More on Bounding Introspection in Modal Nonmonotonic Logics

Abstract

By \( {\user1{\mathcal{L}}} \) we denote the set of all propositional formulas. Let \( {\user1{\mathcal{C}}} \) be the set of all clauses. Define \( {\user1{\mathcal{C}}}_{0} = {\user1{\mathcal{C}}} \cup {\left\{ {\neg L\eta :\eta \in {\user1{\mathcal{C}}}} \right\}} \) . In Sec. 2 of this paper we prove that for normal modal logics \( {\user1{\mathcal{S}}} \), the notions of \( {\left( {{\user1{\mathcal{S}}},{\user1{\mathcal{C}}}_{0} } \right)} \)-expansions and \( {\user1{\mathcal{S}}} \)-expansions coincide. In Sec. 3, we prove that if I consists of default clauses then the notions of \( {\user1{\mathcal{S}}} \)-expansions for I and \( {\left( {{\user1{\mathcal{S}}},{\user1{\mathcal{C}}}} \right)} \)-expansions for I coincide. To this end, we first show, in Sec. 3, that the notion of \( {\user1{\mathcal{S}}} \)-expansions for I is the same as that of \( {\left( {{\user1{\mathcal{S}}},{\user1{\mathcal{L}}}} \right)} \)-expansions for I.