Luca Bourroux

Jenny Benois-Pineau

Romain Giot

Multi Layered Feature Explanation Method for Convolutional Neural Networks ⋆

Keywords: XAI, Features attribution, ResNet, Gaze Fixation Density Maps

The most popular methods for Artificial Intelligence such as Deep Neural Networks are, for the vast majority, considered black boxes. It is necessary to explain their decisions to understand the input data which influence most the result. Methods presented in this paper aim at an explanation in image classification tasks: which data in the input are the most important for the result. We further extend the Feature Explanation Method (FEM) from our previous work, transforming it into a multi-layered FEM (MLFEM). The evaluation of the method is designed by comparison of explanation maps with human Gaze Fixation Density maps (GFDM). We show that proposed MLFEM outperforms FEM and popular DNN explanation methods in terms of classical comparison metrics with GFDM.

Introduction and related work

Deep learning (DL) approaches have become indispensable in data analysis and classification. Although the results of DL models have been exemplary, they lack transparency, which prevents a wide use of them in critical applications such as medical image analysis or security.

The need for explanations of Deep Neural Networks (DNN) decisions has led to an active research in eXplainable Artificial Intelligence (XAI). In the field of pattern recognition for images and videos, explanation of a DNN's decision consists in identifying the set of input pixels that have contributed the most into the decision [START_REF] Ayyar | White box methods for explanations of convolutional neural networks in image classification tasks[END_REF]. A famous example of decisions on a wrong data is given by Ribeiro et al. [START_REF] Ribeiro | Why should i trust you? explaining the predictions of any classifier[END_REF]: here, a trained classifier wrongly used the presence of snow as the distinguishing feature between the "Wolf" and "Husky" classes. One of the ways to automatically verify if the DL classifier builds its decision on the adequate pattern in the input data is to compare the set of "important" pixels with human observations of visual content; we use this approach in this paper.

Ayyar et al. [START_REF] Ayyar | White box methods for explanations of convolutional neural networks in image classification tasks[END_REF] analyzed the variety of explanations methods highlighting important patterns from the input image. Following their proposed taxonomy, we can identify "black-box" and "white-box" families of explanation methods. The "black-box" methods remain model agnostic and are applicable to any classifier, as they identify important pixels in images by masking different parts of the input and tracking induced decisions (e.g., LIME [START_REF] Ribeiro | Why should i trust you? explaining the predictions of any classifier[END_REF] method). "White-box" methods, on the contrary, use the internal architecture of DNNs and can be subdivided into several groups. The first one concerns the methods based on linearization of the Deep-CNN. One of the first was the so-called Deconvolution Network (DeconvNet) proposed by Zeiler et al. [START_REF] Zeiler | Visualizing and understanding convolutional networks[END_REF]. The principle here was to build the mapping of the output score to the input space using reverse filters or "deconvolution" and thus identifying the important pixels. The methods based on gradient back-propagation such as a popular GradCam [START_REF] Selvaraju | Grad-cam: Visual explanations from deep networks via gradient-based localization[END_REF] or its further improved versions, (e.g., smoothgrad [START_REF] Smilkov | Smoothgrad: removing noise by adding noise[END_REF] or integrated gradients [START_REF] Sundararajan | Axiomatic attribution for deep networks[END_REF]) proceed by propagation of gradients from the last layers to the input with regard to the changed input. The important input information is located where the gradients are strong. The Layered Relevance Propagation method (LRP) [START_REF] Bach | On pixel-wise explanations for non-linear classifier decisions by layer-wise relevance propagation[END_REF] is also based on the same idea of back propagation, but without the need of gradient computations. Here, the relevance of neurons from the last decision layer through receptive fields in previous layers using the principle of conservation of the relevance at each layer allows identifying important input neurons-pixels. Feature Explanation Method (FEM) [START_REF] Ahmed Asif Fuad | Features Understanding in 3D CNNs for Actions Recognition in Video[END_REF] (Section 2.1) is also based on backpropagation, but it is not linear in the sense that with the help of statistical filtering of the features of the last convolutional layer it identifies the most important ones. The various features maps are usually depicted using a heatmap overlaying the input image, explaining important regions in the input to the user.

Propagating information through subsequent layers, the Deep NNs loose high resolution information due to the cascaded convolutions and subsampling. Hence, it is logical to explore the DNN classifier a bit more, preserving important details in each conv layer which finally bring the classification decision. We assert that the fusion of explanations from several layers is a key to improve the final explanation. For this reason, we propose an extension of FEM, the Multi-Layered FEM (MLFEM), that relies on information fusion on feature importance from different layers. We study different strategies of fusion and benchmark them against Gaze Fixation Density Maps resulting from psycho-visual experiments on image databases.

