*(Editor’s note: This is the fourth installment of a five-part series on building an artificial general intelligence).*

Artificial intelligence research has focused almost exclusively on one kind of problem, one that, in Judea Pearl’s words, can be characterized as curve fitting.

Playing chess or ‘go,’ solving the Towers of Hanoi problem or recognizing pictures of cats can all be solved by optimizing a set of parameters. These are problems that are easy to specify and evaluate. There is a clear winner and clear loser in chess or ‘go’.

It is easy to know that one has solved the Towers of Hanoi problem when the discs are all moved from the first to the third spindle. It is easy to evaluate the correctness of classifying pictures of cats. Even autonomous driving is relatively easy to evaluate by the number of miles driven and mishaps that occur.

Problems like these can be called “path problems.” The goal of the learning process is to find a path through the space of all possible parameters that arrives at a solution. That is what optimization does – it adjusts the parameter values such that the output of the model more closely approximates the desired objective.

A classic example of this lies within the hobbits and orcs problem.

Three hobbits and three orcs arrive at the side of the river where they find a small boat. All six want to cross the river, but if the orcs on either side of the river outnumber the hobbits, they will eat the hobbits. Otherwise, everyone is trustworthy. The boat will hold only two creatures at a time. How do they get across?

This is a well-structured problem. The rules and moves are clearly defined along with the goal. The parameters of the model correspond to the probability or ordering for each creature to cross the river. It is a well-studied problem in cognitive psychology and in machine learning.

There is a lot to recommend using such well-structured problems as the basis of research. If one has a dissertation to write, grant to propose or a project to get funded, it is important to be able to know that progress is being made. But, it is a fundamental mistake to think that these problems are representative of the kind of problems that an artificial general intelligence agent would have to be able to solve.

Another class of problems involves some level of “insight.” In order to solve the problem, the individual must change how he or she thinks about it. Once that change occurs, however, the solution of the problem is often easy and almost instantaneous. For example, the mutilated checkerboard problem is one that can be addressed as an “insight problem.”

A checkerboard has 64 squares arranged in an alternating black-red pattern. Imagine that you have a set of 32 dominoes, each of which covers exactly two squares of the checkerboard. But, what if you cut off the squares in the upper left corner and the lower right corner of the board. Now, there are 62 squares. Can you cover the board with 31 dominoes?

A computer could try to solve this problem by listing out all of the possible arrangements of the dominoes on the mutilated checkerboard. If all of the possible arrangements have been tried without finding a solution, it could then conclude that there is no solution. There are 6,728 arrangements of dominoes on a full checkerboard, so trying a few thousand layouts should be reasonable. If a designer were to set up the problem to find the solution this way, it would pre-suppose that the designer already knew the answer.

If we remove the squares at diagonally opposite corners, those two squares must be of the same color. Knowing this, it is obvious that the dominoes cannot cover the mutilated checkerboard because there are more red squares than black ones and every domino must cover one red and one black square.

An expert might recognize this issue as a “parity problem” and solve it with equally high speed. By representing this as a parity problem, the person can solve regular 8 x 8 checkerboards, or any other size. If relying on brute force, on the other hand, the computer would have to try 53,060,477,521,960,000 combinations of layouts before reaching a conclusion with just a 12 x 12 variant of the checkerboard. Even a fast computer might take some time to evaluate all of those positions.

Another insight problem is the Curt and Goldie problem. According to this, Curt and Goldie are found dead on the floor of a two-story frame house in a small town near a railroad track. The door is locked, and no one has been in the room. The area surrounding the bodies is wet and there are a few shards of glass and an old table nearby. How did they die?

The two-strings problem (Maier, 1931) is one of the first insight problems to receive much scientific attention. You are in a room with the task of tying two strings together. The strings hang from the ceiling, but they are too far apart to allow you to reach both of them at the same time. In the room is a table, a wrench, a screwdriver and a lighter. How can you tie the two strings together?

Like the usual tasks undertaken for AI research, these tasks are reasonably well-defined. Solutions are easy to evaluate, but unlike the standard tasks, there is no clear path from the problem statement to the solution. Rather, they require a change in representation to solve.

The checkerboard problem is best solved when represented as a parity problem.

The Curt and Goldie problem is solved when one recognizes that Curt and Goldie are goldfish. A passing train knocked their goldfish bowl from the old table.

The two strings problem is solved when one of the objects on the table is used as a weight to swing the string like a pendulum over to where both strings can be reached.

Insight problems cannot be solved with the current approach to machine learning because their solutions do not depend on “curve fitting,” that is, on adjusting the parameters of a model. Instead, one has to typically overcome a common model that views Curt and Goldie as people, and the things on the table as tools. Once the solver adopts the right representation, however, solving the problems is typically rapid, requiring very little additional effort.

As a field, we have become more proficient at solving path problems, because we have become more efficient at solving the insight problems that find the right representation to use to make the path problems amenable to solution through parameter adjustment. It is the designers who have found the right insights. Machine learning depends on the success of the designers’ ability to find these insights. We do not know how to get computers to design new solutions that have not already been listed out for them.

Arguably, insight problems are the more critical ones to solve if we are to achieve artificial general intelligence, but we currently have no way to even address them using the present machine learning paradigm. That will have to change.