  Generalized circuits are an important tool in the study of the computational
complexity of equilibrium approximation problems. However, in this paper, we
reveal that they have a conceptual flaw, namely that the solution concept is
not monotonic. By this we mean that if $\varepsilon < \varepsilon'$, then an
$\varepsilon$-approximate solution for a certain generalized circuit is not
necessarily also an $\varepsilon'$-approximate solution. The reason for this
non-monotonicity is the way Boolean operations are modeled. We illustrate that
non-monotonicity creates subtle technical issues in prior work that require
intricate additional arguments to circumvent. To eliminate this problem, we
show that the Boolean gates are a redundant feature: one can simulate stronger,
monotonic versions of the Boolean gates using the other gate types. Arguing at
the level of these stronger Boolean gates eliminates all of the aforementioned
issues in a natural way. We hope that our results will enable new studies of
sub-classes of generalized circuits and enabler simpler and more natural
reductions from generalized circuits to other equilibrium search problems.
