Monitorability of omega-regular languages

Arguably, ω-regular languages play an important rôle as a specification formalism in many approaches to systems monitoring via runtime verification. However, since their elements are infinite words, not every ω-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 ω-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 ∈ L, or for any infinite word w, we have that uvw 6∈ 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 lineartime 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 ω-regular languages in general, i.e., whether they are given by an LTL formula, a nondeterministic Büchi automaton, or even by an ω-regular expression.