Order of Magnitude Comparisons of Distance

TL;DR

This paper introduces a polynomial-time algorithm for efficiently solving a specific class of order of magnitude distance constraints, demonstrating that such reasoning can be computationally tractable.

Contribution

It presents the first polynomial-time algorithm for solving order of magnitude distance constraints and proves the decidability of the first-order theory over these constraints.

Findings

Algorithm solves constraints in polynomial time

Decidability of the first-order theory over these constraints

Applicable to both finite and infinite scale differences

Abstract

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.