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921 Representing Hypotheses
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Hypotheses in GAS are often represented by bit strings, so that they can be easily manipulated by genetic operators such as mutation and crossover The hypotheses represented by these bit strings can be quite complex For example, sets of if-then rules can easily be represented in this way, by choosing an encoding of rules that allocates specific substrings for each rule precondition and postcondition Examples of such rule representations in GA systems are described by Holland (1986); Grefenstette (1988); and DeJong et al (1993) To see how if-then rules can be encoded by bit strings, first consider how we might use a bit string to describe a constraint on the value of a single attribute To pick an example, consider the attribute Outlook, which can take on any of the three values Sunny, Overcast, or Rain One obvious way to represent a constraint on Outlook is to use a bit string of length three, in which each bit position corresponds to one of its three possible values Placing a 1 in some position indicates that the attribute is allowed to take on the corresponding value For example, the string 010 represents the constraint that Outlook must take on the second of these values, or Outlook = Overcast Similarly, the string 011 represents the more general constraint that allows two possible values, or (Outlook = Overcast v Rain) Note 111 represents the most general possible constraint, indicating that we don't care which of its possible values the attribute takes on Given this method for representing constraints on a single attribute, conjunctions of constraints on multiple attributes can easily be represented by concatenating the corresponding bit strings For example, consider a second attribute, Wind, that can take on the value Strong or Weak A rule precondition such as
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(Outlook = Overcast
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Rain) A (Wind = Strong)
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can then be represented by the following bit string of length five:
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Outlook 01 1 Wind 10
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Rule postconditions (such as PlayTennis = yes) can be represented in a similar fashion Thus, an entire rule can be described by concatenating the bit strings describing the rule preconditions, together with the bit string describing the rule postcondition For example, the rule
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IF Wind = Strong
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would be represented by the string
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Outlook 111
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THEN PlayTennis = yes
Wind 10
PlayTennis 10
where the first three bits describe the "don't care" constraint on Outlook, the next two bits describe the constraint on Wind, and the final two bits describe the rule postcondition (here we assume PlayTennis can take on the values Yes or No) Note the bit string representing the rule contains a substring for each attribute in the hypothesis space, even if that attribute is not constrained by the rule preconditions This yields a fixed length bit-string representation for rules, in which substrings at specific locations describe constraints on specific attributes Given this representation for single rules, we can represent sets of rules by similarly concatenating the bit string representations of the individual rules In designing a bit string encoding for some hypothesis space, it is useful to arrange for every syntactically legal bit string to represent a well-defined hypothesis To illustrate, note in the rule encoding in the above paragraph the bit string 111 10 11 represents a rule whose postcondition does not constrain the target attribute PlayTennis If we wish to avoid considering this hypothesis, we may employ a different encoding (eg, allocate just one bit to the PlayTennis postcondition to indicate whether the value is Yes or No), alter the genetic operators so that they explicitly avoid constructing such bit strings, or simply assign a very low fitness to such bit strings In some GAS, hypotheses are represented by symbolic descriptions rather than bit strings For example, in Section 95 we discuss a genetic algorithm that encodes hypotheses as computer programs
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