Monitoring Discounted Sum Properties
For runtime verification practitioners, it provides the first principled framework for monitoring discounted sums with finite resources, addressing a fundamental trade-off between volatility and over-aggregation.
The paper studies runtime monitoring of discounted sums, proving that exact monitoring requires infinite memory, but introducing ε-approximately sound monitoring with finite memory bounds. It extends to stochastic processes with optimal statistical guarantees and demonstrates practical applicability in algorithmic fairness.
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.