Symbolic Backwards-Reachability Analysis for Higher-Order Pushdown Systems

Higher-order pushdown systems (PDSs) generalise pushdown systems through the use ofhigher-order stacks, that is, a nested “stack of stacks” structure. These systems may be usedto model higher-order programs and are closely related to the Caucal hierarchy of infinitegraphs 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 push-down 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 con-figurations is reachable is regular and computable in n-EXPTIME. In fact, the problem isn-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 μ-calculus model-checking, the compu-tation of winning regions of reachability games and determining whether the word languageaccepted by a higher-order pushdown automata is non-empty. Correction, Jan 2011: we have recently discovered that this application to LTL is erroneous. In particular, the proof of Proposition 5.1 is mistaken. For example, 〈p, [[ab]]〉 →֒ 〈q, [[ab][ab]]〉 →֒ 〈q, [[b][ab]]〉 →֒ 〈p, [[ab]]〉 forms a loop without a repeating head. A correct and optimal algorithm for linear-time model checking was published in FSTTCS 2010 [12].