An Introduction to Genetic Algorithms - Boente

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COMPUTER EXERCISES 1. 2. 3. 4. 5. 6. 7. * 8. * Chapter 2: Genetic Algorithms in Problem Solving Implement a genetic programming algorithm and use it to solve the "6−multiplexer" problem (Koza 1992). In this problem there are six Boolean−valued terminals, {a0, a1, d0,d1, d2, d3}, and four functions, {AND, OR, NOT, IF}. The first three functions are the usual logical operators, taking two, two, and one argument respectively, and the IF function takes three arguments. (IF X Y Z) evaluates its first argument X. If X is true, the second argument Y is evaluated; otherwise the third argument Z is evaluated. The problem is to find a program that will return the value of the d terminal that is addressed by the two a terminals. E.g., if a0 = 0 and a1 = 1, the address is 01 and the answer is the value of d1. Likewise, if a0 = 1 and a1 = 1, the address is 11 and the answer is the value of d3. Experiment with different initial conditions, crossover rates, and population sizes. (Start with a population size of 300.) The fitness of a program should be the fraction of correct answers over all 2 6 possible fitness cases (i.e., values of the six terminals). Perform the same experiment as in computer exercise 1, but add some "distractors" to the function and terminal sets—extra functions and terminals not necessary for the solution. How does this affect the performance of GP on this problem? Perform the same experiment as in computer exercise 1, but for each fitness calculation use a random sample of 10 of the 2 6 possible fitness cases rather than the entire set (use a new random sample for each fitness calculation). How does this affect the performance of GP on this problem? Implement a random search procedure to search for parse trees for the 6−multiplexer problem: at each time step, generate a new random parse tree (with the maximum tree size fixed ahead of time) and calculate its fitness. Compare the rate at which the best fitness found so far (plotted every 300 time steps—equivalent to one GP generation in computer exercise 1) increases with that under GP. Implement a random−mutation hill−climbing procedure to search for parse trees for the 6−multiplexer problem (see thought exercise 2). Compare its performance with that of GP and the random search method of computer exercise 4. Modify the fitness function used in computer exercise 1 to reward programs for small size as well as for correct performance. Test this new fitness function using your GP procedure. Can GP find correct but smaller programs by this method? Repeat the experiments of Crutchfield, Mitchell, Das, and Hraber on evolving r = 3 CAs to solve the problem. (This will also require writing a program to simulate cellular automata.) 62
9. * 10. * Compare the results of the experiment in computer exercise 7 with that of using random−mutation hill climbing to search for CA lookup tables to solve problem. (See Mitchell, Crutchfield, and Hraber 1994a for their comparison.) Perform the same experiment as in computer exercise 7, but use GP on parse−tree representations of CAs (see thought exercise 1). (This will require writing a program to translate between parse tree representations and CA lookup tables that you can give to your CA simulator.) Compare the results of your experiments with the results you obtained in computer exercise 7 using lookup−table encodings. Figure 2.29 gives a 19−unit neural network architecture for the "encoder/decoder" problem. The problem is to find a set of weights so that the network will perform the mapping given in table 2.2—that is, for each given input activation pattern, the network should copy the pattern onto its output units. Since there are fewer hidden units than input and output units, the network must learn to encode and then decode the input via the hidden units. Each hidden unit j and each output unit j has a threshold Ãj. If the incoming activation is greater than or equal to Ãj, the activation of the unit is set to 1; otherwise it is set to 0. At the first time step, the input units are activated according to the input activation pattern (e.g., 10000000). Then activation spreads from the input units to the hidden Figure 2.29: Network for computer exercise 10. The arrows indicate that each input node is connected to each hidden node, and each hidden node is connected to each output node. Table 2.2: Table for computer exercise 10. Input Pattern Output Pattern 10000000 10000000 01000000 01000000 00100000 00100000 00010000 00010000 00001000 00001000 00000100 00000100 00000010 00000010 Chapter 2: Genetic Algorithms in Problem Solving 00000001 00000001 units. The incoming activation of each hidden unit j is given by ia i wi, j, where ai is the activation of input unit i and wi, j is the weight on the link from unit i to unit j. After the hidden units have been activated, they in 63
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An Introduction to Genetic Algorith
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Table of Contents Chapter 4: Theore
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Chapter 1: Genetic Algorithms: An O
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Chapter 1: Genetic Algorithms: An O
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1.4 SEARCH SPACES AND FITNESS LANDS
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potential energy is a measure of ho
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A simple method of implementing fit
Page 16 and 17: from an initial state to a goal. Fo
Page 18 and 19: Robert Axelrod of the University of
Page 20 and 21: Chapter 1: Genetic Algorithms: An O
Page 26 and 27: instances of H at time t, and let
Page 28 and 29: What is the total payoff after 10 g
Page 30 and 31: 6. * c. d. a. b. Chapter 1: Genetic
Page 32 and 33: As a simple example, consider a pro
Page 34 and 35: Chapter 2: Genetic Algorithms in Pr
Page 36 and 37: it to get one fitness case correct:
Page 38 and 39: Evolving Cellular Automata Chapter
Page 42 and 43: up most of the computation time. Ch
Page 46 and 47: the prospect of using GAs to automa
Page 48 and 49: where Ã is the standard deviation
Page 54 and 55: the brain. In a feedforward network
Page 58 and 59: Figure 2.21: An illustration of Mil
Page 62 and 63: experiments with grammatical encodi
Page 64 and 65: function of the average error of th
Page 70 and 71: simulated generations, and such sim
Page 72 and 73: environments, they can fairly quick
Page 74 and 75: Chapter 3: Genetic Algorithms in Sc
Page 76 and 77: Figure 3.6: A schematic illustratio
Page 82 and 83: an initial population in which the
Page 88 and 89: these random effects, Bedau and Pac
Page 90 and 91: 3. * 4. * the female child in the n
Page 92 and 93: calculating the observed average fi
Page 94 and 95: (4.1) The goal is to find n = n* th
Page 96 and 97: competition in the d *···* parti
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Page 100 and 101: Steepest−ascent hill climbing (SA
Page 102 and 103: (If the algorithm spends only 1/m o
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Page 106 and 107: (4.7) Chapter 4: Theoretical Founda
Page 108 and 109: 2. 3. Calculate the fitness f(x) of
Page 110 and 111: Defining ri,j(k) and is somewhat tr
Page 112 and 113: Results of the Formalization How ca
Page 114 and 115: Chapter 4: Theoretical Foundations
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Chapter 4: Theoretical Foundations
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Figure 4.5: Predicted and observed
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COMPUTER EXERCISES 1. 2. 3. 4. 5. 6
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prone to rather arbitrary orderings
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Inversion works by choosing two poi
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The one hitch is that, as was seen
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used in the work of Tanese (1989),
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Two individuals are chosen at rando
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strategies community, in which para
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* 10. * 11. * 12. * Chapter 4: Theo
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Chapter 6: Conclusions and Future D
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Chapter 6: Conclusions and Future D
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Appendix A: Selected General Refere
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Evolution Artificielle Foundations
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Kaufmann. Appendix B: Other Resourc
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Morgan Kaufmann. Appendix B: Other
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Appendix B: Other Resources Forrest
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Appendix B: Other Resources Grefens
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Kirkpatrick, S., Gelatt, C.D., Jr.,
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Appendix B: Other Resources Mitchel
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Appendix B: Other Resources Roughga
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Appendix B: Other Resources Vose, M
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An Introduction to Genetic Algorithms - Boente

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