Solution of Algebraic Equations by Using Autonomous Computational Methods
Andrew Pownuk
Department of Mathematical Sciences University of Texas at El Paso, El Paso, Texas, USA [email protected], https://andrew.pownuk.com
AMS Fall Central Sectional Meeting September 12-13, 2020
1 / 63 Outline
1 Game Theory
2 Autonomous Computational Methods
3 Algebraic Equations
4 Generalizations and Possible Limitations
5 Autonomous Computational Methods
6 Machine Learning
7 Conclusions
2 / 63 Checkers (Game Theory)
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods Machine Figure: Game tree Learning
Conclusions Arthur Samuel in 1959 applied alphabeta pruning in order to create a program that learned how to play checkers. Alphabeta pruning is a search algorithm that seeks to decrease the number of nodes that are evaluated by the minimax algorithm in its search tree.
3 / 63 Game Theory ( tic-tac-toe)
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
Figure: Game tree
4 / 63 Game Theory (GO)
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
Figure: Game tree
5 / 63 Game Theory (Winning Strategy)
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
Figure: Game tree
6 / 63 Basic Mathematical Operations
Game Theory
Autonomous Computational Addition Methods 1 + 1 = 2 Algebraic Equations 1 + 0 = 1 Generalizations and Possible Limitations 2 + (−1) = 1
Autonomous Computational Multiplication Methods 1 ∗ 0 = 0 Machine Learning i ∗ i = −1 Conclusions Algebra a + b = b + a a(b + c) = ab + ac etc.
7 / 63 Self Adaptive Computational Methods
Game Theory Autonomous Product rule (input information) Computational Methods 0 0 0 Algebraic (f ∗ g) = f ∗ g + f ∗ g Equations
Generalizations and Possible After calculations (new theorem created automatically) Limitations 0 0 0 Autonomous ((f ∗ g) ∗ h) = (f ∗ g) ∗ h + (f ∗ g) ∗ h Computational Methods 0 0 0 0 Machine ((f ∗ g) ∗ h) = (f ∗ g + f ∗ g ) ∗ h + (f ∗ g) ∗ h Learning 0 0 0 0 Conclusions (f ∗ g ∗ h) = (f ∗ g) ∗ h + (f ∗ g ) ∗ h + (f ∗ g) ∗ h New theorem can be used in exactly the same way like the original theorem.
(f ∗ g ∗ h)0 = f 0 ∗ g ∗ h + f ∗ g 0 ∗ h + f ∗ g ∗ h0
8 / 63 Automated Theorem Proving
Game Theory
Autonomous Computational Methods Lagrange Theorem. Algebraic Equations Function f (x) is continuous in the interval [a, b] Generalizations and Possible Function f (x) is differentiable in the interval (a, b) Limitations f (b)−f (a) 0 Autonomous then exists c ∈ (a, b) such that = f (c) Computational b−a Methods
Machine Learning Function f (x)g(x) is continuous in the interval [a, b] Conclusions Function f (x)g(x) is differentiable in the interval (a, b) then exists c ∈ (a, b) such that f (b)g(b)−f (a)h(a) 0 0 b−a = f (c)g(c) + f (c)g (c)
9 / 63 Differentiation
Game Theory
Autonomous Computational Methods
Algebraic Equations Input information Generalizations 0 0 0 and Possible (f (x) + g(x)) = f (x) + g (x) Limitations (f (x) − g(x))0 = f 0(x) − g 0(x) Autonomous Computational 0 0 0 Methods (f (x)g(x)) = f (x)g(x) + f (x)g (x) f 0(x)g(x)−f (x)g 0(x) Machine (f (x)/g(x))0 = Learning g 2(x) 0 0 0 Conclusions (f (g(x))) = f (g(x))g (x) arithmetic operations
10 / 63 Differentiation
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Automatically generate formulas out of: Limitations List of arithmetical operations Autonomous Computational List of elementary functions and constants Methods Machine Information about the numer of arithmetical operations Learning and functions compositions Conclusions
11 / 63 Differentiation (Latex source)
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
12 / 63 Differentiation (step 1)
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
13 / 63 Differentiation (step 2)
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
14 / 63 Differentiation (step 3)
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
15 / 63 Differentiation (step 4)
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
16 / 63 Differentiation (step 5)
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
17 / 63 Set theory
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible A ∪ B = B ∪ A Limitations C Autonomous A ∩ A = ∅ Computational Methods (A ∪ B) ∩ (B ∪ A)C = B ∩ BC Machine Learning ((A ∪ B) ∩ (B ∪ A)C ) ∩ ((A ∪ B) ∩ (B ∪ A)C )C = ∅ Conclusions etc.