The reminder of the paper is organized as follows. Section 2 presents our contribution -MLFEM, Section 3 gives the evaluation methodology, while results are reported in Section 4. Conclusion and perspectives are drown in section 5.

Multi Layered Feature Explanation Method

Our method MLFEM is built upon FEM method [START_REF] Ahmed Asif Fuad | Features Understanding in 3D CNNs for Actions Recognition in Video[END_REF]. The latter relies on the analysis of activations at the last conv layer of a CNN classifier. As each layer of a CNN embeds information at a different scale, we assume that computing FEM at several layers and fusing them would improve the quality of the feature attribution; this is the main idea of MLFEM. In the following, we briefly review FEM and describe the adaptations for MLFEM.

Reminder of Feature Explanation Method (FEM)

Feature Explanation Method (FEM) [START_REF] Ahmed Asif Fuad | Features Understanding in 3D CNNs for Actions Recognition in Video[END_REF] is a recent algorithm used to produce an explanation map of the decision of a CNN. In opposite to other methods of the literature, it is class agnostic and does not need to provide a class of interest. FEM makes two hypotheses. The first is that strong features at the last convolutional layer will contribute the most in the final decision of the CNN in a classification task when pushed through fully connected layers. The second hypothesis will help us to select strong features. It assumes that the features in each feature map follow a Gaussian Distribution; this is a simplification hypothesis, but in case of large feature maps it could hold. At the last layer of convolution the strongest features are the representations of the most relevant regions in the input image. This comes from the interpretation of the "convolutional" part of a deep CNN as of a multiscaled pyramid with filtering, non-linear input signal transformations and subsampling [START_REF] Zemmari | Deep Learning in Mining of Visual Content[END_REF].

By analyzing only the last layer of the CNN part of the model, one can get input pixels which contributed the most into the final decision. As FEM uses activations of the last layer of the CNN after the non-linearity (most likely ReLU) only positive features will be picked up. As FEM is class agnostic, it is not a problem if it emphasizes on some high activations that are negatively weighted in the next layer. Indeed the features remain important for the overall classification wether they vote for or against a specific class.

The last convolutional layer produces activations of size (W ×H) for D feature maps f i . D binary maps, b i,i=1...D corresponding to each of the D feature maps f i are computed by selecting their strongest features: f i (x, y) ≥ µ i + kσ i . Mean µ i and standard deviation σ i are individually estimated for each feature map f i . In the same manner as the most important features are selected in each feature map f i , the contribution of each map into decision is also weighted by a map-importance weight corresponding to µ i . This will give us a saliency map s = b i µ i of dimension (W × H) that is then upscaled to the resolution of the input image by linear interpolation. A min-max normalization is finally used to bring the domain from R + to [0, 1] and obtain the final normalized map of feature importance S.

Principles of Multi Layered FEM (MLFEM)

In fact, FEM as presented previously can be applied on any layer of a CNN. We can merely pretend to truncate the model at a particular layer and see that FEM would work as is. The application of FEM on a CNN consisting of L convolutional layers will yield L different feature importance maps. As all importance maps are interpolated in FEM method we finally have L maps of the input resolution. The information provided by the maps is layer-dependent and it is interesting to fuse them. Now the question is how to obtain a single Fig. 1: FEM applied on every convolutional block of a typical ResNet50 architecture. Resolution is higher for the first layers. heat-map for the input highlighting the pixels which have contributed into the Netwok decision the most.

Let M be our convolutional neural network, with l = 1, ..., L convolutional layers. Let us denote F l a feature tensor obtained at each convolutional layer after the positive non-linearity (ReLU).

Let us denote by H(F l ) the operator which implements FEM method yielding to a normalized importance map S l of features F l . The multi layered FEM pixel importance map is obtained by fusing all the importance maps S l with a fusion operator : S = L l=1 S l . The combination of L different maps can be done in a recursive manner. For each intermediary map S l we will construct S as the combination of S and the current intermediary map S l . We will then move along for each l < L. An alternative is to devise a fusion operator that takes a variable number of arguments to produce a single final feature importance map.

Intuition behind Multi-layerd FEM The reasoning for applying multiple times the same explanation method at different network's layers is the following. The network passes the input image through multiple layers of convolution. Convolution layers produce results that are position invariant. They are meant to pick up on spatially-local feature. With each step deeper in the network, the convolutional layer picks up on a more and more abstract concepts (see Figure 1). The very first layers are generally performing edge detection while the later ones extract abstract concept like "face", "car" etc.. So different information is available at different points in the network. By combining the different activation maps at different points in the network, we can reconstruct a heatmap that takes advantage of all this scattered information.

