ADAPTING ISLAM TO EUROPE: The Albanian Example

Nathalie Clayer (CNRS-EHESS, Paris)

This paper discusses the ways in which local actors adapted Islam to Europe, specifically in Southeastern Europe, from the beginning of the twentieth century until today. I will analyse some of the discourses and actions which are explicitly presented by local actors themselves as efforts to adapt Islam to Europe. Albania lends itself well for such a discussion, since it was the only European country with a Muslim majority, before the recent creation of Bosnia-Herzegovina and Kosovo as independent states. I will concentrate on inter-war Albania in order to subsequently highlight some features of the post-socialist situation.

To analyze the various means of adaptation, it is necessary to take into account the various actors who envision different ways of adaptation and eventually implement them. In particular, we must consider the fact that there are discursive and non discursive dimensions to their ways of adapting Islam to Europe. It is also important to look at the context, not only the national one, but also the international one, and to consider all the possible interactions with trends of thought existing in the Muslim world (and in the non-Muslim world) in a given period.

One of these trends, Islamic reformism, emerged before World War I partly as a reaction to the political superiority of the West, to Western imperialism and to the activities of Christian missionaries in the fields of religion and education. 1 Thus, Islam came to be redefined by Islamic reformists in relation with the "West", "Western civilization", "Europe", "modernity" or "modernization", especially around some key issues like science, education, secularism and women. The building of a "European Islam", which occurs later in Southeastern Europe, is linked to this movement, but also has special features which differ according to regional contexts and periods.

However, in Southeastern Europe during the twentieth and twenty-first centuries, all Muslims are not necessarily motivated by the building of a "European" or a "modern Islam". Instead, some are guided by a search for a genuine, a "true Islam" also subjectively defined Others, probably the most numerous, are worried about the search for salvation, for baraka (blessing or divine gift), for intercession, in order to face the troubles in their daily lives, while still others do not worry at all about adapting their Islamic beliefs and practices.

Finally, outsiders can also have a part in the building of a "European Islam", which has been especially true for Islam in the Balkans. As early as the nineteenth century, some European sources already present an image of a different Balkan area, one with a "non-fanatic", "non-fundamentalist" Islam. 2

The issue of the adaptation of Islam in inter-war Albania

In inter-war Albania, the national and international context was characterized by the building of a nation-state, the establishment of an authoritarian regime (as happened in other European countries and in Turkey at the same time), and by a deep economic crisis. The period was also one that saw the development of Communism in Europe (after the October Revolution), while colonialism was still at work, and Italy was exerting an ever greater pressure on Albania. In matters of religion, secularist trends were very strong among European and Turkish intellectuals and politicians, even if the First World War led to a reinforcement of the position of religion in the European public space, at least for a short period. 3 In the Muslim World, the press was the main channel of expression of reformist and nationalist trends. 4 As for Albania, the state was declared to be without any official religion. However, beyond this official separation of church and state (as in Turkey), 5 the state exerted close control over religious matters. In this context, in the issue of the adaptation of Islam to the new situation of the country, as an independent state in Europe fighting to keep its sovereignty, three main periods or phases become evident. During these specific times, debates and reforms led by different actors took specific forms.

