Approximation Fixpoint Theory (AFT) and Justification Theory (JT) are two frameworks to unify logical formalisms. AFT studies semantics in terms of fixpoints of lattice operators, and JT in terms of so-called justifications, which are explanations of why certain facts do or do not hold in a model. While the approaches differ, the frameworks were designed with similar goals in mind, namely to study the different semantics that arise in (mainly) non-monotonic logics. The first contribution of our current paper is to provide a formal link between the two frameworks. To be precise, we show that every justification frame induces an approximator and that this mapping from JT to AFT preserves all major semantics. The second contribution exploits this correspondence to extend JT with a novel class of semantics, namely : we formally show that ultimate semantics can be obtained in JT by a syntactic transformation on the justification frame, essentially performing a sort of resolution on the rules.

中文翻译：

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将证明理论嵌入到近似不动点理论中

近似不动点理论（AFT）和论证理论（JT）是统一逻辑形式主义的两个框架。 AFT 根据格算子的不动点来研究语义，而 JT 根据所谓的理由来研究语义，即解释为什么某些事实在模型中成立或不成立。尽管方法不同，但框架的设计目标相似，即研究（主要）非单调逻辑中出现的不同语义。我们当前论文的第一个贡献是提供两个框架之间的正式联系。准确地说，我们证明每个理由框架都会产生一个近似器，并且从 JT 到 AFT 的映射保留了所有主要语义。第二个贡献利用这种对应关系，用一类新颖的语义来扩展 JT，即：我们正式证明，可以通过对理由框架进行句法转换来在 JT 中获得最终语义，本质上是对规则进行某种解析。