conference paper published yes Thomas A Henzinger author 40876CD8-F248-11E8-B48F-1D18A9856A870000-0002-2985-7724 Nicolas Adrien Mazzocchi author b26baa86-3308-11ec-87b0-8990f34baa85 Naci E Sarac author 8C6B42F8-C8E6-11E9-A03A-F2DCE5697425 GradSch department ToHe department FOSSACS: Foundations of Software Science and Computation Structures Vigilant Algorithmic Monitoring of Software project Safety and liveness are elementary concepts of computation, and the foundation of many verification paradigms. The safety-liveness classification of boolean properties characterizes whether a given property can be falsified by observing a finite prefix of an infinite computation trace (always for safety, never for liveness). In quantitative specification and verification, properties assign not truth values, but quantitative values to infinite traces (e.g., a cost, or the distance to a boolean property). We introduce quantitative safety and liveness, and we prove that our definitions induce conservative quantitative generalizations of both (1)~the safety-progress hierarchy of boolean properties and (2)~the safety-liveness decomposition of boolean properties. In particular, we show that every quantitative property can be written as the pointwise minimum of a quantitative safety property and a quantitative liveness property. Consequently, like boolean properties, also quantitative properties can be min-decomposed into safety and liveness parts, or alternatively, max-decomposed into co-safety and co-liveness parts. Moreover, quantitative properties can be approximated naturally. We prove that every quantitative property that has both safe and co-safe approximations can be monitored arbitrarily precisely by a monitor that uses only a finite number of states. https://research-explorer.ista.ac.at/download/12467/12468/qsl.pdf application/pdfno https://research-explorer.ista.ac.at/download/12467/13153/2023_LNCS_HenzingerT.pdf application/pdfno Springer Nature2023Paris, France eng 0302-9743 1611-3349 9783031308284 2301.11175 00128860930001710.1007/978-3-031-30829-1_17 13992349-370 Henzinger TA, Mazzocchi NA, Sarac NE. 2023. Quantitative safety and liveness. 26th International Conference Foundations of Software Science and Computation Structures. FOSSACS: Foundations of Software Science and Computation Structures, LNCS, vol. 13992, 349–370. Henzinger, Thomas A, Nicolas Adrien Mazzocchi, and Naci E Sarac. “Quantitative Safety and Liveness.” In <i>26th International Conference Foundations of Software Science and Computation Structures</i>, 13992:349–70. Springer Nature, 2023. <a href="https://doi.org/10.1007/978-3-031-30829-1_17">https://doi.org/10.1007/978-3-031-30829-1_17</a>. Henzinger TA, Mazzocchi NA, Sarac NE. Quantitative safety and liveness. In: <i>26th International Conference Foundations of Software Science and Computation Structures</i>. Vol 13992. Springer Nature; 2023:349-370. doi:<a href="https://doi.org/10.1007/978-3-031-30829-1_17">10.1007/978-3-031-30829-1_17</a> T. A. Henzinger, N. A. Mazzocchi, and N. E. Sarac, “Quantitative safety and liveness,” in <i>26th International Conference Foundations of Software Science and Computation Structures</i>, Paris, France, 2023, vol. 13992, pp. 349–370. Henzinger, T. A., Mazzocchi, N. A., &#38; Sarac, N. E. (2023). Quantitative safety and liveness. In <i>26th International Conference Foundations of Software Science and Computation Structures</i> (Vol. 13992, pp. 349–370). Paris, France: Springer Nature. <a href="https://doi.org/10.1007/978-3-031-30829-1_17">https://doi.org/10.1007/978-3-031-30829-1_17</a> T.A. Henzinger, N.A. Mazzocchi, N.E. Sarac, in:, 26th International Conference Foundations of Software Science and Computation Structures, Springer Nature, 2023, pp. 349–370. Henzinger, Thomas A., et al. “Quantitative Safety and Liveness.” <i>26th International Conference Foundations of Software Science and Computation Structures</i>, vol. 13992, Springer Nature, 2023, pp. 349–70, doi:<a href="https://doi.org/10.1007/978-3-031-30829-1_17">10.1007/978-3-031-30829-1_17</a>. 124672023-01-31T07:23:56Z2025-09-09T12:21:08Z