Black-box Testing Liveness Properties of Partially Observable Stochastic Systems

TL;DR

This work addresses black-box testing of finite-state probabilistic systems against ω-regular specifications under partial observability. It introduces restart-based strategies that guarantee termination and violation-tail detection when violations have positive probability, and proves asymptotically optimal bounds on the time to the last restart via progress radius and progress probability. A key contribution is reducing general ω-regular testing to Rabin languages and delivering a memory-efficient strategy 𝔖[f] parameterized by a growth function with limsup f(n)=∞, achieving logarithmic memory. Empirically, the approach scales to large benchmarks, often surpassing fully observable strategies and enabling violation discovery in systems where prior methods fail.

Abstract

We study black-box testing for stochastic systems and arbitrary $ω$-regular specifications, explicitly including liveness properties. We are given a finite-state probabilistic system that we can only execute from the initial state. We have no information on the number of reachable states, or on the probabilities; further, we can only partially observe the states. The only action we can take is to restart the system. We design restart strategies guaranteeing that, if the specification is violated with non-zero probability, then w.p.1 the number of restarts is finite, and the infinite run executed after the last restart violates the specification. This improves on previous work that required full observability. We obtain asymptotically optimal upper bounds on the expected number of steps until the last restart. We conduct experiments on a number of benchmarks, and show that our strategies allow one to find violations in Markov chains much larger than the ones considered in previous work.

Figures (6)

• Figure 1: A family of partially observable Markov chains
• Figure 2: Strategy $\mathfrak{S}[f]$ for the Rabin language $\mathcal{R}_k$ and a function $f \colon \mathbb{N} \rightarrow \mathbb{N}$.
• Figure 3: Left: Intuitively, $r_\gamma$ and $r_\beta$ denote an upper bound of how hard it is to reach a BSCC. Right:$R_\gamma$ and $R_\beta$ measure how hard it is to reach a state with label $\bm{e}_i$/$\bm{f}_i$ inside the BSCCs
• Figure 4: A Markov chain (left), and (a finite prefix of) one of its runs for $n=2$ and $f(n)=n$ (right). After $2$ steps, the run has reached a good BSCC $\mathcal{B}$. After $4$ steps, $\sigma$ checks whether to restart, but decides against it because of $s_\text{goal}$ in step $4$. After 8 steps it checks again, restarting this time. This restart is covered by the third case of the case distinction, with $k=4$. The run must have visited $s_\text{goal}$ in step 4, because otherwise the minimal $k$ would be $3$ or less.
• Figure 5: Two families of Markov chains. The initial state is $s_\text{start}$. The good runs are those that visit $s_\text{goal}$ infinitely often. For the top chains, $P_\text{good}=q$, $R_\textnormal{m} = M$, and $P_\textnormal{m} = p$. For the bottom chains, $P_\text{good}=p^M$, $R_\textnormal{m} =M$, $P_\textnormal{m} =p^M$.

Theorems & Definitions (38)

• Example 1.1
• Definition 3.1: Black-box testing strategies
• Definition 3.2
• Lemma 3.2
• proof
• Definition 4.1: Good-reachability and good-witness radii
• Definition 4.2: Progress radius
• Definition 4.3
• Lemma 4.3
• Lemma 4.3: Restarting probability