Fusion operators

We can apply quite a number of usual operators in data fusion. In our present work, we have appealed to the algebraic fusion operators, and to the fusion by a convolutional neural network trained with regard to the ground truth obtained from human observers of the content. ,a). The add operation in fact constructs a geometric sequence. Σα(1 -α) l-1 S l . The normalization operator can also be applied, either at the end of the fusions or interleaved between each binary operator.

By the fact that add and top are not commutative they take advantage of the structure of data, namely, of the fact that the first operand of the operator is the cumulated map in the recursive fusion approach and can be more or less taken into account regulated by the parameter α.

Fusion by a convolutional neural network The idea here consists in training a light CNN m which input is the set of feature importance maps S l from all layers of the CNN model M to be explained, interpolated to the resolution of input images of M . The training of m is fulfilled with regard to the ground truth expressing human perception of the visual content in the classification task. This perception is measured by Gaze Fixation Density Maps (GFDM) obtained in a psycho-visual experiment when human subjects observe images to classify them. We refer the reader to [START_REF] Obeso | Visual vs internal attention mechanisms in deep neural networks for image classification and object detection[END_REF] for a detailed explanation of such an experiment. Human gaze fixations from a number of observers are recorded for each image by an eye-tracking device.

Then on each fixation (u, v) in the image plane, a 2D-Gaussian surface N (u,v,Σ) is centered with the mean vector µ = (u, v) T and a diagonal covariance matrix Σ with equal σ 2 values on the principal diagonal. The scale parameter σ is defined from the geometry of the experiment to represent the projection of the fovea, into the image plane. Summing up and normalizing multi-Gaussian surface from different observers for the same image, its GFDM G is obtained. An example of such maps on the dataset from [START_REF] Obeso | Visual vs internal attention mechanisms in deep neural networks for image classification and object detection[END_REF] is illustrated in figure 2.

We call this CNN-based fusion operator N ET . As a light fusion CNN m we use a simple architecture. The input tensor consisting of intermediate importance maps S l is pushed through three successive convolution layers with pooling. They have the total effect of pooling the input tensor by a factor of 4 and multiplying the depth by a factor of 8. Lastly, a weighted sum is computed to output a final Fig. 2: Example of GFDMs on Mexculture database [START_REF] Obeso | Visual vs internal attention mechanisms in deep neural networks for image classification and object detection[END_REF] predicted 2D map. The loss function is the Euclidean loss, which is the mean square error between G and the application of N ET operator to the input set of of importance maps S l . The fusion network m is trained on GFDMs of a training set of a given dataset. We present our datasets in section 4.

Implementation of MLFEM on CNN classifiers

The MLFEM is a white-box method and can be applied to any architecture of a CNN in visual classification tasks. Here we present the implementation of MLFEM for ResNet50 [START_REF] Rousseau | Residual networks as flows of diffeomorphisms[END_REF] architecture.

ResNet50 contains 16 residual blocks; as a block is the natural unit of choice to apply FEM, we apply it to the output of each of them, after the activation function. This gives us 16 different applications of FEM to fuse. The goal is to maximize the different semantic meaning that one can extract: for ResNet50 we take advantage of the structure of the network.

ResNet50 used in our experiments is trained with the Adam optimizer [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF], with a binary cross entropy for binary classification tasks and a categorical crossentropy loss for multi-label classification tasks.

Evaluation of MLFEM explanations

Methodology. In [START_REF] Ayyar | White box methods for explanations of convolutional neural networks in image classification tasks[END_REF] and [START_REF] Jouis | Anchors vs attention: Comparing XAI on a real-life use case[END_REF] the methodology of evaluation of explanation methods was proposed. It consists in the comparison of the pixel importance maps obtained by the network sensing with the maps expressing human perception of the same visual content. This perception is expressed by gaze fixation density maps we have presented in section 2.3. Today, this methodology is possible thanks to public databases with the recorded gaze fixation of observers, like [START_REF] Obeso | Visual vs internal attention mechanisms in deep neural networks for image classification and object detection[END_REF][START_REF] Jiang | Salicon: Saliency in context[END_REF][START_REF] Borji | CAT2000: A large scale fixation dataset for boosting saliency research[END_REF] and we will apply it to our MLFEM method with different fusion operators and compare with different explanation methods which generate pixel importance maps as MLFEM does.

