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Uniqueness of system integration scheme of artificial intelligence technology in fractional differential mathematical equation

Publicado en línea: 15 Jul 2022
Volumen & Edición: AHEAD OF PRINT
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Recibido: 19 Feb 2022
Aceptado: 26 Apr 2022
Detalles de la revista
License
Formato
Revista
eISSN
2444-8656
Primera edición
01 Jan 2016
Calendario de la edición
2 veces al año
Idiomas
Inglés
Introduction

Fractional differential equation is a generalization of integer order calculus. It discusses the theory of any non integer order differential and integral equation [1]. In recent years, fractional differential equations have been widely concerned and studied by many mathematicians, and gradually become a hot issue, which is inseparable from the in-depth development of the theory of fractional differential equations and its application in various fields. Fractional calculus has a history of 321 years. It comes from the discussion in the correspondence between g. W. Leibniz (16951679) and L. Euler (1730). At that time, calculus had just been born for more than ten years, and fractional calculus was not supported by physical background, so it was not until the middle of the 20th century that fractional calculus began to make a breakthrough. Mirzoev, K.A. and others said that, especially in recent decades, fractional calculus has an application background in many fields, which makes it develop more and more rapidly. Due to the mature development of integer order differential equations, the fractional order properties and the existence and multiplicity of solutions discussed by scientific researchers in the initial stage use the method of integer order to deal with related problems [2]. Many researchers have studied the existence and number of solutions of fractional differential equations under different orders and different boundary value conditions, as well as the eigenvalue problem of fractional differential equations. The existence and uniqueness often use some fixed point theorems. Increasing decreasing operator and variational method are also widely used in fractional differential equations. Based on the existing research, this paper discusses the uniqueness of solutions of fractional differential equations under specific order and boundary value conditions. The first example of applying fractional differential equations is the Liouville potential equation in 1832. Since then, fractional differential equations will be considered when establishing equations. Therefore, they are more and more widely used, especially in the fields of electrical conduction and diffusion processes of biological systems in recent years. Dai, Q. and others said that a new technological revolution and new industrial transformation are being carried out globally. A new Internet plus big data + AI + era is coming. Its main manifestations are: facing the global development needs of “innovation, green, openness, sharing and individuality”, as well as the rapid development of new Internet technology, new information technology, new artificial intelligence technology, new energy technology, new material technology and new biotechnology, especially, New Internet technologies (Internet of things, Internet of vehicles, mobile Internet, satellite network, heaven earth integrated network, future Internet, etc.), new information technologies (cloud computing, big data, 5g, high-performance computing, modeling / simulation, quantum computing, blockchain technology, etc.), And the rapid development of a new generation of artificial intelligence technology (based on big data intelligence, swarm intelligence, man-machine hybrid intelligence, cross media reasoning, autonomous intelligence, etc.) is triggering major changes in new models, new means and new ecosystems in the fields of national economy, national economy, people's livelihood and national security [3]. Damag, F.H. and others said that at the same time, the new information environment, technology and the new goal of human social development, which are undergoing major changes, are giving birth to a new evolutionary stage of artificial intelligence technology and applications: with the rapid development of mobile terminals, new Internet, sensor networks, Internet of vehicles, wearable devices and sensing devices, The network has begun to connect people, machines and things in the world unprecedented, and quickly reflect their needs, knowledge and abilities; Big data has become a strategic resource that can not be ignored by human society; The significant improvement of high-performance computing power provides the guarantee for the implementation of artificial intelligence; The breakthrough of artificial intelligence model and algorithm represented by deep learning and the cross integration and interaction of data and knowledge in society, physical space and information space have promoted the formation of new computing paradigm; These have greatly promoted the major changes in the new information environment and new technology [4]. Abbas, S. and others said that on the other hand, based on the needs of the new era of “innovation, green, openness, sharing and individuality”, the fields of Intelligent City, intelligent manufacturing, intelligent medical treatment, intelligent transportation, intelligent logistics, intelligent robot, unmanned driving, smart phone, intelligent toy, intelligent society and intelligent economy are also developing rapidly, Their changes in modes, means and business formats urgently need the new development of a new generation of artificial intelligence technology and application [5].

