A Note on Monitors and B\"uchi automata

When a property needs to be checked against an unknown or very complex system, classical exploration techniques like model-checking are not applicable anymore. Sometimes a~monitor can be used, that checks a given property on the underlying system at runtime. A monitor for a property $L$ is a deterministic finite automaton $M_L$ that after each finite execution tells whether (1) every possible extension of the execution is in $L$, or (2) every possible extension is in the complement of $L$, or neither (1) nor (2) holds. Moreover, $L$ being monitorable means that it is always possible that in some future the monitor reaches (1) or (2). Classical examples for monitorable properties are safety and cosafety properties. On the other hand, deterministic liveness properties like "infinitely many $a$'s" are not monitorable. We discuss various monitor constructions with a focus on deterministic omega-regular languages. We locate a proper subclass of of deterministic omega-regular languages but also strictly large than the subclass of languages which are deterministic and codeterministic, and for this subclass there exists a canonical monitor which also accepts the language itself. We also address the problem to decide monitorability in comparison with deciding liveness. The state of the art is as follows. Given a B\"uchi automaton, it is PSPACE-complete to decide liveness or monitorability. Given an LTL formula, deciding liveness becomes EXPSPACE-complete, but the complexity to decide monitorability remains open.