Evaluation Metrics Evaluating the relevance of pixel importance explanation maps is an open problem. There is no widely agreed upon metrics for assessing their quality. We propose to employ metrics widely used in psychovisual community for comparison of saliency maps [START_REF] Meur | Methods for comparing scanpaths and saliency maps: Strengths and weaknesses[END_REF]: the Pearson Correlation Coefficient and the Similarity metric. Pearson Correlation Coefficient(PCC) is defined as:

corr(x, y) = W u H v (x u,v -x)(y u,v -y) W u H v (x u,v -x) 2 W u H v (y u,v -y) 2
with x u,v being the value of the pixel at position (u, v) in one saliency map and y u,v in another.

The similarity metric is defined as such:

sim(x, y) = min(x u,v , y u,v )
Design of experiments. To evaluate the proposed MLFEM method, we will perform three kinds of i) overall method comparison, ii) dependence on correct or wrong classification results, iii) sensitivity to clutter in the image.

Overall method comparison. We will compare pixel importance maps generated by MLFEM with different fusion methods and the reference methods such as FEM and GradCam [START_REF] Selvaraju | Grad-cam: Visual explanations from deep networks via gradient-based localization[END_REF] which remains the most popular in explanation methods generating pixel importance maps.

Dependence on correct/wrong prediction. We divide the test dataset into images that were correctly categorized by the neural network M and those that were not. We will analyze if a drop in correlation with the ground trouth GFDMs is observed.

In the case of a drop, we can deduce that the convolutional part of the network did not pick up on the relevant features of the image. The fully connected part of the network will then not have the correct information in feature space to classify the image.

If the metrics do not change between correctly and wrongly classified images, we can presume that the fully connected part of M , given supposedly correct feature space information, was not able to classify the images. We can then add more fully connected layers to help the network categorize the inputs.

Sensitivity to clutter. In case the image is cluttered, the GFDMs are dispersed as human attention is attracted by multiple singularities/objects in the image. It is reasonable to expect that the CNN allocates more importance to a strongest relevant region in the image. In this case, the similarity metrics between our explanation map and GFDMs will be lower than for images with low clutter effects.

Experiments and Results

Datasets

We have chosen three different datasets to work on. The first one is MexCulture [START_REF] Obeso | Visual vs internal attention mechanisms in deep neural networks for image classification and object detection[END_REF]. It is composed of 284 images from four classes supplied with GFDMs. It is a subset of 12000 images of the Mexican architectural style: Modern, Pre-Hispanic and Colonial, there is also a rejection class. The dataset contains 2000 Fig. 3: Sample of the data in Salicon database (with GFDM overlaid) images for each category for training and 2000 other images for validation. The GFDM was constructed by instructing the subject to categorize the presented images. We have illustrated the GFDMs in figure 2.

The second one is Salicon [START_REF] Jiang | Salicon: Saliency in context[END_REF]. It is composed of 15000 images of different (up to 80) categories, 10000 of them are supplied with GFDMs It is common to have multiple categories present for each image, hence these images are more cluttered than those from MexCulture dataset. To construct the GFDMs, the subjects were free to "look around" the image and expressed their attention by mouse pointing. We illustrate the GFMDs from this dataset in figure 3.

Finally, we use Cat2000 [START_REF] Borji | CAT2000: A large scale fixation dataset for boosting saliency research[END_REF] as a third dataset. It is composed of 4000 images, 2000 with GFDM, each divided into 20 equally populated categories. The GFDMs were obtained from gaze fixations and the images are less cluttered.

Results

Overall method comparison. We compute two similarity metrics PCC and sim for each classified image of each dataset for every method (FEM, GradCam noted CAM and MLFEM with NET, ADD and TOP variantes) with regard to the GFDMs. To compare the methods between them, we compute a 2 × 2 matrix, comparing the number of times that a method m a was better at explaining classification than a method m b in terms of higher value of each metric. This will give us 6 matrices, 2 for each dataset, one for the comparison using PCC and another for the sim metric, see figures 4a, 4b, 4c.

In figures 5a, 5b distribution of the metrics in Salicon and Cat2000 datasets is plotted for every method compared to the GFDMs.

We can see here in figures 4a and 5a that the proposed MLFEM method is better suited for the explanation of ResNet 50 on the Salicon dataset. We achieve a mean correlation coefficient of, 0.70 whereas the GradCam method only achieves 0.37. We can also see that without resorting to learned fusion operator, FEM and ADD/TOP are better with, respectively, 0.38, 0.43, 0.41 values of PCC. The sim metric behavior is the same. The trained N ET fusion operator is the best in terms of both metrics. In Mexculture dataset, the N ET operator is at least as good as other operators, see figure 4c. For Cat2000 dataset the conclusion is the same as for Salicon, as illustrated in 4b and 5b.

