Too often, IA researchers describe the successes of their system, but they keep a lower profile when they get poor outcomes. I do not want to comply to this bad habit. I describe here CAIA’s behavior when success does not come, in the present case, the magic square problem. An order N Magic Square has N horizontal lines, and N vertical lines. Therefore, there are N2 squares, and each one contains a number between 1 and N2, so that each number appears only once. Moreover, the sum of the verticals, the horizontals, and the two diagonals, must have the same unknown value, SUM.
First, CAIA finds symmetries, which are permutations of the unknowns which lead to an identical set of constraints. Seven symmetries are evident for all magic squares: three rotations and four reflections with respect to horizontal, vertical, and both diagonals. CAIA knows nothing about geometry, but it can find all symmetries, both geometric and non-geometric. To do that, it uses the general component that solves constraint satisfaction problems. I had only to specify one particular problem: finding all the permutations of the unknowns leading to the initial set of constraints.
I hoped that it would find the seven geometrical symmetries. However, CAIA was far beyond any of my expectations: it found a huge number of symmetries that I never imagined existed. Even for N=4, it finds 31 symmetries! Some are completely unexpected, for instance, one of them transforms the first horizontal into the second vertical: both contain the same values, naturally not in the same order.
Finding this is not evident: almost 400 years ago, Bernard Frénicle de Bessy performed an outstanding study of the N=4 magic square. In particular, he defined a standard representation of the solutions, which is always used. However, he has found 880 basic solutions; multiplied by 8, this gives a total of 7040 solutions. CAIA found only 220 basic solutions; multiplied by 32, we find again 7040. When N grows, the number of symmetries increases at a startling rate: for N=10, CAIA finds 15,359 symmetries! On this point, CAIA’s behavior is flawless, although a lot of computer time is needed: 31,193 seconds for all the symmetries when N=10.
In the following step, CAIA tries to find other constraints. For the magic problem, it adds that the sum of the unknowns is equal to the sum of the first N2 integers, since there is a bijection. After that, it can find the value of SUM. For us, humans, this is easy because we are seeing the square: it is clear that the sum of the N horizontal lines is N*SUM. It is also evident that this sum contains all the squares, so its value is the sum of all the numbers between 1 et N2, that is N2*(N2+1)/2. Therefore, we have SUM=N*(N2+1)/2.
I did not give CAIA this formula, it has to find the value of SUM from the set of constraints. For this task, it is rather good: for N=16, it finds SUM=2056, after many substitutions in the 34 initial constraints, plus the bijection constraint. It would have been better to prove the general formula, but it is difficult for a blind system, which cannot see the square as we do. It does not find other constraints, and I do not believe that constraints with less than N unknowns exist for this problem. In total, for this task, CAIA has done well overall, although it could not set out the general formula.
The last part is to find the solutions, and this is the point where CAIA fails: it has been designed for finding all the solutions of a problem. For this reason, it is very interesting to find symmetries. However, it makes no sense to find all the solutions when there are billions of them. In its normal working, it considers the formulation of the problem, and particularly it finds new constraints, easier to use than the initial ones. In some cases, (as for N=3), it directly finds the solution. When there is no new constraint, it is often better to stop trying to be clever, but to write cleverly a combinatorial program that generates all the solutions.
For a magic square, this works well for the small values of N: for N=8, it generates the first 1,000 solutions in 4 seconds. Unfortunately, the search space dramatically increases with N: the number of unknowns grows as N2, and the number of possible values for each unknown also grows as N2. Very soon CAIA is completely overcome. The method still works for N=9: 10 seconds for the first 1,000 solutions. Unfortunately, there is a sharp discontinuity for N=10: after 14 hours, it has not yet found a solution.
The reason for this failure is that it is unthinkable that, given the huge number of solutions, one can ever find all of them when N is not small. It is sufficient to find one of them. We could think that all we have to do is to launch the search for solutions, and stop when one is found. In reality, it is often better to use other methods for scanning the search space, pinpointing the areas where the probability of finding a solution is high. For instance, one can find solutions for the set of equations when the unknowns may have continuous values. Then, one can look for the combinations of integer values close to the values of these solutions.
Another heuristic may be used, such as to make the initial problem more constrained, adding new constraints so that it would be easier to find solutions for this new problem. If it has solutions, they would also be solutions of the initial problem. When there are zillions of solutions, we can add constraints that remove some of them. In a future post, we will see how this could be very efficient for finding solutions for some magic square problems.
To sum up, although a problem may have many solutions, CAIA experiences serious difficulties when all its constrains has many unknowns, each unknown having many possible values; this is normal, the search space is huge. In this situation, it can, nonetheless, have excellent results when it can generate new constraints with few unknowns; this happened with the magic cube, where it found constraints with only three unknowns. If this is impossible, it must use completely different methods, good for finding a needle in a huge haystack. CAIA improves the combinatorial search with a meta-combinatorial search generating new constraints; this is sometimes not enough for solving this kind of problem, some modules must be added to CAIA.