Default logic

When an intelligent system (either computer–based or human) tries to solve a problem, it may be able to rely on complete information about this problem, and its main task is to draw the correct conclusions using classical reasoning. In such cases classical predicate logic may be sufficient. However in many situations the system has only incomplete information at hand, be it because some pieces of information are unavailable, be it because it has to respond quickly and does not have the time to collect all relevant data. Classical logic has indeed the capacity to represent and reason with certain aspects of incomplete information. But there are occasions in which additional information needs to be “filled in” to overcome the incompleteness, for example because certain decisions must be made. In such cases the system has to make some plausible conjectures, which in the case of default reasoning are based on rules of thumb, called defaults. For example, an emergency doctor has to make some conjectures about the most probable causes of the symptoms observed. Obviously it would be inappropriate to await the results of possibly extensive and time–consuming tests before beginning with the treatment. When decisions are based on assumptions, these may turn out to be wrong in the face of additional information that becomes available; for example, medical tests may lead to a modified diagnosis. The phenomenon of having to take back some previous conclusions is called nonmonotonicity; it says that if a statement φ follows from a set of premises M and M ⊆ M ′, φ does not necessarily follow from M ′. Default Logic, originally presented in [Reiter, 1980], provides formal methods to support this kind of reasoning. Default Logic is perhaps the most prominent method for nonmonotonic reasoning, basically because of the simplicity of the notion of a default, and because defaults prevail in many application areas. However there exist several alternative design decisions which have led to variations of the initial idea; actually we can talk of a family of default reasoning methods because they share the same foundations. In this paper we present the motivations and basic ideas of some of the most important default logic variants, and compare them both with respect to interconnections and the fulfillment of some properties. The key idea underlying all default reasoning methods is the use of rules allowing for “jumping to conclusions” even in the absence of certain information. In other words, this is the proposed solution to the problem of reasoning with incomplete information, as described above. In addition, default reasoning can be viewed as an approach to reasoning with inconsistent information. It is well documented that classical “collapses” in the