Dependence on correct/wrong prediction. For Salicon, we are in a multi-label classification task with objects of several classes in the same image. We count the number of times when every class present in an image is detected by the network: i.e., the corresponding output neuron has an activation value larger than 0.5. Fig. 5: Distribution of the different metrics for each explanation method for the Salicon and Cat2000 dataset.

as a correct prediction. When at least one class is not correctly predicted, this image is considered wrongly classified. We do not see any significant effect of the correct/wrong categorization of the image on the quality of the explanation map (Figures 6a and6b diplay the average PCC and Similarity metrics for both correctly and wrongly classifed samples). For Salicon dataset (Figure 6a), we have a small drop in the quality of explanation for miss-classified images (in red), a 3% drop in the PCC and no change in the sim metric. For Cat2000, figure 6b, a small increase of the quality of explanation for miss-classified images is observed. The PCC is 4% better, but there is practically no change for the similarity metric. Sensitivity to clutter. This experiment is conducted on Salicon dataset with different classes of objects present in the same image. We divide images into ten categories, with ith category containing images that have i different classes present in it. Then we can plot the mean of the different metrics as a function of the number of classes appearing in an image.

We can see in Figure 7 that our hypothesis holds: the quality of our explanation drops when the number of classes present in an image increases. However, even in high clutter situation, our proposed method with N ET fusion gives the best scores. The N ET method only loses 11% of its performance measured by PCC, where the FEM method loses about 33%.

The similarity metric, however, shows a lower dependence in regard to the number of classes present in an input image. The implication of this merits further research in future works.

In this work, we have extended the method FEM for the explanation of decisions of CNN classifiers by introducing a Multi-Layered strategy: MLFEM. We showed its performance on the ResNet as the latter is nowadays the most efficient CNN classifier. Nevertheless, the method remains generic and applicable to any CNN whether it is residual or not.

We studied different fusion strategies of individual importance maps from each CNN layer. The evaluation has been performed accordingly to the evaluation method designed by us similarly with [START_REF] Jouis | Anchors vs attention: Comparing XAI on a real-life use case[END_REF] which consists in the comparison of explanation maps with gaze fixation density maps (GFDM) of human observers. In terms of comparison metrics of explanation maps and GFDM, MLFEM achieves better performance than the similar state-of-the-art method GradCam and the original FEM. Over GradCam, we got an improvement of 89% for PCC on Salicon and 241% on Cat2000; the improvement of similarity metric is of 30% on Salicon and 51% on Cat2000. We got an improvement over FEM of 84% for PCC on Salicon and 20% on Cat2000; for similarity metric, the improvement is of 27% on Salicon and 9% on Cat2000.

We note that amongst the proposed fusion strategies, the "learnt" fusion operator achieves the best results according to the considered comparison metrics. Hence, the proposed MLFEM method better explains the decisions of the CNN classifier ResNet with regard to the human perception of the content in the classification tasks.

In the future works, it may be interesting to apply and adapt MLFEM to newly appeared transformer networks and to use it with another kind of data than images. In such a case, we will face the problem of definition of the ground truth for the evaluation methodology proposed. Hence, the proposed method opens multiple research questions which have to be addressed in the future.

  Algebraic fusion operators applied in our work are presented below. They are applied individually to the element of the maps. -The max operator max(a, b) is the result of the max operation applied element wise to a and b. max u,v (a, b) = max(a u,v , b u,v ) -The weighted addition add(a, b) is the result of the addition of a and b given a factor α. add u,v (a, b) = α • a u,v + (1 -α) • b u,v -The top operator. It is defined in relation to the add operator, taking only the top 50% features of b. b ′ i = b i if (b i > µ(b)), 0 otherwise, top(a, b) = add(a, b') -The f em operator. We can produce the same result as the FEM method would by using this fusion operator. f em(a, b) = b it is also a special case of the add operator with α = 0 The maximum is commutative: max(a, b) = max(b, a), but the add and so the top operators are not: add(a, b) ̸ = add(b, a), top(a, b) ̸ = top(b

Fig. 4 :

 4 Fig. 4: Number of images with a better explanation by method m a -column than with method m b -line -on the Salicon, Cat2000 and MexCulture datasets.

Fig. 6 :Fig. 7 :

 67 Fig. 6: Metrics for correctly classified (green) and wrongly classified images (red) for the Salicon and Cat2000 dataset with NET, CAM, FEM, ADD, TOP.


