Default Logics with Preference Order: Principles and Characterisations

Abstract

Practical use of default logics requires mechanisms to select the more suitable extensions from out of the several often allowed by a classical default theory. An obvious solution is to order defaults in a preference hierarchy, and use this ordering to select preferred extensions. The literature contains many suggestions on how to implement such a scheme. The problem is that they yield different results: all authors agree that preferred extensions employ preferred defaults, but this apparent agreement hides differences in lower level decisions. While motivations for these are rarely discussed, their consequences for overall behaviour are wide-ranging. This paper points towards standardisation, discussing principles that ought to hold and then working top-down to determine lower level details. We present characterisations, uncover anomalies of existing approaches, and suggest repairs.

We build on work by Brewka and Eiter [4], who first identified some of the desiderata discussed here. A slightly modified version of the notion of preferred extension proposed by these authors, and one by Delgrande and Schaub [5], are identified as the most and least inclusive notions of extension satisfying all desiderata. We point out that these two (in the literature previously termed “descriptive” and “prescriptive”, respectively) differ along two rather independent dimensions, and two additional notions are then identified, representing the remaining possibilities.