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Genetic Programming Theory and Practice XV pp 17–37Cite as

Schema Analysis in Tree-Based Genetic Programming

Part of the Genetic and Evolutionary Computation book series (GEVO)

Abstract

In this chapter we adopt the concept of schemata from schema theory and use it to analyze population dynamics in genetic programming for symbolic regression. We define schemata as tree-based wildcard patterns and we empirically measure their frequencies in the population at each generation. Our methodology consists of two steps: in the first step we generate schemata based on genealogical information about crossover parents and their offspring, according to several possible schema definitions inspired from existing literature. In the second step, we calculate the matching individuals for each schema using a tree pattern matching algorithm. We test our approach on different problem instances and algorithmic flavors and we investigate the effects of different selection mechanisms on the identified schemata and their frequencies.

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Notes

  1. 1.

    The terms root parent and non-root parent refer to the two parents involved in a crossover operation: the root parent passes on to the child its entire rooted tree structure, with the exception of the subtree swapped by crossover at an arbitrary location (called a cutpoint) from the non-root parent.

  2. 2.

    To maintain low computational times, certain compromises had to be made in terms of population size and number of generations.

References

  1. Affenzeller, M., Winkler, S., Wagner, S., Beham, A.: Genetic Algorithms and Genetic Programming: Modern Concepts and Practical Applications. Numerical Insights. CRC Press, Singapore (2009)

    CrossRef  Google Scholar 

  2. Altenberg, L., et al.: The evolution of evolvability in genetic programming. Advances in genetic programming 3, 47–74 (1994)

    Google Scholar 

  3. Banzhaf, W.: Genetic programming and emergence. Genetic Programming and Evolvable Machines 15(1), 63–73 (2014). https://doi.org/10.1007/s10710-013-9196-7

    CrossRef  Google Scholar 

  4. Banzhaf, W., Leier, A.: Evolution on neutral networks in genetic programming. In: Genetic programming theory and practice III, pp. 207–221. Springer (2006)

    Google Scholar 

  5. Burke, E., Gustafson, S., Kendall, G.: A survey and analysis of diversity measures in genetic programming. In: Proceedings of the 4th Annual Conference on Genetic and Evolutionary Computation, pp. 716–723. Morgan Kaufmann Publishers Inc. (2002)

    Google Scholar 

  6. Burke, E.K., Gustafson, S., Kendall, G.: Diversity in genetic programming: An analysis of measures and correlation with fitness. IEEE Transactions on Evolutionary Computation 8(1), 47–62 (2004)

    CrossRef  Google Scholar 

  7. Götz, M., Koch, C., Martens, W.: Efficient algorithms for descendant-only tree pattern queries. Inf. Syst. 34(7), 602–623 (2009). https://doi.org/10.1016/j.is.2009.03.010

    CrossRef  Google Scholar 

  8. Holland, J.H.: Adaptation in Natural and Artificial Systems. The University of Michigan Press (1975)

    Google Scholar 

  9. Hu, T., Banzhaf, W., Moore, J.H.: Population Exploration on Genotype Networks in Genetic Programming. In: Proceedings of the 13th International Conference on Parallel Problem Solving from Nature – PPSN XIII, 2014, pp. 424–433. Springer International Publishing, Cham (2014)

    Google Scholar 

  10. Koza, J.R.: Genetic Programming: On the Programming of Computers by Means of Natural Selection. MIT Press, Cambridge, MA, USA (1992)

    MATH  Google Scholar 

  11. Krawiec, K., Wieloch, B.: Functional modularity for genetic programming. In: Proceedings of the 11th Annual Conference on Genetic and Evolutionary Computation, GECCO ’09, pp. 995–1002. ACM, New York, NY, USA (2009). http://doi.acm.org/10.1145/1569901.1570037

  12. Poli, R.: Hyperschema theory for gp with one-point crossover, building blocks, and some new results in ga theory. In: Genetic Programming, Proceedings of EuroGP 2000, pp. 15–16. Springer-Verlag (2000)

    Google Scholar 

  13. Poli, R.: Exact schema theory for genetic programming and variable-length genetic algorithms with one-point crossover. Genetic Programming and Evolvable Machines 2(2), 123–163 (2001). https://doi.org/10.1023/A:1011552313821

    CrossRef  Google Scholar 

  14. Poli, R.: A simple but theoretically-motivated method to control bloat in genetic programming. In: Proceedings of the 6th European Conference on Genetic Programming, EuroGP’03, pp. 204–217. Springer-Verlag, Berlin, Heidelberg (2003). http://dl.acm.org/citation.cfm?id=1762668.1762688

    MATH  Google Scholar 

  15. Poli, R., Langdon, W.B., Dignum, S.: Generalisation of the limiting distribution of program sizes in tree-based genetic programming and analysis of its effects on bloat. In: in GECCO 2007: Proceedings of the 9th Annual Conference on Genetic and Evolutionary, pp. 1588–1595. ACM Press (2007)

    Google Scholar 

  16. Poli, R., McPhee, N.F.: General schema theory for genetic programming with subtree-swapping crossover: Part I. Evolutionary Computation 11(1), 53–66 (2003).

