Quantitative Monitoring of Signal First-Order Logic

2026-02-28

AI Summary

This work addresses the limitations of existing Signal First-Order Logic (SFO), which lacks quantitative semantics and online monitoring capabilities, hindering the verification of complex real-time properties in hybrid systems. We introduce, for the first time, a robustness-based quantitative semantics for SFO and define its past-time fragment. To enable efficient online monitoring, we propose a “pastification” transformation that converts bounded-response formulas into equisatisfiable past-time formulas. Building on this foundation, we develop the first publicly available prototype system supporting full SFO, enabling quantitative runtime verification of properties beyond the expressiveness of Signal Temporal Logic. Experimental evaluation across multiple benchmarks demonstrates that our approach is both practical and efficient.

Technology Category

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

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Data transparency and provenance

Abstract

Runtime monitoring checks, during execution, whether a partial signal produced by a hybrid system satisfies its specification. Signal First-Order Logic (SFO) offers expressive real-time specifications over such signals, but currently comes only with Boolean semantics and has no tool support. We provide the first robustness-based quantitative semantics for SFO, enabling the expression and evaluation of rich real-time properties beyond the scope of existing formalisms such as Signal Temporal Logic. To enable online monitoring, we identify a past-time fragment of SFO and give a pastification procedure that transforms bounded-response SFO formulas into equisatisfiable formulas in this fragment. We then develop an efficient runtime monitoring algorithm for this past-time fragment and evaluate its performance on a set of benchmarks, demonstrating the practicality and effectiveness of our approach. To the best of our knowledge, this is the first publicly available prototype for online quantitative monitoring of full SFO.

Problem

• Signal First-Order Logic
• runtime monitoring
• quantitative semantics
• hybrid systems
• real-time specifications

Innovation

• Signal First-Order Logic
• quantitative semantics
• runtime monitoring
• pastification
• robustness