Twenty-Five Years of Preferred Subtheories

In the seminal paper [6], Gerd Brewka argued that ranking a set of default rules without prerequisites, and selecting extensions according to a lexicographic refinement of the inclusion ordering proves to be a natural, simple and efficient way of dealing with the multiple extension (or “subtheories”) problem. This natural idea has been reused, discussed, revisited, reinvented, adapted many times in the AI community and beyond. Preferred subtheories do not only have an interest in default reasoning, but also in reasoning about time, reasoning by analogy, reasoning with compactly represented preferences, judgment aggregation, and voting. They have several variants (but arguably not so many). In this short paper I will say as much as I can about preferred subtheories in sixteen pages. 1 Prioritized Default Theories and Preferred Subtheories Preferred subtheories were introduced in [6] as a way of representing and exploiting priorities between default rules. Their starting point was the THEORIST system [28] for default reasoning. In Poole’s system – equivalent to the restriction of Reiter’s default logic to normal defaults without prerequisites – a default theory is a set of facts F plus a set of hypotheses Δ (both composed of logical formulas) and an extension is the set of logical consequences of a set-inclusion maximal subset D of Δ such that D ∪ F is consistent. In spite of the (apparently drastic) restriction to normal defaults without prerequisites, this system is able to deal adequately with many of the standard default reasoning examples from the literature, but not all, because of the impossibility of expressing priorities between defaults. Let me reuse this example from [6], suggested to Gerd Brewka by Ulrich Junker. “Usually one has to go to a meeting. This rule does not apply if somebody is sick, unless he only has a cold. The rule is also not applicable if somebody is on vacation.” As shown by [6] (Section 3), given that the person is sick, the natural writing of this example in Poole’s system generates two extensions: one where she has to attend the meeting and one where she does not. In order to avoid this, one would need to “(...) look down in the hierarchy of exceptions and block defaults lower in the hierarchy. (...) the number of defaults may increase heavily in cases where more exceptions and exceptions of exceptions are involved” T. Eiter et al. (Eds.): Brewka Festschrift, LNAI 9060, pp. 157–172, 2015. c © Springer International Publishing Switzerland 2015