Schema Disruption in Chromosomes That Are Structured as Binary Trees

Abstract

Abstract. We are interested in schema disruption behavior when chromosomes are structured as binary trees. We give the definition of the disruption probability dp(H) of a schema H, and also the relative diameter rel∆(H) of H. We show that in the general case that dp(H) can far exceed rel∆(H), but when the chromosome is a complete binary tree then the inequality dp(H) ≤ rel∆(H) holds almost always. Thus the more compactly the tree chromosome is structured, the better is the behavior to be expected from geneticism.

Schema disruption in tree-structured chromosomes

“…(Tree radius ρ = 1 amounts to trivialities.) Proof: This result is the natural extension of Proposition 3 in Greene [3], and its proof.…”

“…Similar to our analysis of the failure reported in Case I, in fact there is no failure of relation dp(H) ≤ rel∆ (H) for these values. (3) Invariably the first inequality fails when d = ρ-1 and δ = 2d. This can give a genuine failing of inequality dp(H) ≤ rel∆(H).…”

“…Non-linear bit arrangements and in particular tree-structured arrangements, and a schema theory for such, have been studied by others. Principally this has come from those in the Genetic Programming (GP) community, although Greene in [2] and [3] has investigated non-linear bit arrangements in the abstract. In GP approaches, individuals are programs, specifically functions, realized as expression trees.…”