1923-1925: Islam, progress and civilisation

During the first phase, the terms of the debate were, above all, "progress", "civilisation", "culture", "West" and "East", "foreigners", and to a lesser extent "Europe". Hafiz Ali Korça, a spokesman for the group of ulamas leading the Islamic religious institutions (whose discourse was very anti-missionary, anti-Semite and anticommunist), and his colleagues writing in the journal of the Community insisted primarily on the search for the true religion, the religion which prevailed when the Muslim civilisation was at its height, with famous sources of knowledge, notably the madrasas of Baghdad and Andalusia. They wanted to prove to the detractors of Islam that this religion didn"t exclude progress. They also wanted to persuade Muslims to return to a true Islam which would allow progress and civilisation. 6 Hafiz Ali Korça and his circle were not forging a "European Islam", since they were in general associating Europe and the West with irreligion (pafesija). 7 If, for them, the West was associated with civilization, this civilization was only the technical one and not the true civilization. The East, on the other hand, was the true spiritual civilization. Japan was an example to follow, since this country was supposed to have taken from the West only its technical civilization. Furthermore, Europe was not a good reference; it could only be a chaotic reference, since there were in fact different civilizations, corresponding to the different countries. 8 All these arguments led Hafiz Ali Korça to conclude that Islam itself did not need to be reformed. It was a modern and healthy religion, adapted to science, progress and civilisation. If the Muslims were not in a good position, it was because they were not good Muslims, because they were not involved, because 6 Numerous articles published by Hafiz Ali Korça, who was the main contributor to the official journal of the Albanian Islamic institutions at that time, touched on these themes. See, for example, A., "Si ishim e si jemi", Zani i naltë, I/5, February 1924, p. 152-158. Other ulamas contributed to diffuse the same message. See for example, Naili V., "Myslimanizma edhe qytetrimi", Zani i naltë, I/4, January 1924, p. 118-121. 7 A.V., "Filosofi mbi fêt", Zani i naltë, I/4, January 1924, p. 121-123. Europe was also negatively associated with the violence of inquisition (see A., "Filosofi mbi fêt", Zani i naltë, I/5, February 1924, p. 149-151). 8 H. A., "Qytetrimi i vërtetë e Qytetrimi mjestëror", Zani i naltë, I/6, March 1924, p. 182-187. they had not kept alive the earlier knowledge. [START_REF] Ali | Muslimanizma dhe riforma[END_REF] Therefore, the solution for Hafiz Ali Korça and the leading group of Albanian Islamic institutions of that time was to establish a good madrasa, with modern methods of teaching and a program including contemporary sciences. With this madrasa, they hoped to prevent the young Albanian Muslims from going to Europe to study, so that they would not be cut off from their religion and the nation. [START_REF]Medreseja e përgjithshme (Universite) e Komunitetit Mysliman[END_REF] In reality, one of the problems for these ulamas was that ever fewer young boys came to study in the madrasas, especially from the south of the country. [START_REF] Nathalie | Investing in the intellectual capital: The kadı of Gjirokastër and Libohova and their descendants[END_REF] In 1924-1925, two such reformed madrasas opened in Tirana and Shkodra, where the pupils were supposed to receive the same education as in the other schools of the country, with the addition of religious subjects. French was to be taught as a language of business. 12 Hafiz Ali Korça and his reformist colleagues also attempted to improve the cultural level by other means: the publication of social, moral, economic, technical and historical papers in the journal published by the Islamic religious Community, as well as the translation and publication of books. These leading ulamas were also interested in improving the organization of the Community and to increase the number of religious specialists (imams, hatibs) in the villages. [START_REF]Veprimet e Keshillit të nalte të Sheriatit dhe nevojat e trupit musliman[END_REF] This vision, especially the idea that Islam did not need to be reformed and that the true civilization was to be found in the East, triggered a controversy. Safet Butka, a young Albanian student in Austria, rejected this theory which, in his view, placed Albania outside Europe and portrayed the Albanians only as Muslims. 