Modal Interpretations of Default Logic

In the paper we study a new and natural modal interpretat ion of defaults. We show that under this interpretat ion there are whole families of modal nonmonotonic logics that accurately represent default reasoning. One of these logics is used in a definition of possible-worlds semantics for default logic. This semantics yields a characterization of default extensions similar to the characterization of stable expansions by means of autoepistemic interpretat ion. We also show that the disjunctive information can easily be handled if disjunction is represented by means of modal disjunctive defaults — modal formulas that we use in our interpretat ion. Our results indicate that there is no single modal logic for describing default reasoning. On the contrary, there exist whole ranges of modal logics, each of which can be used in the embedding as a "host" logic. 1 I n t r o d u c t i o n The default logic of Reiter [1980] is a nonmonotonic formalism based on the paradigm of "negation as failure to prove" and is defined by means of a certain fixed-point construction. It is a formalism in the language of propositional calculus (or, in a more general variant, in the language of first-order logic). In 1982, McDermott [1982], building on the jo in t work w i th Doyle [1980], introduced a large class of modal nonmonotonic logics. He proposed a general scheme which, also using "negation as failure to prove" and a fixed-point construction, assigns to each monotone modal logic its nonmonotonic variant. The autoepistemic logic of Moore [1985], an important modal formalism, belongs to the McDermott-Doyle's family of logics (see [Shvarts, 1990]). In recent years there have been numerous attempts to explain and exploit the nature of the relationship between the default logic and modal nonmonotonic logics. There are two main reasons behind the interest in this particular research area. First ly, modal nonmonotonic logics often have clear, intui t ive semantics (for example, list semantics [Moore, 1985], possible-world semantics [Moore, 1984; Konolige, 1988], or preference semantics [Shoham, 1987]), and the default logic lacks one. By embedding the default logic into a modal nonmonotonic logic w i th an elegant semantics, insights into semantic aspects of default logic can be gained. Secondly, the automated inference methods for the "host" modal nonmonotonic logic could be used as a uniform tool for handl ing default theories. The default logic was first embedded into a variant of autoepistemic logic by Konolige [1988]. Marek and Truszczynski [1989; 1990] proposed to embed default logic into the nonmonotonic variant of the logic of necessttation N — the modal logic that does not contain any modal axiom schemata and uses modus ponens and necessitation as inference rules. Recently, L in and Shoham [1990] defined yet another, this time bimodal, nonmonotonic logic, which they called the logic of grounded knowledge (denoted G K ) , and provided an interpretation of the default logic w i th in logic G K . Each of these approaches has some disadvantages. We discuss them briefly in the next section. In this paper we consider another interpretat ion of defaults in the modal language. It is somewhat related to the approach of Siegel [1990] to modal nonmonotonic logic. We argue that our translation avoids the problems of the translations used in the earlier approaches. We show that under this new interpretation, default logic can be faithfully embedded into any of the whole range of modal nonmonotonic logics. As a consequence, possible-worlds semantics for these modal logics yield possible-worlds semantics for default logic, and their automated proof methods (whenever exist) can be applied to default theories. For example, in Section 4, we choose logic S4F, that has a particularly well-structured possible-world semantics, to obtain possible-world semantics for default logic. An important feature of the new translation is that it easily lends itself to an extension suitable for default *This work was partially supported by Army Research Office under grant DAAL03-89-K-0124, and by National Science Foundation and the Commonwealth of Kentucky EPSCoR program under grant R I I 8610671. I would like to thank Vladimir Lifschitz who informed me about the work of Siegel after seeing the preliminary version of my paper.