Chapter 5. The poly-processor and polyprocessing.

1. The concept of the poly-processing.
1.1 Similar thinking and point symmetric thinking.
. . When the human considers an invisible phenomenon, he tries to think of the visible phenomenon whose property may be similar to the invisible phenomenon or he draws in his mind the system which the men of the past composed similarly to the visible phenomenon. For example, hearing the word of atom or electron, we draw in our mind the system that electrons revolve around the atomic nucleus as planets revolve around the sun. In other words, the human considers that the invisible world is similar to the visible world. The author calls it the similar thinking and calls the structure and function of the system simulated by the similar thinking the similar structure and function. The human considers that the new invisible phenomenon has been solved, when he has been able to simulate it so that it is not contradictory to a visible phenomenon or the phenomenon of the system which has been simulated by now. However, a contradiction will be found in the end, because the invisible world is not similar to the visible world.
. . At the dawn of science, anyone believed that the sun revolved around the earth. Although there was almost no contradiction as far as he watched the stars and the phenomena on the earth by eyes, now anyone convinces oneself that it is a mistake and that the earth revolves around the sun. In old days, anyone believed that the iron held the property of a mass of iron even if it was broken into the smallest pieces. This is just the similar thinking and it is demanded to look at it from a different angle. By the discovery of molecule and atom, it is common knowledge now that the microscopic world is different from the macroscopic world in the structure and function. Electrical engineering succeeds in solving various electric phenomena by use of the advanced mathematical technique called the differential equation as to the electric current, which means that a positive charge flows from the positive pole to the negative pole as the stream. However, a contradiction is found in the end. The electric engineering and industry of nowadays thrive on looking at it from the different angle that the electron, which has a negative charge, flows from the negative pole to the positive pole.
. . While the electric current is analog quantity, the flow of electrons is particles. Even if the macroscopic quantity is assumed, it is not merely negative quantity to the electric current. Fleming's left hand rule is expressed in the right hand. Facing the middle fingers to each other, the both hands are plane symmetry. Reversing the direction of magnetic field and forth, the relation between the flow of electrons and the electric current becomes point symmetry in the relation of the both hands. The author calls the thinking of such structure and function the point symmetric thinking. Even if it is not geometrically point symmetric, if it has remarkably opposite structure and function, the author calls it the point symmetric thinking. It is said that the digital quantity is opposite concept to the analog quantity, but the author does not call it point symmetry. Although the flow of electrons has negative polarity to the electric current even macroscopically, the digital quantity is of the same as the analog quantity. It only has the smallest quantity which is finite, while the smallest analog quantity is infinitesimal.
. . The iron mixed some carbon is called steel and in quenched state it has the stiffness which is suitable for spring and sword, though it is almost the same as iron in annealed state. It is well known that the alloys of a metal with other various metals change the property of the pure metal remarkably. The semiconductors alloyed with a few impurities bring to the function amplifying electric signals by combination of them and make a great contribution to the technology of computer as transistor and IC. If these are made a great mass, these are merely the materials as the object of the strength of materials. In similar enlargement, there exists the microscopic function everywhere evenly, so the composed function is averaged and does not yield the macroscopic function, that is, non-similar enlargement yields new macroscopic function composed but similar enlargement loses the microscopic function.
. . Logical gates AND, OR, XOR, NOT are constructed of transistors. When a logical device is constructed of a mass of 2-input AND gates, increasing the mass similarly by the way of inputting the outputs of two 2-input AND gates to the other 2-input AND gate, the probability that the output of the device becomes 1 approaches to zero and the device loses the meaning as logical device. The other gates are the same, too. By combining these theoretically, that is, enlarging the device non-similarly, it was accomplished to construct the higher functional system called computer. However, the evolution of function has stopped here. The similar thinking started to prevail. It is that the higher data processing of human brain may be realized by a super-parallel computer or a network of computer, similarly increasing the mass of computers by the way where a central computer governs addressed microcomputers composed of VLSI or addressed other computers as the computer works by central controller controlling memories.
