[Paper Review] Model checking probabilistic pushdown automata

Javier Esparza, Antonı́n Kučera+1 more|arXiv (Cornell University)|Jul 13, 2004

Formal Methods in Verification32 references124 citations

TL;DR

This paper establishes the decidability of qualitative and quantitative model checking for probabilistic pushdown automata (pPDA) across multiple probabilistic logics, including PCTL and properties expressible via generalized random walks or deterministic Büchi automata. The key contribution is a formal framework enabling verification of probabilistic systems with complex control and stack-based state transitions.

ABSTRACT

We consider the model checking problem for probabilistic pushdown automata (pPDA) and properties expressible in various probabilistic logics. We start with properties that can be formulated as instances of a generalized random walk problem. We prove that both qualitative and quantitative model checking for this class of properties and pPDA is decidable. Then, we show that model checking for the qualitative fragment of the logic PCTL and pPDA is also decidable. Moreover, we develop an error-tolerant model checking algorithm for general PCTL and the subclass of stateless pPDA. Finally, we consider the class of properties definable by deterministic Buchi automata, and show that both qualitative and quantitative model checking for pPDA is decidable.

Motivation & Objective

• To address the challenge of verifying probabilistic systems with recursive control flow and unbounded stack usage.
• To determine whether model checking for pPDA is decidable under various probabilistic logics, particularly PCTL and properties expressible as generalized random walks.
• To develop an error-tolerant algorithm for general PCTL model checking in the restricted case of stateless pPDA.
• To extend decidability results to properties definable by deterministic Büchi automata.

Proposed method

• Reduces model checking problems for pPDA to generalized random walk problems over multidimensional state spaces.
• Applies decidability results from stochastic processes to prove the existence of solutions for qualitative and quantitative queries.
• Leverages the structure of pPDA to define recursive equations that capture reachability and recurrence probabilities.
• Employs automata-theoretic techniques to reduce properties defined by deterministic Büchi automata to acceptance conditions over infinite runs.
• Designs an error-tolerant algorithm that approximates PCTL model checking results for stateless pPDA with bounded error.
• Uses probabilistic bisimulation and abstraction techniques to reduce state space explosion in verification.

Experimental results

Research questions

• RQ1 Is model checking decidable for pPDA and properties expressible as generalized random walks?
• RQ2 Can qualitative and quantitative model checking be decided for pPDA using PCTL?
• RQ3 Is there a practical algorithm for PCTL model checking in stateless pPDA with tolerance for approximation errors?
• RQ4 Can properties defined by deterministic Büchi automata be verified over pPDA with both qualitative and quantitative guarantees?

Key findings

• Qualitative and quantitative model checking for pPDA and generalized random walk properties is decidable.
• Model checking for the qualitative fragment of PCTL over pPDA is decidable.
• An error-tolerant algorithm is developed for general PCTL model checking in stateless pPDA, enabling practical verification with bounded error.
• Both qualitative and quantitative model checking is decidable for pPDA when properties are defined by deterministic Büchi automata.