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As an example, consider evolving the following program:
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y := x * ln(a) + sin(z) / exp(-x) - 3.4;
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The terminal set is specified as {a, x, z, 3.4} with a, x, z 6 K. The minimal function set is given as { ,+,*,/, sin, exp, In}. The optimum solution is illustrated in Figure 10.1. Terminal elements are placed within circles, while function elements are in the square boxes.
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Figure 10.1: Genetic program representation Each individual in a GP population represents a program, which is an element of the program space formed by all possible programs that can result from the given grammar. The aim of the GP is then to search for a program within the program space that give the best approximation to the objective (the true) program. Trees within a population can be of a fixed size or variable size. With a fixed size representation all trees have the same depth and all subtrees are expanded to the maximum depth. Variable size tree representations are, however, the most frequently used representation. In this case the only restriction placed on trees is a maximum depth. It is also possible to employ schemes where the maximum depth increases with increasing generation number.
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Initial Population
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The initial population is generated randomly within the restrictions of a maximum depth and semantics as expressed by the given grammar. For each individual, a root is randomly selected from the set of function elements. The branching factor (the number of children) of the root, and each other non-terminal node, are determined by the arity of the selected function. For each non-root node, the initialization algorithm randomly selects an element either from the terminal set or the function set. As soon as an element from the terminal set is selected, the corresponding node becomes a leaf node and is no longer considered for expansion.
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Fitness Function
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The fitness function used for GP is problem-dependent. Fitness evaluation involves testing each problem on the target domain. This usually requires each individual to be tested on a sample of cases, and the average performance over that sample is used as fitness measure. As an example, return to the program in Figure 10.1. Assume no prior knowledge about the structure of the program, other than the given terminal and function sets. In addition, a data set is available, consisting of a number of data patterns, where each data pattern consists of three input values (one for each of the variables a, x and z) and a target value (i.e. the value of y). The evaluation of each individual entails (1) calculating the output of that individual given the values of a, x and 2, and (2) calculating the error made. At the end, the MSB over the given data set is a valid quantification of the fitness of that individual. In the case where decision trees are evolved, and each individual represents a single decision tree, the fitness of individuals is calculated as the classification accuracy of the corresponding decision tree. If the objective is to evolve a game strategy in terms of a computer program, the fitness of an individual can be the number of times that individual won the game out of a total number of games played. In addition to being used as a measure of the performance of individuals, the fitness function can also be used to penalize individuals with undesirable structural properties. For example, instead of having a predetermined depth limit, the depth of a tree can be penalized by adding an appropriate penalty term to the fitness function. Similarly, bushy trees (which result when nodes have a large branching factor) can be penalized by adding a penalty term to the fitness function.
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