  We study the role of perfect completeness in probabilistically checkable
proof systems (PCPs) and give a new way to transform a PCP with imperfect
completeness to a PCP with perfect completeness when the initial gap is a
constant. In particular, we show that $\text{PCP}_{c,s}[r,q] \subseteq
\text{PCP}_{1,1-\Omega(1)}[r+O(1),q+O(r)]$, for $c-s=\Omega(1)$. This implies
that one can convert imperfect completeness to perfect in linear-sized PCPs for
$NTIME[O(n)]$ with a $O(\log n)$ additive loss in the query complexity $q$. We
show our result by constructing a "robust circuit" using threshold gates. These
results are a gap amplification procedure for PCPs (when completeness is
imperfect), analogous to questions studied in parallel repetition and
pseudorandomness.
  We also investigate the time complexity of approximating perfectly
satisfiable instances of 3SAT versus those with imperfect completeness. We show
that the Gap-ETH conjecture without perfect completeness is equivalent to
Gap-ETH with perfect completeness, i.e. we show that Gap-3SAT, where the gap is
not around 1, has a subexponential algorithm, if and only if, Gap-3SAT with
perfect completeness has subexponential algorithms. We also relate the time
complexities of these two problems in a more fine-grained way, to show that
$T_2(n) \leq T_1(n(\log\log n)^{O(1)})$, where $T_1(n),T_2(n)$ denote the
randomized time-complexity of approximating MAX 3SAT with perfect and imperfect
completeness, respectively.
