Crossover Bias in Genetic Programming
Path length, or search complexity, is an understudied property of trees in genetic programming. Unlike size and depth measures, path length directly measures the balancedness or skewedness of a tree. Here a close relative to path length, called visitation length, is studied. It is shown that a population undergoing standard crossover will introduce a crossover bias in the visitation length. This bias is due to inserting variable length subtrees at various levels of the tree. The crossover bias takes the form of a covariance between the sizes and levels in the trees that form a population. It is conjectured that the crossover bias directly determines the size distribution of trees in genetic programming. Theorems are presented for the one-generation evolution of visitation length both with and without selection. The connection between path length and visitation length is made explicit.
KeywordsPath Length Genetic Program Binary Tree Internal Node Catalan Distribution
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- 1.Altenberg, L.: The evolution of evolvability in genetic programming. In: Kinnear Jr., K.E. (ed.) Advances in Genetic Programming, pp. 47–74. MIT Press, Cambridge (1994)Google Scholar
- 2.Altenberg, L.: The Schema Theorem and Price’s Theorem. In: Whitley, L.D., Vose, M.D. (eds.) Foundations of Genetic Algorithms 3, Estes Park, Colorado, USA, 31 July–2 Aug. 1994, pp. 23–49. Morgan Kaufmann, Seattle (1995)Google Scholar
- 3.Langdon, W.B., Soule, T., Poli, R., Foster, J.A.: The evolution of size and shape. In: Spector, L., Langdon, W.B., O’Reilly, U.-M., Angeline, P.J. (eds.) Advances in Genetic Programming 3, pp. 163–190. MIT Press, Cambridge (1999)Google Scholar
- 4.McPhee, N.F., Poli, R.: A schema theory analysis of the evolution of size in genetic programming with linear representations. In: Miller, J., Tomassini, M., Lanzi, P.L., Ryan, C., Tetamanzi, A.G.B., Langdon, W.B. (eds.) EuroGP 2001. LNCS, vol. 2038, pp. 18–125. Springer, Heidelberg (2001)Google Scholar
- 7.Smits, G., Kotanchek, M.: Pareto-front exploitation in symbolic regression. In: O’Reilly, U.-M., et al. (eds.) Genetic Programming Theory and Practice II, vol. 17, Ann Arbor, 13-15 May 2004, Kluwer, Dordrecht (2004)Google Scholar