Research Methods

It is generally believed that artificial intelligence technology is an AI that can imitate all human behaviors, that is, machines can perform all tasks that human beings can complete. At present, there are two methods to develop artificial intelligence technology: one is computer science oriented, which emphasizes simulating human external behavior through machine programming without investigating what its internal driving force is, which is the research method of symbolism and behaviorism; The other is guided by neuroscience, emphasizing the need to understand the internal driving force of human behavior, and then design an artificial neural network to realize these neuropsychological functions, so that the machine can simulate the human brain as accurately as possible. This is the research method of connectionism and functionalism. However, both methods of realizing artificial intelligence technology by programming and simulating human brain face great difficulties [6]. First of all, many complex human behaviors cannot be mechanically simulated and executed by the machine completely through manual programming, but must be adapted and mastered by the machine itself through machine learning. Just as alphago can invent some go strategies that human beings have never found, its chess power is far beyond the level that human beings can achieve. This shows that machine learning method has more development potential than manual programming method. Machine learning has become the mainstream of the development of computer science. Secondly, at present, neuroscience's understanding of the human brain is still in its infancy, which is still far from completely understanding the working mechanism of the brain. To sum up, we propose a new research method for the development of artificial intelligence technology, which is guided by the mathematical basis of artificial intelligence technology. It is no longer just to simulate human behavior patterns and brain mechanisms, but to find out whether there is a common mathematical principle between human cognition and machine learning? Finally, artificial intelligence technology is realized by powerful mathematical means. The form of hardware and software of human brain and computer is completely different, but the mathematical principles they follow are isomorphic [7]. Therefore, as long as we can understand the abstract mathematical structure of artificial intelligence technology, we can construct the mathematical model of artificial intelligence technology.

The following is the mathematical model of artificial intelligence technology and the uniqueness of the solution of fractional differential mathematical equation: D0+αu(t)=f(t,u(t),u(t)),0<t<1,u(0)=u(1)=u(0)=u(1)=0 D_{{0^ +}}^\alpha u\left(t \right) = f\left({t,u\left(t \right),u\left(t \right)} \right),\,0 < t < 1,u\left(0 \right) = u\left(1 \right) = {u^{'}}\left(0 \right) = {u^{'}}\left(1 \right) = 0

Table 1 Properties of independent variables

The purpose of this paper is to establish a Logistic model to judge the credit risk of an enterprise based on the financial data provided by the enterprise and some qualitative indicators to measure the nature of the enterprise, which is represented by the probability P value. The cut-off point selected is 0.5, that is, when the value is greater than 0.5, the default probability of the enterprise is small and the enterprise is considered as a good customer, and the code of the explained variable is y=1; when the p-value is less than 0 and 5, the default probability of the enterprise is large and the enterprise is considered as a bad customer. In this case, the code of the explained variable is y=0. The explanatory variables are the 20 indicators in the table. The Logistic regression model thus established is as follows: LnP1P=α+β1X1+β2X2++β19X19+β20X20 {\rm{Ln}}{P \over {1 - P}} = \alpha + {\beta _1}{X_1} + {\beta _2}{X_2} + \ldots + {\beta _{19}}{X_{19}} + {\beta _{20}}{X_{20}}

Where, is Riemann Liouville fractional differential, f: (0,1] × [0,+∞) × [0, + ∞) → [0, + ∞) is continuous, and are not reduced relative to the first part and not increased relative to the second part in. The existence and uniqueness of the solution are discussed by using the fixed point theorem in ordered metric space. Based on the inspiration of an article, this paper mainly discusses the uniqueness of the solution of fractional differential mathematical equations. The equations are as follows: D0+αu(t)=f1(t,u(t),v(t)),0<t<1 D_{{0^ +}}^\alpha u\left(t \right) = {f_1}\left({t,u\left(t \right),v\left(t \right)} \right),\,0 < t < 1

Where, ; is Riemann Liouville fractional differential, is continuous,. We study this problem based on the coupled fixed point theory on ordered metric space.

If hC ([0,1]), α < 3 ≤ 4 D0+αu(t)=h(t),0<t<1,u(0)=u(1)=u(0)=u(1)=0, \matrix{{D_{{0^ +}}^\alpha u\left(t \right) = h\left(t \right),\,0 < t < 1,} \hfill \cr {u\left(0 \right) = u\left(1 \right) = u{'}\left(0 \right) = u{'}\left(1 \right) = 0,} \hfill \cr}

Then the unique solution of the equation is, where G(t,s)={(ts)α1+(1s)α2tα2[(st)+(α2)(1t)s]Γ(α)0st1tα2(1s)α2[(st)+(α2)(1t)s]Γ(α)0ts1 G\left({t,s} \right) = \left\{{\matrix{{{{{{\left({t - s} \right)}^{\alpha - 1}} + {{\left({1 - s} \right)}^{\alpha - 2}}{t^{\alpha - 2}}\left[{\left({s - t} \right) + \left({\alpha - 2} \right)\left({1 - t} \right)s} \right]} \over {\Gamma \left(\alpha \right)}}} \hfill & {0 \le s \le t \le 1} \hfill \cr {{{{t^{\alpha - 2}}{{\left({1 - s} \right)}^{\alpha - 2}}\left[{\left({s - t} \right) + \left({\alpha - 2} \right)\left({1 - t} \right)s} \right]} \over {\Gamma \left(\alpha \right)}}} \hfill & {0 \le t \le s \le 1} \hfill \cr}} \right.

