Abstract
Holland's fundamental theorem of genetic algorithms (the “schema theorem”) provides a lower bound on the sampling rate of a single hyperplane during genetic search. However, the theorem tracks the change in representation for a single hyperplaneas if its representation is independent of other hyperplanes. Hyperplane samples are clearly interdependent and interactions in the hyperplane samples means that Holland's notion of “implicit parallelism” does not universally hold when conflicting hyperplanes interact. A set of equations are defined which allows one to model the interaction of strings, or of schemata representing hyperplanes at order-N and hyperplanes less than order-N. These equations do not account for the effects of higher order hyperplanes or co-lateral competitions. Nevertheless, these equations can serve to better describe the interaction of primary hyperplane competitors.
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References
C. Bridges and D. Goldberg, An analysis of reproduction and crossover in a binary-coded genetic Algorithm,Proc. 2nd Int. Conf. on Genetic Algorithms and Their Applications, ed. J. Grefenstette (Lawrence Erlbaum, 1987).
R. Das and D. Whitley, The only challenging problems are deceptive: global search by solving order-1 hyperplanes,Proc. 4th Int. Conf. on Genetic Algorithms, eds. R. Belew and L. Booker (Morgan Kaufmann, 1991).
S. Forrest and M. Mitchell, Relative building block fitness and the building block hypothesis,Foundations of Genetic Algorithms 2, ed. D. Whitley (Morgan Kaufmann, 1992)
D. Goldberg, Simple genetic algorithms and the minimal, deceptive problem, in:Genetic Algorithms and Simulated Annealing, ed. L. Davis (Morgan Kaufmann, 1987) pp. 74–88.
D. Goldberg,Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley, 1989).
D. Goldberg, Genetic algorithms and Walsh functions: Part I, a gentle introduction, Complex Systems 3 (1989) 129–152.
D. Goldberg, B. Korb and K. Deb, Messy genetic algorithms: motivation, analysis, and first results, Complex Systems 3 (1989) 493–530.
J. Holland,Adaptation in Natural and Artificial Systems (University of Michigan Press, Ann Arbor, 1975).
G. Liepins and M. Vose, Representation issues in genetic algorithms, J. Exp. Theor. Art. Int. 2 (1990) 101–115.
M. Mitchell, S. Forrest and J. Holland, The royal road for genetic algorithms: fitness landscapes and GA performance,Proc. 1st European Conf. on Artificial Life (MIT Press, 1992).
D. Schaffer, Some effects of selection procedures on hyperplane sampling by genetic algorithms, in:Genetic Algorithms and Simulated Annealing, ed. L. Davis (Morgan Kaufmann, 1987) pp. 89–103.
D. Whitley, Fundamental principles of deception in genetic search,Foundations of Genetic Algorithms, ed. G. Rawlins (Morgan Kaufmann, 1989).
J. Grefenstette and J. Baker, How genetic algorithms work: a critical look at implicit parallelism,Proc. 3rd Int. Conf. on Genetic Algorithms, ed. D. Schaffer (Morgan Kaufmann, 1989).
M. Vose, Models of genetic algorithms, in:Foundations of Genetic Algorithms 2, ed. D. Whitley (Morgan Kaufmann, 1992).
M. Vose and G. Liepins, Punctuated equilibria in genetic search, Complex Systems 5 (1991) 31–44.
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Whitley, D., Das, R. & Crabb, C. Tracking primary hyperplane competitors during genetic search. Ann Math Artif Intell 6, 367–388 (1992). https://doi.org/10.1007/BF01535526
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DOI: https://doi.org/10.1007/BF01535526