Nested Words for Order-2 Pushdown Systems

TL;DR

This paper introduces order-2 nested words to model linear behaviors of collapsible higher-order pushdown systems, enabling decidable model checking of MSO and PDL under certain bounds.

Contribution

It proposes a novel nested word structure for order-2 CPDS and establishes decidability results for model checking with an under-approximation.

Findings

Decidability of satisfiability and model checking for MSO and PDL with bounds.

Model checking complexity is ExpTime-Complete for PDL.

Allows unbounded stack height while maintaining decidability.

Abstract

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.