14 Morover, he rejected Islam and the Orient (associated with five centuries of Ottoman domination) and, instead, proposed a combination of Albanianism and the West. 15 In many texts, Hafiz Ali Korça and his colleagues were addressing young people, such as Safet Butka, who were, according to them, touched by irreligion and the wish to imitate the West. 16 If some young people were opposed to Hafiz Ali Korça for promoting the rejection of Islam and of the East, Baba Ali Tomori, a leading Bektashi promoting the reform of Bektashism, also strongly criticized him, claiming that "Our faith [Islam] can only live on European soil after undergoing reforms". So he was clearly advocating an adaptation to Europe. Nevertheless, the solution for him was the promotion of the non-Sunni, Bektashi version of Islam (in a way assimilated to Shiism, but with its mystical component), a solution already envisaged by some foreign observers as early as the nineteenth century (such as Dora d"Istria and Margaret Hasluck). 17 Indeed, for Baba Ali Tomori, it was necessary to return to the original Islam of the Prophet, which he presented as a liberal, democratic and perfect Islam. Because for this Bektashi baba the problem was that after the time of the Prophet, it was Abu Bekir, Omar and Osman who usurped the rights of Ali, the designated successor of the Prophet, and that subsequently other Sunni leaders did "bad things". Baba Ali Tomori felt that only when these bad people were made to disappear from the East, when Atatürk would accomplish his reforms, when the Arabs would unite with the Persians around the belief of Imam Jafer Sadik, when the philosophy of Muhidin Arabi would be considered, only then would the East be proud. 18 However, he gave no details on the process of implementing the secularizing reforms of Atatürk while at the same time adopting a mystical non-Sunni Islam, which he also wanted to reform. 19 It was precisely during this period, between 1923 and 1925, that the leading Albanian ulamas like Hafiz Ali Korça were confronted with reforms implemented by the government. Sometimes reforms were only proposed by intellectuals and parliamentary deputies who were trying to build a European state, similar to what Atatürk was doing in Turkey. These reforms did not exactly correspond to the changes envisioned by Hafiz Ali Korça and his colleagues. The administrators, 15 Safet Butka, "Letër e hapur", Shqiptari i Amerikës (Korçë), n°108, 17 June 1924, p. 4 and n°109, 21 June 1924, p. 3. 16 See "Nji bisedë me nji jeun shqiptar mysliman", Zani i naltë, I/11, August 1924, p. 338-342; II/1, March 1925, p. 402-405; II/2, April 1925, p. 430-433 ; II/3, May 1925, p. 471-473. 17 See Nathalie Clayer, "Der Balkan, Europa und der Islam", p. 309-312. 18 Varf Ali Tomori, "Myslimanizmi dhe Orienti", Shqiptari i Amerikës, n°120, 2 August 1924, p. 2-3. 19 The proposed reform consisted of a rational organization and the training of clerics. intellectuals and deputies generally wanted to nationalise Islam and banish it from the public sphere. In reaction, in January 1924, the heads of the Islamic institutions requested that the Constituent Assembly give more rights in legal matters and in the administration of vakfs to the Community. Furthermore, they wanted the Community to be subsidized and that vakfs seized by the state should be returned. The Islamic Community should also have the right to open schools, if the teaching of religion in state schools was not provided anymore. 20 The strong tendency to suppress religion courses in state schools, in particular, gave rise to constant reactions from the part of the leaders of the Islamic institutions. On these occasions they used the reference to Europe of their adversaries who were advocating the progressive separation between school and religion in the name of Europeanization. Hafiz Ali Korça and his colleagues were publicizing that in Italy, Bulgaria, Greece, and elsewhere in Europe, the authorities had realized the importance of religion for the preservation of moral principles, especially in response to the danger that Bolshevism represented. Thus, for them, the European model did not lead to the separation of school and religion; on the contrary, the courses of religion should remain in school programs. 21 On the other hand, this view of education endorsed by Ali Korça and the leading ulamas of the Islamic Community was strongly opposed by other ulamas.

