Levesque introduces the problem of how to represent knowledge (or as he puts it, beliefs) in computers as an issue of how to deal with the fact that this knowledge can occur in inherently intractable forms. This intractability derives from the required size of knowledge bases (KBs) and the subsequent combinatorial explosion, in the worst case, of reasoning (i.e. searching) them. Levesque goes on to argue that if you are going to use these forms, that there is nothing that can be done to avoid the combinatorial explosion. You can't avoid it by eliminating the worst case conditions that lead to the combinatorial explosion as there will always be some worst case.

In addition to the intractability problem Levesque suggest that there is another significant problem of reasoning about knowledge and it is that at some level of a problem that knowledge from a higher level yet is not required (i.e. the recursive application of knowledge must top out some where). While this seems intuitively obvious it is not clear to me if this ever comes up in practice.

To solve this problem Levesque proposes to restrict the form of the knowledge in the KB to that which can be handled in a tractable manner. This parallels the technique in SAT, an NP-complete problem in general, to HORN-SAT which can be solved in polynomial time. To accomplish this Levesque uses the concept of a "vivid" representation for the knowledge. This vivid form of representation is one where each piece of knowledge in the domain of interest (the domain of interest is an important distinction as the specificity of the domain forces the removal of extraneous information and disambiguates the meanings of symbols, at least to within the domain of interest) is a concrete or atomic level fact that is a projection from the KB to the "real" world of the domain. (The origin of the term "vivid" seems to come from Levesque's response to why "thinking may sometimes feel like much more than symbol manipulation [for humans]" in that the symbols being manipulated correspond so strongly, i.e. vividly, with actual or real objects that we forget the fact that we are manipulating symbols.)

The natural extension that the vivid form of knowledge representation (KR) suggests is that of sensory percepts (Levesque uses pictures/visuals as an example of sensory percepts) instead of natural language or some variation there of. Levesque suggests that language is a trap of combinatorial explosions because of its ability to easily represent disjunctions (and other logical forms, such as existential quantifiers, etc.) that are impossible, or at least much more constrained in sensory percepts. The vivid form of KR is constrained to be as simple as possible and to explicitly represent all relationships of interest. This restriction, Levesque proposes, allows the tractable searching of a KB where a more complex KR would make it impossible. This seems to be easily supportable and Levesque refers to study where only a small increase in complexity causes a problem to go from easily solvable to impossible. This is also reminiscent of the GSAT paper where the selection of 3-SAT problems for solution were tightly constrained to the border between easily solvable and impossible to solve.

One of the major advantages of the vivid form of KR is that it can easily be placed into a database format. This allows the power of a mature technology to be exploited and solutions found (within the domain of interest, of course) in reasonable time. The unfortunate aspect of the vivid form of KR is that it can somewhat difficult to derive from more traditional forms of KR and can be very large or even infinite (for example if you want to transform the sentence "all humans are mortal" to the U.S. census you would end up with one entry per person in the census, just for this single fact). This suggests that a more compact representation might be used in a main KB with the generation of vivid sub-KBs being formed for specific reasoning tasks.

Another transformation that occurs when the vivid form of a KB is formed is that of making assumptions about default values to form a complete content for the KB. By adding detail information to the vivid KB that corresponds to default knowledge (such as "common sense" information) a consistent and tractable basis for reasoning could be formed. It must be recognized that by introducing this default information the reasoning is no longer logically sound. The assumption that Levesque makes here that any errors in reasoning could be repaired easily by the modification of previously assumed default values to correspond to the "real" cases. This seems reasonable, but a more rigorous demonstration of this is necessary.

Overall I found Levesque's arguments compelling, but lacking in concrete demonstration or proof of their validity. This is not surprising as the paper does not purport to be of a technical nature. A certain amount of hand waving was done, particularly in the realm of details, but this does not detract from the insightful views about the problems with more traditional KR and KBs. The fact that Levesque has presented high quality suggestions for the problems that he raises makes his paper that much more interesting.