[Paper Review] Model Checking Stochastic Branching Processes

Taolue Chen, Klaus Dräger+1 more|arXiv (Cornell University)|Jun 6, 2012

Formal Methods in Verification7 references4 citations

TL;DR

This paper presents PSPACE-complete algorithms for model checking stochastic branching processes against probabilistic tree logics and deterministic parity tree automata, enabling verification of properties like almost-sure satisfaction of LTL-like conditions on all branches. It introduces a tree extension of PCTL (PTTL), proves its model checking is in PSPACE, and shows the qualitative fragment is decidable in polynomial time, unifying verification of Markov chains and nondeterministic transition systems under a single framework for random trees.

ABSTRACT

Stochastic branching processes are a classical model for describing random trees, which have applications in numerous fields including biology, physics, and natural language processing. In particular, they have recently been proposed to describe parallel programs with stochastic process creation. In this paper, we consider the problem of model checking stochastic branching process. Given a branching process and a deterministic parity tree automaton, we are interested in computing the probability that the generated random tree is accepted by the automaton. We show that this probability can be compared with any rational number in PSPACE, and with 0 and 1 in polynomial time. In a second part, we suggest a tree extension of the logic PCTL, and develop a PSPACE algorithm for model checking a branching process against a formula of this logic. We also show that the qualitative fragment of this logic can be model checked in polynomial time.

Motivation & Objective

• To address the lack of formal verification techniques for stochastic branching processes modeling dynamic process creation in parallel programs.
• To unify the verification of probabilistic Markov chains and nondeterministic transition systems under a single framework for random trees.
• To develop a logic (PTTL) suitable for specifying and verifying probabilistic properties of infinite random trees generated by branching processes.
• To establish complexity bounds for model checking, showing PSPACE-completeness for quantitative queries and polynomial time for qualitative ones.
• To extend classical model checking to tree-based probabilistic systems where children of nodes are not independent, requiring novel reductions.

Proposed method

• Reduces model checking of branching processes against a deterministic parity tree automaton to computing the probability of reaching 'good' states along all branches.
• Introduces a transformation of the original branching process into an auxiliary process with extended types to encode truth values of subformulas, enabling reduction to reachability properties.
• Uses a function fϕ,ψ to map types to {0, 1/2, 1} based on whether they satisfy φ, ψ, or their negations, encoding the status of temporal logic formulas.
• Constructs a new branching process Δ′ where each rule is split into three variants based on the value of fϕ,ψ, allowing encoding of AU and AR operators as AF and AG properties.
• Applies known results on PSPACE decidability of reachability probabilities in branching processes (via Propositions 1 and 2) to derive complexity bounds.
• Employs a duality lemma to reduce AG to AF, enabling the use of existing algorithms for probability of eventual reachability.

Experimental results

Research questions

• RQ1 Can the probability that a random tree generated by a stochastic branching process satisfies a given ω-regular property on all branches be computed efficiently?
• RQ2 Is model checking against the qualitative fragment of a new probabilistic tree logic (PTTL) decidable in polynomial time?
• RQ3 How does the complexity of model checking stochastic branching processes compare to classical Markov chains, especially regarding qualitative properties (0 or 1 probability)?
• RQ4 Can temporal logic operators like AU and AR in PTTL be reduced to reachability problems in transformed branching processes?
• RQ5 What are the complexity bounds for model checking when the specification is given as a deterministic parity tree automaton?

Key findings

• Model checking a stochastic branching process against a deterministic parity tree automaton is decidable in PSPACE.
• The problem of comparing the acceptance probability with any rational number is in PSPACE.
• Checking whether the acceptance probability is exactly 0 or 1 can be done in polynomial time.
• The qualitative fragment of the proposed PTTL logic is decidable in polynomial time.
• Model checking PTTL formulas is in PSPACE, and the reduction technique using auxiliary branching processes with truth-value encoding is sound and complete.
• The approach generalizes both Markov chain model checking and non-probabilistic transition system verification, showing that qualitative properties in branching processes are sensitive to exact transition probabilities, unlike in Markov chains.