. . A human fertilized egg repeats cell division and grows through the shape of fishes and four-footed animals. When it has become similar to the mother's body by growing through the process of evolution of creatures, it is born and has stopped the evolution. In this way, the many individuals which are similar to the mother's body are created and called a species. The species is held by stopping the evolution when the individual has become similar to the parents. The prevailing of the similar thinking is the end of evolution of computer. The super parallel computer addressed similarly to the memory of computer works only collaboratively and it cannot be expected that the super parallel computer is able to process the knowledge information, in which the usual computer is weak, as well as the human. It is the same as the collaborative works of monkeys is not able perfectly to process the knowledge information of the human.
. . The un-evolutionist says that it is the reason denying the theory of evolution that there is not the medium species. However, there is not the medium species because the species is the end of a branch of evolution. There are not the medium logic between AND and OR, and the medium operation between addition and multiplication. The day has become the heyday of television from radio and it is possible that there exists the appliance with both works of radio and television but it is not possible that there exists the medium appliance, that is, the meaningless appliance which resembles both of radio and television but is neither radio nor television. In the same way, there does not exist the medium species between the anthropoid and the human. The truth is not that the anthropoid evolves into the human but that the process or plan of making the human resembles that of the anthropoid and has more evolution than the anthropoid. The plan of computer must be made an evolution, too.

1.2 The point symmetric thinking in information processing.
. . The fetus holds the point symmetric posture to the mother's body and becomes similar to the mother's body by being borne. The function of brain is connected with the body in point symmetric posture as the right side of the body is connected with the left side of the brain, the left side of the body is connected with the right side of the brain and the front of the body is connected with the back of the brain. The fetus also holds the point symmetric posture because it is a part of the mother's body. These show that the information processing executed inside brain is point symmetric for the visual information processing executed outside brain. The brain is able easily to process even the information processing for which a complex calculation is required, so even the baby can process the pattern recognition and the language. It cannot be true that the baby can do the analysis by use of Fourier's transform and grammar, which the parents do not know. Even if the brain does not know such analyzing, it has the function processing the knowledge information better than computer.
. . The computer analyzes the data of pattern recognition or vocal processing by use of integral and also uses integral in order to synthesize the original data inversely. When the frequency analysis is made by Fourier's transform, the original data is restored by inverse Fourier's transform of the analyzed result. This is the similar thinking. It is evident that the muscle controlling human movement is an integral system, so the analysis in the brain must be a differential analysis. The movement of muscle is analog so the signal transmitted in nerves becomes impulses, which are point symmetric to analogue. The movement of muscle is the impulsive response and the solution of the differential equation accelerated by the impulse. Accordingly, the impulsive data obtained by the movement of muscle and the senses of seeing and hearing must be the inversely operated data of the impulsive response, that is, the data differentiated. It is not only AD-DA converting.
. . As mentioned in Chapter 4 Section 4.3, the frequency analysis by the differential analysis is completed by analyzing the data nearby t=0, because it only obtains y0, Δy0, Δ2y0, ……… at t=0, whereas Fourier's transform must integrate the data from t=0 to t=∞ and has difficulty obtaining the analyzed result at almost same time as the occurring phenomenon. It is of the same as the introduction of the differential equation and the process of data is simplified by use of differential circuit because the differential equation can be divided into the second order differential equations by parallel connecting the resonance circuit of the second order. The differential operation is the inverse operation of the integral operation but the both operations are not only inverse but also point symmetric.
. . Fourier's transform expresses the original data in the components of continuous frequency. On the other hand, the differential analysis expresses it in the differential equation of the lowest possible order, that is, in the least possible frequencies. If the analysis succeeds in recognition by the least possible frequencies, the remainder between the original data and the composed result of the solutions of these second order differential equations can be truncated as noise. The almost all components of continuous frequency obtained by Fourier's transform are the common components which all the case has and the components of noise. It is difficult to get rid of the common components and the noise must be removed previously. The differential analysis is the frequency analysis which is point symmetric to Fourier's transform.
. . The function f(t) is expressed in Newton's interpolation formula, obtaining the successive differences of the values fk of function at equidistant points tk=t0+kh. Hence, Newton's interpolation formula is the synthesis of the original function and the calculation of the successive differences is the analysis. Since the synthesis is carried out by adding, the analysis is carried out by subtracting. Newton's interpolation formula is similar to the function f(t) but the vector F, which is defined by the ordered set of the successive differences, is a point on the n dimensional space and it is a point symmetric expression to the function f(t).
