Time for Timed Monitorability

2025-04-14

AI Summary

This work addresses the monitorability problem for timing specifications in real-time systems, marking the first extension of discrete-time monitorability theory to real-time temporal logic. We introduce a refined monitorability definition imposing dual bounds—on both event count and elapsed time—and establish the first theoretical framework for monitorability of real-time temporal specifications. We prove that for deterministic Timed Muller automata, both monitorability and the minimal monitoring delay (together with its associated event bound) are decidable and effectively computable; in contrast, for nondeterministic Timed Büchi automata, both properties are undecidable. Our results precisely characterize the theoretical limits of real-time runtime monitoring, thereby providing a formal foundation for feasibility analysis and tool development in online monitoring of real-time systems.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Monitoring is an important part of the verification toolbox, in particular in situations where exhaustive verification using, e.g., model-checking, is infeasible. The goal of online monitoring is to determine the satisfaction or violation of a specification during runtime, i.e., based on finite execution prefixes. However, not every specification is amenable to monitoring, e.g., properties for which no finite execution can witness satisfaction or violation. Monitorability is the question whether a given specification is amenable to monitoring, and has been extensively studied in discrete time. Here, we study, for the first time, the monitorability problem for real-time specifications. For specifications given by deterministic Timed Muller Automata, we prove decidability while we show that the problem is undecidable for specifications given by nondeterministic Timed B""uchi automata. Furthermore, we refine monitorability to also determine bounds on the number of events as well as the time that must pass before monitoring the property may yield an informative verdict. We prove that for deterministic Timed Muller automata, such bounds can be effectively computed. In contrast we show that for nondeterministic Timed B""uchi automata such bounds are not computable.

Problem

• Study monitorability for real-time specifications.
• Decide monitorability for deterministic Timed Muller Automata.
• Compute bounds for monitoring deterministic Timed Muller Automata.

Innovation

• Monitorability for real-time specifications
• Decidability with deterministic Timed Muller Automata
• Computing bounds for monitoring events and time