  We study partial quantifier elimination (PQE) for propositional CNF formulas.
PQE is a generalization of quantifier elimination where one can limit the set
of clauses taken out of the scope of quantifiers to a small subset of target
clauses. The appeal of PQE is twofold. First, PQE can be dramatically simpler
than full quantifier elimination. Second, PQE provides a language for
performing incremental computations. Many verification problems (e.g.
equivalence checking and model checking) are inherently incremental and so can
be solved in terms of PQE. Our approach is based on deriving clauses depending
only on unquantified variables that make the target clauses
$\mathit{redundant}$. Proving redundancy of a target clause is done by
construction of a "certificate" clause implying the former. We describe a PQE
algorithm called $\mathit{START}$ that employs the approach above. To evaluate
$\mathit{START}$, we apply it to invariant generation for a sequential circuit
$N$. The goal of invariant generation is to find an $\mathit{unwanted}$
invariant of $N$ proving unreachability of a state that is supposed to be
reachable. If $N$ has an unwanted invariant, it is buggy. Our experiments with
FIFO buffers and HWMCC-13 benchmarks suggest that $\mathit{START}$ can be used
for detecting bugs that are hard to find by existing methods.
