The Relative Succinctness and Expressiveness of Modal Logics Can Be Arbitrarily Complex

We study the relative succinctness and expressiveness of modal logics, and prove that these relationships can be as complex as any countable partial order. For this, we use two uniform formalisms to define modal operators, and obtain results on succinctness and expressiveness in these two settings. Our proofs are based on formula size games introduced by Adler and Immerman and bisimulations.