1. COMP Part II.
Genetic Algorithms.
Genetic Algorithms

2. Similarity Templates (Schemata).
Topic 5.
Similarity Templates (Schemata).
Genetic Algorithms

3. GAs by John Holland. Holland introduced
a “population” of binary strings which he called “chromosomes”.
The “population” evolves using kind of “natural selection” together with the genetics-inspired operators of crossover, mutation, and inversion.
Bits in a “chromosome” represent genes, and each “gene” is an instance of a particular “allele”, 0 or 1.
The selection operator chooses those chromosomes in the population that will be allowed to reproduce, and on average the fitter chromosomes produce more offspring than the less fit ones.
Crossover exchange subparts of two chromosomes
Mutation randomly changes the allele values of some locations in the chromosome.
Inversion reverses the order of a contiguous section of the chromosome rearranging the genes order.
Genetic Algorithms

4. Basic Structure of a Genetic Algorithm.
Randomly generate initial population of n strings (“chromosomes”)
Evaluate the fitness of each string in the population
3. Repeat the following steps until next generation of n individual strings produced
a. Select pair of parent chromosomes from current population
according to their fitness, i.e. chromosomes with higher fitness
are selected more often
b. Apply crossover (with probability)
c. Apply mutation (with probability of occurrence)
4. Apply generational replacement
5. Go to 2 or terminate if termination condition met
Genetic Algorithms

5. Basic Structure of a Genetic Algorithm.
Randomly generate initial population of n strings (“chromosomes”)
Evaluate the fitness of each string in the population
3. Repeat the following steps until next generation of n individual strings produced
a. Select pair of parent chromosomes from current population
according to their fitness, i.e.
chromosomes with higher fitness
are selected more often
b. Apply crossover (with probability)
c. Apply mutation (with probability of occurrence)
4. Apply generational replacement
5. Go to 2 or terminate if termination condition met
Each iteration in the cycle
produces a new “generation” of chromosomes.
The entire set of generations is called a run.
Typical GA run is from 50 to 500 or more generations.
At the end of a run often there is at least one highly fit chromosome in the population.
Genetic Algorithms

6. Example Implementation of a GA.
Let the length of the string l = 8 , and the number of chromosomes in the population (population size) n = 4.
Fitness function f(x) is equal to the number of ones in the bit string x.
Selection operator Fitness-proportionate selection, i.e. the number of times an individual is expected to reproduce is equal to its fitness fi divided by the average fitness of the population f Ni = fi / f
Crossover probability pc = 0.7
Mutation probability pm = 0.001
Make a run of three generations.
Genetic Algorithms

7. Example of a Genetic Algorithm.
2. Evaluate the fitness of each string in the population
Chromosome Chromosome Fitness = number of ones
index string in the string
f8 = 5
f12 = 4
f13 = 6
f14 = 6
f Average fitness f = ( )/4 = 5.25
The average fitness in the population of possible solutions increases with every generation.
Generation №
Generation 3
5.0
3.0
1.0
Genetic Algorithms

8. Why Genetic Algorithms work?
It might be seen that
similar “looking” chromosomes do have similar fitness values, and therefore
are taken for reproduction equally often.
Chromosome Chromosome Fitness Times to be selected
index string for reproduction
f1 = N1 = 1
f2 = N2 = 2
f3 = N3 = 0
n = f4 = N4 = 1
Generation 0
Genetic Algorithms

9. Why Genetic Algorithms work?
It might be seen that
similar “looking” chromosomes do have similar fitness values, and therefore
are taken for reproduction equally often.
Chromosome Chromosome Fitness Times to be selected
index string for reproduction
f8 = N8 = 1
f9 = N9 = 1
f10 = N10 = 0
f11 = N11 = 2
Generation 2
Genetic Algorithms

10. Similarities in GAs chromosomes.
Similarities among highly fit strings guide the search.
Chromosome Chromosome Fitness
index string
f8 = 5
f12 = 6
f13 = 4
f14 = 6
Generation 3
Genetic Algorithms

11. Similarity Template = Schema.
Holland introduced a
similarity template = schema = building block
to describe subset of strings with similarities at certain positions.
Chromosome Chromosome Fitness
index string
f8 = 5
f12 = 6
f13 = 4
f14 = 6
Generation 3
Genetic Algorithms

