The incorporation of domain knowledge (DK) in AI has been studied for years and turns out to be critical in practice. Functions are also a basic notion for dealing with data science projects and are somehow related to DK. Consider the following scenario. Let \(D(y, z_1, \ldots , z_n)\) be a dataset, Alice a data scientist, Bob a domain expert and \(y = f(z_1, \ldots , z_n)\) a function known to Bob from his background knowledge. Alice is interested in the following simple yet crucial questions: How to define the satisfaction of f in D ? How to measure that satisfaction efficiently? How does this satisfaction relate to the supervised learning task of learning f from D ? It turns out that these problems are related to the study of counterexamples through the use of functional dependencies (FDs) and, in particular, FD measures used to quantify their satisfaction in a dataset such as the \(g_3\) indicator where the equality is replaced by more flexible predicates. In this article, we first examine the complexity of computing \(g_3\) . It is known that \(g_3\) can be computed in polynomial time when using equality, while it becomes NP -hard when using general predicates. Our goal is to refine this dichotomy by studying the impact of the following common properties: reflexivity, transitivity, symmetry, and antisymmetry. We show that symmetry and transitivity together are sufficient to guarantee that the \(g_3\) can be computed in polynomial time. However, removing one of them makes the problem NP -hard. Second, we study the computation of \(g_3\) in the polynomial and NP -hard cases identified previously. We propose different exact and approximate solutions for the computation of \(g_3\) in both cases. We compare these solutions in a detailed experimental study of time performance and approximation accuracy. All the algorithms are also made available via fastg3 , an open-source Python library with an underlying C++ implementation. Finally, we link counterexamples and \(g_3\) to supervised learning with a web application called adesit . adesit is intended to be part of an iterative data refinement process right after data selection and just before the machine learning process itself. It provides a way to evaluate the ability of a dataset to perform well for a given supervised learning problem through statistical and visual exploration. In a last section, we validate our approach by applying it to the industrial problem of air gap monitoring in compact hydro-generators.

Taking domain knowledge into account is a long-standing issue in AI, especially nowadays where huge amounts of data are collected in the hope of delivering new in-sights and value. Let us consider the following scenario. Let D(y, x1, … ,xn) be a dataset, Alice a data scientist, Bob a domain expert and$y$=$f$(x1, … , xn) a function known by Bob from his background knowledge. We are interested in the following simple yet crucial questions for Alice: how to define the satisfaction of f in D and how difficult is it to measure that satisfaction? It turns out that those problems are related to functional dependencies (FDs) and especially FD measurements used to quantify their satisfaction in a dataset such as the g3 indicator. In this paper, we examine the computation of g3 with crisp FDs (aka. exact FDs) and a large class of non-crisp FDs replacing strict equality by more flexible predicates. Interestingly, it is known that the computation of g3 with crisp FDs is polynomial but turns out to be NP-Hard for non-crisp FDs. In this paper, we propose different exact and approximate solutions for the computation of g3 for both types. First, for crisp FDs with very large datasets, we propose solutions based on uniform and stratified random sampling. Second, for non-crisp FDs we present a detailed computation pipeline with various computation optimizations, including approximation algorithms and adaptations of recent developments in sublinear algorithms for NP-Hard problems. We also propose an in-depth experimental study of the algorithms presented in terms of time performances and approximation accuracy. All the algorithms are also made available through FASTG3, an open-source Python library designed to be intuitive and efficient thanks to an underlying C++ implementation.

Thanks to the numerous machine learning tools available to us nowadays, it is easier than ever to derive a model from a dataset in the frame of a supervised learning problem. However, when this model behaves poorly compared with an expected performance, the underlying question of the existence of such a model is often underlooked and one might just be tempted to try different parameters or just choose another model architecture. This is why the quality of the learning examples should be considered as early as possible as it acts as a go/no go signal for the following potentially costly learning process. With ADESIT, we provide a way to evaluate the ability of a dataset to perform well for a given supervised learning problem through statistics and visual exploration. Notably, we base our work on recent studies proposing the use of functional dependencies and specifically counterexample analysis to provide dataset cleanliness statistics but also a theoretical upper bound on the prediction accuracy directly linked to the problem settings (measurement uncertainty, expected generalization...). In brief, ADESIT is intended to be part of an iterative data refinement process right after data selection and right before the machine learning process itself. With further analysis for a given problem, the user can characterize, clean and export dynamically selected subsets, allowing to better understand what regions of the data could be refined and where the data precision must be improved by using, for example, new or more precise sensors.