Cut-free tableau calculi for some propositional normal modal logics

We give sound and complete tableau and sequent calculi for the prepositional normal modal logics S4.04, K4B and G0(these logics are the smallest normal modal logics containing K and the schemata □A → □□A, □A → A and □⋄□A → (□ → □A); □A → □□A and A→□⋄A; □A → □□A and □(□(A→ □A) → A) → □A resp.) with the following properties: the calculi for S4.04 and G0are cut-free and have the interpolation property, the calculus for K4B contains a restricted version of the cut-rule, the so-called analytical cut-rule.In addition we show that G0is not compact (and therefore not canonical), and we proof with the tableau-method that G0is characterised by the class of all finite, (transitive) trees of degenerate or simple clusters of worlds; therefore G0is decidable and also characterised by the class of all frames for G0.