The latter did not intervene in the debate, but we know that they reacted against the new program of the madrasas, a program that they considered too secularized and thus inadequate to train ulamas. 22 During this first phase, education was at the center of the issue of the adaptation of Islam to the new era. For Hafiz Ali Korça and his colleagues, Islamic education should be combined with a scientific modern education in order to attract young boys and their families lured by the West. But other ulamas questioned the centralization of Islamic institutions and thought, on the contrary, that only a strengthened Islamic education could save Islam in the country.

Outside the ranks of the ulamas however, the adaptation of Islam to Europe was already on the agenda: for the most reformist Bektashi Baba, this adaptation would be possible only through the promotion of a reformed mystical Islam and other reforms like those established by Atatürk. Some European governmental policies were already similar to those decreed by the Turkish leader, which was leading to a use of Islam, but also to a progressive rejection of Islam (and religion in general) from the public sphere, particularly in the educational system. Some Young Albanians studying in the West were even more radical and were suggesting a rejection of Islam altogether.

1927-1929: the issue of Muslim reforms

The second decisive moment came in February 1927 with a call for reform by Salih Vuçitern, the Director of the vakf administration. This call was certainly provoked by the disorder and confusion present throughout the vakf administration supervised by him. But more importantly, reform of the madrasas had failed because some ulamas were against education reform, while also objecting to the new organization of the Islamic religious institutions. Salih Vuçitern didn"t refer explicitly to a "European Islam" in his polemic, but the issue of "Islam in Albania" was implicitly formulated as "Islam in Europe". Indeed, when Salih Vuçitern published his call for reform in the journal of the Islamic Community, this journal also began to publish letters and articles from the Ahmadiyya, an Islamic missionary group active in Europe, underlining the conversion of European scholars and publicizing, for example, the new mosque in Berlin. 23 In the framework of the debate, other actors and even some ulamas explicitly refer to the need to adapt Islam to Europe. But let us first examine the reasoning of Salih Vuçitern, who was not a Muslim theologian like Hafiz Ali Korça and didn"t hesitate to speak about "reforms". He urged his co-religionists to wake up, because in the civilized world only those who wake up and progress could survive. For the Albanian Muslims, reforms were particularly vital because they were surrounded by "peoples of culture". In fact, whereas the Sharia itself required Muslims to act, to improve the situation of Islam and Muslims, nothing was done for the mosques, the madrasas, economic activities, mutual aid, etc. Concretely, as the person responsible for the vakf administration, the reforms suggested by Vuçitern concerned the removal of all regional madrasas and of all the superfluous mosques in the country. On the other hand, he proposed the reorganization of the vakfs and the foundation of a 23 See Nathalie Clayer, "La Ahmadiyya lahori et la réforme de l"islam albanais dans l"entre-deuxguerres", dans Véronique Bouillier & Catherine Servan-Schreiber, De l'Arabie à l'Himalaya. Chemins croisés en hommage à Marc Gaborieau, Paris, Maisonneuve et Larose, 2004, p. 211-228. modern education institute, where the contemporary sciences should be taught (with the primary institute in Tirana, and a lower level branch in Shkodra). Indeed, he considered education to be the only means to change the mentality of the people and to change the impression of foreigners (and even of some Muslims) who considered Muslims to be ignorant. Education here was meant mainly as Islamic education, since it was to be transmitted through madrasas, mosques, lectures and a specific journal. [START_REF] Vuçiterni | Deklaratë mbi nevojën e reformave[END_REF] The call of Salih Vuçitern provoked many reactionsboth positive and negativewhich were published in the journal of the Islamic religious institutions. Two ulamas reacted positively. Hafiz Xhafer (from Korça) added the suggestion of also opening a second branch of the new madrasa in the south of the country. Sherif Langu, in view of public opinion and the fact that the time of reform had arrived, asked that the High Council of the Sharia, the highest religious authority, confirm that the proposed reforms were not contradictory to the sacred law. 25 One Muslim intellectual from Shkodra also supported Salih Vuçitern"s propositions, this time by using the argument of adaptation to Europe.

He reasoned that since Albania was an independent country and since it was twentieth century Europe, the Islamic institutions should be organized properly.

More precisely, the problem for him was that the regional madrasas had been incapable of implementing the new education program. Similar to Sherif Langu, he asked the High Council of the Sharia to clearly state its position on the reforms. [START_REF] Shkodrani | Të gjith kemi nji derd![END_REF] Negative reactions came from some ulamas of Shkodra who considered that the proposed measures, notably the closing of superfluous mosques and of provincial madrasas, were against the sacred law. They gave no more precise arguments and criticized Salih Vuçitern rather on form, or on the issue of representation and authority. [START_REF] Vuçiterni | Nji fjalë mbi deklaratën time[END_REF] We can suppose that they felt threatened in their religious and educational prerogatives.