18 / 63 Probability theory
Game Theory
Autonomous Computational Methods
Algebraic Equations Generalizations P(A ∪ B) = P(A) + P(B) − P(A ∩ B) and Possible Limitations P(A ∩ B) Autonomous P(A|B) = Computational P(B) Methods Machine P(A|B)P(B) = P(A ∩ B) Learning Conclusions P(A ∪ B) = P(A) + P(B) − P(A|B)P(B) etc.
19 / 63 Analysis
Game Theory
Autonomous Computational d du Methods EA + n = 0, u(0) = 0, u(L) = 0 Algebraic dx dx Equations L L L Generalizations and Possible Z d du Z Z Limitations EA vdx + nvdx = 0vdx, u(0) = 0, u(L) = 0 dx dx Autonomous Computational 0 0 0 Methods L L Machine Z dv Z du Learning u dx = vdx + u(0)v(L) − u(L)v(L) Conclusions dx dx 0 0 L L Z d du Z du dv du(0) du(L) EA vdx = EA dx+EA v(0)−EA v(L) dx dx dx dx dx dx 0 0 etc. 20 / 63 Algebra
Game Theory
Autonomous Computational Methods
Algebraic Fundamental theorem of Galois theory. Equations Number of automorphisms equals the degree of the extension. Generalizations and Possible Limitations |Gal (K/F )| = [K : F ] Autonomous Computational Methods Field extension K/F .
Machine Degree of the extension [K : F ]. Learning Number of automorphisms in the Galois extension |Gal(K/F )|. Conclusions For example, if we√ know√ that 6 = [K : F ] = 3 2, −3 : , then we know instantly Q Q √ √ 3 that 6 = |Aut (K/F )| = Aut Q 2, −3 /Q etc.
21 / 63 Automated Programming
Game Theory
Autonomous Computational Methods int f(int x) Algebraic Equations { Generalizations return x*x; and Possible Limitations } Autonomous Computational Methods f(f(x)) Machine Learning
Conclusions int ff(int x) { return (x*x)*(x*x); }
22 / 63 Group Axioms
Game Theory A group is a set (mathematics), G together with an Binary Autonomous 00 00 Computational operation · (called the group law of G) that combines any Methods two element elements a and b to form another element, Algebraic Equations denoted a · b. To qualify as a group, the set and operation, Generalizations (G, ·), must satisfy four requirements known as the group and Possible Limitations axioms: Autonomous 1 Computational Closure: For all a, b ∈ G, the result of the operation, a · b, Methods is also in G. Machine 2 Learning Associativity: For all a, b and c in G,(a · b) · c = a · (b · c).
Conclusions 3 Identity element: There exists an element e in G such that, for every element a in G, the equation 1 = e · a = a · e = a holds. Such an element is unique. 4 Inverse element: For each a ∈ G, there exists an element b ∈ G, commonly denoted a1 (or a, if the operation is denoted +, such that a · b = b · a = e, where e is the
23 / 63 identity element. Field Axioms
Game Theory 1 Autonomous Addition is commutative: x + y = y + x, for all x, y ∈ F . Computational 2 Methods Addition is associative: (x + y) + z = x + (y + z), for all
Algebraic x, y, z ∈ F . Equations 3 Existence of additive identity: there is a unique element Generalizations and Possible 0 ∈ F such that x + 0 = x, for all x ∈ F . Limitations 4 Existence of additive inverses: if x ∈ F , there is a unique Autonomous Computational element x ∈ F such that x + (x) = 0. Methods 5 Machine Multiplication is commutative: xy = yx, for all x, y ∈ F . Learning 6 Multiplication is associative: (xy)z = x(yz), for all Conclusions x, y, z ∈ F . 7 Existence of multliplicative identity: there is a unique element 1 ∈ F such that 1 6= 0 and x1 = x, for all x ∈ F . 8 Existence of multliplicative inverses: if x ∈ F and xneq0, there is a unique element (1/x) ∈ F such that x(1/x) = 1. 9 Distributivity: x(y + z) = xy + xz, for all x, y, z ∈ F . 24 / 63 Expression Evaluation
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
25 / 63 Expression Evaluation
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
Expression evaluation is possible without specifying any explicit method for expression evaluation.
