Propositional Dynamic Logic has Craig Interpolation: a tableau-based proof

We show that Propositional Dynamic Logic (PDL) has the Craig Interpolation Property. This question has been open for many years. Three proof attempts were published, but later criticized in the literature or retracted. Our proof is based on the main ideas from Borzechowski (1988). We define a cyclic tableau system for PDL with a loading mechanism to recognize successful repeats. For this system, we show soundness and completeness via a game. To show interpolation, we modify Maehara's method to work for tableaux with repeats: we first define pre-interpolants at each node, and then use a quasi-tableau to define interpolants for clusters (strongly connected components). In different terms, our method solves the fixpoint equations that characterize the desired interpolants, and the method ensures that the solutions to these equations can be expressed within PDL. The proof is constructive and we show how to compute interpolants. We also make available a Haskell implementation of the proof system that provides interpolants. Lastly, we mention ongoing work to formally verify this proof in the interactive theorem prover Lean, and several questions for future work.