Illustrated below is a "traffic light" ...a vertical arrangement of a red, an amber, and a green light each of which can be either on or off. We will use the idea of traffic lights at an intersection to illustrate the relation between the concepts of possibilities, dependencies, and constraints. In addition to the basic traffic light, the illustration below lists two sets of possibilities. One is termed the logical possibilities (or in this case you could think of it as the physically logical possibilities). There are 8 logical possibilities ranging from the case where all three light are on at the same time to the case where all three lights are off at the same time. Note that each object - one of the lights - can be in one of two states - on or off. Since there are three lights, the number of logical possibilities is the number of states raised to the power of the number of objects - in this case 2 to the 3.
In the darker gray box is the set of what I have termed 'designed possibilities.' This is the set of possibilities that any well-behaved traffic light will exhibit. And there are only three such possibilities, the cases where one of the three lights is on and the remaining are off. Finally, the pointers to the circles is meant to convey the obvious, but important fact that the designed possibilities are a subset of the logical possibilities.
Keeping these ideas in mind, pretend that for some reason you can't see a traffic light, but your companion traveling with you can. Further pretend that when you ask your companion to tell you the state of the traffic light, your companion says that it is red. Notice that if you assume that this is a well-behaved traffic light, then you know all you need to know because the only designed possibility in which the red light is on is one where the green and yellow lights are off. You can "infer" this because this dependency between the on\off values of the three lights will always hold for well-behaved traffic lights. Whenever, the value of one thing depends on the value of another we say that there is a dependency between these two things. Notice that in some cases the dependency may be such that knowing one value uniquely determines another value. That is the case here...knowing that the red light is on uniquely determines the value of the remaining lights. However, knowing that the yellow light is off doesn't allow us to infer the value of the green light or the value of the red light. It does, however, let us infer that one of these lights is on and one is off.
Anytime, there is a dependency among the values that different entities can take, we can potentially take advantage of these dependencies to reason about the objects. Working cross word puzzles is one long exercise in attempting to use dependencies to reason to a unique set of values for each square in the puzzle.
|Now let us gain some experience in explicitly thinking about possibilities and dependencies. Consider the standard traffic intersection with a light at each of the four corners. Work out the number of logically possible values that this four light configuration can take on. Next work out the 'designed' possibilities and write out the inferences that can be derived when these dependencies hold. Finally, work out a system that functions in the same manner but now uses only two lights ... a red and a green light on each "traffic light." Finally, decide whether you could reduce the number of lights use to just one...say a green light on each "traffic light."|
|© Charles F. Schmidt|