Runtime Verification (RV) is a lightweight formal technique in which program or system execution is monitored and analyzed, to check whether certain properties are satisfied or violated after a finite number of steps. The use of RV has led to interest in deciding whether a property is monitorable: whether it is always possible for the satisfaction or violation of the property to be determined after a finite future continuation. However, classical two-valued monitorability suffers from two inherent limitations. First, a property can only be evaluated as monitorable or non-monitorable; no information is available regarding whether only one verdict (satisfaction or violation) can be detected. Second, monitorability is defined at the language-level and does not tell us whether satisfaction or violation can be detected starting from the current monitor state during system execution. To address these limitations, this paper proposes a new notion of four-valued monitorability for $\omega$-languages and applies it at the state-level. Four-valued monitorability is more informative than two-valued monitorability as a property can be evaluated as a four-valued result, denoting that only satisfaction, only violation, or both are active for a monitorable property. We can also compute state-level weak monitorability, i.e., whether satisfaction or violation can be detected starting from a given state in a monitor, which enables state-level optimizations of monitoring algorithms. Based on a new six-valued semantics, we propose procedures for computing four-valued monitorability of $\omega$-regular languages, both at the language-level and at the state-level. We have developed a new tool that implements the proposed procedure for computing monitorability of LTL formulas.

中文翻译：

---

$\omega$-regular 语言的四值可监控性

运行时验证 (RV) 是一种轻量级形式技术，在该技术中，程序或系统的执行被监视和分析，以检查在有限的步骤数后是否满足或违反某些属性。RV 的使用引起了人们对决定财产是否可监控的兴趣：在有限的未来延续后，是否总是可以确定财产是否满足或违反。然而，经典的二值可监控性受到两个固有的限制。首先，属性只能被评估为可监控或不可监控；没有关于是否只能检测到一项裁决（满意或违反）的信息。第二，可监控性是在语言级别定义的，它不会告诉我们是否可以在系统执行期间从当前监控状态开始检测到满意或违规。为了解决这些限制，本文提出了 $\omega$ 语言的四值可监控性的新概念，并将其应用于州级。四值可监控性比二值可监控性提供更多信息，因为可以将属性评估为四值结果，表示对于可监控属性，只有满意、只有违规或两者都处于活动状态。我们还可以计算状态级别的弱可监控性，即是否可以从监视器中的给定状态开始检测满意或违规，从而实现状态级别的监控算法优化。基于新的六值语义，我们提出了在语言级别和州级别计算 $\omega$-regular 语言的四值可监控性的程序。我们开发了一种新工具，用于实现建议的 LTL 公式可监控性计算程序。