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| '''Holland's schema theorem''' is widely taken to be the foundation for explanations of the power of [[genetic algorithms]]. It says that short, low-order schemata with above-average fitness increase exponentially in successive generations. The theorem was proposed by [[John Henry Holland|John Holland]] in the 1970s.
| | The name of the author is Figures. Minnesota is where he's been living for years. To collect coins is what his family and him enjoy. She is a librarian but she's usually wanted her own company.<br><br>My web page: [http://ironptstudio.com/rev/239375 ironptstudio.com] |
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| A [[schema (genetic algorithms)|schema]] is a template that identifies a [[subset]] of strings with similarities at certain string positions. Schemata are a special case of [[cylinder set]]s; and so form a [[topological space]].
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| == Description ==
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| For example, consider binary strings of length 6. The schema 1*10*1 describes the set of all strings of length 6 with 1's at positions 1, 3 and 6 and a 0 at position 4. The * is a [[Wildcard character|wildcard]] symbol, which means that positions 2 and 5 can have a value of either 1 or 0. The ''order of a schema'' <math> o(H)</math> 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*10*1 is 4 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. Using the established methods and [[genetic operator]]s of [[genetic algorithms]], the schema theorem states that short, low-order schemata with above-average fitness increase exponentially in successive generations. Expressed as an equation:
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| :<math>\operatorname{E}(m(H,t+1)) \geq {m(H,t) f(H) \over a_t}[1-p].</math>
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| Here <math>m(H,t)</math> is the number of strings belonging to schema <math>H</math> at generation <math>t</math>, <math>f(H)</math> is the ''observed'' fitness of schema <math>H</math> and <math>a_t</math> is the ''observed'' average fitness at generation <math>t</math>. The probability of disruption <math>p</math> is the probability that crossover or mutation will destroy the schema <math>H</math>. It can be expressed as:
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| :<math>p = {\delta(H) \over l-1}p_c + o(H) p_m</math> | |
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| where <math> o(H)</math> is the order of the schema, <math>l</math> is the length of the code, <math> p_m</math> is the probability of mutation and <math> p_c </math> is the probability of crossover. So a schema with a shorter defining length <math> \delta(H) </math> is less likely to be disrupted.<br />An often misunderstood point is why the Schema Theorem is an ''inequality'' rather than an equality. The answer is in fact simple: the Theorem neglects the small, yet non-zero, probability that a string belonging to the schema <math>H</math> will be created "from scratch" by mutation of a single string (or recombination of two strings) that did ''not'' belong to <math>H</math> in the previous generation.
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| ==References==
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| * J. Holland, ''Adaptation in Natural and Artificial Systems'', The MIT Press; Reprint edition 1992 (originally published in 1975).
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| * J. Holland, ''Hidden Order: How Adaptation Builds Complexity'', Helix Books; 1996.
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| [[Category:Genetic algorithms]]
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| [[Category:Theorems in discrete mathematics]]
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The name of the author is Figures. Minnesota is where he's been living for years. To collect coins is what his family and him enjoy. She is a librarian but she's usually wanted her own company.
My web page: ironptstudio.com