12. Similarity Template = Schema.
Definition:
A schema (plural, schemata) is a similarity template describing a subset of strings with similarities at certain string positions Other name for a schema is a building block of a chromosome.
Genetic Algorithms

13. Similarity Template = Schema.
To describe building blocks of binary chromosomes, the following three letter alphabet is used:
{0, 1, *}
Here, * is a do not care symbol.
Genetic Algorithms

14. Similarity Template = Schema.
To describe building blocks of binary chromosomes, the following three letter alphabet is used:
{0, 1, *}
Here, * is a do not care symbol.
Example: **0 is a schema of the chromosome
111100
Genetic Algorithms

15. Similarity Template = Schema.
Example: **0 is a schema of the chromosome
111100
The meaning of the schema is that it works a pattern matching device: a schema matches a particular string iff at every location in the schema a 1 matches 1 in the string, a 0 matches a 0, or a * matches either.
Genetic Algorithms

16. Similarity Template = Schema.
As an example consider strings and schemata of length 5.
Example 1: The schema *0000 matches two chromosomes:
1) and 2)
Example 2: The schema *000* matches four chromosomes:
1) ,
2) ,
3) , and 4)
Genetic Algorithms

17. Similarity Template = Schema.
A question opposite to the previous examples is
How many building blocks there are in a particular chromosome of a fixed length?
Answer: To match a particular chromosome of a fixed length the schema must
a) be of the same length,
b) have the same symbol as the chromosome does or a “*“ - do not care symbol at any particular locus
Genetic Algorithms

18. Similarity Template = Schema.
A question opposite to the previous examples is
How many building blocks there are in a particular chromosome of a fixed length?
Answer: To match a particular chromosome of a fixed length the schema must
a) be of the same length,
b) have the same symbol as the chromosome does or a “*” - do not care symbol at any particular locus
Thus, in a binary chromosome of a length l there are
2l building blocks,
as a symbol at a particular locus in the chromosome must correspond to the same or the do not care symbol in the corresponding locus in the schema.
Genetic Algorithms

19. Similarity Template = Schema.
How many building blocks there are in a particular chromosome of a fixed length?
Answer: in a binary chromosome of a length l there are
2l building blocks,
as a symbol at a particular locus in the chromosome must correspond to the same or the do not care symbol in the corresponding locus in the schema.
Example 3:
The 2bit (l=2) chromosome has 22=4 building blocks
1) *1 ,
2) * ,
3) ** , and 4)
Genetic Algorithms

20. Similarity Template = Schema.
How many building blocks there are in a particular chromosome of a fixed length?
Answer: in a binary chromosome of a length l there are
2l building blocks.
Example 4:
The 3bit (l=3) chromosome has 23=8 building blocks
1) * ,
2) ** ,
3) *** ,
4) *1 ,
5) *01 ,
6) **1 ,
7) *0* , and 8)
Genetic Algorithms

21. Similarity Template = Schema.
Definition:
The ORDER of a schema is the number of non-* symbols it contains.
Example 5:
Order of the schema 10*00*** is 4.
Example 6:
Order of the schema *** is 0.
One might say that the order of the schema *** is lower than the one of the schema 10*00***.
Genetic Algorithms

22. Similarity Template = Schema.
Definition:
The DEFINING LENGTH of a schema is the distance, i.e. the number of positions, between the outermost non-* symbols.
Example 7:
The defining length of the schema 10*00* is 3.
10*00*
outermost non-* symbols
3 positions between the outermost
non-* symbols
Genetic Algorithms

23. Similarity Template = Schema.
Definition:
The DEFINING LENGTH of a schema is the distance, i.e. the number of positions, between the outermost non-* symbols.
Example 7:
The defining length of the schema 10*00* is 3.
Example 8:
The defining length of the schema *11*** is 0.
One might say that the defining length of the schema *11*** is shorter than the one of the schema 10*00*
Genetic Algorithms

24. Schema Theorem.
Highly fit, short defining length, low order schemas propagate from generation to generation and give exponential increase of samples to the observed best.
Genetic Algorithms