Faced with this polemic, the High Council of the Sharia finally decided to issue a legal opinion (fatwa) in November 1927. It considered that ijtihad (rational interpretation) was justified as far as vakfs (pious foundations) were concerned. So Salih Vuçitern"s propositions could and should be followed, under specific conditions: if the mosques and madrasas were in a state of decrepitude, if the ancient vakfs of rulers had no legal property certificates (as the sixteenth century shaykh al-Islam stated), and if it was necessary to transform charity institutions in order to adapt them to the present time. 28 The legal opinion was clearly less controversial than Salih Vuçitern"s call, not least because of its strictly religious nature. However, it was leaving the door open to various interpretations and didn"t put an end to the debate. Salih Vuçitern"s supporters argued that the highest Islamic authority could not be contested, and that the reforms should be implemented, while the anti-reform ulamas continued to oppose this position, certainly because the main problem was in fact the profile of the new madrasas, their place in this new framework and the power balance within the Islamic institutions. 29 However, at the very time when Salih Vuçitern was advocating reforms concerning mosques and madrasas, he was also the spokesman of the Islamic institutions against various aspects of a more far-reaching reform launched by the Albanian government, namely the introduction of the civil code and the abolition of all religious judicial bodies, including the Sharia courts. 30 In the second half of 1928, the government and some intellectuals were willing to support and advance the reforms and their implementation. The political authorities not only carried out the judicial reforms, they were also, in fact, the initiators of the reform of the Islamic institutions. At the end of August the government asked the High Council of the Sharia to form a commission in order to set up a new organization for the Muslim believers"adapted to the need of the time", and functioning with "modern rules". The report, submitted by the commission in January 1929, represented more or less the same spirit of change as had been inspired by Salih Vuçitern: its authors insisted on the suppression of superfluous mosques and masjids, the opening of a new general madrasa with two 28 "Nji pergjegje nga Këshilli i Naltë pyetjeve qi bâhen në Nr 11 të vitit 1927 të revistës "Zani i naltë"", Zani i naltë, V/1, November 1927, p. 409-412. 29 J. Latifi, "Rreth reformave", Zani i naltë, V/2, December 1927, p. 420-424; Shkodrani, Zani i naltë, V/2, December 1927, p. 441-443; T. Myftija and H.I. Karakaçi, Zani i naltë, V/3, January 1928, p. 464-465; Drejtimi, Zani i naltë, V/3, January 1928, p. 465-466; S. Vuçiterni, "Deklarat"e II", Zani i naltë, V/3, January 1928, p. 467-472. 30 Drejtimi, "Kodi civil", Zani i naltë, V/5, March 1928, p. 534. regional branches, and limiting the wearing of religious clothing only to those who were deserving of it and needed to do so. 31 This position was far from the standpoint of leading secularist administrators.

In November-December, one of them, Mehdi Frashëri, a Muslim intellectual promoting the Bektashi version of Islam and one of those responsible for the new civil code, launched a new debate on the "Muslim reforms" in the press. He firmly believed that these reforms were necessary in order for Albanians to be included among "European people". Either the Albanian Muslims would undertake reforms or Islam would have to disappear from the country. The reforms in question should concern the position of women (including the ban of the veil), the access to Islamic sources without the intermediary of an authority, the formation of local Islamic councils by educated people, doctors, lawyers, etc., and the suppression of all madrasas. The reforms also called for the founding of two colleges in the country with a new educational system, as Salih Vuçitern had proposed, but the best students would be sent to Europe to study philosophy and Arabic at the university, and they would become muftis. Mehdi Frashëri also suggested changing the way of performing prayers, etc. 32 The answer of the Islamic religious institutions was given by Ismet Dibra, an ulama active in the new madrasa established in Tirana in 1924-25. The main criticism of the position of Mehdi Frashëri was that ijtihad and reforms should be undertaken only according to Sharia law. Reforming beliefs and the way of performing prayers was, for example, absolutely impossible for Ismet Dibra. 33 He totally agreed with Mehdi Frashëri that in order to survive in Europe the Albanian

