A framework for coalitional normative systems

We propose coalitional normative system (cns), which can selectively restrict the joint behavior of a coalition, in this paper. We extend the semantics of atl and propose Coordinated atl (co-atl) to support the formalizing of cns. We soundly and completely characterize the limitation of the normative power of a coalition by identifying two fragments of co-atl language corresponding to two types of system properties that are unchangeable by restricting the joint behavior of such a coalition. Then, we prove that the effectiveness checking, feasibility and synthesis problems of cns are ptime-complete, np-complete and fnp-complete, respectively. Moreover, we define two concepts of optimality for cns, that is, minimality and compactness, and prove that both minimality checking and compactness checking are conp-complete while the problem of checking whether a coalition is a minimal controllable coalition is dp-complete. The relation between ns and cns is discussed, and it turns out that nss intrinsically consists of a proper subset of cnss and some basic problems related to cns are no more complex than that of ns.