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This paper establishes a comprehensive theory of runtime monitorability forHennessy-Milner logic with recursion, a very expressive variant of the modal$\mu$-calculus. It investigates the monitorability of that logic with alinear-time semantics and then compares the obtained results with ones thatwere previously presented in the literature for a branching-time setting. Ourwork establishes an expressiveness hierarchy of monitorable fragments ofHennessy-Milner logic with recursion in a linear-time setting and exactlyidentifies what kinds of guarantees can be given using runtime monitors foreach fragment in the hierarchy. Each fragment is shown to be complete, in thesense that it can express all properties that can be monitored under thecorresponding guarantees. The study is carried out using a principled approachto monitoring that connects the semantics of the logic and the operationalsemantics of monitors. The proposed framework supports the automatic,compositional synthesis of correct monitors from monitorable properties.

Adventures in Monitorability: From Branching to Linear Time and Back Again