An Analysis of the Max Problem in Genetic Programming

We present a detailed analysis of the evolution of genetic programming (GP) populations using the problem of nding a program which returns the maximum possible value for a given terminal and function set and a depth limit on the program tree (known as the MAX problem). We connrm the basic message of Gathercole and Ross, 1996] that crossover together with program size restrictions can be responsible for premature convergence to a sub-optimal solution. We show that this can happen even when the population retains a high level of variety and show that in many cases evolution from the sub-optimal solution to the solution is possible if suucient time is allowed. In both cases theoretical models are presented and compared with actual runs. Price's Covariance and Selection Theorem is experimentally tested on GP populations. It is shown to hold only in some cases, in others program size restrictions cause important deviation from its predictions .

[1]  Walter Böhm,et al.  Exact Uniform Initialization For Genetic Programming , 1996, FOGA.

[2]  Una-May O'Reilly,et al.  The Troubling Aspects of a Building Block Hypothesis for Genetic Programming , 1994, FOGA.

[3]  Lothar Thiele,et al.  A Mathematical Analysis of Tournament Selection , 1995, ICGA.

[4]  P. Ross,et al.  An adverse interaction between crossover and restricted tree depth in genetic programming , 1996 .

[5]  Riccardo Poli,et al.  Schema Theory for Genetic Programming with One-Point Crossover and Point Mutation , 1997, Evolutionary Computation.

[6]  George R. Price,et al.  Selection and Covariance , 1970, Nature.

[7]  Walter Alden Tackett,et al.  Greedy Recombination and Genetic Search on the Space of Computer Programs , 1994, FOGA.

[8]  L. Altenberg.  The evolution of evolvability in genetic programming , 1994 .

[9]  Lee Altenberg,et al.  The Schema Theorem and Price's Theorem , 1994, FOGA.

[10]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[11]  Una-May O'Reilly,et al.  Genetic Programming II: Automatic Discovery of Reusable Programs. , 1994, Artificial Life.

[12]  Erik D. Goodman,et al.  The royal tree problem, a benchmark for single and multiple population genetic programming , 1996 .

[13]  P.A. Whigham,et al.  A Schema Theorem for context-free grammars , 1995, Proceedings of 1995 IEEE International Conference on Evolutionary Computation.

[14]  William B. Langdon.  Data structures and genetic programming , 1995 .

[15]  Hitoshi Iba,et al.  Random Tree Generation for Genetic Programming , 1996, PPSN.