Holland's schema theorem

Holland's schema theorem gives a lower bound on how many instances of a genetic-algorithm schema are expected in the next generation, based on fitness, crossover, and mutation.

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Schema (genetic algorithm): A pattern of fixed and unspecified positions that represents a subset of encoded candidate solutions. The theorem tracks the expected number of instances matching this pattern.
Genetic algorithm: A population-based search method that uses selection and genetic operators to generate candidate solutions. The theorem describes schema counts across generations in this algorithmic framework.
Building block hypothesis: The hypothesis that genetic algorithms succeed by combining short, low-order, above-average schemata. It turns schema-level survival bounds into a proposed account of search performance.
John Henry Holland: An American computer scientist who developed foundational ideas in genetic algorithms and adaptive systems. He introduced the schema framework and its theorem in genetic algorithm theory.
Schema defining length: The distance between the first and last fixed positions in a schema. Longer defining lengths create more opportunities for crossover to disrupt matching instances.
Fitness (genetic algorithm): A numerical measure used to evaluate candidate solutions and guide their reproduction. Schema fitness is compared with population-average fitness in the bound.
No Free Lunch theorem: A result stating that averaged over all possible problems, no optimization algorithm outperforms others universally. It limits claims that schema-based advantages imply general superiority across problems.
Adaptation in Natural and Artificial Systems: John Holland's 1975 book presenting genetic algorithms and a theory of adaptation in artificial systems. It is the foundational work in which the schema theorem was formulated.
Selection (genetic algorithm): The process of choosing candidate solutions to contribute offspring according to a fitness-based rule. Above-average schema fitness increases its expected representation through selection.
Population (genetic algorithm): A set of candidate solutions maintained and evolved together by a genetic algorithm. Expected schema representation counts instances within this changing set.