g.  Do a Web search until you have collected 100 pictures of men on horseback, using state-of-the-art image recognition software.

h. Using state-of-the-art theorem proving software, find a proof that the medians of a triangle are concurrent.

i. Using state-of-the-art theorem proving software, find a proof of Fermat's little theorem.
It is plausible to suppose that, in many of these cases, you can say reliably that one task will take much longer than another, either by a human judgment or using an expert system. For instance, task (a) is much shorter than any of the others. Task (b) is much shorter than any of the others except (a) and possibly (h).  Task (c) is certainly much shorter than (d), (f), (g), or (i). However, with certain pairs such as (c) and (h) or (c) and
(e) it would be difficult to guess whether one is much shorter than another, or whether they are of comparable difficulty.

You have a number of independent identical computers, of unknown vintage and characteristics, on which you will schedule tasks of these kinds. Note that, under these circumstances, there is no way to predict the absolute time required by any of these tasks within a couple of orders of magnitude. Nonetheless, the comparative lengths presumably still stand.

Given: a particular schedule of tasks on machines, infer what you can about the relative order of completion times. For example, given the following schedule

## Machine M1: Tasks A,B,H,D. Machine M2: Tasks C,I.

it should be possible to predict that (a) and (b) will complete before (c); that (c) will complete before (d); and that (d) will complete before (i); but it will not be possible to predict the order in which (c) and (h) will complete.

In all three examples, the given information has the form "The distance between points W and X is mu less than the distance between Y and Z ". In examples 1 and 2, the points are geometric. In example 3, the points are the start and completion times of the various tasks, and the constraints on relative lengths can be put in the form "The distance from start(a) to end(a) is mu less than the distance from start(c) to end(c)", and so on. In example 3, there is also ordering information: the start of each task precedes its end; the end of (a) is equal to the start of (b); and so on. The problem is to make inferences based on this weak kind of constraint.

It should be noted that these examples are meant to be illustrative, rather than serious applications. Example 1 does not extend in any obvious way to a class of natural, large problems. Example 2 is implausible as a state of knowledge; how does the reasoner find himself knowing just the order-of-magnitude relations among distances and no other geometric information? Example 3 is contrived. Nonetheless, these illustrate the kinds of situations where order-of-magnitude relations on distance do arise; where they express a substantial part of the knowledge of the reasoner; and where inferences based purely on the order-of-magnitude comparisons can yield useful conclusions.

The methods presented in this paper involve construing the relation "Distance D is much shorter than distance E" as if it were "Distance D is infinitesimal as compared to distance E." As we shall see, under this interpretation, systems of constraints over distances can be