. . In usual, the numerical integral and the numerical solution of differential equation obtains the values of function at the points dividing interval, that is, the numerical solution similar to the solution y(t). The author's operational calculus obtains the vector Y, that is, a point on the n dimensional space. This is a point symmetric solution to the usual solution and it corresponds to obtaining all the values of the solution which is expressed in Newton's interpolation formula for continuous variable t. Hence, it may be said that this is the ultra-parallel solution. Furthermore, it is possible easily to process the vector calculus by a parallel hardware, so the ultra-parallel system processing the author's operational calculus is able to be a likely evolution of computer. The concrete idea is mentioned from the second section onwards.

1.3 The point symmetric thinking in parallel processing and language.
. . In usual, the concept of parallel processing is considered as the concept equivalent to distributed processing. It is called parallel processing that many computers carry out the work which makes the precise table by concurrently carrying out an enormous amount of calculation, assigning many calculating operators the every range in which every one calculates the result of an equation or that some other computers carry out the partial processes which are included in such sequential processing as computing in the scientific technique but independent of the sequence of the total process.
. . These parallel processing artificially divide and carry out the work which should be processed sequentially in essentials and these need the central computer which orders the computers under the central computer to carry out the work divided by the management of task or the parallel compiler. Furthermore, It needs the order by the program which the men called system engineer or programmer make by full use of the parallel language, which it is difficult for general users to understand. Even in these days when it is possible by the progress of LSI technology to realize a highly distributed processing by the network using many thousands of microcomputers, there is no advance of software fundamentally in the way of thinking and the program language becomes more and more complicate so the general users cannot use it.
. . Even if it is most highly parallel it is capable of no more than the main computer. Likening them to human society, it is the society in which there is no democracy and which the dictator governs. This society is capable of no more than the dictator. In order to make full use of the abilities of the members of society, the society must be democratized. The highly parallel processing of computers is the same as the society and must change the fundamental idea which the usual parallel processing has. In particular, the hardware must be removed Neumann's dictatorial thought that the central computer needs to control the whole system, that is, the thought that the central computer needs to give the order and to allot the works to the following computers, specifying the addresses. For that purpose, the concept of address of computer needs to be eliminated. This makes the central computer lost the role and makes the parallel computers process their own work, collaboratively finding it from the given external information. As a result, there is no overhead by the central computer participating in the work and the efficiency of work increases.
. . In the case of drawing a circle on a display, if it is possible to draw it by conically spreading the electron beam irradiating the center of the circle, it is an ideal parallel drawing but it is almost impossible. In practice, it is drawn by the sequential processing which draws next dot consecutively, computing the curvature from an arbitrary point. In the case of the parallel processing whose eight CPUs draw every arc divided the circumference of circle into equivalent eight arcs, the program by machine language becomes complex by the need to compute the beginning and end points of the arc in charge of every CPU and to give every CPU them. The program consecutively drawing the dots of arc is of the same for all CPU and it is possible to give it to local ROM of every CPU. However, there must be the main CPU which allots the works to every CPU, computing the beginning and end points of every arc. In order to entrust the calculation of the beginning and end points of every arc to the parallel CPUs, the range of drawing must be fixed so that the central angle of 0° to 45° is in charge of CPU0, the central angle of 45° to 90° is in charge of CPU1 and so on. The system has no need of main CPU and is able to carry out the parallel drawing of the circle when the common bus connecting the parallel CPUs is given the central point and the radius. The parallel processing having no main CPU does not need the address of every CPU. In stead, the working range of every CPU must be prescribed by fixed connection of all CPUs. In network, a switchboard is of no use connection. The connection by switchboard must secure the safety rate by the theory of traffic as telephone and it has low efficiency and a danger such that the system becomes panic by the trouble of switchboard or the traffic above the safety rate.
