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论文推介:秘密逻辑与插值规则

发布时间:2022-10-11 来源:本站原创 作者:本站编辑   浏览次数:

近日,熊作军副教授与托马斯·奥古特尼斯教授合作的论文《秘密逻辑与插值规则》(The Logic of Secrets and the Interpolation Rule)在线发表在了SCI索引杂志《数学与人工智能年鉴》(Annals of Mathematics and Artificial Intelligence)上。

        论文为LAMAS 2020上秘密逻辑(A Logic of Secrets)这一报告的完善与扩张,使用了“带标签的极大一致集”作为构造典范模型的基本要素,给出了一个基于三元典范关系的典范模型,称为“标准模型”,从而证明了极小秘密逻辑系统ICS与自反的秘密逻辑系统ICST的可靠与完全的公理系统。基于“秘密认知”的语义定义,给出了一种表达力“强”于“等值替换规则RE”但弱于“单调性规则Mo”的新的关于秘密信息的推理规则——插值规则(the Interpolation Rule,若有\phi \to \psi,\psi\to\chi皆为定理,则有(S\phi\land S\chi) \to S\psi 为定理)。进一步,我们发现,极小秘密逻辑系统ICS是非正规模态系统ECK的扩张,ICS系统与ICST系统的完全性的证明也为克里普克语义系统下证明ECK(有邻域语义上的完全性)与ECKT(暂无完全性证明)的完全性提供了参考。最后,基于插值规则的不含S公理的极小非正规模态逻辑系统IC是否具有完全性则是一个有趣的问题。

文章pdf链接:https://rdcu.be/cXiqZ 

In this article we formalise the notion of knowing a secret as a modality, by combining standard notions of knowledge and ignorance from modal epistemic logic. Roughly speaking, Ann knows a secreet if and only if she knows it and she knows that everyone else does not know it. The main aim is to study the properties of these secretly knowing modalities. It turns out that the modalities are non-normal, and are characterised by a derivation rule we call Interpolation that is stronger than Equivalence but weaker than Monotonicity. We study the Interpolation rule and position it in the landscape of non-normal modal logics. We show that it, in combination with basic axioms, gives us a complete characterisation of the properties of the secretly knowing modalities under weak assumptions about the properties of individual knowledge, in the form of a sound and complete axiomatisation. This characterisation gives us the most basic and fundamental principles of secretly knowing.

(郭美云/审核,熊作军/供稿、发布)