Not many people have the ability to look at the few fundamental elements of a system (such as constructs of a new computer language) and extrapolate the long-term implications of their use. But rather than jump in and tackle the difficult question of understanding the expressive power and limits of computation, I’ll start by talking about a simpler problem. Don’t panic: it involves crayons and coloring!

This example is going to seem pretty easy on the surface, and you may be familiar with it already. It involves trying to find a way to color in maps so that no areas which share a border are going to be the same color. Here’s an example of a simple coloring with 5 crayons, and then finding a method for doing it with only 4 (taken from Wikipedia’s article on the subject):

Without prior knowledge, the average Joe can only guess at how many crayons he’d need to reliably fill in any map he is given. Just by trying a few cases, he might get an intuition that a small box of different colors will do the trick. But if Joe knew the Devil was going to be presenting him with an arbitrary map and getting his soul if he couldn’t color it properly, he’d probably buy Crayola’s biggest box…120 colors! (Plus, because he doesn’t know for sure that even THAT would be enough, he’d also probably try to round up the discontinued colors off of eBay!)

Yet there is salvation from this kind of paranoia. Early formalisms of the map-coloring-problem from mathematicians comforted us with proof that it *never* took more than 5 different colors…and that 3 will certainly not always be enough. After considerable debate and experimentation over the years, we ultimately found that we only need four unique colors for any map the Devil—or anyone else—makes.

(Note: Depending on your purposes, discovering it was 4 vs. 5 might be somewhat academic. I myself feel that the more crucial step was bringing in math to save us from needing to pack a potentially infinite number of crayons.)

Now…what if the “Devil’s Map” you must solve is instead an arbitrary selection from mankind’s constant creation of new ideas for software? That raises a lot of questions about what a programmer needs in their backpack to be confident they could correctly “color in” any program specification with a working implementation.

(more…)