G (t, s) is called the Green's function of formula 3. For equation 2, it can be written as follows: D0+αu(t)=f(t,u,v) D_{{0^ +}}^\alpha u\left(t \right) = f\left({t,u,v} \right)

Then its solution can be written as follows: (u(t),v(t))=(0tG1(t,s)f1(s,u(s),v(s))ds,0tG2(t,s)f2(s,v(s),u(s)))ds. \left({u\left(t \right),v\left(t \right)} \right) = \left({\int_0^t {{G_1}\left({t,s} \right){f_1}\left({s,u\left(s \right),v\left(s \right)} \right)ds,} \,\int_0^t {{G_2}\left({t,s} \right){f_2}\left({s,v\left(s \right),u\left(s \right)} \right)}} \right)ds.

Therefore, for equation system 2, there are two Green's functions, as follows: G(t,s)={(ts)α1+(1s)α2tα2[(st)+(α2)(1t)s]Γ(α)0st1tα2(1s)α2[(st)+(α2)(1t)s]Γ(α)0ts1 G\left({t,s} \right) = \left\{{\matrix{{{{{{\left({t - s} \right)}^{\alpha - 1}} + {{\left({1 - s} \right)}^{\alpha - 2}}{t^{\alpha - 2}}\left[{\left({s - t} \right) + \left({\alpha - 2} \right)\left({1 - t} \right)s} \right]} \over {\Gamma \left(\alpha \right)}}} \hfill & {0 \le s \le t \le 1} \hfill \cr {{{{t^{\alpha - 2}}{{\left({1 - s} \right)}^{\alpha - 2}}\left[{\left({s - t} \right) + \left({\alpha - 2} \right)\left({1 - t} \right)s} \right]} \over {\Gamma \left(\alpha \right)}}} \hfill & {0 \le t \le s \le 1} \hfill \cr}} \right.

Suppose is a poset, on which the metric D is given, so that is a complete metric space and F is the mapping of.

We say that is directional if exists for any such that.

We call normal if the following conditions are met:

If is an undiminished sequence in x such that, then for all n;

If is an undifferentiated sequence in y such that, then for all n. is called the coupled fixed point of F, if,

We say that f has mixed monotonicity if for all, there is xu,y>vF(x,y)F(u,v). x \le u,y > v \Rightarrow F\left({x,y} \right) \le F\left({u,v} \right).

Set up Φ Representing such a cluster of functions, satisfies: (1) φ Is continuous; (2) φ Is not reduced; (3) ○ is a poset, on which there is a metric d, so that is a complete metric space, F is the mapping of, and has mixed monotonicity on X, so that: ψ(d(F(x,y),F(u,v)))ψ(max(d(x,u),d(y,v)))φ(max(d(x,u),d(y,v))) \psi \left({d\left({F\left({x,y} \right),F\left({u,v} \right)} \right)} \right) \le \psi \left({\max \left({d\left({x,u} \right),d\left({y,v} \right)} \right)} \right) - \varphi \left({\max \left({d\left({x,u} \right),d\left({y,v} \right)} \right)} \right)

For all, where and are normal, there is such that, then F has a coupled fixed point. If the sequences and in X are defined as follows: xn+1=F(xn,yn),yn+1=F(yn,xn),n=0,1,, {x_{n + 1}} = F\left({{x_n},{y_n}} \right),{y_{n + 1}} = F\left({{y_n},{x_n}} \right),n = 0,1, \ldots, Solimnd(xn,x*)=limnd(yn,y*)=0 {\rm{So}}\mathop {\lim}\limits_{n \to \infty} d\left({{x_n},{x^*}} \right) = \mathop {\lim}\limits_{n \to \infty} d\left({{y_n},{y^*}} \right) = 0

If is normal on the basis of lemma 2, we can obtain the uniqueness of the coupled fixed point and.

The main result is that the norm is given in Banach space. Define partial order relation ≤ on e as follows: u,vE,uvu(t)v(t),t[0,1] u,v \in E,u \le v \Leftrightarrow u\left(t \right) \le v\left(t \right),t \in \left[{0,1} \right]

The metric d is defined as follows: can verify that under this metric, is directional and is normal.

