# Towards Logical Specification of Statistical Machine Learning

## TL;DR

This paper presents a novel logical framework for formalizing and analyzing statistical properties of machine learning classifiers, including performance, robustness, and fairness, using epistemic and counterfactual logic.

## Contribution

It introduces a formal model based on Kripke structures and develops logical formulas to express and relate statistical properties of classifiers, including new formalizations of robustness and fairness.

## Key findings

- Relationships among classifier properties are established.
- Robustness-related properties are identified and formalized.
- Counterfactual knowledge is used to formalize fairness.

## Abstract

We introduce a logical approach to formalizing statistical properties of machine learning. Specifically, we propose a formal model for statistical classification based on a Kripke model, and formalize various notions of classification performance, robustness, and fairness of classifiers by using epistemic logic. Then we show some relationships among properties of classifiers and those between classification performance and robustness, which suggests robustness-related properties that have not been formalized in the literature as far as we know. To formalize fairness properties, we define a notion of counterfactual knowledge and show techniques to formalize conditional indistinguishability by using counterfactual epistemic operators. As far as we know, this is the first work that uses logical formulas to express statistical properties of machine learning, and that provides epistemic (resp. counterfactually epistemic) views on robustness (resp. fairness) of classifiers.