Embedding Defeasible Logic into Logic Programs

(c) If D is decisive then the implications (a) and (b) are also true in the opposite direction. That is, if D is decisive, then the stable model semantics of P(D) corresponds to the provability in defeasible logic. However part (c) is not true in the general case, as the following example shows. Example 5. Consider the defeasible theory r 1 :) :p r 2 : p) p In defeasible logic, +@:p cannot be proven because we cannot derive ?@p. However , blocked(r 2) is included in the only stable model of P(D), so def-:p is a sceptical conclusion of P(D) under stable model semantics. If we wish to have an equivalence result without the condition of decisive-ness, then we must use a diierent logic programming semantics, namely Kunen semantics. Theorem 3. (a) D ` +p , P(D) ` K strict-p. (b) D ` ?p , P(D) ` K not strict-p. (c) D ` +@p , P(D) ` K def-p. (d) D ` ?@p , P(D) ` K not def-p. 6 Conclusion We motivated and presented a translation of defeasible theories into logic programs , such that the defeasible conclusions of the former correspond exactly with the sceptical conclusions of the latter under the stable model semantics, if a condition of decisiveness is satissed. If decisiveness is not satissed, we have to use Kunen semantics instead. This paper closes an important gap in the theory of nonmonotonic reasoning, in that it relates defeasible logic with mainstream semantics of logic programming. This result is particularly important, since defeasible reasoning is one of the most successful nonmonotonic reasoning paradigms in applications.