A Non-wellfounded, Labelled Proof System for Propositional Dynamic Logic

We define an infinitary labelled sequent calculus for PDL, \(\mathbf {G3PDL}^{\infty }\). A finitarily representable cyclic system, \(\mathbf {G3PDL}^{\omega }\), is then given. We show that both are sound and complete with respect to standard models of PDL and, further, that \(\mathbf {G3PDL}^{\infty }\) is cut-free complete. We additionally investigate proof-search strategies in the cyclic system for the fragment of PDL without tests.