Cluster expansion and the boxdot conjecture

The boxdot conjecture asserts that every normal modal logic that faithfully interprets T by the well-known boxdot translation is in fact included in T. We confirm that the conjecture is true. More generally, we present a simple semantic condition on modal logics L0 which ensures that the largest logic where L0 embeds faithfully by the boxdot translation is L0 itself. In particular, this natural generalization of the boxdot conjecture holds for S4, S5, and KTB in place of T. 1 The boxdot translation The boxdot translation is the mapping φ 7→ φ · ✷ from the language of monomodal logic into itself that preserves propositional variables, commutes with Boolean connectives, and satisfies (✷φ) · ✷ = · ✷φ · , where · ✷φ is an abbreviation for φ ∧✷φ. It is easy to see that for any normal modal logic L, the set of formulas interpreted in L by the boxdot translation, L · ✷ = {φ : ⊢L φ · }, is also a normal modal logic (nml), and contains the logic T = K ⊕ ✷p → p. The boxdot translation is a faithful interpretation of T in the smallest nml K (i.e., K · ✷ −1 = T), and more generally, in any logic between K and T. The boxdot conjecture, formulated by French and Humberstone [4], states that the converse also holds: L · ✷ −1 = T =⇒ L ⊆ T. French and Humberstone proved the conjecture for logics L axiomatized by formulas of modal degree 1, and Steinsvold [5] has shown it for logics of the form L = K ⊕ ✸✷p → ✷✸p, but the full conjecture remained unsettled. ∗Supported by grant IAA100190902 of GA AV ČR, Center of Excellence CE-ITI under the grant P202/12/G061 of GA ČR, and RVO: 67985840.