Polynomial Bound on Accelerated Label Sizes

Determine whether the label sizes generated by the quantitative reachability procedure for weighted pushdown systems over pumping semirings remain polynomial in the number of control states, which would reduce the procedure's complexity from doubly exponential to singly exponential.

Background

The paper presents a fixpoint algorithm for quantitative reachability in weighted pushdown systems over pumping semirings. The algorithm represents runs using accelerated labelled trees and terminates because tree heights are bounded and only finitely many relevant pumping expressions occur at each height.

The complexity analysis bounds the size of labels by an exponential function of the squared number of control states. This bound leads to a doubly exponential running-time estimate. The authors note that the second exponential arises entirely from the operation that concatenates the labels of two child trees, and that the resulting bound is already far from the known polynomial bound in the Boolean case. Establishing that label sizes are instead polynomial in the number of control states would make the overall procedure singly exponential.

References

Note that our bound is not tight. The second exponential stems entirely from step "(H3)", which concatenates both its children's labels, and already for $\osem{S} = \trivialsem$, the bound is far from the known polynomial one . Whether the label "sizes" stay polynomial in $|Q|$, which would make our procedure singly exponential, remains open.

— Quantitative Verification of Infinite-State Networks  (2610.11991 - Nevatia et al., 8 Oct 2026) in Section 5, subsection “Complexity” (immediately following Proposition 5.1)