Equivalence of monitorability notions (Aceto et al. vs. Bauer et al.)

Determine whether the monitorability definition used by Aceto et al.—a formula is monitorable if and only if a monitor can be synthesised in their modal μ-calculus framework—is equivalent to the monitorability definition used by Bauer et al. for LTL3, which characterises non-monitorability via the existence of ugly prefixes in the good/bad/ugly prefix classification of finite traces.

Background

The paper discusses two approaches to monitorability. Bauer et al. define good and bad prefixes for LTL properties, and introduce ugly prefixes; the presence of ugly prefixes indicates non-monitorability under their LTL3-based perspective.

Aceto et al. present a positive notion of monitorability in a modal μ-calculus framework by synthesising monitors; a formula is monitorable precisely when such a monitor can be constructed. The authors note the resemblance between these semantics and formula progression and conjecture that these monitorability notions coincide.

References

Aceto et al. define monitorable formulae as those for which a monitor can be synthesised — we conjecture that this definition and that of Bauer et al. coincide.

— Semantics for Linear-time Temporal Logic with Finite Observations  (2411.14581 - Amjad et al., 2024) in Section 6.1 (Prefix Characterisations and Monitorability)

In fact, there's a sense in which the `discovery' that $[\mu-]A$ is the most sensitive surrogate for $A$ is less a discovery and more a formal restatement of the problem at hand---monitoring for $[\mu-]A$ just is checking one's message to see whether it could only have arisen, given $\mu$, from a message that verified $A$. But we already knew that that's what we wanted to do. What we don't know is what doing it entails, and $[\mu-]A$ doesn't give us any hints.

— A Formal Framework for Noisy Runtime Verification  (2609.10462 - Logan et al., 9 Sep 2026) in Section “Whence Monitors?”