COMP305. Part II. Genetic Algorithms. Genetic Algorithms

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

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

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

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

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

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 8 11100110 f8 = 5 12 00101110 f12 = 4 13 11011110 f13 = 6 14 01111110 f14 = 6 f Average fitness f = (5+4+6+6)/4 = 5.25 The average fitness in the population of possible solutions increases with every generation. 0 1 2 3 Generation № Generation 3 5.0 3.0 1.0 Genetic Algorithms

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 1 00000110 f1 = 2 N1 = 1 2 11101110 f2 = 6 N2 = 2 3 00100000 f3 = 1 N3 = 0 n = 4 00110010 f4 = 3 N4 = 1 Generation 0 Genetic Algorithms

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 8 11100110 f8 = 5 N8 = 1 9 11001110 f9 = 5 N9 = 1 10 00100110 f10 = 3 N10 = 0 11 00111110 f11 = 5 N11 = 2 Generation 2 Genetic Algorithms

Similarities in GAs chromosomes. Similarities among highly fit strings guide the search. Chromosome Chromosome Fitness index string 8 11100110 f8 = 5 12 11011110 f12 = 6 13 00101110 f13 = 4 14 01111110 f14 = 6 Generation 3 Genetic Algorithms

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 8 11100110 f8 = 5 12 11011110 f12 = 6 13 00101110 f13 = 4 14 01111110 f14 = 6 Generation 3 Genetic Algorithms

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

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

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: 111**0 is a schema of the chromosome 111100 Genetic Algorithms

Similarity Template = Schema. Example: 111**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

Similarity Template = Schema. As an example consider strings and schemata of length 5. Example 1: The schema *0000 matches two chromosomes: 1) 00000 and 2) 10000 Example 2: The schema *000* matches four chromosomes: 1) 00000 , 2) 10000 , 3) 00001 , and 4) 10001 Genetic Algorithms

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

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

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 11 has 22=4 building blocks 1) *1 , 2) 1* , 3) ** , and 4) 11 Genetic Algorithms

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 101 has 23=8 building blocks 1) 10* , 2) 1** , 3) *** , 4) 1*1 , 5) *01 , 6) **1 , 7) *0* , and 8) 101 Genetic Algorithms

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

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

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

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