Attribute Grid Computer Based on Qualitative Mapping for Artificial Intelligence

. new kind of Computer, Attribute Grid Computer based on Qualitative Mapping and the Mutual conversion relation between Probability and the conversion Degree Function is discussed in this paper.


Introduction
In recently years, with the breakthrough of Alpha GO and the neural network based on convolution in pattern recognition, Deep Learning has become a research hot topic. in fact, some of basic and very important problems in it have not been resulted yet.First of all, the basic operation of Neural Unit is Classification, so it is called a classifier in general textbooks, the recognition of the pattern is implemented in the neural network by an iterative algorithm, such that the function of pattern recognition of Deep learning by adjusting the connection weight parameters between different levels and different classifiers (neurons), and there are a lot of uncertainty, such as probability and fuzziness and so on.In [1,2], a new kind of Computer, Attribute Grid Computer(AGC) based on Qualitative Mapping(QM), it is shown that some of artificial methods such as Expert System, Artificial Neural Network and Support Vector Machine can be fused and unified together can be fused in the framework of qualitative criterion transformation of QM and AGC.The basic operation of QM is covering, its mechanism is the conversion from quantity of attribute into quality of attribute.What is the principle of pattern recognition?Why did the Neural Network and AGC can recognize a pattern?What relation between classification and covering is?Whether does there is any linking between the probability and fuzziness in ANN and AGC?In this paper, the qualitative the envelope of qualitative criteria is subdivided more detail, such that the probability of each classified sample falling into subdivision grid can be counted respectively.In this way, not only any classified samples can be recognized by the Grid-based GAC in detail, but also the indicate linking between the probability and the degree of (fuzzy) conversion can be given.
For the sake of discussion, first of all, let us give the definitions of qualitative mapping and Attribute Grid Computer.
2 Qualitative Mapping of Conjunction Property Judgment be the conjunction attribute of object u whose n factor attributes are ai (u), i=1,...,n, x=(x1,...,xn), the quantity vector of a(u), xiXiR, the quantity of a i (u), p i (u) the quality or property of a i (u), let , the mapping :X→{0,1} P u is called the Qualitative Mapping (QM) whose criterion is the [,], if for any xX, there is [,] and the conjunction property For conveniently discussion, we introduce the definition of trivial artificial neuron.Definition 2 Let ai(u) be the proterty of object u, i=1,...,n,xiXi are the quantitive attribute of ai(u).pij(u)is the jth qualitative attribute of ai(u).j=1,…,m, |[ij,ij]Xi is the qualitative criterion of pij(u), ={[ij,ij]} is the cluster of qualitative criterion, which satisfies: [ij,ij][il,il]=,l=1,...,m,lj, and be the conjugate property of ai(u), x=(x1,...,xn)X=X1…Xn R n , is a quantitative attribute of a(u), ik{1,...n}, jl{1,.. Here,(i1j1,...,ikjl,...,injm) is a combination of ik and jl , and v=v(i1j1,...,ikjl,..., injm) is its order number.Since for every ik,jl has m different choices, we have m n combinations in total.So, v{1,…,m n }.Let , and be the grid constructed by m n different n dimensional hyper rectangular parallelepiped.Thus, qualitative mapping  with qualitative criterion ([v,v]) can be written as: X→{0,1}.For any xX, there exists property Here, (2) is a qualitative mapping to judge whether the property pv(x,u) of an object u with vector x is true or not.The Grid of 3-Dimension Qualitative Criterion Because the input of the qualitative mapping is a n dimension grid, and the output is a truth value of the property p(u).From the view of point of the mathematics, the computing of is a conversion from quantity x into the quality p(u), so we called it the qualiattive mapping from quanlity into quality.

Attribute Grid Computer based on Qualitative Mapping
It is obvious that, according to the relation between the input x and the output p(x) of qualitative mapping (2), a Qualitative Mapping Logical Unit, or Electro-circuit Unit can be easely designed, the feature extraction and feature conjunction of attribute of object can be implemented by it.
An example of 2-array Qualitative Mapping Unit for the Judging of truth value of 2-array conjunction property whose input are 2 varibles, the output are 9 conjunction properties, the qualitative criteia is a 33 grid, is shown in fig.4, there is a number of feedback circuits which aim is for the adjusting of the qualitative mapping criterion.
By a number of conjunction or disjunction of Qualitative Mapping Units, An Attribute Computing Network can be integrated, not only a series of Artificial Intelligent approachs, such as the Expert System, Artificial Neural Network, Support Vector Machine can be simulated by the Attribute Computing Network, but also they are transformed each other by the adjusting of integration mode (conjunction or disjunction), and hierarchy construction, the feedback learning of connection weight and etc.

