Bootstrapping CAIA Part III

 For the realization of a General Problem Solving system, a number of meta-problems arise. The main idea of bootstrapping is that these meta-problems will be solved by the system itself, in the same way as it solves the problems for which it was designed. CAIA is quite an elaborate system; however, it is not yet able to solve every type of problem, and particularly many kinds of meta-problems. I gradually increase its capacities so that it can solve a wider range of problems. For instance, a useful feature for solving more mathematical problems has been the introduction of unknowns with an infinite set of possible values.

It turns out that two families of meta-problems can be solved by CAIA, using its current formalism:

To find the symmetries from the study of the formulation of a problem. We have already considered several times why this capacity is interesting.

To add new elements to a particular family of problems. A researcher has to test his/its system with problems at various levels of difficulty. Unfortunately, there are often too few of them, or they are not difficult enough: as they have an entertaining goal, very complex problems, necessary for checking the system, may be absent. Therefore, I have associated to several families of problems, the definition of a meta-problem that creates new problems in this family.

Unfortunately, many meta-problems have a definition completely different from the problems currently solved by CAIA, and also by most of the general systems. Some of the meta-problems encountered when one wants to solve a problem are:

How to cope with a new result: keep it or eliminate it?

Is it better to backtrack immediately, or to wait for a little, hoping to find a useful result?

If one decides to backtrack, which set of choices will be considered?

How to select the next rule and the elements to which it applies?

What is the probability that a particular step will succeed?

What would be the interest of the result of a derivation if it succeeds?

In order to succeed a bootstrap, these meta-problems must also be solved by CAIA. For the present time, it solves them by using methods that I have found, and it can neither create, nor modify them. Therefore, the next step is to give it the capacity to solve meta-problems such those I have just mentioned.

Fortunately, one can use methods similar to those already successful for more usual problems. For instance, one can consider two levels of backtrack. At the lowest level, one successively examines the situation after adding one constraint from the set that contains the various possibilities. At the highest level, one successively examines the sets of constraints that are known at this step of the resolution: a meta-backtrackis a backtrack on the backtracks that can be considered.

Looking again at the Triplets problem with N=12, one quickly finds two possibly interesting backtracks. They are defined by the set of constraints that one has to consider successively:

On the one hand: 12Q+12C=24A or 23A or 22A….or A, 24 choices in all.

On the other hand: (B even and C even and R even, and P even) or (B even and C odd and R odd and P even) or…(B odd and C odd and R odd and P odd), 11 choices in all (5 of the 16 possible choices have been eliminated).

Choosing the best way to backtrack is a meta-problem, which can be solved by opting for the first one (a linear equality is better than a parity constraint) or for the second one ( only 11 branches are better than 24, one unknown is also better than three; moreover, at each step, one adds four new constraints rather than only one). However, it is also possible to meta-backtrack: one considers both, then one focuses its efforts on the one whose results are more promising (in this problem, the equality constraints). This allows to solve the problem successfully, even when the evaluation of these backtracks is unsatisfactory.

Many meta-problems may be solved by a set of judgements, whose synthesis will give a value allowing to rate and rank the candidates. This is often sufficient when some uncertainty is allowed because there are not too many possibilities to consider, and computers are very fast. When the decision is very important such as choosing a particular backtrack, these judgements are sometimes too approximate. Either one tries to improve the quality of these judgements, or one meta-backtracks.

Naturally, there remains another meta-problem: to find the judgements that one has to use for solving each kind of meta-problem. For the preceding example, the basic elements for these judgements were general: it is better that there are not too many branches in a backtrack, it is better to add each time several constraints rather than only one, an equality constraint is more selective than a parity constraint, adding a constraint with one unknown is better than adding one with several unknowns. They allow a pre-selection which will be improved by the meta-backtrack. In this particular case, one is not very far from the end of the ascent of the meta-levels necessary for the successful completion of the bootstrap. Many other kinds of meta-problems have to be solved, but bootstrapping AI is perhaps not as hard as it sounded.

 

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