  We study linear time model checking of collapsible higher-order pushdown
systems (CPDS) of order 2 (manipulating stack of stacks) against MSO and PDL
(propositional dynamic logic with converse and loop) enhanced with push/pop
matching relations. To capture these linear time behaviours with matchings, we
propose order-2 nested words. These graphs consist of a word structure
augmented with two binary matching relations, one for each order of stack,
which relate a push with matching pops (or collapse) on the respective stack.
Due to the matching relations, satisfiability and model checking are
undecidable. Hence we propose an under-approximation, bounding the number of
times an order-1 push can be popped. With this under-approximation, which still
allows unbounded stack height, we get decidability for satisfiability and model
checking of both MSO and PDL. The problems are ExpTime-Complete for PDL.
