Symbolic Backwards-Reachability Analysis for Higher-Order Pushdown Systems

TL;DR

This paper presents a symbolic backwards-reachability analysis method for higher-order pushdown systems, extending existing techniques to handle more complex models and providing computational complexity results.

Contribution

It introduces a regularity-preserving backwards-reachability analysis for higher-order APDSs and proves its n-EXPTIME-completeness, enabling advanced verification tasks.

Findings

The set of configurations from which a regular set is reachable is regular.

The backwards-reachability problem for higher-order APDSs is n-EXPTIME-complete.

Applications include model-checking and reachability game analysis.

Abstract

Higher-order pushdown systems (PDSs) generalise pushdown systems through the use of higher-order stacks, that is, a nested "stack of stacks" structure. These systems may be used to model higher-order programs and are closely related to the Caucal hierarchy of infinite graphs and safe higher-order recursion schemes. We consider the backwards-reachability problem over higher-order Alternating PDSs (APDSs), a generalisation of higher-order PDSs. This builds on and extends previous work on pushdown systems and context-free higher-order processes in a non-trivial manner. In particular, we show that the set of configurations from which a regular set of higher-order APDS configurations is reachable is regular and computable in n-EXPTIME. In fact, the problem is n-EXPTIME-complete. We show that this work has several applications in the verification of higher-order PDSs, such as linear-time model-checking, alternation-free mu-calculus model-checking and the computation of winning regions of reachability games.