Attribute Grid Computer for Pattern Recognition
The recognition of some of patterns which varies with time t or variable x, such as Electrocardiograph etc., can be considered as the recognition of graph of a function y=f(x).So, it is a basic problem whether a method or a model of recognition of graph of a function y=f(x) could be found out or not.Let ECGu be the Electrocardiograph of u, since it could be considered as a function from interval [t 0 ,t m ] to current set Y y:[t 0 ,t m ]→Y, for any t[t 0 ,t m ], there is a y u Y, such that t→yu(t), the coordinate of any point of ECGu is (t,yu(t)).Let t=tj,j=0,…m, be a sampling serial of [t0,tm], then a m+1dimansion vector yu(t0,…, tm)=(yu(t0),…,yu(tm)) could be got by the m+1 values of function y=y u (t).
Distinguishing between normal and abnormal ECG is a typical classification operation.And from the abnormal or faulty electrocardiogram, identify what disease is the patient suffering from?It is a diagnostic or identification operation.Pattern recognition is the most basic function of the human brain.The success of pattern recognition based on the deep learning algorithm based on neural network is considered to be a breakthrough achievement of artificial intelligence.Since the basic function of artificial neurons is classification, it is also called a classifier in general textbooks.Therefore, deep learning realizes the function of pattern recognition by adjusting the connection weight parameters between different levels and different classifiers (neurons).

Relation Between Probability and (Fuzzy) Degree of Conversion Function
Let Y={  =   (),  = 1, … , } ⊆  be a set of N normal electrocardiograms   (), as shown in Figure 6 Since the coordinate generation transformation :  → H(  ) can be regarded as a transformation of the continuous function  = () into an m-dimensional vector or point in the Hilbert space, the analysis will be performed by each moment. =   Sampling line {  ( =   )} The m-dimensional coordinate system H(  ) is a Hilbert space.The transform F is transformed from the pattern space  to the Hilbert space H(  ).
Then, the so-called judgment problem of whether the electrocardiogram  = () is normal, under the Hilbert transform :  → H(  ), is converted into a set of m values constituting the electrocardiogram  = () {(  )} , whether the vector y=(y_1,...,y_m) corresponding to the space H(  ) belongs to the qualitative criteria [, ], i.e:  = ( 1 , … ,   ) ∈ [, ] problem.Therefore, there are propositions: In order to obtain a finer recognition algorithm than the classification, one of the simplest ways is to give a refinement map or operator ， such that the electrocardiograph ECGu, Let the electrocardiogram set Y= {  =   (),  = 1, … , } ⊆  (envelope diagram) be divided into an n× as shown in Fig. 9(a) Grid of m small lattices, namely: Then, the envelope refinement map (=  1 ∘ ) induces a probability map : Ω → H(  ), and the normal ECG falls into the grid of space H(  ), i.e: (16) Obviously, the Gauss function ( 15) is a fuzzy membership function.

Fig. 2 .
Fig.2 .Logic Computing Unit and Attribute Grid Computer Induced by Qualitative Mapping It is shown that since qualitative mapping and the artificial neuron can be defined each other, and a series of artificial intelligent approachs can be fused into the qualitative mapping by varied transformation of qualitative criterion, the Attribute Network Computing based on Qualitative Mapping proposed in here is a mathematical model in which a lot of intelligent methods have be fused.

Fig. 3 .
Fig.3 .Competition Between Pattern of Compuing Values of Function y=f(x) and its GraphIt is shown in this paper, the computing values y'(xj) of function y=f(x) at point xj equals not to the value f(x j ), that is :y'(x j )f(x j ), because the number of memory a computer is finite.But the computing value y'(xj) has be taken as the function value f(xj), indeed the pattern constructed by the set of {(xj,y'(xj)}, P({(xj,y'(xj)}) has be taken to be the image of function f(x).Why does it can do?What the principle we can do that is?If there is the principle and it could be used for general pattern recognition, the problem is very important for us.

Fig. 4 .
Fig.4 .Envelope of College of Normal electrocardiogram ECG Convert in Hilbert Space In fig.4,we show that in the new Coordinate System or the Hilbert space, whose axis respectively sampling y|t=t1, because the qualitative criterion is the hypercube [,]=[1,1]… [m,m], such that whether ECGu of u in the time t=tj is normal or not can be represented by following qualitative mapping.
Fig 11. one point of deviant cardiograph breaks through the strip of the cardiograph Fig.12 The data of deviant cardiograph and the identification by qualitative mapping.
in the refinement mapping The image under ( =  1 ∘ ), because the set Y ' =⋃ ⋃ [   ,    ]  =1  =1 The topology of the electrocardiogram set X.If Ω = {| [   ,    ] ⊆Y} is the normal electrocardiogram falling into the interval [   ,    ] The set of random events, N is the total number of normal ECGs in Y,    is the time  =   The normal ECG falls on the grid [   ,    ] ⊆ [  ,   ] The probability that the normal ECG   () falls into the sub-grid [  ,   ] is: