We study partial quantifier elimination (PQE) for propositional CNF formulas.
In contrast to full quantifier elimination, in PQE, 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, it 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 redundant. Proving redundancy
of a target clause is done by construction of a ``certificate'' clause implying
the former. We describe a PQE algorithm called START that employs
the approach above. We apply START to generating properties of a
design implementation that are not implied by specification. The existence of
an unwanted property means that this implementation is buggy. Our
experiments with HWMCC-13 benchmarks suggest that START can be used
for generating properties of real-life designs