  We study the synthesis problem for systems with a parameterized number of
processes. As in the classical case due to Church, the system selects actions
depending on the program run so far, with the aim of fulfilling a given
specification. The difficulty is that, at the same time, the environment
executes actions that the system cannot control. In contrast to the case of
fixed, finite alphabets, here we consider the case of parameterized alphabets.
An alphabet reflects the number of processes that are static but unknown. The
synthesis problem then asks whether there is a finite number of processes for
which the system can satisfy the specification. This variant is already
undecidable for very limited logics. Therefore, we consider a first-order logic
without the order on word positions. We show that even in this restricted case
synthesis is undecidable if both the system and the environment have access to
all processes. On the other hand, we prove that the problem is decidable if the
environment only has access to a bounded number of processes. In that case,
there is even a cutoff meaning that it is enough to examine a bounded number of
process architectures to solve the synthesis problem.
