1. A Probabilistic Functional Crossover Operator
for Genetic Programming
Josh Bongard
Morphology, Evolution and Cognition Laboratory
Department of Computer Science
University of Vermont

2. Crossover operators
Headless chicken crossover (Jones, 1995)
Cross an existing tree with a new, randomly-created one
Size fair crossover (Langdon, 1999)
Cross subtrees with probability proportional to their size similarity
Deterministic homologous crossover (D’haeseleer, 1994)
Cross subtrees based on the similarity of their relative position within their tree
Probabilistic homologous crossover (Langdon, 1999)
Cross subtrees with prob. prop. to their relative positions within their trees

3. Crossover operators
Semantically Driven Crossover (Beadle and Johnson, 2008)
Cross trees, and only retain children if they differ semantically from parents
Enzyme genetic programming (Lones and Tyrrell, 2001)
Constructive linear GP: elements attach to one another;
Crossover: if donated elements attach to parent, keep them; otherwise, discard.

4. of Natural Selection (1930): “Probability of a mutation
Snipping * +
sin
0.8 x / y
(dx/dt=)
(dy/dt=)
(dx/dt=) *
(dy/dt=) /
[-24.0,48.3] +
sin x y
[-0.03,0.05]
0.8 x x
[0.8,0.8]
[0.3,0.35]
[0.3,0.35]
Bongard, J and Lipson, H (2007). Proceedings of the National Academy of Sciences, 104(24):
Ronald Fisher:
population biology,
modern statistics:
The Genetical Theory
of Natural Selection (1930):
“Probability of a mutation
being favorable is inversely
proportional to its magnitude.”
(dx/dt=) *
(dy/dt=) / +
0.04 x y
0.8 x

5. + Deterministic Functional Crossover (GPTP 2009) [] [0.1, 0.9] y
[-0.3,
0.6]
[0.0,
0.9] x +
[-0.3,
0.6]
[0.3,
0.3] x
0.3

6. Probabilistic Functional Crossover (GECCO 2010)x y
[-0.3,
0.6]
[0.1,
0.9]
[-0.83,
2.72] x y
0.424
0.1
2.0003

7. Probabilistic Functional Crossover (GECCO 2010)x y
[-0.3,
0.6]
[0.1,
0.9]
[-0.83,
2.72] x y
0.424
0.1
2.0003 x y
0.424/
2.5243
0.1
/2.5243
2.0003/
( )

8. Probabilistic Functional Crossover (GECCO 2010)x y
[-0.3,
0.6]
[0.1,
0.9]
[-0.83,
2.72] x y
0.424
0.1
2.0003 x y
0.424/
2.5243
0.1
/2.5243
2.0003/
( ) x y

9. Probabilistic Functional Crossover (GECCO 2010)x y
[-0.3,
0.6]
[0.1,
0.9]
[-0.83,
2.72] x y
0.424
0.1
2.0003
0.208 x
0.833 y
0.961 x y
0.424/
2.5243
0.1
/2.5243
2.0003/
( ) x y

10. Results

11. Results

12. Results

13. Results

14. Results

15. Results

16. Results
~9.2%
~4.5%
~11.2%

17. Results
Random crossover
D-FXO
P-FXO

18. Results
Random crossover
D-FXO
P-FXO