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void draw(){ } int gen = 0; //marks the number of generations void keyPressed(){ for(int x=5; x<width-5; x++) //go through all pixels for(int y=5; y<height-5; y++){ int k=0; for(int i=-1; i<=1; i++) //go through all 8-neighbors for(int j=-1; j<=1; j++){ if(i==0 && j==0)continue; //skip if the same one color c = get(x+i,y+j); //if the neighbor is black if(red(c)==0.)k++; //add it to the counter } if(k==3) //if all black neighbors are exactly 3 memory[x][y]=0; //become black else if(k>=5) //if all black neighbors are greater or equal to 5 memory[x][y]=255; //become white } for(int x=0; x<width; x++) for(int y=0; y<height; y++) set(x,y,color(memory[x][y])); //copy memory[] to image println(gen++); }
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We first define an image called MyImage, where we will draw the cellular automata (CA). Then we define an array called memory to hold temporary information about each CA. In the setup() section, we first create the image and then randomly color each pixel either black or white (lines 6 12). In the keyPressed() section, we loop through all pixels of the image MyImage, and for each pixel we loop in its 8-neighborhood (lines 24 26) and count the number of black pixels (using the counter k). Then we apply the following rules: if the number of black neighbors is exactly 3, then we set the memory[][] array (which functions as a copy of the image) to 0 (i.e., black); otherwise, if the number of black neighbors is greater or equal to 5, then we set the memory[] [] array to 255 (i.e., white). When done, we copy the memory[][] elements to the image and draw it. We repeat the process for every mousePressed, which corresponds to the generations. The result of this process is shown in Figures 7-11 and 7-12.
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Figure 7-11: Cellular automata progression on 1 till 50 generations
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Figure 7-12: Thirty generations later on a 400 400 pixelmap
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7.6 Evolutionary Algorithm
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An evolutionary (a.k.a genetic) algorithm is a search technique for optimizing or solving a problem. Its mechanism is based on evolutionary biology, using terms and processes such as genomes, chromosomes, crossover, mutation, or selection. The evolution starts from a population of completely random individuals and happens in generations. In each generation, the fitness of the whole population is evaluated, multiple individuals are stochastically selected from the current population (based on their fitness), and modified (mutated or recombined) to form a new population, which becomes current in the next iteration of the algorithm. For example, let s say that a secret number is to be guessed: 001010. First, we try out some initial guesses (just generated randomly). These guesses are known as genomes. a. 110100 b. 111101 c. 011011 d. 101100 We then evaluate how good the guesses are (this is called their fitness score). The scores for each of the guesses are given below and represent the number of digits that are exactly correct. a. 110100 score 1 b. 111101 score 1 c. 011011 score 4 d. 101100 score 3 We then select the best and combine them with other good solutions. This is known as reproduction. We shall combine solutions c and d. We may split the solutions (cross them over) any way we like. Below we use four digits of one solution and two of another. This creates the four new solutions shown here: Crossover: take first 2 digits of c + last 4 of d: e 011100 Crossover: take first 2 digits of d + last 4 of c: f 101011 Crossover: take first 4 c + last 2 d: g 011000 Crossover: take first 4 d + last 2 c: h 101111 Now we get the fitness of each of the new solutions. Their scores follow.
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Remember this is just the number of digits in exactly the right place. e. 011100 score 3 f. 101011 score 4 g. 011000 score 4 h. 101111 score 3 We then select the best of these new solutions for reproduction. We shall choose g + f. This time, we cross them over by taking five digits of one number and one of the other. Crossover 1st f + last 5 g: i 111000 Crossover 1st g + last 5 f: j 001011 Crossover first 3 f + last 3 g: k 101000 Crossover first 3 g + last 3 f: l 011011 Random change of one of k s digits: m 001010 Now we determine the fitness for each of these new solutions. These scores follow and are based on the number of digits that are exactly right. Note that we must get all six digits right. We are, therefore, hoping for a score of 6. Luckily solution m gets a score of 6 and is the correct answer; it matches our secret number. We have managed to find the answer in 13 guesses. This is better than complete random guesses (the average chance of success of which is about 64). i. 111000 score 3 j. 001011 score 5 k. 101000 score 4 l. 011011 score 4 m. 001010 score 6 = solution in 13 guesses What we have done is to start off with some guesses of a hidden number (a string of six 1s and 0s), selected the fittest (closest to the number), and crossed them over (mixed them with each other) to produce (reproduce) some new solutions. We kept doing this until we got an ideal solution (all six digits correct). While genetic algorithms appear to solve a predefined problem (i.e., guess the specific string 001010), they can also be used to address a problem whose solution is not known in advance but can be described through attributes or other indirect characteristics. In that sense, the operation of matching can be applied
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