Approximating Weakly Preferred Semantics in Abstract Argumentation through Vacuous Reduct Semantics

We consider the recently introduced vacuous reduct semantics in abstract argumentation that allows the composition of arbitrary argumentation semantics through the notion of the reduct. We show that by recursively applying the principle of vacuous reduct semantics we are able to cover a broad range of semantical approaches. Our main result shows that we can recover the weakly preferred semantics as the unique solution of a fixed point equation involving an infinite application of the vacuous reduct semantics based only on the very simple property of conflict-freeness. We also conduct an extensive study of the computational complexity of the recursive application of vacuous reduct semantics, which shows that it completely covers each level of the polynomial hierarchy, depending on the recursion depth.