Schema Theorem

Holland's Schema Theorem states that \[E[m(s,t+1)] \geq \frac{\hat{u}(s,t)}{\bar{f}(t)}m(s,t) \left(1 - p_{c}\frac{d(s)}{l-1}\right) (1 - p_{m})^{o(s)} \] where
- $m(s,t) =$ instances of schema $s$ in population at time $t$.
- $\bar{f}(t) =$ average fitness of population at time $t$.
- $\hat{u}(s,t) =$ average fitness of instances of $s$ at time $t$.
- $p_c =$ probability of single point crossover operator.
- $p_m = $ probability of mutation operator.
- $l = $ length of single bit strings.
- $o(s)$ number of defined (non *) bits in $s$.
- $d(s) = $ distance between leftmost, rightmost defined bits in $s$.
More fit schemas will tend to grow in influence, especially those containing a small number of defined bits and when these bits are close together.
It is incomplete, e.g., it fails to consider the possitive effects of crossover and mutation.