We study a modification of the Quantifier Elimination (QE) problem called Partial QE (PQE) for propositional CNF formulas. In PQE, only a small subset of target clauses is taken out of the scope of quantifiers. The appeal of PQE is twofold. First, it provides a language for performing $\mathit{incremental}$ computations. Many verification problems (e.g. equivalence checking and model checking) are inherently incremental and so can be solved in terms of PQE. Second, PQE can be dramatically simpler than QE. We perform PQE by adding a set of clauses depending only on unquantified variables that make the target clauses redundant. Proving redundancy of a target clause is done by derivation of a "certificate" clause $\mathit{implying}$ the former. We implemented this idea in a PQE algorithm called $\mathit{START}$. It bears some similarity to a SAT-solver with conflict driven learning. A major difference here is that $\mathit{START}$ backtracks as soon as a target clause is proved redundant (even if no conflict occurred). We experimentally evaluate $\mathit{START}$ on a practical problem. We use this problem to compare PQE with QE and QBF solving.

中文翻译：

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通过证书条款消除部分量词

我们研究了对命题 CNF 公式称为部分 QE (PQE) 的量词消除 (QE) 问题的修改。在 PQE 中，只有一小部分目标从句被排除在量词的范围之外。PQE 的吸引力是双重的。首先，它提供了一种用于执行 $\mathit{incremental}$ 计算的语言。许多验证问题（例如等效性检查和模型检查）本质上是增量的，因此可以通过 PQE 解决。其次，PQE 可以比 QE 简单得多。我们通过添加一组子句来执行 PQE，这些子句仅依赖于使目标子句冗余的未量化变量。证明目标子句的冗余是通过推导“证书”子句 $\mathit{implying}$ 来完成的。我们在名为 $\mathit{START}$ 的 PQE 算法中实现了这个想法。它与具有冲突驱动学习的 SAT 求解器有一些相似之处。这里的一个主要区别是 $\mathit{START}$ 只要证明目标子句是多余的（即使没有发生冲突）就会回溯。我们在一个实际问题上通过实验评估 $\mathit{START}$。我们使用这个问题来比较 PQE 与 QE 和 QBF 求解。