Frédéric Gabriel

Methodological Reflections on Approaching Phenomena of Autocephaly 1

Presentation of: Marie-Hélène Blanchet, Frédéric Gabriel, Laurent Tatarenko (éd.), Autocéphalies. L'exercice de l'indépendance dans les Églises slaves orientales (IX e -XXI e siècle)

Rome, École française de Rome, 2021. At the Oberseminar zur Geschichte Russlands und Ostmitteleuropas in der Vormoderne

Ludwig-Maximilians-Universität München Prof. Dr. Julia Herzberg June 2, 2022

First of all, I'd like to highlight the method we chose to use: this volume aims to historicize and de-confessionalize a subject that is often only addressed in the context of controversy, and that is built upon contemporary political and identity-based claims. Autocephaly is a phenomenon that reveals the dynamics of a community that is in the process of being formed, a community that wants to access self-government. When we talk about autocephaly, what we are really talking about is government, to the point that this form was central to the emergence of the Bulgarian and Serbian Empires in the Middle Ages, and later to the nationalisms of the nineteenth century in the Slavic world. In this context, the government sees itself as jointly civil and ecclesiastical, with all the dissensions that can occur between these two domains, but also in relation to external authorities: the Byzantine or Ottoman Empires, the ancient Eastern Patriarchates, the Roman Church, neighboring states, etc. This is why autocephaly is still a highly controversial subject, and why these debates often prevent a historicized and sufficiently objective grasp of the subject. This has been our goal here: the non-partisan and above all comparative examination of an ecclesial form that reveals profound tensions, and that is too rarely studied with sufficient distance from these controversies. I would also like to insist on the semantic trap represented by the terms we commonly use to designate a transhistorical reality. They may mislead us into thinking its essence has remained unchanged over the ages. Diverse phenomena may continue to be referred to by a single term, even though the objective of many official discourses is to establish a natural continuity between different periods and heterogeneous institutional forms, especially since autocephaly is not separable from the very delicate notions of people, ethnos, nation and state, all of which also evolve over time. This is why it is imperative to observe and compare practices of autocephaly over the long term, all the while remaining careful not to shift from the notion of a government to that of a norm.

Why we chose this long millennium, from the ninth to the twenty-first century, is explained by our purpose: in the nineteenth century, the many claims of autocephaly were repeatedly presented not as the creation of a Church, but as the reactivation of medieval autocephaly, which was legitimized by its antiquity, and which coincided with the heroicizing of political independence. We may suspect that the institutional, demographic and geopolitical parameters are very different, but this appeal to the past is central to the consciousness of those who form the Church and it must be taken into account. Faced with these mechanisms, historians are not the guardians of an illusory axiological neutrality, but at the very least they do not presume to judge past events, and their role is not to exalt a great identity-based narrative.

The ninth and tenth centuries may be seen as the inaugural era of autocephaly in the Slavic world, where this form of Church was particularly prominent. Bulgaria is an exemplary case in that it shows us that autocephaly itself is difficult to date, and even to define, simply because, for a long time, it was not theorized or canonically described by any higher dogmatic authority: at most there are some indications of it among Byzantine canonists in the twelfth century.

Concretely, each time an autocephalous Church was created, it was the result of negotiations that were highly dependent on the specificities of the geopolitical situation. Even today, specialists of medieval Bulgaria have differing arguments and hypotheses that only partially overlap. The differences in dating are indicative of the various possible interpretations of (incomplete) sources and of political and identity issues, but also of the long-lasting absence of any predetermined codification of autocephaly. It would therefore be wrong to systematize the processes of autocephaly a posteriori and even more wrong to classify historical cases according to this systematization. These first instants of Bulgarian history can teach us about the decisive role of sovereigns in the emergence of something resembling autocephaly: on the one hand, sovereigns intervened in the choice of the head of "their own" autocephalous Church and managed to avoid appealing to the historical patriarchy on which the constituency in question depended, and on the other hand, they worked to ensure that this constituency was accorded a level of dignity in accordance with that independence.

But Günter Prinzing noted how, despite the existence of autocephaly, the archbishops of Ohrid remained connected to the Byzantine emperor. It was the emperor who validated their election, and most of the hierarchs of the archdiocese were Byzantines. When the Byzantine Empire annexed Bulgaria in 1018 and recognized a specific status for Ohrid as an autocephalous archbishopric, the emperor reserved the right to choose the name of the archbishop to be elected from a short list submitted to him by the Synod of Ohrid. Even if we consider that the former Bulgarian autocephalous Church thus became an autocephalous archbishopric, it still had a special status that implied a relationship of subordination towards the Byzantine emperor.

