Order of Magnitude Comparisons of Distance
  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 approximate. 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 decidable.
