Monitorability of $\omega$-regular languages

TL;DR

This paper investigates the concept of monitorability for omega-regular languages in runtime verification, providing complexity results for LTL-defined languages and extending these findings to general omega-regular languages.

Contribution

It offers the first precise complexity characterization of the monitorability problem for omega-regular languages, especially those specified by LTL formulas.

Findings

Complexity of monitorability for LTL formulas determined.

Results extend to general omega-regular languages.

Monitorability is more expressive than safety languages.

Abstract

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.