Muslims had to adopt what Mehdi Frashëri called « the principles of the present

European life », but what he himself somewhat differently mentioned as « all the present sciences which are the fruit of the progress ». But he totally disagreed with the position that it was proven that Islam and Muslims could not progress. Islam, like other religions, was not a hindrance to progress, since Europeans and Americans firmly believed in progress, while also believing that the Bible was God"s word and that Christ was raised from the dead. Hafiz Ismet Dibra also 31 Arkivi Qendror i Shtetit (Tirana), Fondi Ministria e Drejtësisë (n°155), Viti 1928, dosja VIII-77, fl. 4 (letter of the Minister of Justice to the Head of the High Council of the Sharia, Tirana, 31/8/28); fl. 38-48, results of the Commission, with a project of statutes, Tirana, 22/1/1929. The last measure was meant to eliminate non-controlled religious specialists. 32 Mehdi Frashëri, "Reformat Myslimane në Shqipëri," in Gazeta e re (Tiranë), n°17, 20/11/1928, p. 1 ; n°26, 1/12/1928, p. 2 ; n°27, 2/12/1928, p. 4 ; n°30, 6/12/1928, p. 2 ; n°31, 7/12/1936, p. 3 ; n°35, 12/12/1928, p. 2 ; n°36, 13/12/1929, p. 2 ; n°37, 14/12/1937, p. 2 ; n°39, 16/12/1928, p. 2. 33 In addition, the traditional way of performing prayers was scientifically thought to be very good for the health of the participants. naturally refuted all the allegations concerning the law level of education of the ulamas leading the religious institutions and their supposed stand against reforms such as the civil code. 34 The reactions of some ulamas working in the central Islamic institutions (including Hafiz Ismet Dibra) were largely ignored. The civil code was implemented in the spring of 1929, and in June a Muslim congress was convened in Tirana, officially by the Islamic institutions, but in fact by the government. New statutes were enacted, which were very different from those suggested by the commission set up by the Islamic institutions. All the regional madrasas were effectively closed, and only one general madrasa was subsequently opened in Tirana, without any branches in the province. Only the needed mosques were to be maintained by the Islamic institutions. Behxhet Shapati, a lawyer who was not part of the ulama milieu, became head of the new Community. The government allowed the independence of Bektachism in 1929, but it granted only autonomy from the Islamic Community, bowing to strong pressure from Muslim parliamentary deputies. The same year, political authorities banned the veil and the fereje (dustcoat). Cultural associations were established in different parts of the country at a later date. 35 It can be said that these reforms were enacted under political constraint, which the administrators of the various Islamic institutions would have not carried out in such a way, even if some of the reforms were thought to be desirable by some leading ulamas. There are, in fact, two ways of viewing the adaptation of Islam to Europe at that time: both views recognized that education, rational interpretation and rational organization were the primary means toward reform. 36 The themes of Europe and the compatibility of Islam and Europe were increasingly present in the official discourse of the Islamic Community, through the publication of texts of the Ahmadiyya and other proselyte Muslim groups in the West. 37 However, the ulamas and some Muslim intellectuals were unwilling to detach the interpretation of the reforms from Sharia principles and the authority of the ulamas. This detachment had been proposed by Mehdi Frashëri, who envisioned even more radical reforms than those instituted by the government. 34 Zani i naltë, V/10-11, September 1928, p. 678-690; V/12, December 1928, p. 726-734. 35 See Nathalie Clayer, "Behind the veil. The reform of Islam in inter-war Albania or the search for a "modern" and "European" Islam", in Nathalie Clayer and Eric Germain (eds.), Islam in Inter-War Europe, London, Hurst, 2008, p. 128-155. 36 However, at the same time, traditionalist trends become stronger within the Islamic institutions, because of the integration of ulamas who had previously protested the reforms. 37 Nathalie Clayer, "La Ahmadiyya". improvement of the position of women, but also supported the secularization of Islamic institutions. The government officially followed this direction but in a less radical way: it suppressed the Sharia courts (a consequence of the adoption of the civil code); it organized and controlled the Islamic institutions, secularizing only their head with the nomination of Behxhet Shapati; it gave autonomous status to Bektashism; and it officially banned the veil, a measure which was not really effective.

1935-1937: The issue of the veil

The third decisive period when we can observe discrepancies between differing intentions aimed at adapting Islam to Europe took place during the years 1935-1937. At that time, the central issue was no longer education, but rather the issue of the veil and the position of women 38 . In fact, these questions had already been debated at the beginning of the 1920s and, following the Congress of 1929, the new head of the Islamic Community was persuaded to ask the government to ban the veil. Indeed, the Ministerial Council banned the veil and the ferece, a ban which was not really enforced, except among civil servants. 39 During 1935 to 1937, when the country was plunged into a deep economic and political crisis, in particular because of tense relations with Italy, the debate gained new impetus. Mehdi Frashëri, citing the reforms in Turkey, declared that Islam did not give a human position to women. Young journalists/intellectuals also severely criticized the veil, which they associated with the Ottoman past, with backwardness, fanaticism and Asia. They argued that it was necessary to ban the veil in order for Albania to be included as part of Europe (which seemed natural because of the geographic position of the country) as well as to save the entire nation from socio-economic decline.

