conference paper published yes Filip Cano Cordoba author 708cad98-e86a-11ef-8098-bdae2d7c6af10000-0002-0783-904X Thomas A Henzinger author 40876CD8-F248-11E8-B48F-1D18A9856A870000-0002-2985-7724 Konstantin Kueffner author 8121a2d0-dc85-11ea-9058-af578f3b45150000-0001-8974-2542 Naci E Sarac author 8C6B42F8-C8E6-11E9-A03A-F2DCE5697425 ToHe department GradSch department CONCUR: Conference on Concurrency Theory Vigilant Algorithmic Monitoring of Software project 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. https://research-explorer.ista.ac.at/download/22921/22978/2026_LIPIcsCONCUR_Cano.pdf application/pdfno Schloss Dagstuhl - Leibniz-Zentrum für Informatik2026Liverpool, United Kingdom eng Runtime VerificationProbabilistic SystemsQuantitative VerificationApproximate Monitoring 1868-8969 9783959774475 2606.2597910.4230/LIPIcs.CONCUR.2026.22 391 Cano Cordoba F, Henzinger TA, Kueffner K, Sarac NE. Monitoring discounted sum properties. In: <i>37th International Conference on Concurrency Theory</i>. Vol 391. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2026. doi:<a href="https://doi.org/10.4230/LIPIcs.CONCUR.2026.22">10.4230/LIPIcs.CONCUR.2026.22</a> Cano Cordoba, Filip, Thomas A Henzinger, Konstantin Kueffner, and Naci E Sarac. “Monitoring Discounted Sum Properties.” In <i>37th International Conference on Concurrency Theory</i>, Vol. 391. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026. <a href="https://doi.org/10.4230/LIPIcs.CONCUR.2026.22">https://doi.org/10.4230/LIPIcs.CONCUR.2026.22</a>. Cano Cordoba, F., Henzinger, T. A., Kueffner, K., &#38; Sarac, N. E. (2026). Monitoring discounted sum properties. In <i>37th International Conference on Concurrency Theory</i> (Vol. 391). Liverpool, United Kingdom: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href="https://doi.org/10.4230/LIPIcs.CONCUR.2026.22">https://doi.org/10.4230/LIPIcs.CONCUR.2026.22</a> F. Cano Cordoba, T. A. Henzinger, K. Kueffner, and N. E. Sarac, “Monitoring discounted sum properties,” in <i>37th International Conference on Concurrency Theory</i>, Liverpool, United Kingdom, 2026, vol. 391. Cano Cordoba, Filip, et al. “Monitoring Discounted Sum Properties.” <i>37th International Conference on Concurrency Theory</i>, vol. 391, 22:1-22:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026, doi:<a href="https://doi.org/10.4230/LIPIcs.CONCUR.2026.22">10.4230/LIPIcs.CONCUR.2026.22</a>. F. Cano Cordoba, T.A. Henzinger, K. Kueffner, N.E. Sarac, in:, 37th International Conference on Concurrency Theory, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026. Cano Cordoba F, Henzinger TA, Kueffner K, Sarac NE. 2026. Monitoring discounted sum properties. 37th International Conference on Concurrency Theory. CONCUR: Conference on Concurrency Theory, LIPIcs, vol. 391, 22:1-22:19. 229212026-09-13T22:01:54Z2026-09-22T06:02:06Z