unsatisfiability of , there is no need to check . As a result, while there are two possible way to expand , one of these two ways is always sufficient to prove unsatisfiability if the formula is unsatisfiable. For example, one may expand the tableau by considering an arbitrary world where holds. If this expansion leads to unsatisfiability, the original formula is unsatisfiable. However, it is also possible that unsatisfiability cannot be proved this way, and that the world where holds should have been considered instead. As a result, one can always prove unsatisfiability by expanding either only or only; however, if the wrong choice is made the resulting tableau may not be closed. Expanding either subformula leads to tableau calculi that are complete but not proof-confluent. Searching as described in the "Searching for a closed tableau" may therefore be necessary. Depending on whether the precondition and consequence of a tableau expansion rule refer to the same world or not, the rule is called static or transactional. While rules for propositional connectives are all static, not all rules for modal connectives are transactional: for example, in every modal logic including axiom T, it holds that implies in the same world. As a result, the relative (modal) tableau expansion rule is static, as both its precondition and consequence refer to the same world. Formula-deleting tableau A method for avoiding formulae referring to different worlds interacting in the wrong way is to make sure that all formulae of a branch refer to the same world. This condition is initially true as all formulae in the set to be checked for consistency are assumed referring to the same world. When expanding a branch, two situations are possible: either the new formulae refer to the same world as the other one in the branch or not. In the first case, the rule is applied normally. In the second case, all formulae of the branch that do not also hold in the new world are deleted from the branch, and possibly added to all other branches that are still relative to the old world. As an example, in S5 every formula that is true in a world is also true in all accessible worlds (that is, in all accessible worlds both and are true). Therefore, when applying , whose consequence holds in a different world, one deletes all formulae from the branch, but can keep