The closed cone is defined as follows: here 0 represents the zero function. We call the coupled upper and lower solutions of formula 2 if: u(t)0tG(t,s)f1(s,u(s),u+(s))ds,0t1 {u^ -}\left(t \right) \le \int_0^t {G\left({t,s} \right){f_1}\left({s,{u^ -}\left(s \right),{u^ +}\left(s \right)} \right)} ds,\,0 \le t \le 1 u+(t)0tG(t,s)f2(s,u+(s),u(s))ds,0t1 {u^ +}\left(t \right) \le \int_0^t {G\left({t,s} \right){f_2}\left({s,{u^ +}\left(s \right),{u^ -}\left(s \right)} \right)} ds,\,0 \le t \le 1

If 0< σ< 1, 3< α ≤ 4, is continuous, is continuous on. Assumed existence λ As follows: makes, for, satisfy: where η and γ Respectively, η: ,. Assuming that formula 2 has a coupled upper and lower solution, formula 2 of the boundary value problem has a unique positive solution, and the sequences and are defined as follows: u0=u,un+1=0tG(t,s)f(s,un(s),vn(s))ds,n=0,1,, {u_0} = {u^ -},{u_{n + 1}} = \int_0^t {G\left({t,s} \right)f\left({s,{u_n}\left(s \right),{v_n}\left(s \right)} \right)ds,\,n = 0,1, \ldots,} v0=v+,vn+1=0tG(t,s)f(s,vn(s),un(s))ds,n=0,1,, {v_0} = {v^ +},{v_{n + 1}} = \int_0^t {G\left({t,s} \right)f\left({s,{v_n}\left(s \right),{u_n}\left(s \right)} \right)ds,\,n = 0,1, \ldots,}

Uniformly converges to u*.

If the universal learning machine is the cornerstone of artificial intelligence technology, the universal understanding machine is the key to understand the profound philosophies of Pythagoras, Kant and Godel. Carefully, it may be found that the three milestones from general-purpose computer to general-purpose learning machine and then to general-purpose understanding machine are the three technical bottlenecks that must be crossed to realize artificial intelligence technology [8].

With the help of Figure 1, it can be preliminarily predicted that the cornerstone of general-purpose computer is data structure and algorithm, the cornerstone of general-purpose learning machine is information structure and search, and the cornerstone of general-purpose understanding machine is knowledge structure and reuse. From the mathematical basis of artificial intelligence technology, algorithms or operations or batch processing, search or batch processing, reuse or targeted batch processing are interlinked in essence. The fundamental difference lies in: data structure, information structure and knowledge structure. The essence is the fundamental difference between data, information and knowledge. Strictly speaking, computing, learning and understanding are the embodiment of some kind of general intelligence, and each has its own merits. For their technical realization, scientific discussion and even philosophical reflection, it can enlighten the whole academic circle. Turing test is the standard to judge whether the machine has AI. Turing only used the expression of whether machines can think, but did not involve the expression of “intelligence” of “thinking like people”. In 1950, Turing put forward the test, that is, Turing test includes two objective “intelligent” criteria: five minute test time and discrimination probability of no more than 70%. In 1966, wizhauman's chat robot adopted a cunning strategy. It meets the two objective criteria of Turing test, but it can not show that the machine has “intelligence”. Therefore, Turing test should exclude cunning strategy [9]. In 2011, the computer system Watson won in the quiz game program. If according to the initial definition of Turing in 1950, Watson passed the higher-level Turing test - Man-machine duel of intelligence quiz; If Watson's designers understand the Turing test as thinking like people, Watson has not passed the Turing test. The three types of formal understanding models not only resolve the two extreme conflicts between Turing machine and Turing test and Chinese house and Chinese Room (Turing test in Chinese), but also sublimate people's understanding of computer, learning machine and understanding machine from the perspective of generalized text. At the same time, they also get an example of indirect formal standard.

Figure 1

Predictable three milestones in the development of artificial intelligence technology

As can be seen from Figure 3, there are several milestones in the development of general-purpose computer, including digitization, symbolization, structure and objectification, and each step of its development has a solid foundation. The development from special-purpose learning machine to general-purpose learning machine has encountered a development bottleneck in strong AI or artificial intelligence technology: natural language understanding or expert knowledge expression or software pattern recognition (the latter has sprung up in the field of statistics based machine learning and deep learning, and has produced special-purpose learning machines in specific fields). The general understanding machine or indirect formal understanding model has found another way to open up the indirect formal way of the eight formal systems from the way of digitization and weak AI (strong human intelligence). In order to connect rational reflection and empirical prediction, a dual formal approach is explored [10].

