Monday, September 10, 2007

The 91 problem

Bongard Problem 91 is a kind of boundary problem. You have to discover the criteria that separate boxes by one line. In this case, the line marked with, well, 91. This example was used in last class to show the phenomena of inconsistency and it called special attention to me. Let me tell you why.

If you still haven't got the solution, here it is: 3 on the left and 4 on the right. Does it make any sense? Do you see the solution? Good. It means that you have already solved the puzzle of "inconsistency": On the left, three squares, on the right, four edges of a square, not "one square". But, is this really an inconsistency? Perhaps not, perhaps these are just squares treated in different ways, and this is a kind of ambiguity. Squares can be represented in different ways, not in any way of course, but in some ways. The game is then formulated as an optimization problem that takes into account n different and possible representations of each box and "force fits" a separation between them.

Force-fitting? Yup. Imagine that we are anxious to win the game as fast as we can. Our nature tells us to get the most relaxation constraints. To reach the goal, you only have to ignore some details. Yep, those details that anyone will ever note--let's try and see if it is acceptable by others.

Another important question has to be responded in order to finish this thought. Would a "Neural Network" solve that problem? The answer is yes, fairly simple. We have 12 examples (boxes) for training and at least 2400 pixels for each example. So NNet will find that one pixel no one is seeing, but it is there, and draw a line to separate the samples. It might work, but this type of criteria makes no sense to us. Did you remember from who we have to gain acceptance? (To me that brings to mind the Lost's series. Jack and Kate in the future... crazy, don't you agree?)