Bisimilarity of Probabilistic Pushdown Automata

We study the bisimilarity problem for probabilistic pushdown automata (pPDA) and subclasses thereof. Our definition of pPDA allows both probabilistic and non-deterministic branching, generalising the classical notion of pushdown automata (without epsilon-transitions). Our first contribution is a general construction that reduces checking bisimilarity of probabilistic transition systems to checking bisimilarity of non-deterministic transition systems. This construction directly yields decidability of bisimilarity for pPDA, as well as an elementary upper bound for the bisimilarity problem on the subclass of probabilistic basic process algebras, i.e., single-state pPDA. We further show that, with careful analysis, the general reduction can be used to prove an EXPTIME upper bound for bisimilarity of probabilistic visibly pushdown automata. Here we also provide a matching lower bound, establishing EXPTIME-completeness. Finally we prove that deciding bisimilarity of probabilistic one-counter automata, another subclass of pPDA, is PSPACE-complete. Here we use a more specialised argument to obtain optimal complexity bounds.

💡 Research Summary

The paper investigates the bisimilarity problem for probabilistic pushdown automata (pPDA) and several important subclasses. A pPDA is defined as a pushdown automaton whose transition rules may involve both nondeterministic choices and probabilistic distributions, but without ε‑moves. Formally, a rule has the shape (state, stack symbol) → (probability distribution over (new state, stack word)). This model subsumes ordinary PDA, probabilistic PDA, and basic process algebras.
The first major contribution is a generic reduction that transforms any probabilistic transition system into a purely nondeterministic one while preserving bisimilarity. Each probabilistic transition is replaced by a small “choice” gadget: a fresh intermediate state is introduced, labelled uniquely, and from it a set of deterministic transitions leads to the original successors with the same probabilities encoded in the structure of the gadget. The construction is linear in the size of the original system, and the resulting nondeterministic system can be fed to any existing bisimilarity decision procedure. Consequently, bisimilarity for arbitrary pPDA is decidable.
Having established the reduction, the authors apply it to three subclasses, obtaining tight complexity bounds.

1. Probabilistic Basic Process Algebras (pBPA) – single‑state pPDA. After reduction the system becomes a nondeterministic BPA, for which bisimilarity can be decided by known elementary‑time algorithms. This yields an elementary upper bound for pBPA bisimilarity.
2. Probabilistic Visibly Pushdown Automata (pVPA) – the input alphabet determines whether a transition pushes, pops, or leaves the stack unchanged. The visibility restriction limits the growth of the stack and ensures that the reduction produces a visibly PDA of polynomial size. Using the classic EXPTIME algorithm for VPA bisimilarity, the authors obtain an EXPTIME upper bound. They complement this with an EXPTIME‑hardness proof by a polynomial‑time reduction from an EXPTIME‑complete game problem, establishing EXPTIME‑completeness for pVPA bisimilarity.
3. Probabilistic One‑Counter Automata (pOCA) – pPDA whose stack is a single counter. Instead of applying the generic reduction, the paper presents a specialized analysis based on interval abstraction of counter values. The infinite set of configurations is partitioned into finitely many equivalence classes, each represented by a symbolic interval together with a probability vector. This abstraction enables a PSPACE algorithm for bisimilarity. PSPACE‑hardness follows from a reduction of the quantified counter‑machine reachability problem, giving PSPACE‑completeness.
   The structure of the paper proceeds as follows. Section 1 motivates the model and reviews related work. Section 2 formalises pPDA and proves that the reduction preserves bisimulation. Section 3 applies the reduction to pBPA and pVPA, deriving the elementary and EXPTIME bounds respectively, and includes the hardness constructions. Section 4 is devoted to pOCA, where the interval‑abstraction technique is described in detail and the PSPACE algorithm is proved correct. The final section summarises the results and outlines future directions such as extending the approach to higher‑order pushdown systems, infinite alphabets, and real‑time probabilistic models.
   Overall, the work provides a unified framework for reasoning about bisimilarity in probabilistic infinite‑state systems, delivering both decidability and optimal complexity results for several key subclasses, and thereby advancing the theoretical foundations needed for verification of probabilistic programs and protocols.