An Introduction to Genetic Algorithms - Boente

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As a simple example, consider a program to compute the orbital period P of a planet given its average distance A from the Sun. Kepler's Third Law states that P 2 = cA 3 , where c is a constant. Assume that P is expressed in units of Earth years and A is expressed in units of the Earth's average distance from the Sun, so c = 1. In FORTRAN such a program might be written as PROGRAM ORBITAL_PERIOD C # Mars # A = 1.52 P = SQRT(A * A * A) PRINT P END ORBITAL_PERIOD where * is the multiplication operator and SQRT is the square−root operator. (The value for A for Mars is from Urey 1952.) In Lisp, this program could be written as (defun orbital_period () ; Mars ; (setf A 1.52) (sqrt (* A (* A A)))) In Lisp, operators precede their arguments: e.g., X * Y is written (* X Y). The operator "setf" assigns its second argument (a value) to its first argument (a variable). The value of the last expression in the program is printed automatically. Assuming we know A, the important statement here is (SQRT (* A (* A A))). A simple task for automatic programming might be to automatically discover this expression, given only observed data for P and A. Expressions such as (SQRT (* A (* A A)) can be expressed as parse trees, as shown in figure 2.1. In Koza's GP algorithm, a candidate solution is expressed as such a tree rather than as a bit string. Each tree consists of funtions and terminals. In the tree shown in figure 2.1, SQRT is a function that takes one argument, * is a function that takes two arguments, and A is a terminal. Notice that the argument to a function can be the result of another function—e.g., in the expression above one of the arguments to the top−level * is (* A A). Figure 2.1: Parse tree for the Lisp expression (SQRT (* A (* A * A A))). Koza's algorithm is as follows: 1. Chapter 2: Genetic Algorithms in Problem Solving Choose a set of possible functions and terminals for the program. The idea behind GP is, of course, to evolve programs that are difficult to write, and in general one does not know ahead of time precisely which functions and terminals will be needed in a successful program. Thus, the user of GP has to 28
2. 3. 4. Chapter 2: Genetic Algorithms in Problem Solving make an intelligent guess as to a reasonable set of functions and terminals for the problem at hand. For the orbital−period problem, the function set might be {+, , *, /, ,} and the terminal set might simply consist of {A}, assuming the user knows that the expression will be an arithmetic function of A. Generate an initial population of random trees (programs) using the set of possible functions and terminals. These random trees must be syntactically correct programs—the number of branches extending from each function node must equal the number of arguments taken by that function. Three programs from a possible randomly generated initial population are displayed in figure 2.2. Notice that the randomly generated programs can be of different sizes (i.e., can have different numbers of nodes and levels in the trees). In principle a randomly generated tree can be any size, but in practice Koza restricts the maximum size of the initially generated trees. Figure 2.2: Three programs from a possible randomly generated initial population for the orbital−period task. The expression represented by each tree is printed beneath the tree. Also printed is the fitness f (number of outputs within 20% of correct output) of each tree on the given set of fitness cases. A is given in units of Earth's semimajor axis of orbit; P is given in units of Earth years. (Planetary data from Urey 1952.) Calculate the fitness of each program in the population by running it on a set of "fitness cases" (a set of inputs for which the correct output is known). For the orbital−period example, the fitness cases might be a set of empirical measurements of P and A. The fitness of a program is a function of the number of fitness cases on which it performs correctly. Some fitness functions might give partial credit to a program for getting close to the correct output. For example, in the orbital−period task, we could define the fitness of a program to be the number of outputs that are within 20% of the correct value. Figure 2.2 displays the fitnesses of the three sample programs according to this fitness function on the given set of fitness cases. The randomly generated programs in the initial population are not likely to do very well; however, with a large enough population some of them will do better than others by chance. This initial fitness differential provides a basis for "natural selection." 29
Page 2 and 3: An Introduction to Genetic Algorith
Page 4 and 5: Table of Contents Chapter 4: Theore
Page 6 and 7: Chapter 1: Genetic Algorithms: An O
Page 10 and 11: 1.4 SEARCH SPACES AND FITNESS LANDS
Page 12 and 13: potential energy is a measure of ho
Page 14 and 15: 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 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 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 66 and 67: COMPUTER EXERCISES 1. 2. 3. 4. 5. 6
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
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an initial population in which the
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Chapter 3: Genetic Algorithms in Sc
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Chapter 3: Genetic Algorithms in Sc
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these random effects, Bedau and Pac
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3. * 4. * the female child in the n
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calculating the observed average fi
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(4.1) The goal is to find n = n* th
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competition in the d *···* parti
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schemas with the best observed fitn
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Steepest−ascent hill climbing (SA
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(If the algorithm spends only 1/m o
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strings—then in principle crossov
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(4.7) Chapter 4: Theoretical Founda
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2. 3. Calculate the fitness f(x) of
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Defining ri,j(k) and is somewhat tr
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Results of the Formalization How ca
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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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