# Fidex and FidexGlo: From Local Explanations to Global Explanations of Deep Models +
## 1 Introduction  
Within artificial intelligence (AI), XAI stands for explainable AI and includes methods that aim to explain the responses of black-box models. The number of publications in XAI is constantly increasing. Extensive reviews of XAI methods have been presented in [1, 2, 3, 4, 5]. Deep learning models are characterized by many neurons grouped together in many successive layers. Even though they have been very successful over the last decade, they remain complex and difficult to explain. Rule-based systems offer transparency and clarity in decision-making. Models defined by explicit rules enable human users to trace each decision back to the specific conditions that triggered it. Unlike black-box machine learning models, the behavior of a rule-based system is comprehensible not only to AI practitioners but also to a broader audience, including domain experts. In certain areas ofapplication, such as medicine, it is essential to explain the decisions made by deep models so that physicians and patients can have more confidence in them. The precursor to deep connectionist models is the multilayer perceptron (MLP). When it was introduced, researchers raised the question of explainability and developed new techniques to provide explanations using propositional rules. A nomenclature of all methods to explain the responses of neural networks by propositional rules was introduced by [PERSON] et al. (2016). Before this century, decision trees (DTs) were a direct competitor to MLPs in classification problems. Their answers are easy to explain and accurate in many applications. Furthermore, models that are combined to form an ensemble are often more accurate than a single model. Several algorithms for learning ensembles were proposed, such as bagging ([PERSON] et al., 2016) and boosting ([PERSON] et al., 2016), and have been applied to DTs and MLPs. However, even DTs lose their explainability in ensemble combination. A key element of explanatory approaches in the context of deep models is the focus on the neighborhood of a sample ([PERSON] et al., 2016). Such approaches are defined as local. A global technique, on the other hand, considers all the data that define a problem. In addition, many other image classification techniques visualize areas that are most relevant to the result ([PERSON] et al., 2016). The discretized interpretable multilayer perceptron (DIMLP) is a neural network model from which responses can be explained by propositional rules. The rules are generated by an algorithm that determines axis-parallel discriminative hyperplanes ([PERSON] et al., 2014). The same algorithm was used to generate rules from ensembles of DIMLPs ([PERSON] et al., 2014). It was later shown that ensembles of decision trees (DTs) are functionally special cases of DIMLP ensembles, so rule extraction was applied to random forests (RFs) and gradient-boosted (GB) trees ([PERSON] et al., 2014). One question that arose at this point was how to apply rule extraction to deep models, but the complexity of the DIMLP rule extraction algorithm, which is global, was not favorable in practice (polynomial of degree four with respect to the size of the input vectors ([PERSON] et al., 2014)). We present two novel algorithms that have been recently introduced by the authors in ([PERSON] et al., 2014) to explain the responses of a number of black-box machine learning models for data classification problems. The first is Fidex, which is local and thus applied to a single sample. The second, called FidexGlo, is global and uses Fidex. Both algorithms generate explanations by means of propositional rules. In our framework, the discriminative boundaries are parallel to the input variables and their location is precisely determined. Fidex is a heuristic algorithm that, at each step, establishes where the best hyperplane is that has increased fidelity the most. Fidelity is a measure of the extent to which the explanation is in agreement with the model response.