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.

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A nondeterministic automaton is history-deterministic if its nondeterminism can be resolved by only considering the prefix of the word read so far. Due to their good compositional properties, history-deterministic automata are useful in solving games and synthesis problems. Deciding whether a given nondeterministic automaton is history-deterministic (the problem) is generally a difficult task, which might involve an exponential procedure, or even be undecidable, for example for pushdown automata. Token games provide a PTime solution to the problem of Büchi and coBüchi automata, and it is conjectured that 2-token games characterise for all $$\omega $$ ω -regular automata. We extend token games to the quantitative setting and analyze their potential to help deciding for quantitative automata. In particular, we show that 1-token games characterise for all quantitative (and Boolean) automata on finite words, as well as discounted-sum ($${\mathsf {DSum}}$$ DSum ) automata on infinite words, and that 2-token games characterise of $${\mathsf {LimInf}}$$ LimInf and $${\mathsf {LimSup}}$$ LimSup automata. Using these characterisations, we provide solutions to the problem of $${\mathsf {Inf}}$$ Inf and $${\mathsf {Sup}}$$ Sup automata on finite words in PTime, for $${\mathsf {DSum}}$$ DSum automata on finite and infinite words in NP$$\cap $$ ∩ co-NP, for $${\mathsf {LimSup}}$$ LimSup automata in quasipolynomial time, and for $${\mathsf {LimInf}}$$ LimInf automata in exponential time, where the latter two are only polynomial for automata with a logarithmic number of weights.

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.

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