Effective quantitative convergence bounds for decisive pWSTS

Characterize the effective decisive probabilistic well-structured transition systems for which, given an upward-closed target U and an initial state x, one can compute a decreasing function f_{U,x}: N a0a0→ [0,1] converging to zero and satisfying P_x(τ_{U ∪ \widetilde U} > n) ≤ f_{U,x}(n) for every n, together with an effectively computable threshold for each precision ε.

Background

The paper’s infinite-horizon approximation procedure terminates for effective pWSTS that are decisive with respect to the upward-closed target, but decisiveness provides no computable a priori bound on the exploration depth. The authors therefore ask for a stronger quantitative form of decisiveness: an effectively computable decreasing upper bound on the probability that a run has not yet reached either the target or a state from which the target is unreachable.

Such a bound would provide an effectively computable exploration threshold for every desired precision and would make the quantitative behavior of decisive probabilistic infinite-state systems more effectively tractable.

References

Several questions remain open. The first is to characterize the effective decisive pWSTS for which, given an upward-closed target U and an initial state x, one can compute a decreasing function f_{U,x} : \mathbb{N} \to [0,1] converging to zero and satisfying

\mathbb{P}x!\left(\tau{U \cup \widetilde U} > n\right) \leq f_{U,x}(n)

for every n, together with an effectively computable threshold for each precision \varepsilon.

— Quantitative coverability for probabilistic well-structured transition systems  (2609.26312 - Faure et al., 22 Sep 2026) in Section 6, “Conclusion and open problems” (Section \ref{sec:ccl})