AI 中文总结
本文提出基于模态逻辑拓扑语义的鲁棒分类逻辑框架，引入含鲁棒性模态和条件连接词的逻辑语言，给出公理化，还介绍最小鲁棒模型，为分析、解释和构建鲁棒分类行为提供形式工具。
AI 中文摘要
鲁棒分类通常被理解为分类器在输入数据的小扰动（通常是对抗性的）下的稳定性。在本文中，我们提出了一个基于模态逻辑拓扑语义的鲁棒分类逻辑框架。评估点是表示机器可读对象的特征向量，公式表达明确的分类。鲁棒性在几何上被解释为局部真值持久性：如果一个分类在某点的某个非空开邻域内都成立，那么它在该点是鲁棒的。基于此观点，我们引入一种逻辑语言，带有在S4拓扑空间上解释的鲁棒性模态以及一个对鲁棒性敏感的条件连接词。这个条件连接词捕捉鲁棒区域之间的全局包含关系以及分类器的其他属性。我们提供了所得逻辑的可靠且完备的公理化。最后，我们引入最小鲁棒模型，一种从指定的鲁棒性约束生成模型的构造方法，它产生了用于分析、解释和构建鲁棒分类行为的形式工具。
英文摘要
Robust classification is commonly understood as the stability of a classifier under small perturbations (often adversarial) of input data. In this paper, we propose a logical framework for robust classification grounded in topological semantics for modal logic. Evaluation points are feature vectors representing machine-readable objects, and formulas express explicit classifications. Robustness is interpreted geometrically as local truth persistence: a classification is robust at a point if it holds throughout some non-empty open neighbourhood of that point. Building on this perspective, we introduce a logical language with a robustness modality interpreted over S4 topological spaces, together with a robustness-sensitive conditional connective. This conditional connective captures global inclusion relations between robust regions and other properties of the classifier: it holds at a point when the neighbourhood witnessing the robustness of one formula is contained in the truth set of another. In this way, robust classifications can be systematically linked to classification conditions. We provide a sound and complete axiomatisation of the resulting logic. Finally, we introduce Minimal Robust Models, a constructive method for generating models from specified robustness constraints, which yields formal tools for analysing, explaining, and structuring robust classification behaviour.
CommentsIn Proceedings LSFA 2026, arXiv:2607.15904
Journal refEPTCS 449, 2026, pp. 185-201
