Stratification for Variants of Default Logic

Default logic (DL) (Reiter 1980) is one of the most prominent approaches to nonmonotonic reasoning. One of the main problems with its applicability is that it is computationally harder than classical logic (Gottlob 1992). (Cholewinski 1994,1995) introduced and studied stratification of default theories to increase the efficiency of default reasoning. The idea is to to split the knowledge into smaller parts, and to apply reasoning in a local way. This paper shows how stratification can work for some important variants of Default Logic. These variants are Justified Default Logic (JDL), Rational Default Logic (RDL), and Constrained Default Logic (CDL). The main idea of stratification is that if a default makes use of information from another default, then we should seek to apply it first. Let T = (W, 0) be a Default Theory, where W is an empty set of formulae and D the set of the two defaults S = 7 and 6’ = y. In this case we should apply S before S’ because 6’ is not applicable to Th( IV) i . A function p assigning a natural number to every default from D is called a stratification function iff for any S, S’ E D the condition “ If Prop(cons(S)) fl Prop(S’) # 8 then p(S) 5 ~(6’) (1)2” holds. By assigning a number to each default, p decomposes D into layers (strata) D1, . . . , D,+ of ascending value under p. A default theory T = (IV, D) is called stratified iff W is consisent , Prop(W) fl Prop(cons( D) U just(D)) = 8, and there exists a stratification function for D. (Cholewinski 1994) shows that the decomposition of a set of defaults according to a stratification function preserves the extensions. For stratification to work properly for default logic variants, we need to add new conditions for stratification functions. Justified Default Logic avoids running into failure by retracting a final, “fatal” step. The