### Default reasoning by deductive planning

Michael Thielscher, Torsten Schaub
1995 Journal of automated reasoning
This paper deals with the automation of reasoning from incomplete information by means of Default Logics. We provide proof procedures for Default Logics' major reasoning modes, namely credulous and skeptical reasoning. We start by reformulating the task of credulous reasoning in Default Logics as deductive planning problems. This interpretation supplies us with several interesting and valuable insights into the proof theory of Default Logics. Foremost, it allows us to take advantage of the
more » ... number of available methods, algorithms, and implementations for solving deductive planning problems. As an example, we demonstrate how credulous reasoning in certain variants of Default Logic is implementable by means of a planning method based on equational logic programming. In addition, our interpretation allows us to transfer theoretical results, like complexity results, from the field of planning to that of Default Logics. In this way, we have isolated two yet unknown classes of default theories for which deciding credulous entailment is polynomial. Our approach to skeptical reasoning relies on an arbitrary method for credulous reasoning. It does neither strictly require the inspection of all extensions nor the computation of entire extensions to decide whether a formula is skeptically entailed. Notably, our approach abstracts from an underlying credulous reasoner. In this way, it can be used to extend existing formalisms for credulous reasoning to skeptical reasoning. From the very beginning, an important task was the development of proof theories in order to automate reasoning in Default Logic, preferably by adopting and extending methods known from classical automated deduction. However, Reiter himself observed that automating the reasoning process in the entire framework is problematic because full-fledged Default Logic lacks the formal property of semi-monotonicity. This property, however, is indispensable for proving in a local fashion, since it allows us to restrict our attention to those parts of a given theory that are related to what shall be proved. For this purpose, Reiter defines in [51] a restricted class of default theories, called normal theories, that are provably semi-monotonic in general. Using this observation, he develops in [51] a first proof theory for this restricted class based on the resolution principle. Nonetheless, it became apparent soon that many interesting problems cannot be encoded via normal default theories [53] . Moreover, it turned out that apart from semi-monotonicity, other desirable properties are not present in the original approach. This insight prompted several authors to develop modifications of the first approach to Default Logic, e.g. Lukaszewicz' Justified Default Logic 1 [39], Brewka's Cumulative Default Logic [8], or Constrained Default Logic [17]. These three variants turn out to be semi-monotonic even in case of arbitrary default theories. This is why they are of great interest, especially for automating default reasoning. In what follows, however, we will mainly focus on the finally mentioned dialect. The choice of Constrained Default Logic as our prime exemplar is of course not an arbitrary one. Constrained Default Logic enjoys several desirable computational properties needed for reasonable proof procedures. Moreover, it has recently been shown in [18] that in certain fragments of Constrained Default Logic reasoning is significantly easier than in Reiter's Default Logic. All this renders our exemplar a prime candidate for computational purposes. However, we show also how our results can be directly applied to Reiter's original definition in case of normal default theories. Moreover, we illustrate how similar results can be obtained for Lukaszewicz' variant, while we do not explicitly consider Cumulative Default Logic due to its tight relationship to Constrained Default Logic (see [58, 17] for details). An important characteristic feature of Constrained Default Logic, Lukaszewicz' variant, and classical Default Logic restricted to normal theories, is that extensions can be generated in a truly iterative way instead of using the usual fixpoint construction. This observation shall be the starting point of our analysis. During the last decade, several calculi designed for classical logic have been applied to define proof theories for (variants of) Default Logic, e.g. the resolution principle as in [51, 4] , the tableaux method [65] as in [61, 62] , or the connection method [6] as in [55, 60] . The aim of this paper is not to provide just another specific implementation technique. Rather, we propose a new view on the reasoning task in Default Logics by regarding it as a problem solving task or, more specifically, as a planning problem. This view appears to be very natural and straightforward as soon as extensions can be generated truly iteratively. We claim that this interpretation reflects adequately the nature of what distinguishes Default Logics from classical logic, namely the additional expressive power provided by default rules. Any formalism for proving in default theories that employs methods known from classical automated deduction has to comply with three substantial differences between a default δ = α : β ω (which allows to conclude ω by default if α holds and β can be consistently assumed) and its classical counterpart, viz. the implication α → ω . First of all, the consistency requirement given by the so-called justification β may suppress the application of δ . Second, as δ is a rule instead of a formula, it is impossible to apply it the other way round, i.e. via contraposition. For instance, we are allowed to conclude ¬a from ¬b given a → b but not by using the default a : b b . Third, the so-called prerequisite α