Holland's schema theorem facts for kids

Holland's schema theorem, also known as the fundamental theorem of genetic algorithms, is a big idea about how genetic algorithms work. It was created by John Holland in the 1970s.

Imagine you have a group of solutions to a problem, like different ways to build a bridge. A genetic algorithm tries to find the best solution by letting these solutions "evolve" over time, much like living things evolve in nature.

The Schema Theorem says that certain "good" patterns, called schemata, tend to become more common over many generations. These patterns are like blueprints for parts of the solutions. If a pattern is short, simple, and leads to better solutions, it will spread quickly through the population.

What is a Schema?

A schema (say: SKEE-muh) is like a special code or template. It helps us find groups of solutions that share certain features.

Let's use an example with binary strings, which are like codes made of 0s and 1s. Imagine a code that is 6 digits long.

• The schema 1*10*1 describes all codes that have a 1 at the first spot, a 1 at the third spot, a 0 at the fourth spot, and a 1 at the sixth spot.
• The `*` symbol is a "wildcard." This means the second and fifth spots can be either a 0 or a 1.

Schema Details

• The order of a schema is how many fixed (non-wildcard) positions it has. For 1*10*1, the order is 4 (because of the 1st, 3rd, 4th, and 6th positions).
• The defining length is the distance between the first and last fixed positions. For 1*10*1, the first fixed position is 1 and the last is 6, so the defining length is 5 (6 minus 1).
• The fitness of a schema is how good, on average, the solutions are that match that schema. "Fitness" here means how well a solution solves the problem.

Holland's theorem explains that if a schema has:

• A short defining length (the fixed parts are close together).
• A low order (not too many fixed parts, so it's more general).
• An above-average fitness (it leads to good solutions).

Then, this schema will likely increase its numbers very quickly in the next generations of the genetic algorithm. This happens because the genetic algorithm's operations, like genetic operators (which are like breeding and small changes), tend to keep these good, short patterns together.

The theorem is an "inequality" because it mostly focuses on how schemata are broken apart. It doesn't fully account for new schemata being created randomly, which is a small but possible event.

Why the Theorem is Important

The Schema Theorem helps us understand why genetic algorithms can be powerful tools for solving complex problems. It shows that these algorithms naturally favor and spread useful building blocks (schemata) within the solutions.

However, it's important to know that the theorem works best when we imagine a very large group of solutions. In real-world genetic algorithms, where the group size is limited, things can sometimes be a bit different.

Limitations of the Theorem

This picture shows a landscape with many "peaks" or best solutions. Genetic algorithms might get stuck on one peak and miss others.

While the Schema Theorem is a great idea, it doesn't always perfectly explain everything that happens in real genetic algorithms.

• Sometimes, if the starting group of solutions is small, the algorithm might accidentally focus on patterns that aren't actually the best. This is like a small group of people all liking the same thing, even if it's not the best choice overall.
• In problems where there are many "best" solutions (like finding the highest points on a bumpy map), a genetic algorithm might get stuck finding just one of these best solutions and miss the others. This is called genetic drift, where the population slowly moves towards one option without a strong reason.

The Schema Theorem is a general rule. It doesn't tell us why genetic algorithms work well for some problems but not for others. It's a fundamental concept, but it's not the whole story of how these smart computer programs solve problems.

• J. Holland, Adaptation in Natural and Artificial Systems, The MIT Press; Reprint edition 1992 (originally published in 1975).
• J. Holland, Hidden Order: How Adaptation Builds Complexity, Helix Books; 1996.