26 / 63 Computational Graph
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning Conclusions COCONUT Project COntinuous CONstraints - Updating the Technology. https://www.mat.univie.ac.at/ neum/glopt/coconut/
27 / 63 Solution of Algebraic Equations
Game Theory
Autonomous Computational step-1 Methods x = (x + (2 + x)) Algebraic Equations step-3 Generalizations x = (x + 2 + x) and Possible Limitations step-4 Autonomous x = (x + 2 + x) Computational Methods step-5 Machine x = x + 2 + x Learning step-6 Conclusions x = x + 2 + x step-7 x = 2 + 2 ∗ x step-9 x + (−1) ∗ x + (−1) ∗ x = 2
28 / 63 Solution of Algebraic Equations
Game Theory
Autonomous Computational Methods Algebraic step-10 Equations x + (−2) ∗ x = 2 Generalizations and Possible step-11 Limitations (−1) ∗ x = 2 Autonomous Computational step-12 Methods x = (2/(−1)) Machine Learning step-13 Conclusions x = (−2) step-14 x = ((−2)/1)
29 / 63 Automatically Generated Latex Report
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
30 / 63 Automatically Generated PDF Reports
Game Theory
Autonomous Computational Hundred/thousand/unlimited number of examples: Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
http://andrew.pownuk.com/research/AlgebraicEquations/
31 / 63 Simplification of the Solution
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
32 / 63 Simplification of the Solution
Game Theory
Autonomous Computational Methods
Algebraic Finding optimal form of the computational process is one Equations or the main goal of this project. Generalizations and Possible Quality of the simplification depend on the amount of Limitations knowledge available in the system and processing power. Autonomous Computational Methods For small problem the simplification can be found uniquely. Machine For more complex problem simplification of the expressions Learning never stops. Conclusions It is possible to use new computational results in order future calculations (self-adaptivity). In this way the system can generate knowledge in autonomous way.
33 / 63 Finite Number of Steps
Game Theory
Autonomous Computational Methods Algebraic 1 Equations Limits limn→∞ n = 0 Generalizations Convergence of infinite series and Possible ∞ Limitations P 1 n=1 n2 (p = 2 > 1 series converges). Autonomous Computational If function f is integrable, then the sequence of Riemann Methods PN R b sum n=1 f (xn)∆xn converges to a f (x)dx (for Machine Learning appropriate partitions). Conclusions Presented methodology can be applied to many scientific theories (mathematics, engineering, chemistry, biology etc.) with finite number of steps.
34 / 63 What is Special About This Research?
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible All calculations are done in fully autonomous way. Limitations
Autonomous Computational Methods
Machine Learning Why autonomous Conclusions calculations?
35 / 63 What is Special About This Research?
Game Theory
Autonomous Computational Methods
Algebraic Equations All calculations are done in fully autonomous way.
Generalizations and Possible Limitations Why autonomous calculations? Autonomous Computational Methods
Machine Learning Correctness Conclusions and Scalability.
36 / 63 Scalability
Game Theory Autonomous Moscoso del Pradon uses his method to determine how Computational Methods much information the brain can process during lexical Algebraic Equations decision tasks. The answer? No more than about 60 bits
Generalizations per second (Emerging Technology, New Measure of and Possible Limitations Human Brain Processing Speed, MIT Technology Review,
Autonomous August 25, 2009). Computational Methods Japanese Fugaku is the world’s most powerful 15 Machine supercomputer, reaching 415.53 petaFLOPS (10 ) on the Learning LINPACK benchmarks. Conclusions Most PC are in the GigaFLOPS (109) range. 54-qubit processor, named Sycamore from Google can calculate some tasks in 200 seconds. The worlds fastest supercomputer needs 10000 years to produce a similar output (experiment done in 2019).
37 / 63 Is Mathematics Invented or Discovered?
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
38 / 63 Mathematical Realism
Game Theory
Autonomous Computational Methods Mathematical entities exist independently of the human Algebraic mind. Equations
Generalizations Humans do not invent mathematics, but rather discover and Possible Limitations it, and any other intelligent beings in the universe would
Autonomous presumably do the same. Computational Methods
Machine Learning
Conclusions
39 / 63 Logicism
Game Theory
Autonomous Computational Methods
Algebraic Equations Logicism is the thesis that mathematics is reducible to Generalizations logic, and hence nothing but a part of logic. and Possible Limitations Logicists hold that mathematics can be known a priori, Autonomous Computational but suggest that our knowledge of mathematics is just Methods part of our knowledge of logic in general, and is thus Machine Learning analytic, not requiring any special faculty of mathematical
Conclusions intuition. In this view, logic is the proper foundation of mathematics, and all mathematical statements are necessary logical truths.
