1. © Negnevitsky, Pearson Education, 2005 1 Lecture 11 Evolutionary Computation: Genetic algorithms Why genetic algorithm work? Why genetic algorithm work?

2. © Negnevitsky, Pearson Education, 2005 2 Schema Theorem GA techniques have a solid theoretical foundation (Holland, 1975; Goldberg, 1989; Rawlins 1991; Whitley, 1993) GA techniques have a solid theoretical foundation (Holland, 1975; Goldberg, 1989; Rawlins 1991; Whitley, 1993) John Holland introduced the notation of schema which came from Greek word meaning “form”. John Holland introduced the notation of schema which came from Greek word meaning “form”. A schema is a set of bit strings of ones, zeros and asterisks, where each asterisk can assume either value 1 or 0. A schema is a set of bit strings of ones, zeros and asterisks, where each asterisk can assume either value 1 or 0. The ones and zeros represent the fixed positions of a schema, while asterisks represent ‘wild cards’ The ones and zeros represent the fixed positions of a schema, while asterisks represent ‘wild cards’

3. © Negnevitsky, Pearson Education, 2005 3 For example, schema. For example, schema. Stands for a set of 4 bit strings. Stands for a set of 4 bit strings. Each string in this set begins with 1 and ends with 0. Each string in this set begins with 1 and ends with 0. These strings are called instances of the schema. These strings are called instances of the schema. 1**0

4. © Negnevitsky, Pearson Education, 2005 4 Relationship Between a Schema and a Chromosome A chromosome matches a schema when the fixed positions in the schema matches the corresponding positions in the chromosome. e.g the schema H A chromosome matches a schema when the fixed positions in the schema matches the corresponding positions in the chromosome. e.g the schema H Matches the following set of 4-bit chromosomes: Matches the following set of 4-bit chromosomes: These chromosomes of said to be instances of H. These chromosomes of said to be instances of H. 1110 1100 1010 1000

5. © Negnevitsky, Pearson Education, 2005 5 The number of defined bits (non-asterisks) in a schema is called the order. The number of defined bits (non-asterisks) in a schema is called the order. Schema H: Schema H: has two defined bits and thus the order is 2. has two defined bits and thus the order is 2. Order

6. © Negnevitsky, Pearson Education, 2005 6 GAs manipulate schemata when they run. GAs manipulate schemata when they run. If GAs use a technique that makes the probability of reproduction proportional to chromosome fitness, then according to the Schema Theorem, we can predict the presence of a given schema in the next chromosome generation. If GAs use a technique that makes the probability of reproduction proportional to chromosome fitness, then according to the Schema Theorem, we can predict the presence of a given schema in the next chromosome generation. In other words, we can describe the GA’s behaviour in terms of the increase of decrease in the number of instances of the a given schema. In other words, we can describe the GA’s behaviour in terms of the increase of decrease in the number of instances of the a given schema.

7. © Negnevitsky, Pearson Education, 2005 7

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10. 10 Effects caused by the Crossover and Mutation Crossover and mutation can both create and destroy instances of a schema. Crossover and mutation can both create and destroy instances of a schema. Destructive effects (effects that decrease the number of instances of the schema H). Destructive effects (effects that decrease the number of instances of the schema H). –Schema will survive after crossover if at least one of its offspring is also its instance. This is the case when crossover does not occur within the defining length of the schema. This is the case when crossover does not occur within the defining length of the schema.

11. © Negnevitsky, Pearson Education, 2005 11 Defining Length of a Schema The distance between the outermost defined bits of a schema is called defining length. E.g. The distance between the outermost defined bits of a schema is called defining length. E.g. Defining length = 3. Defining length = 3. Defining length = 5. Defining length = 5. Defining length = 7. Defining length = 7. ****1011 *0*1*10* 1******0

12. © Negnevitsky, Pearson Education, 2005 12 If crossover takes place within the defining length, the schema H can be destroyed and offspring that are not instances of H can be created. If crossover takes place within the defining length, the schema H can be destroyed and offspring that are not instances of H can be created. The schema H will not be destroyed if two identical chromosomes cross over, even when crossover occurs within the defining length. The schema H will not be destroyed if two identical chromosomes cross over, even when crossover occurs within the defining length.

13. © Negnevitsky, Pearson Education, 2005 13 The probability that the schema H will survive after crossover can be defined as: The probability that the schema H will survive after crossover can be defined as:

14. © Negnevitsky, Pearson Education, 2005 14 Destructive Effect of mutation It is also clear that the probability of survival under mutation is higher for low-order schemata than for high-order ones. It is also clear that the probability of survival under mutation is higher for low-order schemata than for high-order ones.

15. © Negnevitsky, Pearson Education, 2005 15 Destructive Effects of crossover and Mutation: