一句话：提出 coaxioms 推广推理系统，使递归定义可解释为既非最小也非最大的不动点。
论文针对归纳与共归纳解释无法给出预期含义的递归定义场景，引入一种广义推理系统。除了通常的公理和推理规则，系统允许 coaxioms，即只能在证明树的无限深度处应用的公理。该机制使递归定义可被解释为不一定是最小或最大不动点的固定点，其存在性由经典结果的平滑扩展保证。该框架严格包含标准推理系统及其归纳与共归纳解释，从而支持归纳与共归纳混合的情形下的形式推理。论文还给出了该推广的语义基础与理论性质，展示了其与经典不动点理论的关系。
We introduce a generalized notion of inference system to support more flexible interpretations of recursive definitions. Besides axioms and inference rules with the usual meaning, we allow also coaxioms, which are, intuitively, axioms which can only be applied "at infinite depth" in a proof tree. Coaxioms allow us to interpret recursive definitions as fixed points which are not necessarily the least, nor the greatest one, whose existence is guaranteed by a smooth extension of classical results. This notion nicely subsumes standard inference systems and their inductive and coinductive interpretation, thus allowing formal reasoning in cases where the inductive and coinductive interpretation do not provide the intended meaning, but are rather mixed together.Comment: This is a corrected version of the paper (arXiv:1808.02943v4) published originally on 12 March 2019
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@article{abs-1808-02943,
title = {Coaxioms: flexible coinductive definitions by inference systems},
author = {Francesco Dagnino},
journal = {Logical Methods in Computer Science},
volume = {Volume 15, Issue 1},
year = {2020},
doi = {10.23638/lmcs-15(1:26)2019}
}