Schemata as Building Blocks: Does Size Matter?

TL;DR

This paper rigorously analyzes the schema theorem and building block hypothesis using an exact evolution equation, revealing conditions under which larger schemata are favored and providing proofs and numerical evidence across different landscapes.

Contribution

It introduces a new schema theorem based on effective fitness and clarifies the conditions favoring large schemata, extending understanding of genetic algorithm dynamics.

Findings

Higher than average effective fitness schemata increase exponentially over time.

Large schemata are favored when schema reconstruction outweighs destruction.

Proves Geiringer's theorem using the evolution equation.

Abstract

We analyze the schema theorem and the building block hypothesis using a recently derived, exact schemata evolution equation. We derive a new schema theorem based on the concept of effective fitness showing that schemata of higher than average effective fitness receive an exponentially increasing number of trials over time. The building block hypothesis is a natural consequence in that the equation shows how fit schemata are constructed from fit sub-schemata. However, we show that generically there is no preference for short, low-order schemata. In the case where schema reconstruction is favoured over schema destruction large schemata tend to be favoured. As a corollary of the evolution equation we prove Geiringer's theorem. We give supporting numerical evidence for our claims in both non-epsitatic and epistatic landscapes.