We introduce FO-AR, an approximation-refinement approach for first-order theorem proving based on counterexample-guided abstraction refinement. A given first-order clause set N is transformed into an over-approximation $$N^{\prime }$$ N ′ in a decidable first-order fragment. That means if $$N^{\prime }$$ N ′ is satisfiable so is N . However, if $$N^{\prime }$$ N ′ is unsatisfiable, then the approximation provides a lifting terminology for the found refutation which is step-wise transformed into a proof of unsatisfiability for N . If this fails, the cause is analyzed to refine the original clause set such that the found refutation is ruled out for the future and the procedure repeats. The target fragment of the transformation is the monadic shallow linear fragment with straight dismatching constraints, which we prove to be decidable via ordered resolution with selection. We further discuss practical aspects of SPASS-AR, a first-order theorem prover implementing FO-AR. We focus in particularly on effective algorithms for lifting and refinement.

中文翻译：

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SPASS-AR：基于对一元浅线性片段的近似-细化的一阶定理证明器

我们介绍了 FO-AR，这是一种基于反例引导的抽象细化的一阶定理证明的近似细化方法。给定的一阶子句集 N 在可判定的一阶片段中被转换为过度近似 $$N^{\prime }$$ N ′。这意味着如果 $$N^{\prime }$$ N ′ 是可满足的，那么 N 也是可满足的。然而，如果 $$N^{\prime }$$ N ′ 是不可满足的，那么该近似为找到的反驳提供了一个提升术语，它逐步转化为 N 不可满足的证明。如果失败，则分析原因以改进原始子句集，以便将来排除找到的反驳并重复该过程。转换的目标片段是具有直线不匹配约束的一元浅线性片段，我们证明可以通过带选择的有序解析来判定。我们进一步讨论了 SPASS-AR 的实际应用，它是一个实现 FO-AR 的一阶定理证明器。我们特别关注用于提升和细化的有效算法。