Strong Negation and Equivalence in the Safe Belief Semantics

The safe belief semantics uses intermediate logics to definean extension of answer sets to all propositional formulas, butonly considering one kind of negation. In this work we extendsafe beliefs adding the strong negation connective. The mainfeature of our extension is that strong negation can occur beforeany formula, and not only at the atomic level. We give resultsconcerning the relation between strong negation extensions ofintermediate logics and safe beliefs and consider the way inwhich strong negation can be eliminated from any formula whilepreserving its semantics. We also propose two new notions ofequivalence: substitution equivalence and contextualized equivalence.We prove that they are both more general than strong equivalenceand, for propositional formulas where strong negation may occurat the non-atomic level, substitution equivalence captures anotion of equivalence that cannot be captured by strong equivalencealone.