An Introduction to Genetic Algorithms - Boente

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it to get one fitness case correct: the case where all the blocks were already in the stack in the correct order. Thus, this program's fitness was 1. (EQ (MS NN) (EQ (MS NN) (MS NN))) Chapter 2: Genetic Algorithms in Problem Solving "Move the next needed block to the stack three times." This program made some progress and got four fitness cases right, giving it fitness 4. (Here EQ serves merely as a control structure. Lisp evaluates the first expression, then evaluates the second expression, and then compares their value. EQ thus performs the desired task of executing the two expressions in sequence—we do not actually care whether their values are equal.) By generation 5, the population contained some much more successful programs. The best one was (DU (MS NN) (NOT NN)) (i.e., "Move the next needed block to the stack until no more blocks are needed"). Here we have the basics of a reasonable plan. This program works in all cases in which the blocks in the stack are already in the correct order: the program moves the remaining blocks on the table into the stack in the correct order. There were ten such cases in the total set of 166, so this program's fitness was 10. Notice that this program uses a building block—(MS NN)—that was discovered in the first generation and found to be useful there. In generation 10 a completely correct program (fitness 166) was discovered: (EQ (DU (MT CS) (NOT CS)) (DU (MS NN) (NOT NN))). This is an extension of the best program of generation 5. The program empties the stack onto the table and then moves the next needed block to the stack until no more blocks are needed. GP thus discovered a plan that works in all cases, although it is not very efficient. Koza (1992) discusses how to amend the fitness function to produce a more efficient program to do this task. The block stacking example is typical of those found in Koza's books in that it is a relatively simple sample problem from a broad domain (planning). A correct program need not be very long. In addition, the necessary functions and terminals are given to the program at a fairly high level. For example, in the block stacking problem GP was given the high−level actions MS, MT, and so on; it did not have to discover them on its own. Could GP succeed at the block stacking task if it had to start out with lower−level primitives? O'Reilly and Oppacher (1992), using GP to evolve a sorting program, performed an experiment in which relatively low−level primitives (e.g., "if−less−than" and "swap") were defined separately rather than combined a priori into "if−less−than−then−swap" Under these conditions, GP achieved only limited success. This indicates a possible serious weakness of GP, since in most realistic applications the user will not know in advance what the appropriate high−level primitives should be; he or she is more likely to be able to define a larger set of lower−level primitives. Genetic programming, as originally defined, includes no mechanism for automatically chunking parts of a program so they will not be split up under crossover, and no mechanism for automatically generating hierarchical structures (e.g., a main program with subroutines) that would facilitate the creation of new high−level primitives from built−in low−level primitives. These concerns are being addressed in more recent research. Koza (1992, 1994) has developed methods for encapsulation and automatic definition of functions. Angeline and Pollack (1992) and O'Reilly and Oppacher (1992) have proposed other methods for the encapsulation of useful subtrees. Koza's GP technique is particularly interesting from the standpoint of evolutionary computation because it allows the size (and therefore the complexity) of candidate solutions to increase over evolution, rather than keeping it fixed in the standard GA. However, the lack of sophisticated encapsulation mechanisms has so far 32
limited the degree to which programs can usefully grow. In addition, there are other open questions about the capabilities of GP. Does it work well because the space of Lisp expressions is in some sense "dense" with correct programs for the relatively simple tasks Koza and other GP researchers have tried? This was given as one reason for the success of the artificial intelligence program AM (Lenat and Brown 1984), which evolved Lisp expressions to discover "interesting" conjectures in mathematics, such as the Goldbach conjecture (every even number is the sum of two primes). Koza refuted this hypothesis about GP by demonstrating how difficult it is to randomly generate a successful program to perform some of the tasks for which GP evolves successful programs. However, one could speculate that the space of Lisp expressions (with a given set of functions and terminals) is dense with useful intermediate−size building blocks for the tasks on which GP has been successful. GP's ability to find solutions quickly (e.g., within 10 generations using a population of 300) lends credence to this speculation. GP also has not been compared systematically with other techniques that could search in the space of parse trees. For example, it would be interesting to know if a hill climbing technique could do as well as GP on the examples Koza gives. One test of this was reported by O'Reilly and Oppacher (1994a,b), who defined a mutation operator for parse trees and used it to compare GP with a simple hill−climbing technique similar to random−mutation hill climbing (see computer exercise 4 of chapter 1) and with simulated annealing (a more sophisticated hill−climbing technique). Comparisons were made on five problems, including the block stacking problem described above. On each of the five, simulated annealing either equaled or significantly outperformed GP in terms of the number of runs on which a correct solution was found and the average number of fitness−function evaluations needed to find a correct program. On two out of the five, the simple hill climber either equaled or exceeded the performance of GP. Though five problems is not many for such a comparison in view of the number of problems on which GP has been tried, these results bring into question the claim (Koza 1992) that the crossover operator is a major contributor to GP's success. O'Reilly and Oppacher (1994a) speculate from their results that the parse−tree representation "may be a more fundamental asset to program induction than any particular search technique," and that "perhaps the concept of building blocks is irrelevant to GP." These speculations are well worth further investigation, and it is imperative to characterize the types of problems for which crossover is a useful operator and for which a GA will be likely to outperform gradient−ascent strategies such as hill climbing and simulated annealing. Some work toward those goals will be described in chapter 4. Some other questions about GP: Chapter 2: Genetic Algorithms in Problem Solving Will the technique scale up to more complex problems for which larger programs are needed? Will the technique work if the function and terminal sets are large? How well do the evolved programs generalize to cases not in the set of fitness cases? In most of Koza's examples, the cases used to compute fitness are samples from a much larger set of possible fitness cases. GP very often finds a program that is correct on all the given fitness cases, but not enough has been reported on how well these programs do on the "out−of−sample" cases. We need to know the extent to which GP produces programs that generalize well after seeing only a small fraction of the possible fitness cases. To what extent can programs be optimized for correctness, size, and efficiency at the same time? Genetic programming's success on a wide range of problems should encourage future research addressing these questions. (For examples of more recent work on GP, see Kinnear 1994.) 33
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 32 and 33: As a simple example, consider a pro
Page 34 and 35: Chapter 2: Genetic Algorithms in Pr
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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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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