🤖 AI Summary
This work addresses the formalization and verification of robustness in machine learning classifiers against adversarial perturbations. It introduces a novel logical framework that integrates topological semantics with modal logic, grounded in S4 topological spaces, wherein robustness is interpreted as the persistence of local truth. The core contributions include the introduction of a robustness modality and a new robustness-sensitive conditional connective to precisely capture inclusion relations among robust regions and classification conditions. Furthermore, the paper proposes a constructive method for generating minimal robust models and establishes a complete axiomatization for the proposed logic. It also provides a formal toolchain that automatically synthesizes models from robustness constraints, enabling structured analysis, interpretation, and modeling of classifier robustness.
📝 Abstract
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.