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Runtime verificationis a lightweight method for monitoring the formal specification of a system during its execution. It has recently been shown that a given state predicate can be monitored consistently by a set of crash-prone asynchronousdistributedmonitors observing the system, only if each monitor can emit verdicts taken from alarge enoughfinite set. We revisit this impossibility result in the concrete context of linear-time logic (ltl) semantics for runtime verification, that is, when the correctness of the system is specified by anltlformula on its execution traces. First, we show that monitors synthesized based on the 4-valued semantics ofltl(rv-ltl) may result in inconsistent distributed monitoring, even for some simpleltlformulas. More generally, given anyltlformula φ, we relate the number of different verdicts required by the monitors for consistently monitoring φ, with a specific structural characteristic of φ called itsalternation number. Specifically, we show that, for everyk ≥ 0, there is anltlformula φ with alternation number kthat cannot be verified at runtime by distributed monitors emitting verdicts from a set of cardinality smaller thank+ 1. On the positive side, we define a family of logics, calleddistributedltl(abbreviated asdltl), parameterized byk≥ 0, which refinesrv-ltlby incorporating2k+ 4 truth values. Our main contribution is to show that, for everyk≥ 0, everyltlformula φ with alternation number kcan be consistently monitored by distributed monitors, each running an automaton based on a (2 ⌈k/2 ⌉ +4)-valued logic taken from thedltlfamily.