Clausal Tableaux for Hybrid PDL

PDL (propositional dynamic logic) [6, 11, 8] is an expressive modal logic invented for reasoning about programs. It extends basic modal logic with expressions called programs. Programs describe relations from states to states and are used to express modalities. Programs are composed with the operators familiar from regular expressions. In addition, they may employ formulas so that conditionals and while loops can be expressed. Fischer and Ladner [6] show the decidability of PDL using a filtration argument. They also prove that the satisfiability problem for PDL is EXPTIME-hard. Pratt [15] shows that PDL satisfiability is in EXPTIME using a tableau method with an and-or graph representation. Goré and Widmann [7] address the efficient implementation of Pratt-style decision procedures. We consider PDL extended with nominals [12, 13], a logic we call hybrid PDL or HPDL. Nominals are atomic formulas that hold exactly for one state. Nominals equip PDL with equality and are the characteristic feature of hybrid logic [2]. The satisfiability problem of HPDL is in EXPTIME [13, 16]. We are interested in a tableau system for HPDL that can serve as a basis for gracefully degrading decision procedures. We found it impossible to extend one of the existing tableau methods for PDL [15, 5, 1, 7] to nominals. The difficulties are in the correctness proofs. For Pratt-like methods [15, 7], the problem stems