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Representation and Terminology
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Imagine that you have lost your car keys. You know that they are somewhere in your house, which looks like this:
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10: AI-Based Problem Solving
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You are standing at the front door (where the X is). As you begin your search, you check the living room. Then you go down the hall to the first bedroom, through the hall to the second bedroom, back to the hall, and to the master bedroom. Not having found your keys, you backtrack further by going back through the living room. Finally, you find your keys in the kitchen. This situation is easily represented by a graph, as shown in Figure 10-1. Representing search problems in graphical form is helpful because it provides a convenient way to depict the way a solution was found. With the preceding discussion in mind, consider the following terms, which will be used throughout this chapter:
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Node Terminal node Search space Goal Heuristics Solution path A discrete point A node that ends a path The set of all nodes The node that is the object of the search Information about whether any specific node is a better next choice than another A directed graph of the nodes visited en route to the goal
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In the example of the lost keys, each room in the house is a node; the entire house is the search space; the goal, as it turns out, is the kitchen; and the solution path is shown in Figure 10-1. The bedrooms, kitchen, and the bath are terminal nodes because they lead nowhere. Heuristics are not represented on a graph. Rather, they are techniques that you might employ to help you better choose a path.
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Figure 10-1
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The solution path to find the missing keys
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Combinatorial Explosions
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Given the preceding example, you may think that searching for a solution is easy you start at the beginning and work your way to the conclusion. In the extremely simple case of the lost keys, this is not a bad approach because the search space is so small. But for many problems (especially those for which you would want to use a computer) the number of nodes in the search space is very large, and as the search space grows, so does the number of possible paths to the goal. The trouble is that often, adding another node to the search space adds more than one path. That is, the number of potential pathways to the goal can increase in a nonlinear fashion as the size of the search space grows. In a nonlinear situation, the number of possible paths can quickly become very large. For instance, consider the number of ways three objects A, B, and C can be arranged on a table. The six possible permutations are
A A B B C C B C C A B A C B A C A B
You can quickly prove to yourself that these six are the only ways that A, B, and C can be arranged. However, you can derive the same number by using a theorem from the branch of mathematics called combinatorics the study of the way things can be combined. According to the theorem, the number of ways that N objects can be arranged is equal to N! (N factorial). The factorial of a number is the product of all whole numbers equal to or less than itself down to 1. Therefore, 3! is 3 2 1, or 6. If you had four objects to arrange, there would be 4!, or 24, permutations. With five objects, the number is 120, and with six it is 720. With 1000 objects the number of possible permutations is huge! The graph in Figure 10-2 gives you a visual feel for what is sometimes referred to as a combinatoric explosion. Once there are more than a handful of possibilities, it very quickly becomes difficult to examine (indeed, even to enumerate) all the arrangements. This same sort of combinatorial explosion can occur in paths through search spaces. Because of this, only the simplest of problems lend themselves to exhaustive searches. An exhaustive search is one that examines all nodes. Thus, it is a brute-force technique. Brute force always works but is not often practical for large problems because it consumes too much time, too many computing resources, or both. For this reason, AI-based search techniques were developed.
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