An Introduction to Genetic Algorithms - Boente

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Chapter 2: Genetic Algorithms in Problem Solving Why does Æd, for example, perform relatively well on the task? In figure 2.8d it can be seen that, although the patterns eventually converge to fixed points, there is a transient phase during which spatial and temporal transfer of information about the density in local regions takes place. This local information interacts with other local information to produce the desired final state. Roughly, Æd successively classifies "local" densities with a locality range that increases with time. In regions where there is some ambiguity, a "signal" is propagated. This is seen either as a checkerboard pattern propagated in both spatial directions or as a vertical black−to−white boundary. These signals indicate that the classification is to be made at a larger scale. (Such signals for resolving ambiguities are precisely what was lacking in the majority rule on this task.) Note that regions centered about each signal locally have The consequence is that the signal patterns can propagate, since the density of patterns with is neither increased nor decreased under the rule. The creation and interactions of these signals can be interpreted as the locus of the computation being performed by the CA—they form its emergent program. The above explanation of how Æd performs the task is an informal one obtained by careful scrutiny of many space−time patterns. Can we understand more rigorously how the rules evolved by the GA perform the desired computation? Understanding the results of GA evolution is a general problem—typically the GA is asked to find individuals that achieve high fitness but is not told how that high fitness is to be attained. One could say that this is analogous to the difficulty biologists have in understanding the products of natural evolution (e.g., us). We computational evolutionists have similar problems, since we do not specify what solution evolution is supposed to create; we ask only that it find some solution. In many cases, particularly in automatic−programming applications, it is difficult to understand exactly how an evolved high−fitness individual works. In genetic programming, for example, the evolved programs are often very long and complicated, with many irrelevant components attached to the core program performing the desired computation. It is usually a lot of work—and sometimes almost impossible—to figure out by hand what that core program is. The problem is even more difficult in the case of cellular automata, since the emergent "program" performed by a given CA is almost always impossible to extract from the bits of the rule table. A more promising approach is to examine the space−time patterns created by the CA and to "reconstruct" from those patterns what the algorithm is. Crutchfield and Hanson have developed a general method for reconstructing and understanding the "intrinsic" computation embedded in space−time patterns in terms of "regular domains," "particles" and "particle interactions" (Hanson and Crutchfield, 1992; Crutchfield and Hanson 1993). This method is part of their "computational mechanics" framework for understanding computation in physical systems. A detailed discussion of computational mechanics and particle−based computation is beyond the scope of this chapter. Very briefly, for those familiar with formal language theory, regular domains are regions of spacetime consisting of words in the same regular language—that is, they are regions that are computationally simple. Particles are localized boundaries between regular domains. In computational mechanics, particles are identified as information carriers, and collisions between particles are identified as the loci of important information processing. Particles and particle interactions form a high−level language for describing computation in spatially extended systems such as CAs. Figure 2.9 hints at this higher level of description: to produce it we filtered the regular domains from the space−time behavior of a GA−evolved CA to leave only the particles and their interactions, in terms of which the emergent algorithm of the CA can be understood. The application of computational mechanics to the understanding of rules evolved by the GA is discussed further in Crutchfield and Mitchell 1994, in Das, Mitchell, and Crutchfield 1994, and in Das, Crutchfield, Mitchell, and Hanson 1995. In the last two papers, we used particles and 40
Chapter 2: Genetic Algorithms in Problem Solving Figure 2.9: A space−time diagram of a GA−evolved rule for the task, and the same diagram with the regular domains filtered out, leaving only the particles and particle interactions (two of which are magnified). (Reprinted from Crutchfield and Mitchell 1994 by permission of the authors.) particle interactions to describe the temporal stages by which highly fit rules were evolved by the GA. Interestingly, it turns out that the behavior of the best rules discovered by the GA (such as Æd) is very similar to the behavior of the well−known Gacs−Kurdyumov−Levin (GKL) rule (Gacs, Kurdyumov, and Levin, 1978; Gonzaga de Sá and Maes 1992). Figure 2.10 is a space−time diagram illustrating its typical behavior. The GKL rule (ÆGKL) was designed by hand to study reliable computation and phase transitions in one−dimensional spatially extended systems, but before we started our project it was also the rule with the best−known performance (for CAs with periodic boundary conditions) on the task. Its unbiased performance is given in the last row of table 2.1. The difference in performance between Æd and ÆGKL is due to asymmetries in Æd that are not present in ÆGKL. Further GA evolution of Æd (using an increased number of ICs) has produced an improved version that approximately equals the performance of the ÆGKL. Rajarshi Das (personal communication) has gone further and, using Figure 2.10: Space−time diagram for the GKL rule with the aforementioned particle analysis, has designed by hand a rule that slightly outperforms ÆGKL. The discovery of rules such as ÆbÆd is significant, since it is the first example of a GA's producing sophisticated emergent computation in decentralized, distributed systems such as CAs. It is encouraging for 41
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 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
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Page 72 and 73: environments, they can fairly quick
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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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