Abstract
The representation used in grammatical evolution (GE) is non-uniformly redundant as some phenotypes are represented by more genotypes than others. This article studies how the non-uniform redundancy of the GE representation depends on various types of grammars. When constructing the phenotype tree from a genotype, the used grammar determines B avg, the average branching factor. B avg measures the expected number of non-terminals chosen when mapping one genotype codon to a phenotype tree node. First, the paper illustrates that the GE representation induces a bias towards small trees. This bias gets stronger with lower B avg. For example, when using a grammar with B avg = 0.5, 75% of all genotypes encode a phenotype tree of size one (codon length 10, two bits per codon, no wrapping, and random bit initialisation). Second, for B avg ≥ 1, the expected size of a phenotype tree is infinite. The resulting bias towards invalid trees increases with higher B avg. For example, for a grammar with B avg = 2.25, around 75% of all genotypes encode invalid trees. In summary, the GE encoding is strongly non-uniformly redundant and the bias depends on B avg. As a compromise between the different biases, the results of this study suggest setting B avg ≈ 1.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
πGE uses a flexible mapping where the genome defines not only the application of rules as in standard GE, but also specifies which non-terminal is decoded next. This implies that the order of non-terminal expansions is itself evolved in πGE [10].
References
R.A. Caruana, J.D. Schaffer, Representation and hidden bias: gray vs. binary coding for genetic algorithms, in Proceedings of the Fifth International Conference on Machine Learning, ed. by J. Laird (Morgan Kaufmann, San Mateo, CA, 1988), pp. 153–161
D. Fagan, M. O’Neill, E. Galván-López, A. Brabazon, S. McGarraghy, An analysis of genotype-phenotype maps in grammatical evolution, in Proceedings of the 13th European Conference on Genetic Programming, EuroGP 2010, ed. by A.I. Esparcia-Alcázar, A. Ekárt, S. Silva, S. Dignum, A.B. Uyar. Lecture Notes in Computer Science, vol. 6021 (Springer, Berlin, 2010), pp. 62–73
R. Harper, GE, explosive grammars and the lasting legacy of bad initialisation, in Proceedings of the IEEE Congress on Evolutionary Computation 2010, CEC 2010 (IEEE Press, New York, 2010), pp. 1–8
E. Hemberg, N. McPhee, M. O’Neill, A. Brabazon, Pre-, in- and postfix grammars for symbolic regression in grammatical evolution, in IEEE Workshop and Summer School on Evolutionary Computing, ed. by T.M. McGinnity. (IEEE, New York, 2008), pp. 18–22
J.R. Koza, Genetic Programming: On the Programming of Computers by Means of Natural Selection (MIT Press, Cambridge, MA, 1992)
S. Luke, Two fast tree-creation algorithms for genetic programming. IEEE Trans. Evol. Comput. 4(3), 274–283 (2000)
M.A. Montes de Oca, Exposing a bias toward short-length numbers in grammatical evolution, in Proceedings of the 11th European Conference on Genetic Programming, EuroGP 2008, ed. by M. O’Neill, L. Vanneschi, S. Gustafson, A.I. Esparcia Alcázar, I. De Falco, A. Della Cioppa, E. Tarantino. Lecture Notes in Computer Science, vol. 4971 (Springer, Berlin, 2008), pp. 278–288
E. Murphy, E. Hemberg, M. Nicolau, M. O’Neill, A. Brabazon, Grammar bias and initialisation in grammar based genetic programming, in Proceedings of the 15th European Conference on Genetic Programming, EuroGP 2012, ed. by A. Moraglio, S. Silva, K. Krawiec, P. Machado, C. Cotta. Lecture Notes in Computer Science, vol. 7244 (Springer, Berlin, 2012), pp. 85–96