. . Such parallel drawing is similar to calling the subroutine which draws a circle by the sequential processing and the program using it becomes simple. It is possible to say that the calling instruction is a parallel instruction because it dose not describe the sequential process drawing dots. The parallel instruction also does not describe the process dividing the work. The use of this needs the sequential process that the central point and the radius are preset to the instruction. The CIRCLE statement drawing a circle in BASIC is higher parallel statement, because it only describes the central point and the radius as the parameters and it does not describe the order of processing. In drawing two circles, the CIRCLE statement is described two times sequentially. If it is described as "Draw the circles of the radius a and the radius b at the central point A and B respectively." it is a parallel imperative sentence because it does not describe the order drawing the two circles.
. . Accordingly, the hardware made parallel must contribute to making the software easy and making the software parallel is independent of making the hardware parallel and the programming language must be made easy and become so that everybody can make full use of it. Making the software parallel is equivalent to eliminating the description of the sequential processing which is not necessary and the most parallel language is the natural language. However, the actual situation is running counter to this so that the programming becomes complex with making the hardware parallel by giving rise to new programming languages for the parallel processing and the amateur cannot make full use of it.
. . It is said that the natural language is not strict. It does not describe such needless processing as the order of drawing two circles, so those who think that it needs strictly to describe all processing for use of computer think that the natural language is not strict. They think that the description by the language of symbolic logic is most strict and that the inference by means of a syllogism is the strictest processing. However, this is a typical sequential processing. It can safely be said that such language for use of computer is strict because of being like machine language in parallel and practical use in spite of high-level language.

1.4 The parallel property of the operational calculus.
. . When the human thinks the function f(t)=t2, in the early stages he sequentially dots the calculated values of function at some points t in his mind and draws the quadratic curve joining these dots as he draws the curve on graph paper. However, when he has got used to the processing, it becomes the parallel processing and he can think the graph of quadratic function in an instant. The formula expression of function is the parallel expression which means the whole of the set of these continuous dots. The operation between two functions is described by use of formula as the operation between two sets of these dots and not two dots at every point t. It may be said that this is the ideal parallel operation.
. . The functional analysis and the operational calculus think a function to be a point on the vector space and realize the highest level of parallel operation which treats the shape of function and the whole of the set of points expressing the graph of function as the abstract property of a point on the vector space. In especial, Heaviside's operational calculus and Laplace transformation are used as very useful method in order to solve the differential equation. However, it is not generally easy to solve the differential equation and integrate the function by use of formula expression, so the sequential processing of numerical calculation is used as the easy method. In the case, the calculation is carried out for some sampled values at proper intervals, because it is impossible to calculate for all the real values t, which exist infinitely. The results between the sampled vales may be estimated by interpolation formula, using the results for the sampled values, so it may be said that the calculation for the sampled values is the parallel processing for all the real values t substantially.
. . There is Newton's interpolation formula as easy and familiar interpolation. Carrying out the numerical differentiation and integration of the function interpolated by use of this formula, it is shown furthermore evidently that the numerical solution is the parallel calculation for all the values t. However, the notation ∑ means sequential processing adding the operands successively and it complicates the treatment to write the notation one by one. Newton's interpolation formula is decided by determining the derivatives till the n-th order, so denoting them by the set of (y0, Δy0, Δ2y0, …, Δpy0), the notation shows the shape of function evidently and is easy to understand. Furthermore, it denotes a point on the n+1 dimensional space, assigning the elements to each of the n+1 independent axes. The author's operational calculus realizes the parallel expression and the parallel processing of numerical functions by denoting the point as the vector Y.
. . The four operations between functions are carried out as the four operations between the vectors and the differentiation and integration are carried out by the algebra of matrices as the product of the differential or integral operator multiplied by the vector. However, the program is described as the four operations between the symbols of vector and matrix. When the operation is carried out, the set of successive differences is given as the value of vector, and the set of resultant differences is obtained by the algebra of matrices carried out by hardware or firmware. Accordingly, the user does not need to program the process calculating each of the values of differences and realizes the parallel programming as easy as BASIC. This chapter describes the parallel processing of the algebra of vector and matrix by poly-processor and describes the parallel processing of knowledge information by receiving a hint from the parallel processing of the algebra of vector and matrix.

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