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We investigate the complexity of modal satisfiability for a family of multi-modal logics with interdependencies among the modalities. In particular, we examine four characteristic multi-modal logics with dependencies and demonstrate that, even if we restrict the formulae to be diamond-free and to have only one propositional variable, these logics still have a high complexity. This result identifies and isolates two sources of complexity: the presence of axiom   $D$   for some of the modalities and certain modal interdependencies. We then further investigate and characterize the complexity of the diamond-free, 1-variable fragments of multi-modal logics in a general setting.

Modal logics with hard diamond-free fragments