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Gene Expression Programming with Pre-Order Traversals: Enhancements in Genetic Algorithms

This presentation explores the integration of Pre-Order Traversals in Gene Expression Programming (GEP). Stefan Kahrs from the University of Kent at Canterbury discusses the evolution of grammar in genetic algorithms, addressing crossover breeding and the problems associated with it. Kahrs introduces innovative solutions to optimize the genetic information flow between genotype and phenotype, leveraging the structural advantages of Pre-Order Traversals to enhance breeding efficiency. The talk covers practical examples, including strategies to prevent lopsided trees and effectively generate initial populations for improved performance.

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Gene Expression Programming with Pre-Order Traversals: Enhancements in Genetic Algorithms

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  1. Gene Expression Programming with Pre-Order Traversals Stefan Kahrs University of Kent at Canterbury Kahrs - GEP pre-order

  2. Overview • intro • gene expression programming and grammar evolution • breeding by crossover, meaning • change to pre-order traversal • a problem with that • ... and its solution Kahrs - GEP pre-order

  3. breed fitness criterion mutate select Genetic Programming population of programs Kahrs - GEP pre-order

  4. GEP and GE • separate genotype and phenotype • genetic information (determining an individual) is given as the list of nodes of the syntax tree in level-order (GEP) orpre-order (GE) • from this a syntax tree is recovered • the list can have additional node symbols that do not contribute to the tree (allows for mutation) Kahrs - GEP pre-order

  5. Why GEP/GE? • separation of genotype/phenotype allows the manipulation of program individuals (optimisation) without interfering with breeding • avoids program bloat, program sizes are bounded Kahrs - GEP pre-order

  6. + sin * x 2 + x y Example, level-order traversal +, sin, *, x, 2, +, x, y Kahrs - GEP pre-order

  7. + sin + * x y x 2 Example, pre-order traversal +, sin, *, x, 2, +, x, y Kahrs - GEP pre-order

  8. Cross-over breeding • take two individuals A and B • throw a dice to determine a crossover pointk • create a new individual whose first k symbols are the first k symbols of A, and its remaining ones are all but the first k symbols of B Kahrs - GEP pre-order

  9. In Pictures but it does not quite work that way, in general Kahrs - GEP pre-order

  10. Arity • we can extend the notion of arity from symbols to sequences of symbols arity() = 1 arity(s·f) = arity(s)-1+arity(f), if arity(s)>0 arity(s·f) = 0, otherwise • explanation: top-down tree traversal, placing f in the unfinished tree closes an open place and opens arity(f) new ones Kahrs - GEP pre-order

  11. Example • take our previous sequence +, sin, *, x, 2, +, x, y • take the initial segment of the first 5 symbols, i.e. +, sin, *, x, 2 • its arity is 1 • this means: if we draw the incomplete tree of this segment there is 1 hole left to be filled with a subtree • this incomplete tree segment would be part of the offspring if this individual fathered with crossover point 5 Kahrs - GEP pre-order

  12. Procreation with different arity at crossover point Kahrs - GEP pre-order

  13. + + sin * sin * x 2 sin + 2 * x y 1 x y Trees with arity 1 and 2 at crossover point 5 Kahrs - GEP pre-order

  14. Offspring, father left tree The subtree that has been placed in the gap is not a subtree of the mother, though it is composed from nodes from the mother + sin * x 2 * x y Kahrs - GEP pre-order

  15. So... • if the arities are the same at cross-over point then subtrees of the mother are preserved • otherwise, mum is atomised and reassembled • making the fresh subtrees fairly random • problems: • asymmetric contribution in procreation • level of randomness that is hard to control Kahrs - GEP pre-order

  16. A solution • replace level-order traversal with pre-order traversal (which GE uses anyway) • why? in pre-order traversal subtrees occur as consecutive subsequences • notion of arity (of sequences) is unchanged! Kahrs - GEP pre-order

  17. + + sin * sin * x 2 sin + 2 * x y 1 x y Crossover at 6, different arities Kahrs - GEP pre-order

  18. * 1 y Crossover at 6, different arities + the subtrees pasted into the tree are either from the mother, or the dormant part of the mother (here: the z) sin * x 2 + z Kahrs - GEP pre-order

  19. A problem • how do we generate the initial population? • with level-order traversal we place (random) non-nullary symbols at the beginning of the sequence, nullary symbols at the end – the result are balanced trees • if we do this with pre-order traversal we get lop-sided trees (more lists than trees) Kahrs - GEP pre-order

  20. Alternatively • we can meddle with the probabilities to prevent lop-sidedness, i.e. same probability for opening/closing a branch • problem: this gives us microbes (very small trees) and these buggers breed as if the end of the world is nigh Kahrs - GEP pre-order

  21. Solution • level-order traversal works well at the beginning, pre-order traversal works well subsequently, so: why not use them this way! • create the initial population by reading a sequence as a level-order traversal • then traverse those trees pre-orderly, and overwrite the genome • from then on, use pre-order only Kahrs - GEP pre-order

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