Defining length

In the field of genetic algorithms, a schema (plural: schemata) is a template that represents a subset of potential solutions. These templates use fixed symbols (e.g., `0` or `1`) for specific positions and a wildcard or "don't care" symbol (often `#` or `*`) for others.

The defining length of a schema, denoted as L(H), measures the distance between the outermost fixed positions in the template. According to the Schema theorem, a schema with a shorter defining length is less likely to be disrupted by the genetic operator of crossover.^[1][2] As a result, short schemata are considered more robust and are more likely to be propagated to the next generation.^[3]

In genetic programming, where solutions are often represented as trees, the defining length is the number of links in the minimum tree fragment that includes all the non-wildcard symbols within a schema H.^[4]

Example

The defining length is calculated by subtracting the position of the first fixed symbol from the position of the last one. Using 1-based indexing for a string of length 5:

• The schema `1##0#` has its first fixed symbol (`1`) at position 1 and its last fixed symbol (`0`) at position 4. Its defining length is 4 − 1 = 3.
• The schema `00##0` has its first fixed symbol at position 1 and its last at position 5. Its defining length is 5 − 1 = 4.
• The schema `##0##` has only one fixed symbol at position 3. The first and last fixed positions are the same, so its defining length is 3 − 3 = 0.