Figure 2

General artificial intelligence

Figure 3

Development of computable, learnable and understandable programming language and eight formal systems of its processing

As can be seen from Figure 4, not only the software patterns based on grammar rules can be automatically identified, but also the software patterns based on probability statistics can be automatically identified, and can be automatically learned, that is, autonomous learning that can be formally understood under specific conditions. For example, the “Ming” in Li Bai's poem (unstructured text) is not only incomprehensible by computers, but also difficult to understand by special learning machines based on pattern recognition, In our formal understanding model, that is, the structured text library of the understanding machine, the “bright moon” and “looking at the bright moon” combined with the “adjective a” and “verb V” given by the thinking map based on expert knowledge can accurately and pertinently reuse expert knowledge; at the same time, the three words “doubt, hope and thought” randomly selected by expert users can be combined with another thinking map based on expert knowledge, From the full-text scene of the whole poem, we can instantly understand the intention of the experts, so as to help the readers, that is, the users of software assisted reading, have a feeling of sudden enlightenment or enlightenment (not only deepen the learning understanding, but also increase the learning interest, enlighten the thinking and significantly improve the memory effect) [11]. Readers with rich imagination will also get inspiration from it, and make the following analogy: not only can each poem be refined with such language points, knowledge points and original points, but also we can infer the combination of the super Chinese character chessboard language chessboard in the background of our learning and understanding system, the visual knowledge chessboard knowledge menu in the foreground human-computer interaction interface and the original chess soul thinking map, The rudiments of learning machine and understanding machine can be predicted. This is beyond the imagination of the pioneers of computer and AI such as Turing and Godel. We can understand this with a little review (although they have also made independent exploration on learning machine and AI).

Figure 4

Application example of twin Turing machine constructed based on dual formal approach

Results analysis and discussion

In this paper, fractional differential equations with two-point boundary values are discussed: Dα1u(t)=f1(t,Dm1u(t),Dn1v(t)),Dα2v(t)=f2(t,Dm2u(t),Dn2v(t)), \matrix{{{D^{\alpha 1}}u\left(t \right) = {f_1}\left({t,{D^{m1}}u\left(t \right),{D^{n1}}v\left(t \right)} \right),} \hfill \cr {{D^{\alpha 2}}v\left(t \right) = {f_2}\left({t,{D^{m2}}u\left(t \right),{D^{n2}}v\left(t \right)} \right),} \hfill \cr}

Where 0 < αi ≤ 1, 0 < miα1, 0 < niα2, i = 1,2, u(0) = v(0) = 0, 0 ≤ ta, a > 1, Dα1 represents Riemann Liouville fractional differential.

According to the definition of fractional integral and differential, we have the following properties: IαIβf(t)=Iα+βf(t),DαIαf(t)=f(t) I^\alpha}{I^\beta}f\left(t \right) = {I^{\alpha + \beta}}f\left(t \right),{D^\alpha}{I^\alpha}f\left(t \right) = f\left(t \right)

Next, we prove the uniqueness of 17 nonlinear fractional differential equations [12, 13, 14]. The initial boundary value problem is equivalent to the following equations: u(t)=1Γ(α1)0t(ts)α11f1(s,Dm1u(s),Dn1v(s))ds,v(t)=1Γ(α2)0t(ts)α21f2(s,Dm2u(s),Dn2v(s))ds, \matrix{{u\left(t \right) = {1 \over {\Gamma \left({{\alpha _1}} \right)}}\int_0^t {{{\left({t - s} \right)}^{{\alpha _1} - 1}}\,{f_1}\left({s,{D^{{m_1}}}u\left(s \right),\,{D^{{n_1}}}v\left(s \right)} \right)ds,}} \hfill \cr {v\left(t \right) = {1 \over {\Gamma \left({{\alpha _2}} \right)}}\int_0^t {{{\left({t - s} \right)}^{{\alpha _2} - 1}}\,{f_2}\left({s,{D^{{m_2}}}u\left(s \right),\,{D^{{n_2}}}v\left(s \right)} \right)ds,}} \hfill \cr}

It is proved that both sides of equation 17 are carried out simultaneously α 1 and α Two integrals, we get: u(t)=C1tα11+1Γ(α1)0t(ts)α11f1(s,Dm1u(s),Dn1v(s))ds,v(t)=C2tα21+1Γ(α2)0t(ts)α21f2(s,Dm2u(s),Dn2v(s))ds, \matrix{{u\left(t \right) = {C_1}{t^{{\alpha _1} - 1}} + {1 \over {\Gamma \left({{\alpha _1}} \right)}}\int_0^t {{{\left({t - s} \right)}^{{\alpha _1} - 1}}\,{f_1}\left({s,{D^{{m_1}}}u\left(s \right),\,{D^{{n_1}}}v\left(s \right)} \right)ds,}} \hfill \cr {v\left(t \right) = {C_2}{t^{{\alpha _2} - 1}} + {1 \over {\Gamma \left({{\alpha _2}} \right)}}\int_0^t {{{\left({t - s} \right)}^{{\alpha _2} - 1}}\,{f_2}\left({s,{D^{{m_2}}}u\left(s \right),\,{D^{{n_2}}}v\left(s \right)} \right)ds,}} \hfill \cr}

Since, C1 = C2 = 0, the proof ends.

