RASC 2006Kahrs - GEP pre-order Gene Expression Programming with Pre-Order Traversals Stefan Kahrs...

RASC 2006 Kahrs - GEP pre-order

Gene Expression Programming with Pre-Order

TraversalsStefan Kahrs

University of Kent at Canterbury


Overview

• intro• gene expression programming and

grammar evolution• breeding by crossover, meaning• change to pre-order traversal• a problem with that• ... and its solution


Genetic Programming

population of programs

fitness criterionbreed

mutate

select


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) or pre-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)


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


Example, level-order traversal

+

sin *

x 2 +

x y

+, sin, *, x, 2, +, x, y


Example, pre-order traversal

+, sin, *, x, 2, +, x, y

+

sin

*

x 2

+

x y


Cross-over breeding

• take two individuals A and B• throw a dice to determine a

crossover point k• 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


In Pictures

but it does not quite work that way, in general


Arity

• we can extend the notion of arity from symbols to sequences of symbols

arity() = 1arity(s·f) = arity(s)-1+arity(f), if arity(s)>0arity(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


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


Procreation with different arity at crossover point


Trees with arity 1 and 2 at crossover point 5

+

sin *

x 2 +

x y

+

* sin

sin 2 *

x y 1


Offspring, father left tree

+

sin *

x 2 *

x y

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


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


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!


Crossover at 6, different arities

+

sin *

x 2 +

x y

+

* sin

sin 2 *

x y 1


Crossover at 6, different arities

+

sin *

x 2 +

z*

y 1

the subtrees pasted into the tree are either from the mother, or the dormant part of the mother (here: the z)


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)


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


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

RASC 2006Kahrs - GEP pre-order Gene Expression Programming with Pre-Order Traversals Stefan Kahrs...

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