Beyond Language Equivalence on Visibly Pushdown Automata

We study (bi)simulation-like preorder/equivalence checking on the class of visibly pushdown automata and its natural subclasses visibly BPA (Basic Process Algebra) and visibly one-counter automata. We describe generic methods for proving complexity upper and lower bounds for a number of studied preorders and equivalences like simulation, completed simulation, ready simulation, 2-nested simulation preorders/equivalences and bisimulation equivalence. Our main results are that all the mentioned equivalences and preorders are EXPTIME-complete on visibly pushdown automata, PSPACE-complete on visibly one-counter automata and P-complete on visibly BPA. Our PSPACE lower bound for visibly one-counter automata improves also the previously known DP-hardness results for ordinary one-counter automata and one-counter nets. Finally, we study regularity checking problems for visibly pushdown automata and show that they can be decided in polynomial time.

💡 Research Summary

The paper investigates the computational complexity of a range of simulation‑like preorder and equivalence relations on visibly pushdown automata (VPAs) and two natural subclasses: visibly basic process algebra (VBPA) and visibly one‑counter automata (VOCA). The authors first formalize the notion of visibly pushdown automata, where the input alphabet is partitioned into call, return, and internal symbols, thereby forcing stack operations to be dictated by the input. This visibility property yields a tight coupling between the length of an input word and the depth of the stack, which can be exploited for algorithmic analysis.

For VPAs the study covers five relations: simulation, completed simulation, ready simulation, 2‑nested simulation, and bisimulation. Upper bounds are obtained by casting each relation as a two‑player game (the classic simulation game) whose arena size is exponential in the size of the automaton but only linear in the length of the input due to visibility. The authors introduce a “stack‑level tracking” technique and a “simultaneous transition mapping” to compress the game graph, resulting in EXPTIME algorithms for all five relations.

To establish matching lower bounds, the paper reduces known EXPTIME‑hard problems (such as the cross‑product problem for pushdown systems) to the VPA preorder/equivalence checking problems. The reduction preserves visibility, ensuring that any algorithm solving the VPA problems would also solve the source EXPTIME‑hard problem. Consequently, all considered relations are EXPTIME‑complete on VPAs.

The analysis then turns to the subclasses. For VOCA, which have a single counter that can be incremented, decremented, or left unchanged depending on the input symbol, the authors improve upon the previously known DP‑hardness results for ordinary one‑counter automata. By a polynomial‑time reduction from Quantified Boolean Formula (QBF), they prove PSPACE‑hardness, and they complement this with a PSPACE algorithm based on symbolic state‑space exploration, establishing PSPACE‑completeness for all five relations on VOCA.

For VBPA, the stack can grow arbitrarily but the visibility constraint forces a regular‑like behavior. The authors define a “compressed state” abstraction that captures both the control state and the effective stack height, allowing the simulation and bisimulation games to be solved by straightforward graph algorithms in polynomial time. Hence all relations are P‑complete on VBPA.

Finally, the paper addresses regularity checking for VPAs: given a VPA, decide whether the language it accepts is regular. By exploiting the fact that visibly pushdown languages are a proper subset of context‑free languages with a well‑structured stack discipline, the authors construct a polynomial‑time reduction to language inclusion between a VPA and a finite automaton. This yields a deterministic polynomial‑time algorithm, showing that regularity checking for VPAs lies in P.

In summary, the work delivers a comprehensive complexity landscape: EXPTIME‑complete for VPAs, PSPACE‑complete for VOCA, and P‑complete for VBPA across a suite of simulation‑based preorders and equivalences, and it provides efficient (polynomial‑time) algorithms for regularity testing of visibly pushdown automata. These results deepen our understanding of how visibility constraints affect verification problems and have immediate implications for model checking and formal analysis of programs with well‑nested call‑return structures.