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{{Evolutionary algorithms}}

{{merge from|Propagation of schema|date=December 2011}}

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|last=Holland |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?id=JE5RAAAAMAAJ |accessdate=1322 JulyApril 20102014}}</ref> In other words, schemata can be used to generate a [[Topological space|topology]] on a space of strings.

== Description ==

For example, consider binary strings of length 6. The schema 1**0*1 describes the set of all words of length 6 with 1's at the first and sixth positions and a 0 at the fourth position. The * is a [[Wildcard character|wildcard]] symbol, which means that positions 2, 3 and 5 can have a value of either 1 or 0. The ''order of a schema'' is defined as the number of fixed positions in the template, while the ''[[defining length]]'' <math> \delta(H) </math> is the distance between the first and last specific positions. The order of 1**0*1 is 3 and its defining length is 5. The ''fitness of a schema'' is the average fitness of all strings matching the schema. The fitness of a string is a measure of the value of the encoded problem solution, as computed by a problem-specific evaluation function.

===Length===

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===Disruption===

If the child of an individual that matches schema H does not ''itself'' match H, the schema is said to have been ''disrupted''.<ref name="UCL1">< /ref>

==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 do 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 arXiv |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.

In the following definitions <math> \Sigma </math> denotes an alphabet, <math> \Sigma^l </math> denotes all words of length <math> l </math> over the alphabet <math> \Sigma </math>, <math> \Sigma_* </math> denotes the alphabet <math>\Sigma</math> with the extra symbol <math>*</math>. <math>\Sigma_*^l</math> denotes all schema of length <math> l </math> over the alphabet <math> \Sigma_* </math> as well as the empty schema <math> \epsilon_* </math>.

For any schema <math>s \in \Sigma^l_*</math> the following operator <math>{\uparrow}s</math>, called the <math>expansion</math> of <math>s</math>, which maps <math>s</math> to a subset of words in <math>\Sigma^l </math>:

<math display="block">{\uparrow}s := \{b \in \Sigma^l | b_i = s_i \mbox{ or } s_i = * \mbox{ for each } i \in \{1,...,l\}\} </math>

Where subscript <math>i</math> denotes the character at position <math>i</math> in a word or schema. When <math> s= \epsilon_*</math> then <math>{\uparrow}s = \emptyset</math>. More simply put, <math>{\uparrow}s</math> is the set of all words in <math>\Sigma^l</math> that can be made by exchanging the <math>*</math> symbols in <math>s</math> with symbols from <math>\Sigma</math>. For example, if <math>\Sigma=\{0,1\}</math>, <math>l=3</math> and <math>s=10*</math> then <math>{\uparrow}s=\{100,101\} </math>.

Conversely, for any <math>A \subseteq \Sigma^l</math> we define <math>{\downarrow}{A}</math>, called the <math>compression</math> of <math>A</math>, which maps <math>A</math> on to a schema <math>s\in \Sigma_*^l</math>:

<math display="block">{\downarrow}A:= s</math>

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]], [[Antisymmetric relation|antisymmetry]] and [[transitive relation|transitivity]] 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>.

The compression and expansion operators form a [[Galois connection]], where <math>\downarrow</math> is the lower adjoint and <math>\uparrow</math> the upper adjoint.<ref name="Fletcher"/>

== 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"/>

For example, let <math>A = \{110, 100, 001, 000\}</math>. The schematic completion of <math>A</math>, results in the following set:

<math display="block">\mathcal{S}(A) =

\{001, 100, 000, 110, 00*, *00, 1*0, **0, *0*, ***, \epsilon_*\}</math>

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

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

{{clear}}

==See also==

*[[Holland's schema theorem]]

*[[PropagationFormal ofconcept schemaanalysis]]

==References==

{{Reflist}}

{{Evolutionary computation}}

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[[Category:Genetic algorithms]]

[[Category:Genetic programming]]