Monitoring Discounted Sum Properties

2026-06-24

AI Summary

This work addresses the fundamental trade-off in runtime monitoring of quantitative signals, where instantaneous noise is high while long-term averaging obscures local structure. It introduces, for the first time, a formal framework for discounted monitoring, defining ε-approximately reliable monitoring in deterministic settings and (ε,δ)-reliability in stochastic ones, thereby establishing statistical optimality and fundamental limits on precision. The authors develop a memory-efficient monitoring algorithm by integrating statistical hypothesis testing with a novel arithmetic specification language supporting multiple discounting schemes, an affine register machine model, and both synchronous and asynchronous semantics. Theoretical analysis yields explicit, tight lower bounds on memory and observation requirements. Empirical evaluation demonstrates the approach’s effectiveness in practical scenarios such as algorithmic fairness.

Abstract

Runtime monitoring of quantitative signals faces a fundamental trade-off between volatility and over-aggregation: instantaneous observations are noisy, while long-run averages obscure local structure. Localisation measures such as discounted averages offer a principled middle ground, yet remain poorly understood in runtime verification. This paper studies discounted sums from a monitoring perspective, in both deterministic and stochastic settings. We formalize the discounted monitoring problem and show that exact, sound monitoring of discounted sums cannot be achieved with finite memory. To overcome this impossibility, we introduce $\varepsilon$-approximately sound monitoring, deriving explicit bounds on memory and observation requirements. We then extend the framework to stochastic processes via expected discounted sums, defining pointwise and uniform $(\varepsilon,δ)$-soundness notions, establishing statistical optimality, and proving impossibility beyond a precision threshold. We also formalize the resource complexity of deterministic discounted monitoring via affine register machines and prove a tight worst-case lower bound. Finally, we present a specification language for arithmetic expressions over multiple discounted sums with synchronous and asynchronous semantics, and evaluate our approach on practical scenarios including algorithmic fairness.

Problem

• runtime monitoring
• discounted sums
• quantitative signals
• finite memory
• stochastic processes

Innovation

• discounted sums
• runtime monitoring
• approximate soundness
• stochastic processes
• resource complexity