40 / 63 Logicism
Game Theory Autonomous Rudolf Carnap (1931) presents the logicist thesis in two parts: Computational Methods The concepts of mathematics can be derived from logical Algebraic Equations concepts through explicit definitions. Generalizations and Possible The theorems of mathematics can be derived from logical Limitations axioms through purely logical deduction. Autonomous Computational Methods Hilbert’s program.
Machine Learning Hilbert proposed to ground all existing theories to a finite,
Conclusions complete set of axioms, and provide a proof that these axioms were consistent. Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic.
41 / 63 Logicism
Game Theory
Autonomous Computational Methods G¨odel’sincompleteness theorems: Algebraic Equations The first incompleteness theorem states that no consistent Generalizations and Possible system of axioms whose theorems can be listed by an Limitations effective procedure (i.e., an algorithm) is capable of Autonomous Computational proving all truths about the arithmetic of the natural Methods numbers. Machine Learning The second incompleteness theorem, an extension of the Conclusions first, shows that the system cannot demonstrate its own consistency. Tarski’s undefinability theorem: arithmetical truth cannot be defined in arithmetic.
42 / 63 Formalism
Game Theory
Autonomous Computational Methods
Algebraic Formalism holds that mathematical statements may be Equations thought of as statements about the consequences of Generalizations and Possible certain string manipulation rules. Limitations
Autonomous According to formalism, mathematical truths are not Computational Methods about numbers and sets and triangles and the likein fact,
Machine they are not ”about” anything at all. Learning (Deductivism) Formalism need not mean that Conclusions mathematics is nothing more than a meaningless symbolic game. It is usually hoped that there exists some interpretation in which the rules of the game hold.
43 / 63 Conventionalism
Game Theory
Autonomous Poincar´e’suse of non-Euclidean geometries in his work on Computational Methods differential equations convinced him that Euclidean
Algebraic geometry should not be regarded as a priori truth. He held Equations that axioms in geometry should be chosen for the Generalizations and Possible results they produce, not for their apparent coherence Limitations with human intuitions about the physical world. Autonomous Computational Methods
Machine Learning
Conclusions
44 / 63 Mathematical Intuitionism
Game Theory
Autonomous Computational Methods In mathematics, intuitionism is a program of Algebraic Equations methodological reform whose motto is that ”there are no
Generalizations non-experienced mathematical truths” (L. E. J. and Possible Limitations Brouwer). Autonomous A major force behind intuitionism was L. E. J. Brouwer, Computational Methods who rejected the usefulness of formalized logic of any sort Machine for mathematics. His student Arend Heyting postulated an Learning intuitionistic logic, different from the classical Aristotelian Conclusions logic; this logic does not contain the law of the excluded middle and therefore frowns upon proofs by contradiction. The axiom of choice is also rejected in most intuitionistic set theories, though in some versions it is accepted.
45 / 63 Other Theories
Game Theory
Autonomous Computational Methods
Algebraic Equations Etc. Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
46 / 63 Mathematics as a Language of Science
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
47 / 63 Autonomous Computational Methods
Game Theory Current version of the software (Plato, 5th century BC). Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
48 / 63 Autonomous Computational Methods
Game Theory Future version of the software. Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
49 / 63 Mathematics as a Language of Science
Game Theory
Autonomous Computational Significant part of engineering problems can be described by Methods using mathematics (and appropriate software). Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
50 / 63 Mathematics as a Language of Science
Game Theory
Autonomous Computational Significant part of engineering problems can be described by Methods using mathematics (and appropriate software). Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
51 / 63 Solution of Algebraic Equations - - Deterministic Approach
Game Theory
Autonomous Computational Methods Algebraic Linear equation ax = b Equations 2 Generalizations Quadratic equation ax + bx + c = 0 and Possible 3 2 Limitations Cubic equation ax + bx + cx + d = 0
Autonomous x Computational Non-elementary solutions x · e = 1 Methods etc. Machine Learning Appropriate method of solution can be applied to appropriate Conclusions equation.
