  Guarded protocols were introduced in a seminal paper by Emerson and Kahlon
(2000), and describe systems of processes whose transitions are enabled or
disabled depending on the existence of other processes in certain local states.
We study parameterized model checking and synthesis of guarded protocols, both
aiming at formal correctness arguments for systems with any number of
processes. Cutoff results reduce reasoning about systems with an arbitrary
number of processes to systems of a determined, fixed size. Our work stems from
the observation that existing cutoff results for guarded protocols i) are
restricted to closed systems, and ii) are of limited use for liveness
properties because reductions do not preserve fairness. We close these gaps and
obtain new cutoff results for open systems with liveness properties under
fairness assumptions. Furthermore, we obtain cutoffs for the detection of
global and local deadlocks, which are of paramount importance in synthesis.
Finally, we prove tightness or asymptotic tightness for the new cutoffs.
