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.
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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].
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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
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