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

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

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.

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): 9943-9948. 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

+ 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

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

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/ (2.0003+0.424+0.1)

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/ (2.0003+0.424+0.1) 1-0.792 x 1-0.167 y 1-0.039

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/ (2.0003+0.424+0.1) 1-0.792 x 1-0.167 y 1-0.039

Results

Results

Results

Results

Results

Results

Results ~9.2% ~4.5% ~11.2%

Results Random crossover D-FXO P-FXO

Results Random crossover D-FXO P-FXO