Four-valued monitorability of ω-regular languages

The use of runtime verification 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 during system execution. However, classical two-valued monitorability suffers from two inherent limitations, which eventually increase runtime overhead. First, no information is available regarding whether only one verdict (satisfaction or violation) can be detected. Second, it does not tell us whether verdicts can be detected starting from the current monitor state during system execution.

This paper proposes a new notion of four-valued monitorability for ω -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 ω -regular languages, both at the language-level and at the state-level. Experimental results show that our tool implementation Monic can correctly, and quickly, report both two-valued and four-valued monitorability.