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We describe a set of simple features that are sufficient in order to make the satisfiability problem of logics interpreted on trees  Tower -hard. We exhibit these features through an  Auxiliary Logic on Trees  (ALT), a modal logic that essentially deals with reachability of a fixed node inside a forest and features modalities from sabotage modal logic to reason on submodels. After showing that ALT admits a  Tower -complete satisfiability problem, we prove that this logic is captured by four other logics that were independently found to be  Tower -complete: two-variables separation logic, quantified computation tree logic, modal logic of heaps and modal separation logic. As a by-product of establishing these connections, we discover strict fragments of these logics that are still non-elementary.

An Auxiliary Logic on Trees: on the Tower-Hardness of Logics Featuring Reachability and Submodel Reasoning