Advertisement

Size Fair and Homologous Tree Crossovers for Tree Genetic Programming

  • W. B. Langdon
Article

Abstract

Size fair and homologous crossover genetic operators for tree based genetic programming are described and tested. Both produce considerably reduced increases in program size (i.e., less bloat) and no detrimental effect on GP performance.

GP search spaces are partitioned by the ridge in the number of program v. their size and depth. While search efficiency is little effected by initial conditions, these do strongly influence which half of the search space is searched. However a ramped uniform random initialization is described which straddles the ridge.

With subtree crossover trees increase about one level per generation leading to subquadratic bloat in program length.

genetic algorithms genetic programming bloat reduction evolution of shape subquadratic length growth linear depth growth uniform initialization binary tree search spaces 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. Alonso and R. Schott, Random Generation of Trees, Kluwer Academic Publishers: Boston, MA, 1995.Google Scholar
  2. 2.
    P. J. Angeline, “Genetic programming and emergent intelligence,” in Advances in Genetic Pro-gramming, K. E. Kinnear, Jr. (ed.), MIT Press, 1994, chap. 4, pp. 75-98.Google Scholar
  3. 3.
    T. Blickle, “Evolving compact solutions in genetic programming: A case study,” in Parallel Problem Solving From Nature IV. Proceedings of the International Conference on Evolutionary Computa-tion, H.-M. Voigt, W. Ebeling, I. Rechenberg, and H.-P. Schwefel (eds.), Springer-Verlag, Berlin, Germany, 22-26 September 1996, vol. 1141 of LNCS, pp. 564-573.Google Scholar
  4. 4.
    T. Blickle and L. Thiele, “Genetic programming and redundancy,” in Genetic Algorithms within the Framework of Evolutionary Computation (Workshop at KI-94, Saarbrücken), Im Stadtwald, Build-ing 44, D-66123 Saarbrücken, Germany, J. Hopf (ed.), (Max-Planck-Institut fur Informatik MPI-1-94-241), 1994, pp. 33-38.Google Scholar
  5. 5.
    W. Bohm and A. Geyer-Schulz, “Exact uniform initialization for genetic programming,” in Founda-tions of Genetic Algorithms IV, University of San Diego, CA, R. K. Belew and M. Vose eds., Morgan Kaufmann, 3-5 August 1996, pp. 379-407.Google Scholar
  6. 6.
    K. Chellapilla, “Evolving computer programs without subtree crossover,” IEEE Transactions on Evolutionary Computation vol. 1(3) pp. 209-216, September 1997.Google Scholar
  7. 7.
    P. D'haeseleer, “Context preserving crossover in genetic programming,” in Proceedings of the 1994 IEEE World Congress on Computational Intelligence, Orlando, Florida, IEEE Press, 27-29 June 1994, vol. 1, pp. 256-261.Google Scholar
  8. 8.
    P. Flajolet and A. Oldyzko, “The average height of binary trees and other simple trees,” Journal of Computer and System Sciences vol. 25 pp. 171-213, 1982.Google Scholar
  9. 9.
    C. Gathercole and P. Ross, “An adverse interaction between crossover and restricted tree depth in genetic programming,” in Genetic Programming 1996: Proceedings of the First Annual Conference, Stanford University, CA, J. R. Koza, D. E. Goldberg, D. B. Fogel, and R. L. Riolo (eds.), MIT Press, 28-31 July 1996, pp. 291-296.Google Scholar
  10. 10.
    D. C. Hooper, N. S. Flann, and S. R. Fuller, “Recombinative hill-climbing: A stronger search method for genetic programming,” in Genetic Programming 1997: Proceedings of the Second Annual Conference, Stanford University, CA, J. R. Koza, K. Deb, M. Dorigo, D. B. Fogel, M. Garzon, H. Iba, and R. L. Riolo (eds.), Morgan Kaufmann, 13-16 July 1997, pp. 174-179.Google Scholar
  11. 11.
    H. Iba, “Random tree generation for genetic programming,” in Parallel Problem Solving from Nature IV, Proceedings of the International Conference on Evolutionary Computation, Berlin,Germany, H.-M. Voigt, W. Ebeling, I. Rechenberg, and H.-P. Schwefel (eds.), Springer Verlag, 22-26 September 1996, vol. 1141 of LNCS, pp. 144-153.Google Scholar
