Unsafe Grammars and Panic Automata

We show that the problem of checking if an infinite tree generated by a higher-order grammar of level 2 (hyperalgebraic) satisfies a given μ-calculus formula (or, equivalently, if it is accepted by an alternating parity automaton) is decidable, actually 2-Exptime-complete. Consequently, the monadic second-order theory of any hyperalgebraic tree is decidable, so that the safety restriction can be removed from our previous decidability result. The last result has been independently obtained by Aehlig, de Miranda and Ong. Our proof goes via a characterization of possibly unsafe second-order grammars by a new variant of higher-order pushdown automata, which we call panic automata. In addition to the standard pop1 and pop2 operations, these automata have an option of a destructive move called panic. The model-checking problem is then reduced to the problem of deciding the winner in a parity game over a suitable 2nd order pushdown system.