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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 10^{8}, from a fraction of an inch to thousands of miles,
and durations spanning a ratio of about 10^{10}, 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 10^{57}. 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.

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