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. Speci cally, 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 in nite di erence in scale or as referring to a nite di erence in scale, as long as the di erence in scale is greater than the number of variables in the constraint set. We also prove that the rst-order theory over such constraints is decidable.