Moreover, this was a major feature of the balance of power that led to the emergence of Balkan autocephalies: they "benefited" from periods during which the power of Constantinople was diminished and local civil and military structures grew stronger. To put it even more clearly, the claim of autocephaly accompanied attempts to translate a rise in political-military power into an imperial regime (following a Byzantine model). After 1330, this was the case of the Serbian kingdom, which gained territories from the Byzantine Empire and which took advantage of the confrontation between the two Byzantine co-emperors, John VI Kantakouzenos and John V Palaiologos, to proclaim the autocephalous patriarchate of Peć.

In Greek sources, this kind of "patriarchy" is never considered on a strictly equal footing with other historical patriarchates: around 1361 or 1362 Kallistos I claimed that the autocephalous Patriarch of Bulgaria had obligations towards the See of Constantinople. So this was not a case of full autocephaly, but rather of a slightly lower degree of independence. This device was in fact a way for the Patriarchate of Constantinople to avoid completely losing its authority in the Balkans.

We can hypothesize that the existence of the Byzantine Empire rendered complete ecclesiastical independence (almost) impossible. Before the nineteenth century, when ecclesiastical institutionalization ended up distinguishing between the two terms more strictly, autocephaly was close to a more or less advanced autonomy. It was essentially the constitution of a locally designated ecclesiastical hierarchy, which would therefore most likely be Slavophone, while still maintaining a symbolic subjection to Constantinople. The codification of autocephaly as strict independence must be dated to the nineteenth century. In the case of medieval Serbia, we could simply speak of a more advanced form of autonomy than in other Orthodox regions.

On the other hand, the modern period witnessed evolution on this point, because the disappearance of Byzantium made it possible to present the autocephalous Churches as institutions that enjoyed a kind of political independence, reporting only to local sovereigns (for the commemoration of the emperor no longer had any meaning at that point).

These contingent developments are important for our subject, because they show us how complex and shifting the phenomena of autonomy and autocephaly could be depending on the conjuncture, not only with regard to the emergence of autocephalous churches, but also with regard to their disappearance: let us recall the end of the Patriarchate of Tărnovo (in Bulgaria) in 1393, and the end of the Patriarchate of Peć (in Serbia) after 1459, because of the political fall of the civil structure that gave them their consistency. For another decisive sign of the change brought about by the modern era and the disappearance of the Byzantine Empire: the re-establishment of the Patriarchate of Peć in 1557, which separated from the jurisdiction of Ohrid, was not the result of an act of the Patriarchate of Constantinople but rather of the Sultan.

We can thus see how different degrees of independence coexisted and affected various aspects of ecclesiastical government; the modalities of this independence resulted from negotiations and power relations. This point is essential, because-a priori-it does not enter into the internalist considerations of autocephaly, which place it within the framework of the definition of Church unity, the first of its characteristics named in the Nicene-Constantinople Creed. Underneath the apparent simplicity of this qualification lies a remarkable flexibility in the thinking and practices of unity. Autocephaly is one of these practices; originally, it referred to archbishops and metropolitans, elected locally without requiring the validation of a higher authority. The head of the community created a community as a guide for the faithful, a decision-making center, an appeal function, and more broadly, as evidenced by the diptychs, a figure of unification and recognition.

Autocephaly reveals itself both as an original expression of the idea of Church unity, with a legitimacy based on the imperative of external recognition, even belatedly, and at the same time, it generally reflects the centrifugal aspirations of local religious structures. Autocephaly thus integrates into its very origin, an element that tempers the proclaimed independence and motivates a salvo of questions that are the guiding themes of our book: is autocephaly the rigidification of a local Church or an intensification of intra-and inter-ecclesial communion? How should we conceive of primacy in the dialectic created between independence and recognition? With autocephaly, we are talking about a Church-to-Church relationship, but it is an asymmetrical, and therefore hierarchical, relationship.

The old patriarchates tended to regard autocephaly as an exception to the rule produced by a kind of privilege of which they themselves would be the guarantors. From this point of view, it is not possible to describe historical patriarchates as "autocephalous": they appeared from the outset at a supra-hierarchical level, while the Churches that wished to become autocephalous had to demand a change of status. For the same reason, it cannot be said that all Churches were originally autocephalous: while their Sees were more or less independent, the patriarchal echelon appeared only later, and an autocephalous Church is precisely a Church that detaches itself from the patriarchy on which it depends (apart from the case of Cyprus, which was only qualified as an autocephalous Church a posteriori).