If is continuous on W and there are two positive functions satisfies: |f1(t,Dm1u1(t),Dn1v1(t))f1(t,Dm1u2(t),Dn1v2(t))|m(t)(Dm1u1(t),Dn1v1(t))(t,Dm1u2(t),Dn1v2(t)),|f2(t,Dm2u1(t),Dn2v1(t))f2(t,Dm2u2(t),Dn2v2(t))|m(t)(Dm1u1(t),Dn1v1(t))(t,Dm1u2(t),Dn1v2(t)),|f2(t,Dm2u1(t),Dn2v1(t))f2(t,Dm2u2(t),Dn2v2(t))|n(t)(Dm2u1(t),Dn2v1(t))(t,Dm2u2(t),Dn2v2(t)) \matrix{{\left| {{f_1}\left({t,{D^{{m_1}}}{u_1}\left(t \right),{D^{{n_1}}}{v_1}\left(t \right)} \right) - {f_1}\left({t,{D^{{m_1}}}{u_2}\left(t \right),\,{D^{{n_1}}}{v_2}\left(t \right)} \right)} \right|} \hfill \cr {\le m\left(t \right){{\left\| {\left({{D^{{m_1}}}{u_1}\left(t \right),\,{D^{{n_1}}}{v_1}\left(t \right)} \right) - \left({t,{D^{{m_1}}}{u_2}\left(t \right),\,{D^{{n_1}}}{v_2}\left(t \right)} \right)} \right\|}_\infty},} \hfill \cr {\left| {{f_2}\left({t,{D^{{m_2}}}{u_1}\left(t \right),{D^{{n_2}}}{v_1}\left(t \right)} \right) - {f_2}\left({t,{D^{{m_2}}}{u_2}\left(t \right),\,{D^{{n_2}}}{v_2}\left(t \right)} \right)} \right|} \hfill \cr {\le m\left(t \right){{\left\| {\left({{D^{{m_1}}}{u_1}\left(t \right),\,{D^{{n_1}}}{v_1}\left(t \right)} \right) - \left({t,{D^{{m_1}}}{u_2}\left(t \right),\,{D^{{n_1}}}{v_2}\left(t \right)} \right)} \right\|}_\infty},} \hfill \cr {\left| {{f_2}\left({t,{D^{{m_2}}}{u_1}\left(t \right),{D^{{n_2}}}{v_1}\left(t \right)} \right) - {f_2}\left({t,{D^{{m_2}}}{u_2}\left(t \right),\,{D^{{n_2}}}{v_2}\left(t \right)} \right)} \right|} \hfill \cr {\le n\left(t \right){{\left\| {\left({{D^{{m_2}}}{u_1}\left(t \right),\,{D^{{n_2}}}{v_1}\left(t \right)} \right) - \left({t,{D^{{m_2}}}{u_2}\left(t \right),\,{D^{{n_2}}}{v_2}\left(t \right)} \right)} \right\|}_\infty}} \hfill \cr}

Then equation 17 has a unique positive solution if the following equation holds: ρ=k10tm(t)(ts)α11dsΓ(α1)<1,θ=k20tn(t)(ts)α21dsΓ(α2)<1, \matrix{{\rho = {{{k_1}\int_0^t {m\left(t \right){{\left({t - s} \right)}^{{\alpha _1} - 1}}ds}} \over {\Gamma \left({{\alpha _1}} \right)}} < 1,} \hfill \cr {\theta = {{{k_2}\int_0^t {n\left(t \right){{\left({t - s} \right)}^{{\alpha _2} - 1}}ds}} \over {\Gamma \left({{\alpha _2}} \right)}} < 1,} \hfill \cr}

Among them, k1=max{a1m1Γ(2m1),a1n1Γ(2n1)},k2=max{a1m2Γ(2m2),a1n2Γ(2n2)} \matrix{{{k_1} = \max \left\{{{{{a^{1 - {m_1}}}} \over {\Gamma \left({2 - {m_1}} \right)}},{{{a^{1 - {n_1}}}} \over {\Gamma \left({2 - {n_1}} \right)}}} \right\},} \hfill \cr {{k_2} = \max \left\{{{{{a^{1 - {m_2}}}} \over {\Gamma \left({2 - {m_2}} \right)}},{{{a^{1 - {n_2}}}} \over {\Gamma \left({2 - {n_2}} \right)}}} \right\}} \hfill \cr}

