. the other matrix elements are: , if edge ij e exist; otherwise 0 e ij  .Thus the power function matrix is 2.12.2.Flow function matrix

KF M
The matrix elements of the flow function matrix KF M are a. at the first row is the node vector

KC M
The matrix elements of the contradiction function are a. at the first row is nodes vector 2.12.4.Properties of the matrix The elements of the matrices is negative i.e. its direction of edge is wrong in supposed supposition;

 
x,y,z;t A n i 

Acted on Euclidean Space
One kind of game is involved the three-dimension Euclidean space, such as football, basketball and tennis,etc., even wars by human beings.This kind of game is acting on the Euclidean space, its true value function is not only depended on time t, but also depended on the position in Euclidean space.Denote the three-dimensions of Euclidean space by x , y , z , then the logic variable is denoted by   , in them , the Δt denote the sample time interval.Remark3: The detail about the three kinds of games and the discrete time variable system will be explained in succession papers by author.

Fig
Fig.1 the structure of graph the first row is the power function value the first column is transposed vector of the first row; c.The other elements of flow function matrix function input or output the node V i. See the 2.11.1.3.Then the flow function matrix  ; b. at the first column is transposed vector of the first row; c. the other elements of contradiction function matrix M KC are: If the edge ij e exist, then the matrix element the graph G K is compute-able in machine.Now some properties about these matrices is explained as below: a the main diagonal element of these three matrices except first row and first column all are zero; b because of the graph K G is a directed graph, then these three matrices except their first row and first column are negative symmetrical matrix based on the main diagonal line as the axis of symmetry; c Another kind of game is involved topological network space w, such as railway, highway, river channel, etc. Follow 2.13, denote the logic variable of this kind game by And the third kind of game is involved topological network space only by relative time step number s not by absolute time t , such as games on chessboard systems.Follow 2.13.denote the logic variable by ;