  12. 12.
    H. Iba, H. de Garis, and T. Sato, “Genetic programming using a minimum description length principle,” in Advances in Genetic Programming, K. E. Kinnear, Jr. (ed.), MIT Press, 1994, chap. 12, pp. 265-284.Google Scholar
  13. 13.
    J. R. Koza, Genetic Programming: On the Programming of Computers by Means of Natural Selection, MIT Press: Cambridge, MA, 1992.Google Scholar
  14. 14.
    J. R. Koza, Genetic Programming II: Automatic Discovery of Reusable Programs, MIT Press: Cambridge, MA, 1994.Google Scholar
  15. 15.
    W. B. Langdon, “Fitness causes bloat: Simulated annealing, hill climbing and populations,” University of Birmingham, School of Computer Science, Technical Report CSRP-97-22, 2 September 1997.Google Scholar
  16. 16.
    W. B. Langdon, “The evolution of size in variable length representations,” in 1998 IEEE International Conference on Evolutionary Computation, Anchorage, Alaska, IEEE Press, 5-9 May 1998, pp. 633-638.Google Scholar
  17. 17.
    W. B. Langdon, “Linear increase in tree height leads to sub-quadratic bloat,” in Foundations of Genetic Programming, Orlando, Florida, T. Haynes, W. B. Langdon, U.-M. O'Reilly, R. Poli, and J. Rosca (eds.), 13 July 1999 Workshop at GECCO'99.Google Scholar
  18. 18.
    W. B. Langdon, “Scaling of program tree fitness spaces,” Evolutionary Computation vol. 7 (4), 1999.Google Scholar
  19. 19.
    W. B. Langdon, “Size fair and homologous tree genetic programming crossovers,” in Proceedings of the Genetic and Evolutionary Computation Conference, Orlando, Florida, W. Banzhaf, J. Daida, A. E. Eiben, M. H. Garzon, V. Honavar, M. Jakiela, and R. E. Smith (eds.), Morgan Kaufmann, 13-17 July 1999, vol. 2, pp. 1092-1097.Google Scholar
  20. 20.
    W. B. Langdon and R. Poli, “An analysis of the MAX problem in genetic programming,” in Genetic Programming 1997: Proceedings of the Second Annual Conference, Stanford University, CA, J. R. Koza, K. Deb, M. Dorigo, D. B. Fogel, M. Garzon, H. lba, and R. L. Riolo (eds.), Morgan Kaufmann, 13-16 July 1997, pp. 222-230.Google Scholar
  21. 21.
    W. B. Langdon and R. Poli, “Fitness causes bloat,” in Soft Computing in Engineering Design and Manufacturing, London, P. K. Chawdhry, R. Roy, and R. K. Pant (eds.), Springer-Verlag: London, 23-27 June 1997, pp. 13-22.Google Scholar
  22. 22.
    W. B. Langdon and R. Poli, “Fitness causes bloat: Mutation,” in Proceedings of the First European Workshop on Genetic Programming, Paris, W. Banzhaf, R. Poli, M. Schoenauer, and T. C. Fogarty (eds.), Springer-Verlag, 14-15 April 1998, vol. 1391 of LNCS, pp. 37-48.Google Scholar
  23. 23.
    W. B. Langdon and R. Poli, “Genetic programming bloat with dynamic fitness,” in Proceedings of the First European Workshop on Genetic Programming, Paris, W. Banzhaf, R. Poli, M. Schoenauer, and T. C. Fogarty (eds.), Springer-Verlag, 14-15 April 1998, vol. 1391 of LNCS, pp. 96-112.Google Scholar
  24. 24.
    W. B. Langdon and R. Poli, “Boolean functions fitness spaces,” in Genetic Programming, Proceed-ings of EuroGP'99, Goteborg, Sweden, R. Poli, P. Nordin, W. B. Langdon, and T. C. Fogarty (eds.), Springer-Verlag, 26-27 May 1999, vol. 1598 of LNCS, pp. 1-14.Google Scholar
  25. 25.
    W. B. Langdon, Data Structures and Genetic Programming: Genetic Programming + Data Structures = Automatic Programming!, Kluwer: Boston, 1998.Google Scholar
  26. 26.
    W. B. Langdon, T. Soule, R. Poli, and J. A. Foster, “The evolution of size and shape,” in Advances in Genetic Programming 3, L. Spector, W. B. Langdon, U.-M. O'Reilly, and P. J. Angeline (eds.), MIT Press: Cambridge, MA, 1999, chap. 8, pp. 163-190.Google Scholar
  27. 27.
    N. F. McPhee and J. D. Miller, “Accurate replication in genetic programming,” in Genetic Algorithms: Proceedings of the Sixth International Conference ICGA95, Pittsburgh, PA, USA, L. Eshelman (ed.), Morgan Kaufmann, 15-19 July 1995, pp. 303-309.Google Scholar