M. Nicolau, Understanding grammatical evolution: initialisation. Genet. Program. Evolvable Mach. 18(4), 467–507 (2017)
M. O’Neill, A. Brabazon, M. Nicolau, S. Mc Garraghy, P. Keenan, πGrammatical evolution, in Proceedings of the Genetic and Evolutionary Computation Conference, GECCO 2004, ed. by K. Deb, R. Poli, W. Banzhaf, H.G. Beyer, E. Burke, P. Darwen, D. Dasgupta, D. Floreano, J. Foster, M. Harman, O. Holland, P. Lanzi, L. Spector, A. Tettamanzi, D. Thierens, A. Tyrrell. Lecture Notes in Computer Science, vol. 3103 (Springer, Berlin, 2004), pp. 617–629
M. O’Neill, C. Ryan, Genetic code degeneracy: implications for grammatical evolution and beyond, in Advances in Artificial Life: 5th European Conference, ECAL 1999, ed. by D. Floreano, J.D. Nicoud, F. Mondada. Lecture Notes in Computer Science, vol. 1674 (Springer, Berlin, 1999), pp. 149–153
M. O’Neill, C. Ryan, M. Nicolau, Grammar defined introns: an investigation into grammars, introns, and bias in grammatical evolution, in Proceedings of the Genetic and Evolutionary Computation Conference, GECCO 2001, ed. by L. Spector, E.D. Goodman, A. Wu, W. Langdon, H.M. Voigt, M. Gen, S. Sen, M. Dorigo, S. Pezeshk, M.H. Garzon, E. Burke (Morgan Kaufmann, San Francisco, CA, 2001), pp. 97–103
F. Rothlauf, Representations for Genetic and Evolutionary Algorithms, 2 edn. (Springer, Berlin, 2006)
F. Rothlauf, Design of Modern Heuristics: Principles and Application. Natural Computing Series (Springer, Heidelberg, 2011)
F. Rothlauf, D.E. Goldberg, Redundant representations in evolutionary computation. Evol. Comput. 11(4), 381–415 (2003)
C. Ryan, R.M.A. Azad, Sensible initialisation in grammatical evolution, in Proceedings of the Bird of a Feather Workshops, Genetic and Evolutionary Computation Conference, GECCO 2003, ed. by A.M. Barry (AAAI, Chicago, 2003), pp. 142–145.
C. Ryan, J.J. Collins, M.O. Neill, Grammatical evolution: evolving programs for an arbitrary language, in Proceedings of the First European Workshop on Genetic Programming, EuroGP 1998, ed. by W. Banzhaf, R. Poli, M. Schoenauer, T.C. Fogarty (Springer, Berlin, 1998), pp. 83–96
A. Thorhauer, On the non-uniform redundancy in grammatical evolution, in Parallel Problem Solving from Nature – PPSN XIV: 14th International Conference, ed. by J. Handl, E. Hart, P.R. Lewis, M. López-Ibáñez, G. Ochoa, B. Paechter. Lecture Notes in Computer Science, vol. 9921 (Springer, Cham, 2016), pp. 292–302
A. Thorhauer, F. Rothlauf, Structural difficulty in grammatical evolution versus genetic programming, in Proceedings of the 15th Annual Conference Companion on Genetic and Evolutionary Computation, GECCO 2013, ed. by C. Blum (ACM Press, New York, NY, 2013), pp. 997–1004
P.A. Whigham, Search bias, language bias and genetic programming, in Genetic Programming 1996: Proceedings of the First Annual Conference on Genetic Programming, ed. by J.R. Koza, D.E. Goldberg, D.B. Fogel, R.L. Riolo (MIT Press, Cambridge, MA, 1996), pp. 230–237
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Schweim, D., Thorhauer, A., Rothlauf, F. (2018). On the Non-uniform Redundancy of Representations for Grammatical Evolution: The Influence of Grammars. In: Ryan, C., O'Neill, M., Collins, J. (eds) Handbook of Grammatical Evolution. Springer, Cham. https://doi.org/10.1007/978-3-319-78717-6_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-78717-6_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-78716-9
Online ISBN: 978-3-319-78717-6
eBook Packages: Computer ScienceComputer Science (R0)