Conclusions

The applications of the specific algorithms above are undoubtedly limited; we are not aware of any practical problems where solving systems of order-of-magnitude relations on distances is the central problem. However, the potential applications of order-of-magnitude reasoning generally are very widespread. Ordinary commonsense reasoning involves distances spanning a ratio of about 10⁸, from a fraction of an inch to thousands of miles, and durations spanning a ratio of about 10¹⁰, from a fraction of a second to a human lifetime. Scientific reasoning spans much greater ranges. Explaining the dynamics of a star combines reasoning about nuclear reactions with reasoning about the star as a whole; these differ by a ratio of about 10⁵⁷. The techniques needed to compute with quantities of such vastly differing sizes are quite different from the techniques needed to compute with quantities all of similar sizes. This paper is a small step in the development and analysis of such computational techniques.

The above results are also significant in the encouragement that they give to the hope that order-of-magnitude reasoning specifically, and qualitative reasoning generally, may lead to useful quick reasoning strategies in a broader range of problems. It has been often found in AI that moving from greater to lesser precision in the mode of inference or type of knowledge does not lead to quick and dirty heuristic techniques, but rather to slow and dirty techniques. Nonmonotonic reasoning is the most notorious example of this, but it arises as well in many other types of automated reasoning, including qualitative spatial and physical reasoning. The algorithms developed in this paper are a welcome exception to this rule. We are currently studying algorithmic techniques for other order-of-magnitude problems, and are optimistic of finding similar favorable results.