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Setting I: The goal is at G1,the signs at choice point 1 are |
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and so on (i.e., the shortest path is indicated by dark symbols on a light background). |
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Setting II: The goal is at G3, the signs at choice point 1 are |
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and at choice point 2 they are the same as in Setting I except for |
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Fig. 7.
Some searches using the devices of figure 6 in the settings of figure 5 |
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If the shortest path to the goal were always indicated as in Setting I, i.e., with dark symbols on a light background, then the function f'(S) = d3(S) (i.e., w3 = 1, w1 = w2 = w4 = 0) would always suffice for following the path. Notice, however, that in Setting II f' assigns exactly the same set of values (0,1,0) at point 1, indicating that f' does not distinguish the two settings. But, in Setting I f' assigns (1,0,0) at point 2, while in Setting II f' assigns (0,0,0) at point 2. Thus, starting from the same initial state (0,1,0) and invoking the same response hy, f' arrives at two different states. Changing the weight assigned to d3 cannot correct the difficulty. This is a clear indication that the set of detectors (d3 in this case) is inadequate. |
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A quick check of the possibilities shows that consistently correct choices in the two settings can be achieved only by assigning a nonzero weight to d4, which is a nonlinear combination of d1 and d3. The function f''(S) = d1 + d3 - 2d4 then performs correctly in both settings and, in fact, performs consistently with any proper sequence of signs. |
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