### 1.4.1 Defining a Solution

Given an informal description of a problem, before even considering a computer, a knowledge base designer should determine what would constitute a solution. This question arises not only in AI but in any software design. Much of software engineering involves refining the specification of the problem.

Typically, problems are not well specified. Not only is there usually much left unspecified, but also the unspecified parts cannot be filled in arbitrarily. For example, if you ask a trading agent to find out all the information about resorts that may have health issues, you do not want the agent to return the information about all resorts, even though all of the information you requested is in the result. However, if the trading agent does not have complete knowledge about the resorts, returning all of the information may be the only way for it to guarantee that all of the requested information is there. Similarly, you do not want a delivery robot, when asked to take all of the trash to the garbage can, to take everything to the garbage can, even though this may be the only way to guarantee that all of the trash has been taken. Much work in AI is motivated by commonsense reasoning; we want the computer to be able to make commonsense conclusions about the unstated assumptions.

Given a well-defined problem, the next issue is whether it matters if the answer returned is incorrect or incomplete. For example, if the specification asks for all instances, does it matter if some are missing? Does it matter if there are some extra instances? Often a person does not want just any solution but the best solution according to some criteria. There are four common classes of solutions:

Optimal solution
An optimal solution to a problem is one that is the best solution according to some measure of solution quality. This measure is typically specified as an ordinal, where only the order matters. However, in some situations, such as when combining multiple criteria or when reasoning under uncertainty, you need a cardinal measure, where the relative magnitudes also matter. An example of an ordinal measure is for the robot to take out as much trash as possible; the more trash it can take out, the better. As an example of a cardinal measure, you may want the delivery robot to take as much of the trash as possible to the garbage can, minimizing the distance traveled, and explicitly specify a trade-off between the effort required and the proportion of the trash taken out. It may be better to miss some trash than to waste too much time. One general cardinal measure of desirability, known as utility, is used in decision theory.
Satisficing solution
Often an agent does not need the best solution to a problem but just needs some solution. A satisficing solution is one that is good enough according to some description of which solutions are adequate. For example, a person may tell a robot that it must take all of trash out, or tell it to take out three items of trash.
Approximately optimal solution
One of the advantages of a cardinal measure of success is that it allows for approximations. An approximately optimal solution is one whose measure of quality is close to the best that could theoretically be obtained. Typically agents do not need optimal solutions to problems; they only must get close enough. For example, the robot may not need to travel the optimal distance to take out the trash but may only need to be within, say, 10% of the optimal distance.

For some problems, it is much easier computationally to get an approximately optimal solution than to get an optimal solution. However, for other problems, it is (asymptotically) just as difficult to guarantee finding an approximately optimal solution as it is to guarantee finding an optimal solution. Some approximation algorithms guarantee that a solution is within some range of optimal, but for some algorithms no guarantees are available.

Probable solution
A probable solution is one that, even though it may not actually be a solution to the problem, is likely to be a solution. This is one way to approximate, in a precise manner, a satisficing solution. For example, in the case where the delivery robot could drop the trash or fail to pick it up when it attempts to, you may need the robot to be 80% sure that it has picked up three items of trash. Often you want to distinguish the false-positive error rate (the proportion of the answers given by the computer that are not correct) from the false-negative error rate (which is the proportion of those answers not given by the computer that are indeed correct). Some applications are much more tolerant of one of these errors than of the other.

These categories are not exclusive. A form of learning known as probably approximately correct (PAC) learning considers probably learning an approximately correct concept.