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Statistics is the branch of mathematics that is concerned with possibilities. Statistical methods can assign a value to how likely something is. For example, given a half-dozen distance sensors around your robot, you should be able to calculate the probability that you are in a corner, or between two walls. Bayesian analysis is particularly relevant to the pattern-matching problem. In daily life, you are most likely to find Bayesian techniques in your e-mail SPAM filters. Statistics, however, is a complex and arcane field that we cannot do justice to here.
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CHAPTER 17 Intelligent Behavior FUZZY LOGIC
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Fuzzy logic is the insecure cousin of Boolean logic. Boolean thinking is crisp and precise, definite. The temperature is hot or the temperature is cold. Fuzzy logic allows the temperature to be mostly hot but still a little bit cold. Boolean logic can be seen as a graph like Fig. 17-7. The moment the temperature passes the 80 degree mark it is suddenly 100% hot. Pow! Figure 17-8 shows the same thing from a fuzzy perspective. The degree of hotness gradually increases as the temperature goes from 70 to 90 degrees. At 80 degrees, it is about half hot and half cold. Warm, if you will. If you are from any of the Northern states, substitute 70 (or even 60) degrees for our 80 degree Texas definition of warm. When you think in Boolean logic, everything switches 100%. If the temperature is Hot, turn on the fan. On or off. This works well for some environments and it is simple to implement. In fuzzy logic you think in a sliding scale. To the degree that the temperature is Hot, turn on the fan. If the temperature is only a little bit hot, say 75 degrees, the fan turns on low. If the temperature is 100 degrees the fan is spinning at full tilt. This varying output value, from not hot to fully hot, is based on the input s membership in the category hot. There are a number of membership functions, such as those shown in Fig. 17-9.
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Fig. 17-7.
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Boolean hot or cold.
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Fig. 17-8.
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Fuzzy hot or cold.
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Intelligent Behavior
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Fig. 17-9.
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Membership functions.
Fig. 17-10.
Fuzzy operators.
There are fuzzy versions of the Boolean operators, too. Instead of a truth table, we represent them as equations and pictures (Fig. 17-10). The min( ) function takes the lower of both values, while max( ) takes the larger. Fuzzy logic also has some modifiers that affect the shape of the membership function. If you want to know if something is very hot and not just hot, you could shape the Hot membership function using the Very modifier before you check the temperature against it. Some modifiers, also known as hedges, are shown in Fig. 17-11. These assume that the output value for 100% membership in a category is 1.00. The hedges take the membership function s points and raise them to a power. Powers greater than 1.0 tighten the function, making it harder to be a full
CHAPTER 17 Intelligent Behavior
Fig. 17-11.
Fuzzy hedges.
member of the category. Lower powers bloat the membership function, making it easier to qualify. Fuzzy logic is a simple form of pattern matching. It classifies its input into membership categories. These membership values can then be used to weight the actions associated with that category. If several categories are trying to drive the same output, their conflicting control values can be averaged together, using the priorities defined by their membership weights.
NEURAL NETWORKS
Computational neurons, and the collections of neurons known as neural networks, are pattern-matching modules based on biological neurons. Each neuron takes one or more inputs that it compares against its inner template. The output of the neuron is then active to the extent that the input matches that template. Figure 17-12 shows a diagram of a typical computational neuron, known as a McCulloch-Pitts neuron. Though a bit scary-looking, its operation is simple. The input is a vector, or list, of numbers called x, each entry of which is indexed as x(i). Internal to the neuron is a set of weights w, one for each input. The sigma operator is a loop that adds up the results from its equation, which is calculated once for every input index. If there are three inputs, the equivalent equation is: y w 1 x 1 w 2 x 2 w 3 x 3 17-6
y is the output value. This function is called a dot product, and it is one method of calculating how closely two vectors match. It also has further mathematical significance, not relevant to this discussion.
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