Tableau-based translation from first-order logic to modal logic

We define a procedure for translation of a given first-order formula to an equivalent modal formula, if such exists, by using tableau-based bisimulation invariance test. Previously developed tableau procedure tests bisimulation invariance of a given first-order formula, and therefore tests whether that formula is equivalent to the standard translation of some modal formula. Using a closed tableau as the starting point, we show how an equivalent modal formula can be effectively obtained.