@InProceedings{langdon:1997:MAX,
  author =       "W. B. Langdon and R. Poli",
  title =        "An Analysis of the {MAX} Problem in Genetic
                 Programming",
  booktitle =    "Proceedings of the Second International Conference on
                 Genetic Programming GP-97",
  year =         "1997",
  editor =       "John R. Koza and Kalyanmoy Deb and Marco Dorigo and
                 David B. Fogel and Max Garzon and Hitoshi Iba and Rick
                 L. Riolo",
  address =      "Stanford, CA, USA",
  month =        "13-16 " # jul,
  publisher =    "Morgan Kaufmann",
  note =         "to appear",
  keywords =     "genetic algorithms, genetic programming",
  size =         "9 pages",
  abstract =     "We present a detailed analysis of the evolution of GP
                 populations using the problem of finding 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 confirm the
                 basic message of \cite{Gathercole:1996:aicrtd} 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 sufficient 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.",
  notes =        "Considerable update of Langdon97",
}