On a Non-Context-Free Extension of PDL

Over the last 25 years, a lot of work has been done on seeking for decidable nonregular extensions of Propositional Dynamic Logic (PDL). Only recently, an expressive extension of PDL, allowing visibly pushdown automata (VPAs) as a formalism to describe programs, was introduced and proven to have a satisfiability problem complete for deterministic double exponential time. Lately, the VPA formalism was extended to so called k-phase multi-stack visibly pushdown automata (k-MVPAs). Similarly to VPAs, it has been shown that the language of k-MVPAs have desirable effective closure properties and that the emptiness problem is decidable. On the occasion of introducing k-MVPAs, it has been asked whether the extension of PDL with k-MVPAs still leads to a decidable logic. This question is answered negatively here. We prove that already for the extension of PDL with 2-phase MVPAs with two stacks satisfiability becomes Σ1-complete.