The attitude of the representatives of the Islamic Community was not entirely clear in this debate. In general, they primarily reacted by defending what they considered to be not only an attack on the veil, but above all an attack on Islam. They emphasized, for example, the superior position of women in Islam. And, like Hafiz Ali Korça in 1923-1925, they in a way were opposed to Europe when they denounced the young intellectuals as being under European influence when the latter (wrongly) considered that the existence of veiled women was against progress. They felt that the young intellectuals did not understand the issue. At the 38 Atheism was also an increasingly debated theme during the 1930s in Albania. 39 For the whole debate, see Nathalie Clayer, "Behind the veil".

same time, the representatives of the Islamic Community were writing many texts attacking atheism among students, especially among those studying in Europe.

They denounced the degeneration of young people, interested only in material culture.

However, in 1936, young Albanian students pursuing Islamic studies in Lahore intervened a bit differently in the debate, using arguments elaborated by Ahmadi Lahori missionaries (the same ones who were active in Europe): they claimed that the progress of Muslims was closely related to solving the problem of the veil. The students believed that the veil was indeed an obstacle to the education of Muslim women. On the other hand, many ulamas did not intervene in the debate and some felt that non-religiously qualified persons had no legitimate say in the matter.

The adaptation to Europe of inter-war Albania compared to post-socialist

Albania

The issue of the adaptation of Islam to Europe was, in inter-war Albania, closely linked with trends of thought (Islamic reformism, Lahori Ahmadism, Occidentalism, secularism, nationalism, etc.). However, there were concrete issues behind the debate: above all education, as well as the position of women in the society, law, nation building, foreign relations, and religious authority. The different visions concerning the way to adapt Islam to Europetotal rejection, rejection from the public sphere, promotion of another version of Islam (Sufi-Islam, especially Bektashism), rationalization, secularization, reform using a Sharia-centered frameworkalso were elaborated according to the interests and the constraints of the actors participating in the debates and acting in these fields.

In the domain of education for example, the viewpoint of the leaders of the religious institutions was related to the competition between the various education networks. The madrasas were generally in bad condition and there were ever fewer students enrolled (except in Shkodra, which also explains the different position of a group of ulamas from that city). These leaders wanted to attract students, to promote religious education, and to train ulamas in a new way. They also fought to maintain religious education in the state schools. They were under several constraints: the constraints imposed by the Albanian government, who urged them to rationalize their organization even more and to control it; the constraints resulting from the resistance of some ulamas who had different ideas concerning the reform of religious education; and the constraints resulting from the secularization of the elites.

Let us now briefly compare these debates and measures to the debates and measures which can be observed since the fall of the Communist regime. We must first look at the context, which is, of course, quite different for the two periods. On the other hand, the search for a "true Islam" is at the heart of intense debates, which was not the case in inter-war Albania, at least not in the same way.

Indeed, in the Albania of today, (as in the rest of the Muslim world) the influence of neo-salafi trends is very strong, especially among young Muslims who are contesting the official Islamic religious representatives. This trend is particularly strong in Albania, since one of the leading neo-salafist was Sheh Albani, who was of Albanian origin and lived in Jordania and Saudi Arabia before his death in 1999. The debates initially focus on dogmatic issues and questions of practices, for example the way of performing prayer. Indeed, the neo-salafis reject the madhhabs, or Islamic law schools (Hanafism, Hanbalism, Malikism, etc.), and prefer to consider the time of the Prophet as the point of reference. They also accept some ulamas, who have not been heads of Islamic institutions, as their councilors and mentors. In fact, this issue has often been coupled with the issue of authority and the balance of power within the Islamic institutions and between the generations which have received a different Islamic education. Until now there has not really been an open confrontation about Islamic religious education in Albania, as there has been in Bosnia. [START_REF]European Islam" and Islamic Education in Bosnia-Herzegovina[END_REF] However, the confrontation between the older generation and a new generation of ulemas educated in Arab Islamic universities has led the members of the older generation who are at the head of the Islamic institutions to favour Turkish networks, especially those of the Fethullajis and the Süleymanjis, when establishing or managing Islamic schools and cultural centres. [START_REF] Solberg | The role of Turkish Islamic Networks in the Western Balkans[END_REF] The search for a European Islam, or in a larger sense the question of the relation between Europe and Islam, also emerged in the public sphere in Albania (and in Kosovo). In this paper it is not possible to analyze the complex evolutions of the debates and measures related to the issue. What is evident is that the tendency to reject Islam as supposedly incompatible with Europe, as a hindrance to the integration to Europe, is stronger now than it was during the inter-war period. Ismail Kadare was one of the first to defend this stance and many intellectuals share his opinion. [START_REF] Nathalie | L"islam facteur des recompositions internes en Macédoine et au Kosovo[END_REF] What is new is that the debate has recently taken on a pan-Albanian character, with the polemic between Kadare and Rexhep Qosja, a Kosovar academician who defends the multiple denominations of the Albanian nation. In recent years, more and more voices are raised against this rejection of Islam. [START_REF] Puto | Adrian Brisku and Fatos Luboja in the issue XII/23[END_REF] However, a kind of rejection under constraint is also to be seen in the changing of names upon emigration, even if it is often a superficial rejection.

