Schema (genetic algorithms): Difference between revisions

Revision as of 13:37, 10 November 2017

Revision as of 17:39, 28 March 2018

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where <math>s</math> is a schema of length <math>l</math> such that the symbol at position <math>i</math> in <math>s</math> is determined in the following way: if <math>x_i = y_i</math> for all <math>x,y \in A</math> then <math>s_i = x_i</math> otherwise <math>s_i = *</math>. If <math>A = \emptyset</math> then <math>{\downarrow}A = \epsilon_*</math>. One can think of this operator as stacking up all the items in <math>A</math> and if all elements in a column are equivalent, the symbol at that position in <math>s</math> takes this value, otherwise there is a wild card symbol. For example, let <math>A = \{100,000,010\}</math> then <math>{\downarrow}A = **0</math>.

Schemata can be [[partially ordered]]. For any <math>a,b \in \Sigma^l_*</math> we say <math>a \leq b</math> if and only if <math>{\uparrow}a \subseteq {\uparrow}b</math>. It follows that <math>\leq</math> is a [[partial ordering]] on a set of schemata from the [[reflexive operator algebra|reflexivity]]{{disambiguation needed|date=June 2017}}, [[antisymmetry]] and [[transitive relation|transitivity]]{{disambiguation needed|date=June 2017}} of the [[subset]] relation. For example, <math>\epsilon_* \leq 11 \leq 1* \leq **</math>.

This is because <math>{\uparrow}\epsilon_* \subseteq {\uparrow}11 \subseteq {\uparrow}1* \subseteq {\uparrow}** = \emptyset \subseteq \{11\} \subseteq \{11,10\} \subseteq \{11,10,01,00\}</math>.