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 $L_0$ which ensures that the largest logic where $L_0$ embeds faithfully by the boxdot translation is $L_0$ itself. In particular, this natural generalization of the boxdot conjecture holds for S4, S5, and KTB in place of T.Comment: 9 page

A new local invariant and simpler proof of Kepler's conjecture and the least action principle on the crystalformation of dense type

A new locally averaged density for sphere packing in R^3 is defined by a proper combination of the local cell (Voronoi cell) and Delaunay decompositions (\S 1.2.2), using only a single layer of surrounding spheres. Local packings attaining the optimal estimate of such a local invariant must be either the f.c.c. or h.c.p. local packings (Theorem I). The main purpose of this paper is to provide a clean-cut proof of this strong uniqueness result via geometric invariant theory. This result also leads to simple proofs of Kepler's conjecture on sphere packing, least action principle of crystal formation of dense type, and optimal packings with containers (Theorems II-IV). This work provides a much simplified alternative to the author's previous work on Kepler's conjecture and least action principle of crystal formation of dense type which involved a local invariant defined by double layer of surrounding spheres [Hsi].Comment: 69 pages, 22 figure

Cluster expansion and the boxdot conjecture Emil Jeˇrábek ∗

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 ↦ → 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 ·✷−1 = {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 ⊕ ✸ h ✷ i p → ✷ j ✸ k p, but the full conjecture remained unsettled