Note on Undecidability of Bisimilarity for Second-Order Pushdown Processes

Broadbent and Göller (FSTTCS 2012) proved the undecidability of bisimulation equivalence for processes generated by ε-free second-order pushdown automata. We add a few remarks concerning the used proof technique, called Defender’s forcing, and the related undecidability proof for first-order pushdown automata with ε-transitions (Jančar and Srba, JACM 2008). Language equivalence of pushdown automata (PDA) is a well-known problem in computer science community. There are standard textbook proofs showing the undecidability even for ε-free PDA, i.e. for PDA that have no ε-transitions; such PDA are sometimes called real-time PDA. The decidability question of language equivalence for deterministic PDA (DPDA) was a famous long-standing open problem. It was positively answered by Oyamaguchi [3] for ε-free DPDA and later by Sénizergues [4] for the whole class of DPDA. Besides their role of language acceptors, PDA can be also viewed as generators of (infinite) labelled transition systems; in this context it is natural to study another fundamental equivalence, namely bisimulation equivalence, also called bisimilarity. This equivalence is finer than language equivalence, but the two equivalences in principle coincide on deterministic systems. Sénizergues [5] showed an involved proof of the decidability of bisimilarity for (nondeterministic) ε-free PDA but also for PDA in which ε-transitions are deterministic, popping and do not collide with ordinary input a-transitions. It turned out that a small relaxation, allowing for nondeterministic popping ε-transitions, already leads to undecidability [2]. In [2], the authors of this note also explicitly describe a general proof technique called Defender’s forcing. It is a simple, yet powerful, idea related to the bisimulation game played between Attacker and Defender; it was used, sometimes implicitly, also in context of other hardness results for bisimilarity on various classes of infinite state systems. The classical PDA, to which we have been referring so far, are the first-order PDA in the hierarchy of higher-order PDA that were introduced in connection with higher-order recursion schemes already in 1970s. The decidability question for equivalence of deterministic nth-order PDA, where n ≥ 2, seems to be open so far. A step towards a solution was made by Stirling [6] who showed the decidability for a subclass of ε-free second-order DPDA. Recently Broadbent and Göller [1] noted that the results in [2], or anywhere else in the literature, do not answer the decidability question for bisimilarity of ε-free second-order PDA.