Prove: |F1(u1(t),v1(t))F1(u2(t),v2(t))|F(U)U=1Γ(α1)|0t(ts)α11(f1(s,Dm1u1(s),Dn1v1(s))f1(s,Dm1u2(s),Dn1v2(s)))ds|0tm(t)(ts)α11dsΓ(α1)(Dm1u1(s),Dn1v1(s))(Dm1u2(s),Dn1v2(s)) \matrix{{\left| {{F_1}\left({{u_1}\left(t \right),{v_1}\left(t \right)} \right) - {F_1}\left({{u_2}\left(t \right),{v_2}\left(t \right)} \right)} \right|} \cr {F\left(U \right) \subseteq U \cdot = {1 \over {\Gamma \left({{\alpha _1}} \right)}}\left| {\int_0^t {{{\left({t - s} \right)}^{{\alpha _1} - 1}}\left({{f_1}\left({s,{D^{{m_1}}}{u_1}\left(s \right),\,{D^{{n_1}}}{v_1}\left(s \right)} \right) - {f_1}\left({s,{D^{{m_1}}}{u_2}\left(s \right),{D^{{n_1}}}{v_2}\left(s \right)} \right)} \right)ds}} \right|} \cr {\le {{\int_0^t {m\left(t \right){{\left({t - s} \right)}^{{\alpha _1} - 1}}ds}} \over {\Gamma \left({{\alpha _1}} \right)}}\left\| {\left({{D^{{m_1}}}{u_1}\left(s \right),{D^{{n_1}}}{v_1}\left(s \right)} \right) - \left({{D^{{m_1}}}{u_2}\left(s \right),{D^{{n_1}}}{v_2}\left(s \right)} \right)} \right\|\infty} \cr}

However: |Dm1(u1(t)u2(t))|=|1Γ(1m1)0t(ts)m1(u1(s)u2(s))ds|t1m1Γ(2m1)|u1(t)u2(t)|a1m1Γ(2m1)|u1(t)u2(t)|k1|u1(t)u2(t)| \matrix{{\left| {{D^{{m_1}}}\left({{u_1}\left(t \right) - {u_2}\left(t \right)} \right)} \right| = \left| {{1 \over {\Gamma \left({1 - {m_1}} \right)}}\int_0^t {{{\left({t - s} \right)}^{- {m_1}}}\left({{u_1}\left(s \right) - {u_2}\left(s \right)} \right)ds}} \right|} \cr {\le {{{t^{1 - {m_1}}}} \over {\Gamma \left({2 - {m_1}} \right)}}\left| {{u_1}\left(t \right) - {u_2}\left(t \right)} \right|} \cr {\le {{{a^{1 - {m_1}}}} \over {\Gamma \left({2 - {m_1}} \right)}}\left| {{u_1}\left(t \right) - {u_2}\left(t \right)} \right| \le {k_1}\left| {{u_1}\left(t \right) - {u_2}\left(t \right)} \right|} \cr}

Similarity: |Dm1(v1(t)v2(t))|a1m1Γ(21)|v1(t)v2(t)|k1|v1(t)v2(t)| \left| {{D^{{m_1}}}\left({{v_1}\left(t \right) - {v_2}\left(t \right)} \right)} \right| \le {{{a^{1 - {m_1}}}} \over {\Gamma \left({2 - 1} \right)}}\left| {{v_1}\left(t \right) - {v_2}\left(t \right)} \right| \le {k_1}\left| {{v_1}\left(t \right) - {v_2}\left(t \right)} \right|

That, (Dm1u1(t),Dn1v1(t))(Dm1u2(t),Dn1v2(t))k1(u1(t),v1(t))(u2(t),v2(t)) {\left\| {\left({{D^{{m_1}}}{u_1}\left(t \right),\,{D^{{n_1}}}{v_1}\left(t \right)} \right) - \left({{D^{m - 1}}{u_2}\left(t \right),\,{D^{{n_1}}}{v_2}\left(t \right)} \right)} \right\|_\infty} \le {k_1}{\left\| {\left({{u_1}\left(t \right),\,\,{v_1}\left(t \right)} \right) - \left({{u_2}\left(t \right),\,\,{v_2}\left(t \right)} \right)} \right\|_\infty}

