This paper establishes a comprehensive theory of runtime monitorability for Hennessy-Milner logic with recursion, a very expressive variant of the modal µ-calculus. It investigates the monitorability of that logic with a linear-time semantics and then compares the obtained results with ones that were previously presented in the literature for a branching-time setting. Our work establishes an expressiveness hierarchy of monitorable fragments of Hennessy-Milner logic with recursion in a linear-time setting and exactly identifies what kinds of guarantees can be given using runtime monitors for each fragment in the hierarchy. Each fragment is shown to be complete, in the sense that it can express all properties that can be monitored under the corresponding guarantees. The study is carried out using a principled approach to monitoring that connects the semantics of the logic and the operational semantics of monitors. The proposed framework supports the automatic, compositional synthesis of correct monitors from monitorable properties.

Abstract. We examine the determinization of monitors. We demonstrate that every monitor is equivalent to a deterministic one, which is at most doubly exponential in size with respect to the original monitor. When monitors are described as CCS-like processes, this doubly-exponential bound is optimal. When (deterministic) monitors are described as finite automata (as their LTS), then they can be exponentially more succinct than their CCS process form.

Monitorability delineates what properties can be verified at runtime. Although many monitorability definitions exist, few are defined explicitly in terms of the guarantees provided by monitors, i.e., the computational entities carrying out the verification. We view monitorability as a spectrum: the fewer monitor guarantees that are required, the more properties become monitorable. We present a monitorability hierarchy and provide operational and syntactic characterisations for its levels. Existing monitorability definitions are mapped into our hierarchy, providing a unified framework that makes the operational assumptions and guarantees of each definition explicit. This provides a rigorous foundation that can inform design choices and correctness claims for runtime verification tools.

We examine the determinization of monitors for HML with recursion. We demonstrate that every monitor is equivalent to a deterministic one, which is at most doubly exponential in size with respect to the original monitor. When monitors are described as CCS-like processes, this doubly exponential bound is optimal. When (deterministic) monitors are described as finite automata (as their LTS), then they can be exponentially more succinct than their CCS process form.

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