Schema (genetic algorithms): Difference between revisions

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{{Evolutionary algorithms}}

A '''schema''' ({{plural form}}: '''schemata''') is a template in [[computer science]] used in the field of [[genetic algorithm]]s that identifies a [[subset]] of strings with similarities at certain string positions. Schemata are a special case of [[cylinder set]]s;, andforming soa form[[Base (topology)|basis]] for a [[topologicalproduct spacetopology]] on strings.<ref name="Holland1">{{cite book |title=Adaptation in Natural and Artificial Systems|year=1992|edition=reprint|publisher=The MIT Press|author=Holland, John Henry |isbn=9780472084609 |url=https://books.google.com/books/about/Adaptation_in_natural_and_artificial_sys.html?id=JE5RAAAAMAAJ |deadurl=no |accessdate=22 April 2014}}</ref> In other words, schemata can be used to generate a [[Topological space|topology]] on a space of strings.

== Description ==

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==Propagation of schema==

In [[evolutionary computing]] such as [[genetic algorithms]] and [[genetic programming]], '''propagation''' refers to the inheritance of characteristics of one generation by the next. For example, a schema is propagated if individuals in the current generation match it and so do those in the next generation. Those in the next generation may be (but don'tdo not have to be) children of parents who matched it.

== The Expansion and Compression Operators ==

Recently schema have been studied using [[order theory]].<ref name = "Fletcher">

{{cite arxivarXiv |author=Jack McKay Fletcher and Thomas Wennkers |year=2017 |title=A natural approach to studying schema processing |eprint=1705.04536|class=cs.NE }}</ref>

Two basic operators are defined for schema: expansion and compression. The expansion maps a schema onto a set of words which it represents, while the compression maps a set of words on to a schema.

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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}}, [[Antisymmetric relation|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>.

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== The Schematic Completion and The Schematic Lattice ==

[[File:Schematic Lattice.png|thumb|The Schematic lattice formed from the schematic completion on the set <math>A=\{111, 011, 001\}</math>. Here the schematic lattice <math>(\mathcal{S}(A),\leq)</math> is shown as a [[Hasse diagram]].

]]

For a set <math>A \subseteq \Sigma^l</math>, we call the process of calculating the compression on each subset of A, that is <math>\{{\downarrow}X | X \subseteq A\}</math>, the schematic completion of <math>A</math>, denoted <math>\mathcal{S}(A)</math>.<ref name="Fletcher"/>

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The [[poset]] <math>(\mathcal{S}(A),\leq)</math> always forms a [[complete lattice]] called the schematic lattice.

[[File:Schematic Lattice.png|thumb|The Schematic lattice formed from the schematic completion on the set <math>A=\{111, 011, 001\}</math>. Here the schematic lattice <math>(\mathcal{S}(A),\leq)</math> is shown as a [[Hasse diagram]].

]]

The schematic lattice is similar to the concept lattice found in [[Formal concept analysis]].

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{{Reflist}}

{{Evolutionary computation}}

[[Category:Genetic algorithms]]

[[Category:Genetic programming]]