An Interpretation of Default Logic in Minimal Temporal Epistemic Logic

Abstract

Many complex reasoning tasks have to deal with incomplete information. Reasoning systems trying to accomplish such a task contain basic knowledge that can be used to draw some conclusions about the domain, but may also have to make reasonable assumptions to be able to draw further conclusions. These assumptions usually reflect common sense knowledge about the domain, but sometimes have to be retracted in view of new evidence. A characteristic of this type of reasoning is that often several (conflicting) sets of assumptions exist which all lead to a coherent description of the domain. Any formalization of such reasoning processes has to take this into account. One of the logical approaches is Reiter’s default logic (e.g., see [Reiter, 1980; Besnard, 1989; Łukaszewicz, 1990]). A characteristic of default logic is that only after a complete set of (default) assumptions has been chosen, it can be checked whether it indeed gives an acceptable description of the domain. To be more specific, during the reasoning one meets a specific type of conditions (called justifications) in default rules to be applied, that cannot be fulfilled only on the basis of what has been derived until that moment. After application of a default rule, only in the future of the reasoning process it can be verified whether such a condition of the applied default rule will turn out to be justified or to be defeated. Therefore there is an essential temporal element in default reasoning. This suggests that a default rule can be given an interpretation as a temporal rule with one of its conditions referring to the future of the reasoning process. It seems that the process of actually constructing a set of coherent assumptions is reflected in Reiters approach to a certain extent, but without making the essential temporal element explicit. In this paper we will describe an interpretation of default logic in temporal epistemic logic. Notice that by temporal here a reference to internal time is meant: the reasoning is about a domain at a certain fixed point in time, but the reasoning system itself has an internal time over which its knowledge can vary.