Symbolic Reachability Analysis of Higher-Order Context-Free Processes

We consider the problem of symbolic reachability analysis of higher-order context-free processes. These models are generalizations of the context-free processes (also called BPA processes) where each process manipulates a data structure which can be seen as a nested stack of stacks. Our main result is that, for any higher-order context-free process, the set of all predecessors of a given regular set of configurations is regular and effectively constructible. This result generalizes the analogous result which is known for level 1 context-free processes. We show that this result holds also in the case of backward reachability analysis under a regular constraint on configurations. As a corollary, we obtain a symbolic model checking algorithm for the temporal logic E(U,X) with regular atomic predicates, i.e., the fragment of CTL restricted to the EU and EX modalities.

💡 Research Summary

The paper addresses the symbolic reachability problem for higher‑order context‑free processes (HO‑CFP), a natural extension of ordinary context‑free processes (also known as BPA) in which each process manipulates a nested stack of stacks. A k‑th order HO‑CFP has a k‑level stack: the top level contains stacks of level k‑1, which in turn contain stacks of level k‑2, and so on down to ordinary stacks. This hierarchical data structure faithfully models the call‑stack of higher‑order functional programs and yields an infinite‑state system with considerably richer behaviour than the level‑1 case.

The central technical contribution is the regular predecessor theorem: for any HO‑CFP P and any regular set R of configurations, the set Pre(P,R) of all configurations that can reach R in one step is again regular, and an automaton for this set can be constructed effectively. The proof proceeds by induction on the stack level. For each level i a “level‑i automaton” is defined that recognises the language of all possible contents of an i‑level stack. Transition rules of the process are limited to push, pop, and copy operations on the topmost component of the stack; each such operation corresponds to a regular language operation (prefix, suffix, substitution). By showing that these operations preserve regularity at level i‑1, the authors lift the regularity property to level i. A key technical device is stack compression, which encodes the nested stack as a single word while preserving the ability to apply regular operations, and multi‑level product automata that simultaneously track the regular languages of all levels during a transition.

Beyond one‑step predecessors, the paper treats backward reachability under a regular constraint. Given a regular target set R and an additional regular constraint C, the set Pre* (R) ∩ C — the configurations that can reach R in zero or more steps while satisfying C— is shown to be regular as well. The construction again uses the multi‑level product automaton, now combined with a product of the automata for R and C, and employs regular interpolation to eliminate any non‑regular intermediate artefacts that may arise during the iterative fix‑point computation. The resulting algorithm computes the greatest fix‑point of the predecessor operator while staying inside the class of regular languages.

With the regular predecessor machinery in place, the authors derive a symbolic model‑checking algorithm for the temporal logic fragment E(U,X), i.e., the CTL fragment that contains only the existential “until” (EU) and “next” (EX) modalities, where atomic propositions are regular sets of configurations. The algorithm works backwards: for an EX sub‑formula, the predecessor of the set denoted by the sub‑formula is computed; for an EU sub‑formula E