Partial grounded fixpoints

Approximation fixpoint theory (AFT) is an algebraical study of fixpoints of lattice operators. This theory induces all major semantics of logic programming, autoepistemic logic, default logic and abstract argumentation frameworks and unifies these formalisms. Recently, AFT was extended with the notion of a grounded fixpoint. This type of fixpoint formalises common intuitions from various application domains of AFT. The study of groundedness was limited to exact lattice points; in this paper, we extend it to the bilattice: for an approximator A of O, we define A-groundedness. We show that all partial A-stable fixpoints are A-grounded and that the A-well-founded fixpoint is uniquely characterised as the least precise A-grounded fixpoint. We apply our theory to logic programs and show that in this case, groundedness closely relates to unfounded sets.status: publishe