Journal of Artificial Intelligence Research
pp. 1-38. Submitted 4/98; published 1/99.
© 1999 AI Access Foundation
and Morgan Kaufmann Publishers.
All rights reserved.
Order of Magnitude Comparisons of Distance
New York University
New York, NY 10012
Order of magnitude reasoning -- reasoning by rough comparisons
of the sizes of quantities -- is often called ``back of the envelope
calculation", with the implication that the calculations are quick though
This paper exhibits an interesting class of constraint sets in which
order of magnitude reasoning is demonstrably fast.
Specifically, we present a polynomial-time algorithm
that can solve a set of constraints of the
form ``Points a and b are much closer together than points c and d.''
We prove that this algorithm can be applied if
``much closer together'' is interpreted either as referring to an infinite
difference in scale or as referring to a finite
difference in scale, as long as the difference in scale
is greater than the number of variables in the constraint set.
We also prove that the first-order theory over such constraints is