The privilege we have just talked about was called into question by Moscow, which was not a historical patriarchate, but ended up acquiring power and prestige on par with those of Constantinople, if not superior. Each patriarchate claimed the exclusive power to grant or refuse autocephaly to the disputed territory. The patriarchates confronted each other across interposed territories, and the Balkans as well as Ukraine were borderlands and zones of influence.

Historically, autocephaly described relationships between Churches (for the recognition of autocephaly implies that there was also communion). It was not only the establishment in law and in fact of a complete, codified and definitive independence. One constant remains: these relationships between Churches were primarily dependent on political, military and economic circumstances, and their independence often resided at the level of the "administrative" functioning of the local Church, while doctrinal and liturgical communion with other Churches is still attested. But there is another parameter that is not without importance: language and ethnos. The linguistic aspect accompanied and reinforced that of territorial defense: the Slavization of the liturgy and of ecclesiastical personnel followed this dynamic and was a decisive element of the relationship with oneself and with the other Churches. But in the Archdiocese of Ohrid in the medieval period, the geographical distribution of dioceses varied between Albanian, Greek, Serbian and Bulgarian settlement areas, so that very different peoples were part of it: so in this case, we cannot speak of a national Church or a State Church as we can for Peć or Tărnovo. And this last situation was controversial, because it was perceived as an ethnic deviation within the Church, which, by definition, does not take this criterion into account in order to form a community. Thus, for our subject, a major break occurred in the nineteenth century, which could be said to be the century of the invention or reinvention of autocephaly as we know it today, an autocephaly based on a reference to the past, but constituted on essentially ethnic and national bases that had practically nothing to do with the medieval model. In other words, at a time when autocephaly was truly becoming a technical term, with widespread use, its new ethnic-political dimensions made it a problematic notion. After the medieval period of Byzantine control, after the early modern period of Ottoman domination, a third period, which could be easily identified based on practices of autocephaly, was therefore distinguished as the age of the Nations.

Imperial universality was followed by the claim of national legitimacy, which could not long tolerate the tutelage of a patriarchate without an emperor; moreover, this patriarchate was no longer in Christian territory. In short, we can see it as a struggle against the yoke of double domination (the Ottoman power, the Patriarchate of Constantinople), which was still strong in the late eighteenth century. There remained the paradox of granting autocephalous status: during the nineteenth century, it was the Patriarchate of Constantinople that granted these autocephalies, more or less quickly, even if other processes were possible -I am thinking, for example, of the Ottoman Empire, which established the Bulgarian Exarchate in 1870.

There is a major issue linked to autocephaly: the legitimacy of this or that Church, as "opposed" to another Church that is always perceived as dominant. Hence the great difficulty of approaching this subject without falling into positions that disrupt its historicization. Autocephaly can be troubling or disturbing, for it is too intimately linked to political imperatives, and too strongly identified with a community constituted by a specific people; on the one hand it is seen as an ancient form specific to Orthodoxy and rooted in the identity of this or that community; on the other hand, it is perceived as the vector of numerous tensions and open conflicts, and it is even supposed to be at the origin of phyletism. There is an oscillation between autocephaly as a model or as a pathology. For this reason, we did not want to study this notion as a timeless and legitimizing essence, but rather as a testimony to conflicts within the framework of "forming a Church". This is why, in our title, after we wrote autocephalies in the plural, we chose to emphasize the exercise of independence, which is always contingent and singular. While knowledge cannot get by without the use of general categories, nothing prevents us from critically reexamining them by going back to singularities. It is not a question of working from a general definition that predetermines whether or not autocephaly is indeed present, but instead of comparing different formalizations of it. As a result, the term autocephaly appears only lately, and it remains rare and above all very poorly or barely defined. It is only recently, on the scale of Christian history, that it has really attracted the attention of hierarchs, canonists and historians.

How does a Church come into being? In the end, autocephaly might well be one of the answers to this question, an answer that has often been forgotten by Western ecclesiology. The phenomenon of authocephaly shows us what a "Church" is (but not everything that it is): it is an entity that is always dynamic, under construction, highly dependent on geopolitics and in particular on border disputes, on the boundaries of empires and on militant nationalisms, which can be very noticeable in the Balkans. This aspect is quite different from the timeless and hieratic image of "the Church", the one and only. As if unity had to be unique, and as if the concept of autocephaly had to be aligned with this uniqueness through a continuist narrative. However, we are dealing with very different forms of degrees of independence and autocephaly, and these cannot be reduced to a normative framework, which remains a bone of contention even now.