原子逻辑的动作研究
A Study on Actions for Atomic Logics
AI总结：
针对非经典逻辑性质证明工作量大的问题，本文基于动作类重新定义原子逻辑，证明其切割消除等性质，验证与原有原子逻辑的等价性，并结合Coq实现形式化，应用于非结合兰贝克演算。
摘要：
如今存在大量非经典逻辑，每一种都最适合用于抽象领域（如语言学、认识论等）的相关问题推理。为每一种非经典逻辑证明有价值的性质，给逻辑学家和计算机科学家带来了巨大工作量。我们需要一种模块化的方法来解决这一难题。针对该问题，本报告对Guillaume Aucher提出的原子逻辑（Atomic Logics）构建提出了新见解。原子逻辑可表示极具一般性的左、右引入规则，还附带一类基于显示逻辑（display logics）和剩余运算（residuation）的新型规则。本文采用新方法定义原子逻辑，其构建基于一类动作，我们证明了这类动作的切割消除（cut-elimination）性质。我们证明其中部分原子逻辑与Aucher的原子逻辑等价，并对一类原子逻辑证明了切割消除与克雷格插值（Craig Interpolation）性质。全文将所提出的理论应用于非结合兰贝克演算（non-associative Lambek Calculus），同时在证明助手Coq中对原始语法进行了计算机可验证的形式化。
英文摘要：
Nowadays there is a large number of non-classical logics, each one best suited for reasoning about some issues in abstract fields, such as linguistics or epistemology, among others. Proving interesting properties for each one of them supposes a big workload for logicians and computer scientists. We want an approach into this problematic that is modular. To adress this issue, the report shows new insights in the construction of Atomic Logics introduced by Guillaume Aucher. Atomic Logics let us represent very general left and right introduction rules and they come along a new kind of rules based on display logics and residuation. A new approach is taken into the definition of Atomic Logics, which is now built on a class of actions for which we prove cut-elimination. We show that some of them are equivalent to Aucher's Atomic Logics and we prove cut-elimination and Craig Interpolation for a class of them. The introduced theory is applied to the non-associative Lambek Calculus throughout the report. It is accompanied by a computer-checked formalisation of the original syntax in the proof assistant Coq.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