Known equation → Method for solution
52 / 63 Machine Learning in Game Theory
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions
Figure: Machine Learning model of chess (Deepmind AlphaZero)
53 / 63 Problem space
Game Theory Autonomous Trees with up to n internal nodes. Computational Methods Set p1 of unary operators (e.g. cos, sin, exp, log). Algebraic Equations Set p2 of binary operators (e.g. +, −, /, ×). Generalizations and Possible a set of L leaf values containing variables (e.g. x, y, z), Limitations constants (e.g. e, π), integers (e.g. {,-10,..., 10,}). Autonomous Computational The number of binary trees with n internal nodes is given by Methods
Machine the n-th Catalan numbers. Learning A binary tree with n internal nodes has exactly n + 1 leaves. Conclusions Each node and leaf can take respectively p2 and L different values. As a result, the number of expressions with n binary operators can be expressed by:
1 2n 4n E = pnLn+1 ≈ √ (p )nLn+1 n n + 1 n n n π · n 2
54 / 63 Solution of Algebraic Equations - - Machine Learning
Game Theory
Autonomous Computational Methods
Algebraic It is possible to classify equations by using the neural Equations networks. Generalizations and Possible The initial guess provided by the neural networks can be Limitations used to find approximate solution of the problem. Autonomous Computational Methods Training of the neural network can be done in fully
Machine autonomous way. Learning The structure and features of the neural network can be Conclusions crated in fully autonomous way. Initial guess provided by the probabilistic methods have to be verified in the future calculations.
55 / 63 Existing System for Theorem Proving
Game Theory
Autonomous Computational Methods Metamath (http://us.metamath.org/) Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions Mizar system (formal language for writing mathematical definitions and proofs). Step by step solutions in Mathematica (WolframAlpha).
56 / 63 Finite Number of Steps? Logic?
Game Theory
Autonomous Computational Methods
Algebraic The fundamental difference between if-else and machine Equations learning is that if-else models are static deterministic Generalizations and Possible environments and machine learning (ML) algorithms, are Limitations probabilistic stochastic environments. Autonomous Computational Cyc - is the world’s longest-lived artificial intelligence Methods project, attempting to assemble a comprehensive ontology Machine Learning and knowledge base. Douglas Lenat began the project in Conclusions July 1984 at MCC, where he was Principal Scientist 19841994, and then, since January 1995, has been under active development by the Cycorp company, where he is the CEO.
57 / 63 Self-Adaptive Methods
Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Methods
Machine Learning
Conclusions Due to complexity of the problems the process of updating of the knowledge base never stops. The program runs practically indefinitely constantly improving his own computations. Calculations can be speed up by using parallel and distributed computing.
58 / 63 Conclusions
Game Theory
Autonomous Computational Methods
Algebraic Equations Mathematical/scientific knowledge can be treated as Generalizations and Possible independent units that can interact with each-other and Limitations create new, possibly useful knowledge. Autonomous Computational Generation of new knowledge can be fully automated and Methods autonomous. No interaction with humans is necessary. Machine Learning Development of new knowledge is possible in many Conclusions different fields (e.g. engineering, computer science, mathematics etc.).
59 / 63 Conclusions (continued)
Game Theory
Autonomous Computational Methods
Algebraic Equations By using presented methodology it is possible to create Generalizations complex examples and appropriate computational methods and Possible Limitations relevant to many areas of mathematics as well as in other Autonomous areas of science and engineering. Computational Methods Machine learning (the main computational method is Machine Learning NOT based on machine learning) can be used as source of
Conclusions good initial guess for processing mathematical information. Actually there is NO main computational method in the system.
60 / 63 Conclusions (continued)
Game Theory
Autonomous Computational Methods
Algebraic Equations Once the information is available in the system it will Generalizations and Possible NEVER be forgotten and can be used for generation of Limitations new mathematical theorems. From that perspective the Autonomous Computational system can be viewed as self-organizing archive of Methods information. Machine Learning Development of this and similar systems should speed up Conclusions cooperation among scientists around the world (theoretical possibility).
61 / 63 Conclusions (continued)
Game Theory
Autonomous Computational Methods
Algebraic Calculations can be done in distributed way (this option is Equations experimental at this moment). Large number of Generalizations and Possible computers can process the information simultaneously in Limitations order to get the results faster. Calculations do not require Autonomous Computational existence of any centralized system. Methods
Machine Parallel computing can significantly speed up the Learning calculations (future work). Conclusions Autonomous interaction with external sources of information extends internal database of information and should increase productivity of the system (future work).
62 / 63 Game Theory
Autonomous Computational Methods
Algebraic Equations
Generalizations and Possible Limitations
Autonomous Computational Thank You Methods
Machine Learning
Conclusions
63 / 63