Figure 11.1: Finite-state machine in Java

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Figure 11.1: Finite-state machine
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Present state Input symbol Next state Output symbol
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C 0 B ft
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B 1 C a
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C 1 A 7
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A 1 A
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B 0 C 7
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Table 11.1: Response of finite-state machine
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Representation
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Each state can be represented by a 6-bit string. The first bit represents the activation of the corresponding state (0 indicates not active, and 1 indicates active). The second
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CHAPTER 11. EVOLUTIONARY
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PROGRAMMING
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bit represents the input symbol, the next two bits represent the next state, and the last two bits represent the output symbol. Each individual therefore consists of 18 bits. The initial population is randomly generated, with the restriction that the output symbol and next state bits represent only valid values.
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Fitness Evaluation The fitness of each individual is measured as the individual's ability to correctly predict the next output symbol. A sequence of symbols is used for this purpose. The first symbol from the sequence is presented to each individual, and the predicted symbol compared to the next symbol in the sequence. The second symbol is then presented as input, and the process iterates over the entire sequence. The individual with the most correct predictions is considered the most fit individual.
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Mutation The following mutation operations can be applied: The initial state can be changed. A state can be deleted. A state can be added. A state transition can be changed. An output symbol for a given state and input symbol can be changed. These operators are applied probabilistically, in one of the following ways: Select a uniform random number between 1 and 5. The corresponding mutation operator is then applied with probability pm. Generate a Poisson number, with mean A. Select mutation operators uniformly from the set of operators, and apply them in sequence.
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Function Optimization
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The next example application of EP is in function optimization. Consider, for example, finding the minimum of the function sin (2x)e~x in the range [0,2].
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11.3. EVOLUTIONARY PROGRAMMING EXAMPLES
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Representation
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The function has one parameter. Each individual is therefore represented by a vector consisting of one floating-point element (not binary encoded). The initial population is generated randomly, with each individual's parameter Cnx selected such that Cnx ~ Z7(0,2).
Fitness Evaluation
In the case of minimization, the fittest individual is the one with the smallest value for the function being optimized; that is, the individual with the smallest value for the function s i n ( 2 x ) e x . For maximization, it is the largest value. Alternatively, if each individual represents a vector of values for which a minimum has to be found, for example, a weight vector of a NN, then the MSE over a data set can be used as fitness measure.
Mutation
Mutation consists of adding a Gaussian random value to each element of an individual. For this example, each individual Cn is mutated using
where
Cnx ~ N(0, a2nx). The following choices for the variance a2nx
Use a static, small value for a2nx. Let o2nx be large, initially, and decrease the variance with increase in the number of generations. This approach allows an exploration of a large part of the search space early in evolution, while ensuring small variations when an optimum is approached. Small variations near the optimum are necessary to prevent individuals from jumping over the optimum point. anx ~ FEp(Cnx)i that is, the variance is equal to the error of the parent. The larger the error is, the more the variation in the offspring, with a consequent large move of the offspring from the weak parent. On the other hand, the smaller the parent's error, the less the offspring should be moved away from the parent. This is a simple approach to implementing a self-adaptive mutation step size. An alternative self-adaptive mutation step size is having a time-varying step size. To illustrate this, it is best to move to a general case where an individual
CHAPTER 11. EVOLUTIONARY
PROGRAMMING
Cn consists of / genes (parameters). Let ACni be the mutation step size for the i-th gene of individual Cn. Then, the lognormal self-adaptation of AC n i is