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Mathematics Awaits: Commentary on “Genetic Programming and Emergence” by Wolfgang Banzhaf.

### Lee Altenberg

2014. *Genetic Programming and Evolvable Machines* 15(1): 87-89.

#### Abstract

Banzhaf provides a portal to the subject of emergence, noting contentious concepts while not getting sucked into fruitless debate. Banzhaf refutes arguments against downward causation much as Samuel Johnson kicks a stone to refute Berkeley — by pointing to concrete examples in genetic programming, such as the growth of repetitive patterns within programs. Repetitive patterns are theoretically predicted to emerge from the evolution of evolvability and robustness under subtree exchange. Selection and genetic operators are co-equal creators of these emergent phenomena. GP systems entirely formal, and thus their emergent phenomena are essentially mathematical. The emergence of Lagrangian distributions for tree shapes under subtree exchange, for example, gives a glimpse of the possibilities for mathematical understanding of emergence in GP. The mathematics underlying emergence in genetic programming should be pursued with vigor.
**Keywords**: evolvability, robustness, subtree exchange, mathematics, matrix theory, Lagrange distribution

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