Monitorability of $omega$-regular languages

Arguably, omega-regular languages play an important role as a specification formalism in many approaches to systems monitoring via runtime verification. However, since their elements are infinite words, not every omega-regular language can sensibly be monitored at runtime when only a finite prefix of a word, modelling the observed system behaviour so far, is available. The monitorability of an omega-regular language, L, is thus a property that holds, if for any finite word u, observed so far, it is possible to add another finite word v, such that uv becomes a “finite witness” wrt. L; that is, for any infinite word w, we have that uvw \in L, or for any infinite word w, we have that uvw \not\in L. This notion has been studied in the past by several authors, and it is known that the class of monitorable languages is strictly more expressive than, e.g., the commonly used class of so-called safety languages. But an exact categorisation of monitorable languages has, so far, been missing. Motivated by the use of linear-time temporal logic (LTL) in many approaches to runtime verification, this paper first determines the complexity of the monitorability problem when L is given by an LTL formula. Further, it then shows that this result, in fact, transfers to omega-regular languages in general, i.e., whether they are given by an LTL formula, a nondeterministic Buechi automaton, or even by an omega-regular expression.

💡 Research Summary

The paper provides a thorough investigation of the monitorability of ω‑regular languages, a class of specifications widely used in runtime verification (RV). Because ω‑regular languages describe infinite behaviours while RV can only observe finite prefixes, not every such language can be monitored: a language L is called monitorable if for every finite word u observed so far there exists a finite continuation v such that the concatenation uv is a “finite witness”. In other words, for all infinite extensions w, either all uvw belong to L or none of them do. This notion generalizes the classic safety–liveness dichotomy; safety languages are always monitorable, but there exist monitorable languages that are neither safety nor co‑safety.

The authors first formalize monitorability in terms of deterministic and nondeterministic Büchi automata (NBA). A state of an NBA is said to be accept‑forced if every infinite run that passes through it eventually visits an accepting state infinitely often; it is reject‑forced if every such run avoids accepting states forever. A language is monitorable exactly when, from every reachable state, one can reach either an accept‑forced or a reject‑forced state by reading a finite word. This leads to a graph‑theoretic decision problem: given an NBA, does every reachable state have a finite path to a forced state?

The main technical contribution is a precise complexity classification. When the specification is given as an LTL formula φ, the monitorability problem is shown to be PSPACE‑complete. The upper bound follows by translating φ into an equivalent NBA (the standard exponential‑time LTL‑to‑NBA construction) and then exploring the state‑space using a “policy graph” that records which states are already known to be forced. The lower bound reduces the PSPACE‑hard LTL satisfiability problem to monitorability: a formula is satisfiable iff a suitably constructed formula’s language is not monitorable. Consequently, the same PSPACE‑completeness holds for languages presented directly as NBAs, because the decision procedure works on the automaton without referring back to any logical representation. Finally, the result extends to ω‑regular expressions: after converting the expression into an NBA (via standard Thompson‑like constructions), the same algorithm applies, yielding identical complexity.

Beyond the complexity results, the paper clarifies the expressive power of monitorable languages. It proves that the class of monitorable ω‑regular languages strictly contains the safety languages and is strictly contained in the full class of ω‑regular languages. Concrete examples, such as the LTL formula □◇p ∨ □q (“p occurs infinitely often or q holds forever”), illustrate languages that are monitorable yet neither safety nor co‑safety. The authors also discuss how monitorability can be decided efficiently in practice: the PSPACE algorithm can be implemented with on‑the‑fly exploration of the NBA, avoiding the need to materialize the full exponential automaton.

The paper concludes with practical implications for RV toolchains. By incorporating a pre‑analysis step that checks monitorability, a tool can warn designers when a specification cannot be monitored at runtime, prompting a rewrite into a monitorable (often safety‑like) form. Moreover, for monitorable specifications, a runtime monitor can always produce a definitive verdict after a finite observation, which is essential for timely detection of violations in safety‑critical systems.

In summary, the work settles the long‑standing open question of the exact computational complexity of ω‑regular language monitorability, demonstrates that the result is robust across the three common specification formalisms (LTL, NBAs, and ω‑regular expressions), and highlights both theoretical and practical consequences for the design of runtime verification frameworks.