@TechReport{langdon:1996:eGPpRN,
  author =       "W. B. Langdon",
  title =        "Evolution of Genetic Programming Populations",
  institution =  "University College London",
  year =         "1995",
  type =         "Research Note",
  number =       "RN/96/125",
  address =      "Gower Street, London WC1E 6BT, UK",
  month =        sep,
  keywords =     "Genetic Programming, Genetic Algorithms, population
                 variety, diversity, genetic programming, Price's
                 theorem, Fisher's theorem, evolution, automatic code
                 generation",
  url =          "ftp://ftp.cs.bham.ac.uk/pub/authors/W.B.Langdon/papers/WBL.ecj.price.125.ps.gz",
  abstract =     "We investigate in detail what happens as genetic
                 programming (GP) populations evolve. Since we shall use
                 the populations which showed GP can evolve stack data
                 structures as examples, we start in Section 1 by
                 briefly describing the stack experiment
                 \cite{Langdon:1995:GPdata}. In Section 2 we show
                 Price's Covariance and Selection Theorem can be applied
                 to Genetic Algorithms (GAs) and GP to predict changes
                 in gene frequencies. We follow the proof of the theorem
                 with experimental justification using the GP runs from
                 the stack problem. Section 3 briefly describes Fisher's
                 Fundamental Theorem of Natural Selection and shows in
                 its normal interpretation it does not apply to
                 practical GAs. An analysis of the stack populations, in
                 Section 4, explains that the difficulty of the stack
                 problem is due to the presence of ``deceptive'' high
                 scoring partial solutions in the population. These
                 cause a negative correlation between necessary
                 primitives and fitness. As Price's Theorem predicts,
                 the frequency of necessary primitives falls, eventually
                 leading to their extinction and so to the impossibility
                 of finding solutions like those that are evolved in
                 successful runs. Section 4 investigates the evolution
                 of variety in GP populations. Detailed measurements of
                 the evolution of variety in stack populations reveal
                 loss of diversity causing crossover to produce
                 offspring which are copies of their parents. Section 5
                 concludes with measurements that show in the stack
                 population crossover readily produces improvements in
                 performance initially but later no improvements at all
                 are made by crossover. Section 6 discusses the
                 importance of these results to GP in general.",
  notes =        "Abridgement of Chapter 7 of langdon:thesis",
  size =         "48 pages",
}