Quantitative Verification of Infinite-State Networks

2026-10-08

AI Summary

This study addresses the challenge of jointly verifying infinite-state and quantitative behaviors in recursive network protocols by proposing a quantitative verification framework for networks of weighted pushdown systems. Methodologically, it introduces the novel concept of "pumping semirings" to establish algebraic conditions for weight closure computation, alongside a hot extension mechanism that separates accelerated weights to ensure precision. Furthermore, piecewise tree algebras are employed for compositional representation, combined with a termination saturation algorithm to solve quantitative reachability over algebraic domains admitting infinite ascending chains. This work achieves unified computation of upward and downward closures for context-free languages and provides the first exact quantitative safety and reachability analysis for acyclic pushdown system networks, effectively overcoming state-space explosion caused by cyclic pumping.

Abstract

Many network protocols and distributed systems combine recursion, unbounded local state, and quantitative behaviour such as cost, latency, or reliability. Existing decidability results for networks of pushdown systems are largely qualitative, and do not extend to quantitative analyses, which must jointly track traces and weights. We present a framework for the quantitative verification of acyclic networks of weighted pushdown systems. Finitely summarising a recursive network component requires collapsing the infinite family of runs obtained by pumping its nested loops, and doing so \emph{exactly}, rather than by over-approximation, is a standing difficulty. We give a class of weight domains where this is possible: pumping semirings, those in which the weights accumulated by such a family collapse to a closed form; informally, the domain must be unable to count iterations. The class admits domains with infinite ascending chains, such as the arctic semiring and downward-closed languages. Over these, we give a terminating saturation algorithm computing quantitative reachability exactly, resting on two ideas: segment tree algebras, a compositional representation of runs, and thermal extensions, which symbolically separate accelerated weights so that further acceleration remains exact. We use this to compute upward and downward closures of context-free languages uniformly, and to lift the algorithm to acyclic networks over "thin" pumping semirings, yielding the first quantitative safety and reachability analyses for networks of pushdown systems.

Problem

• quantitative verification
• weighted pushdown systems
• infinite-state networks
• pumping semirings
• quantitative reachability

Innovation

• quantitative verification
• weighted pushdown systems
• pumping semirings
• segment tree algebras
• thermal extensions