Then, |F1(u1(t),v1(t))F1(u2(t),v2(t))|=k10tm(1)(ts)α11dsΓ(α1)(u1(t),v1(t))(u2(t),v2(t))=ρ(u1(t),v1(t))(u2(t),v2(t)) \matrix{{\left| {{F_1}\left({{u_1}\left(t \right),\,{v_1}\left(t \right)} \right) - {F_1}\left({{u_2}\left(t \right),\,{v_2}\left(t \right)} \right)} \right|} \cr {= {{{k_1}\int_0^t {m\left(1 \right){{\left({t - s} \right)}^{{\alpha _1} - 1}}ds}} \over {\Gamma \left({{\alpha _1}} \right)}}{{\left\| {\left({{u_1}\left(t \right),\,{v_1}\left(t \right)} \right) - \left({{u_2}\left(t \right),\,{v_2}\left(t \right)} \right)} \right\|}_\infty}} \cr {= \rho {{\left\| {\left({{u_1}\left(t \right),\,{v_1}\left(t \right)} \right) - \left({{u_2}\left(t \right),\,{v_2}\left(t \right)} \right)} \right\|}_\infty}} \cr}

Similarity: |F2(u1(t),v1(t))F2(u2(t),v2(t))|θ(u1(t),v1(t))F2(u2(t),v2(t)) \left| {{F_2}\left({{u_1}\left(t \right),\,{v_1}\left(t \right)} \right) - {F_2}\left({{u_2}\left(t \right),\,{v_2}\left(t \right)} \right)} \right| \le \theta {\left\| {\left({{u_1}\left(t \right),\,{v_1}\left(t \right)} \right) - {F_2}\left({{u_2}\left(t \right),\,{v_2}\left(t \right)} \right)} \right\|_\infty}

Then, |F2(u1(t),v1(t))F2(u2(t),v2(t))|max{ρ,θ}(u1(t),v1(t))(u2(t),v2(t)) \left| {{F_2}\left({{u_1}\left(t \right),\,{v_1}\left(t \right)} \right) - {F_2}\left({{u_2}\left(t \right),\,{v_2}\left(t \right)} \right)} \right| \le \max \left\{{\rho,\theta} \right\}{\left\| {\left({{u_1}\left(t \right),\,{v_1}\left(t \right)} \right) - \left({{u_2}\left(t \right),\,{v_2}\left(t \right)} \right)} \right\|_\infty}

F is a fully continuous operator. According to Banach fixed point theorem, operator f has a unique fixed point in U, which is the unique positive solution of equation 17.

Conclusion

Both calculation and programming belong to the work of arithmetic paradigm. Calculation is to write a recursive function, and programming is to solve this recursive function. In many cases, we can write an algebraic equation, but we can't solve it at all. Therefore, calculation is equivalent to p problem in computational complexity, and programming is equivalent to NP problem. Are these two kinds of problems completely equivalent? So far, there is no answer. Programmable problems are only a small subset of computable problems. If the problem solution cannot be programmed, the machine cannot handle this kind of problem, so it needs to solve this kind of problem through machine learning. Learning is a cognitive process that gradually forms new concepts and knowledge through the accumulation of data and experience. Its core is geometric intuition and logical analysis. Computers cannot program independently because their internal a priori structures are arithmetic paradigms, which are discontinuous and incomplete. Learning function is automatic programming, because its internal a priori structure includes not only arithmetic paradigm, but also geometric paradigm and logical paradigm, which are continuous and complete. The internal mathematical structure of computer and learning machine is fundamentally different. The above is our mathematical problem of learning machine. In order to realize automatic programming, the machine must have corresponding mathematical structure. In addition to the arithmetic structure that Turing machines are good at, there are also familiar geometric and logical structures. Unlike the discrete and incomplete arithmetic structure of Turing computer, the continuous geometric structure and complete logical structure of learning machine are called “Godel machine” 18. Others have proved an important mathematical theorem: as long as there is enough data input, machines with complete logical structure can be programmed automatically. This theorem also applies to continuous geometry Structural machine. It proves the existence of universal learning machine. Therefore, the existence of universal learning machine is a basic mathematical theorem. It is not only the core of universal learning machine, but also the cornerstone of artificial intelligence technology, which is equivalent to the “church Turing topic” in computer. Combined with the basic principles described in Figures 1 to 4, a general intelligent machine with understanding ability can be further conceived and designed.

Figure 1

Predictable three milestones in the development of artificial intelligence technology
Predictable three milestones in the development of artificial intelligence technology

Figure 2

General artificial intelligence
General artificial intelligence

Figure 3

Development of computable, learnable and understandable programming language and eight formal systems of its processing
Development of computable, learnable and understandable programming language and eight formal systems of its processing

Figure 4

Application example of twin Turing machine constructed based on dual formal approach
Application example of twin Turing machine constructed based on dual formal approach

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