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Solution of Algebraic Equations by Using Autonomous Computational Methods

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Solution of Algebraic Equations by Using Autonomous Computational Methods 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 2 (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 2 (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(AjB) = Computational P(B) Methods Machine P(AjB)P(B) = P(A \ B) Learning Conclusions P(A [ B) = P(A) + P(B) − P(AjB)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 jGal (K=F )j = [K : F ] Autonomous Computational Methods Field extension K=F . Machine Degree of the extension [K : F ]. Learning Number of automorphisms in the Galois extension jGal(K=F )j. Conclusions For example, if wep knowp that 6 = [K : F ] = 3 2; −3 : , then we know instantly Q Q p p 3 that 6 = jAut (K=F )j = 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 2 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 2 G, there exists an element b 2 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 2 F . Computational 2 Methods Addition is associative: (x + y) + z = x + (y + z), for all Algebraic x; y; z 2 F . Equations 3 Existence of additive identity: there is a unique element Generalizations and Possible 0 2 F such that x + 0 = x, for all x 2 F . Limitations 4 Existence of additive inverses: if x 2 F , there is a unique Autonomous Computational element x 2 F such that x + (x) = 0. Methods 5 Machine Multiplication is commutative: xy = yx, for all x; y 2 F . Learning 6 Multiplication is associative: (xy)z = x(yz), for all Conclusions x; y; z 2 F . 7 Existence of multliplicative identity: there is a unique element 1 2 F such that 1 6= 0 and x1 = x, for all x 2 F . 8 Existence of multliplicative inverses: if x 2 F and xneq0, there is a unique element (1=x) 2 F such that x(1=x) = 1. 9 Distributivity: x(y + z) = xy + xz, for all x; y; z 2 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
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