A Theory of Sampling for Continuous-time Metric Temporal Logic (0911.5642v2)

Abstract: This paper revisits the classical notion of sampling in the setting of real-time temporal logics for the modeling and analysis of systems. The relationship between the satisfiability of Metric Temporal Logic (MTL) formulas over continuous-time models and over discrete-time models is studied. It is shown to what extent discrete-time sequences obtained by sampling continuous-time signals capture the semantics of MTL formulas over the two time domains. The main results apply to "flat" formulas that do not nest temporal operators and can be applied to the problem of reducing the verification problem for MTL over continuous-time models to the same problem over discrete-time, resulting in an automated partial practically-efficient discretization technique.

• The paper establishes precise conditions under which flat MTL formulas remain invariant when sampled from continuous to discrete time.
• It introduces the concept of non-Berkeleyness to ensure that continuous signals change slowly enough to yield meaningful discrete samples.
• The study offers a practical reduction technique for verifying continuous-time systems using discrete-time tools, enhancing real-time system analysis.

A Theory of Sampling for Continuous-time Metric Temporal Logic

This paper addresses the concept of sampling within the field of Metric Temporal Logic (MTL) and aims to bridge the gap between the satisfiability of MTL formulas over continuous-time models and their counterparts over discrete-time models. The work focuses on discerning the extent to which discrete-time sequences, formed by sampling continuous-time signals, can preserve the semantics of MTL formulas across varying time domains.

Core Contribution

The principal contribution is the establishment of conditions under which MTL formulas remain invariant under sampling. Specifically, the paper targets "flat" formulas, characterized by the absence of nested temporal operators, and proposes an automated technique for reducing continuous-time verification problems to discrete-time counterparts. This is achieved through a specified partial discretization method that is argued to be practically efficient.

Key Findings

• Sampling Invariance: The paper demonstrates that through regular adaptations applied to MTL formulas, the semantics across continuous and discrete-time samples can be consistently related. The adaptations facilitate a compositional symmetry between the formula's interpretations over different time domains.
• Non-Berkeleyness: A new concept akin to non-Zenoness is introduced to ensure dense-time models do not change too quickly relative to sample intervals, thus enabling meaningful sampling of temporal behaviors.
• Discrete vs. Continuous Semantics: By leveraging canonical sampling and adaptation strategies ($_{\phi}, _{\phi}$), the paper shows that flat MTL formulas maintain their core temporal logic properties through sampling, provided certain conditions and constraints are met.

Implications

This research holds significant implications for the practice of model checking and verification of real-time systems. Particularly, it outlines a pathway to utilize current discrete-time verification tools for analyzing continuous-time systems by asserting properties that can be verified discreetly and inferred back to the continuous field.

Speculation on Future Developments

The theoretical findings present an opportunity for the development of new tools that facilitate the verification of real-time systems exhibiting both discrete and continuous behaviors. Practically, such advancements can streamline the implementation of digital systems that interact intricately with continuous processes, paving the way for broader application in embedded systems and control domains.

Conclusion

Overall, the paper contributes a rigorous approach to understanding how MTL, which spans different temporal logic landscapes, can be unified through sampling techniques. Although the results presented are primarily theoretical, they offer a foundational strategy for enhancing practical verification methodologies, particularly in systems where real-time and hybrid dynamics are critical.