Terminating Tableaux for Modal Logic with Transitive Closure

We present a terminating tableau system for the modal logic K∗. K∗ extends the basic modal logic K with a reflexive transitive closure operator for relations and is a proper fragment of propositional dynamic logic. We investigate two different approaches to achieve termination, namely chain-based blocking and pattern-based blocking. Pattern based-blocking has not been applied to a modal logic with a reflexive transitive closure operator. We have a modular completeness proof that adapts to both termination approaches. Extending completeness arguments for a related description logic, we establish a strengthened soundness property of our calculus that we call straightness. Using this property we are able to prove both verification and refutation soundness.