2026-08-25T22:08:25Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:27353 2026-08-24T14:08:31Z ddc:004 open_access

Monitoring Discounted Sum Properties Cano, Filip Henzinger, Thomas A. Kueffner, Konstantin Saraç, N. Ege Runtime Verification Probabilistic Systems Quantitative Verification Approximate Monitoring 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 ε-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 (ε,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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Filip Cano and Thomas A. Henzinger and Konstantin Kueffner and N. Ege Saraç 2026 Is Part Of LIPIcs, Volume 391, 37th International Conference on Concurrency Theory (CONCUR 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CONCUR.2026.22 urn:nbn:de:0030-drops-273533 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2026.22 eng https://creativecommons.org/licenses/by/4.0/legalcode