Abstract
We formalize and describe the mapping process of integer input (genotype) to an output sentence (phenotype) in Grammatical Evolution (GE). The aim is to study the grammatical and search bias which is produced by the mapping. We investigate changes in input and the effect on output and analyze the neighboring solutions as well as the effect of changes and bias in representation. Different types of changes are defined to allow classification of the effects that input changes (operators) have. The changes are a part of identifying what the neighborhood for GE search looks like. We call this disruption in GE. Furthermore, a schema theorem is introduced for investigating preservation of material during application of variation operators, an attempt to identify the population effects.
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- 1.
Canonical GE has a binary chromosome that will be transcribed to integers, \(f:\mathbb {Z}_2 \rightarrow \mathbb {Z}_{2^m},\ f(C_2)=C\) where m is the codon size and C 2 the binary representation of the chromosome. Here we simplify and skip the transcription step, binary-to-integer, and use a sequence of integers instead.
- 2.
\(\mathbb {N}\) refers to the natural numbers, \(\mathbb {Z}\) denotes integers, and \(\mathbb {Z}_n\) integers modulo n.
- 3.
In canonical GE the maximum number of production choices is \(r_{max} = max_{r_{i*} \in R}(|r_{i*}|)\). If r max uses unique identifiers the analysis can be facilitated. The context and analysis should make it obvious when the integer inputs are unique or ambiguous.
- 4.
GE can be seen as a PCFG (Definition 4 on page 4), where all the probabilities are uniform, i.e. the probabilities are determined by the number of production choices for each non-terminal, p ij = 1∕|r i∗|. A bias towards productions r ij can be achieved by multiple identical production choices in the solution grammar rules.
- 5.
It is possible to create crossover operators that only return a single individual.
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Hemberg, E. (2018). Theory of Disruption in GE. 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_5
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