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Douglas Hofstadter on Problem Solving

GodelWhen faced with a complex problem, we often set about solving it by breaking it down into smaller pieces, then solving each piece in turn. But as Douglas Hofstadter notes in Godel, Escher, Bach, a problem can often be decomposed in more than one way. Choose the wrong way and we may find ourselves unable to solve the problem at all:

“There is no guarantee that the method of problem reduction will work. There are many situations where it flops. Consider this simple problem, for instance. You are a dog, and a human friend has just thrown your favourite bone over a wire fence into another yard. You can see your bone through the fence, just lying there in the grass - how luscious! There is an open gate in the fence about fifty feet away from the bone. What do you do? Some dogs will just run up to the fence, stand next to it, and bark; others will dash up to the open gate and double back to the lovely bone. Both dogs can be said to be exercising the problem reduction technique; however, they represent the problem in their minds in different ways, and this makes all the difference. The barking dog sees the subproblems as (1) running to the fence, (2) getting through it, and (3) running to the bone - but that second subproblem is a "toughie", whence the barking. The other dog sees the subproblems as (1) getting to the gate; (2) going through the gate; (3) running to the bone. Notice how everything depends on the way you represent the "problem space" - that is, on what you perceive as reducing the problem (forward motion towards the overall goal) and what you perceive as magnifying the problem (backward motion away from the goal).”

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