    CrossRef  Google Scholar 

  17. Poli, R., McPhee, N.F.: General schema theory for genetic programming with subtree-swapping crossover: Part II. Evolutionary Computation 11(2), 169–206 (2003). https://doi.org/10.1162/106365603766646825

    CrossRef  Google Scholar 

  18. Poli, R., McPhee, N.F.: Covariant parsimony pressure for genetic programming. In: GECCO 2008: Proceedings of the 10th annual conference on Genetic and Evolutionary Computation, pp. 1267–1274. ACM Press (2008)

    Google Scholar 

  19. Poli, R., Vanneschi, L., Langdon, W.B., McPhee, N.F.: Theoretical results in genetic programming: The next ten years? Genetic Programming and Evolvable Machines 11(3–4), 285–320 (2010). http://dx.doi.org/10.1007/s10710-010-9110-5

    CrossRef  Google Scholar 

  20. Stephens, C.R., Waelbroeck, H.: Effective degrees of freedom in genetic algorithms. Physical Review E 57(3), 3251–3264 (1998)

    CrossRef  Google Scholar 

  21. Vladislavleva, E.J., Smits, G.F., Den Hertog, D.: Order of nonlinearity as a complexity measure for models generated by symbolic regression via pareto genetic programming. Evolutionary Computation, IEEE Transactions on 13(2), 333–349 (2009)

    CrossRef  Google Scholar 

  22. Wagner, G.P., Altenberg, L.: Perspective: complex adaptations and the evolution of evolvability. Evolution 50, 967–976 (1996)

    CrossRef  Google Scholar 

  23. Wagner, S., Kronberger, G., Beham, A., Kommenda, M., Scheibenpflug, A., Pitzer, E., Vonolfen, S., Kofler, M., Winkler, S.M., Dorfer, V., Affenzeller, M.: Architecture and design of the heuristiclab optimization environment. Advanced Methods and Applications in Computational Intelligence, Topics in Intelligent Engineering and Informatics 6, 197–261 (2013)

    CrossRef  Google Scholar 

  24. White, D.: An overview of schema theory. Computing Research Repository CoRR abs/1401.2651 (2014). http://arxiv.org/abs/1401.2651

  25. Woodward, J.R.: Modularity in Genetic Programming. Proc. of Genetic Programming: 6th European Conference, EuroGP 2003 Essex, pp. 254–263. Springer (2003). http://dx.doi.org/10.1007/3-540-36599-0_23

    Google Scholar 

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Acknowledgements

The work described in this paper was done within the COMET Project Heuristic Optimization in Production and Logistics (HOPL), #843532 funded by the Austrian Research Promotion Agency (FFG).

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Authors and Affiliations

  1. Heuristic and Evolutionary Algorithms Laboratory, University of Applied Sciences Upper Austria, Hagenberg, Austria

    Bogdan Burlacu, Michael Affenzeller, Michael Kommenda & Stephan Winkler

  2. Institute for Formal Models and Verification, Johannes Kepler University, Linz, Austria

    Bogdan Burlacu, Michael Affenzeller, Michael Kommenda & Stephan Winkler

  3. Heuristic and Evolutionary Algorithms Laboratory, University of Applied Sciences Upper Austria, Hagenberg, Austria

    Gabriel Kronberger

Authors
  1. Bogdan Burlacu
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  2. Michael Affenzeller
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  3. Michael Kommenda
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  4. Gabriel Kronberger
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  5. Stephan Winkler
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Correspondence to Bogdan Burlacu .

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Editors and Affiliations

  1. BEACON Center for the Study of Evolution in Action and Department of Computer Science, Michigan State University, East Lansing, Michigan, USA

    Wolfgang Banzhaf

  2. Institute for Biomedical Informatics, University of Pennsylvania, Philadelphia, Pennsylvania, USA

    Randal S. Olson

  3. Institute for Biomedical Informatics, University of Pennsylvania, Philadelphia, Pennsylvania, USA

    William Tozier

  4. Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan, USA

    Dr. Rick Riolo

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Burlacu, B., Affenzeller, M., Kommenda, M., Kronberger, G., Winkler, S. (2018). Schema Analysis in Tree-Based Genetic Programming. In: Banzhaf, W., Olson, R., Tozier, W., Riolo, R. (eds) Genetic Programming Theory and Practice XV. Genetic and Evolutionary Computation. Springer, Cham. https://doi.org/10.1007/978-3-319-90512-9_2

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  • DOI: https://doi.org/10.1007/978-3-319-90512-9_2

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