  Monitors are a key tool in the field of runtime verification, where they are
used to verify system properties by analyzing execution traces generated by
processes. Work on runtime monitoring carried out in a series of papers by
Aceto et al.$~$has specified monitors using a variation on the regular fragment
of Milner's CCS and studied two trace-based notions of equivalence over
monitors, namely verdict and $\omega$-verdict equivalence. This article is
devoted to the study of the equational logic of monitors modulo those two
notions of equivalence. It presents complete equational axiomatizations of
verdict and $\omega$-verdict equivalence for closed and open terms over
recursion-free monitors. It is also shown that verdict equivalence has no
finite equational axiomatization over open monitors when the set of actions is
finite and contains at least two actions.