  28. 28.
    P. Nordin, W. Banzhaf, and F. D. Francone, “Introns in nature and in simulated structure evolution,” in Bio-Computation and Emergent Computation, Skovde, Sweden, D. Lundh, B. Olsson, and A. Narayanan (eds.), World Scientific Publishing, 1-2 September 1997.Google Scholar
  29. 29.
    P. Nordin, “Evolutionary Program Induction of Binary Machine Code and its Applications,” der Universitat Dortmund am Fachereich Informatik, PhD thesis, 1997.Google Scholar
  30. 30.
    P. Nordin and W. Banzhaf, “Complexity compression and evolution,” in Genetic Algorithms: Proceedings of the Sixth International Conference ICGA95, Pittsburgh, PA, USA, L. Eshelman (ed.), Morgan Kaufmann, 15-19 July 1995, pp. 310-317.Google Scholar
  31. 31.
    P. Nordin, W. Banzhaf, and F. D. Francone, “Efficient evolution of machine code for CISC architectures using instruction blocks and homologous crossover,” in Advances in Genetic Program-ming 3, L. Spector, W. B. Langdon, U.-M. O'Reilly, and P. J. Angeline (eds.), MIT Press: Cambridge, MA, 1999, chap. 12, pp. 275-299.Google Scholar
  32. 32.
    P. Nordin, F. Francone, and W. Banzhaf, “Explicitly defined introns and destructive crossover in genetic programming,” in Advances in Genetic Programming 2, P. J. Angeline and K. E. Kinnear, Jr. (eds.), MIT Press: Cambridge, MA, 1996, chap. 6, pp. 111-134.Google Scholar
  33. 33.
    U.-M. O'Reilly and F. Oppacher, “The troubling aspects of a building block hypothesis for genetic programming,” in Foundations of Genetic Algorithms 3, Estes Park, Colorado, 31 July2 August 1994, L. D. Whitley and M. D. Vose (eds.0, Morgan Kaufmann, 1995, pp. 73-88.Google Scholar
  34. 34.
    R. Poli and W. B. Langdon, “Schema theory for genetic programming with one-point crossover and point mutation,” Evolutionary Computation vol. 6 (3) pp. 231-252, 1998.Google Scholar
  35. 35.
    R. Poli and W. B. Langdon, “Sub-machine-code genetic programming,” in Advances in Genetic Programming 3, L. Spector, W. B. Langdon, U.-M. O'Reilly, and P. J. Angeline (eds.), MIT Press: Cambridge, MA, 1999, chap. 13, pp. 301-323.Google Scholar
  36. 36.
    J. P. Rosca, “Analysis of complexity drift in genetic programming,” in Genetic Programming 1997: Proceedings of the Second Annual Conference, Stanford University, CA, J. R. Koza, K. Deb, M. Dorigo, D. B. Fogel, M. Garzon, H. Iba, and R. L. Riolo (eds.), Morgan Kaufmann, 13-16 July 1997, pp. 286-294.Google Scholar
  37. 37.
    R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, 1996.Google Scholar
  38. 38.
    T. Soule, “Code Growth in Genetic Programming,” University of Idaho, Moscow, Idaho, PhD thesis, 15 May 1998.Google Scholar
  39. 39.
    T. Soule and J. A. Foster, “Code size and depth flows in genetic programming,” in Genetic Programming 1997: Proceedings of the Second Annual Conference, Stanford University, CA, J. R. Koza, K. Deb, M. Dorigo, D. B. Fogel, M. Garzon, H. Iba, and R. L. Riolo (eds.), Morgan Kaufmann, 13-16 July 1997, pp. 313-320.Google Scholar
  40. 40.
    T. Soule and J. A. Foster, “Removal bias: a new cause of code growth in tree based evolutionary programming,” in 1998 IEEE International Conference on Evolutionary Computation, Anchorage, Alaska, IEEE Press, 5-9 May 1998, pp. 781-186.Google Scholar
  41. 41.
    T. Soule, J. A. Foster, and J. Dickinson, “Code growth in genetic programming,” in Genetic Programming 1996: Proceedings of the First Annual Conference, Stanford University, CA, J. R. Koza, D. E. Goldberg, D. B. Fogel, and R. L. Riolo (eds.), MIT Press, 28-31 July 1996, pp. 215-223.Google Scholar
  42. 42.
    B.-T. Zhang and H. Mühlenbein, “Balancing accuracy and parsimony in genetic programming,” Evolutionary Computation vol. 3 (1) pp. 17-38, 1995.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • W. B. Langdon
    • 1
  1. 1.Centrum voor Wiskunde en InformaticaAmsterdam

Personalised recommendations