The promotion of Bektashism by some intellectuals, both Bektashis and foreigners, as a liberal and non-fundamentalist form of Islam is also present in the debates, but I am not sure that its impact is very important, despite efforts of some leading Bektashis to rationalize their faith. [START_REF] Clayer | L"islam balkanique aujourd"hui entre science et recherche de valeurs[END_REF] What is new in this post-communist period is the presence of proselytizing Islamic networks (in this sense the Lahori Ahmadiyya was a precursor). The development of some of these new Islamic trends and networks are also more or less connected with a search for a "European" or at least a "modern" Islam. That is why education according to "modern standards" is often a key way for these networks to establish themselves in the society. One good example is the group of Fethullah Gülen, which has established numerous non-religious schools and madrasas in Albania and elsewhere in the Balkans. One must remember that for young Muslim Albanians the search for a European Islam is also filtered through an Islam defined by Europeans: converts and/or specialists of Islam, like William Chittick, whose books are translated into Albanian. More popularly, a new trend coming from the rest of the Muslim world has appeared. This trend concerns the rationalization of Islam and the proofs of the compatibility of Islam and science: the solution is to no longer adopt Western sciences, but rather to prove that Islam itself is scientific and that there is an Islamic science. 47 To understand more precisely the stakes behind the debates, it is important to analyze the relative position of the different actors. Taking the issue of education for example, it would be interesting to study into details the debate, while considering the various features of the fields of education, of religion, of politics, and of authority. It would be necessary to analyse the offer proposed by the madrasas in this new context, the way they compete with other schools in the country or with a self-taught approach of Islam, as well as the way the terms "Europe" and "modernity" are connected with the promotion of these educational institutions or in the choice by students and their families of these institutions. It would be useful to study the role of the official Islamic religious establishments and of other Islamic networks in the elaboration of courses and way of behaving in the schools, as well as the role of the state in this area and the view of international institutions on the subject. In brief, it is important to understand the interests and the constraints of the different actors.

But even without such a complete study, we can say that the contemporary problem of "adapting Islam to Europe" in Albania has some common features with the problem as it was posed during the inter-war period. However, in many ways, it has not the same significance today, because the local and the international contexts have considerably changed. "Adapting Islam to Europe" is thus very relative, depending on what Islam and Europe represents (rationality, sciences, education, spirituality, secularism, position of women, perversion, morale, democracy, European Union, etc.) and on the balance of powers between various religious and non religious actors.

N.C. 47 Ibid.

Finally

  in 1937, the political authorities again took the initiative for taking measures, and proposed a law banning the veil, allegedly following an initiative of two muftis and of the head of the Islamic Community. The latter issued a fatwa (legal opinion) certifying that it was not haram (forbidden by religion) for a woman to show her face. Eight days later, the parliament approved a law banning face covering. This obliged the leaders of the Islamic Community to justify the lifting of the veil and to join, in a way, the arguments of the young Muslim students in Lahore. The ban, however, was again not strictly implemented, because of the fear of reactions from the population and also because the true aim of the law was to present the image of a Western country to other Western countries, at a time when the sovereignty of Albania was threatened.This time the debate concerning the adaptation of Islam had been launched by non-religious Muslims who, in a way, were promoting the rejection of the veil and the rejection of Islam, at least from the public sphere. The political authorities had taken legal measures in this direction, because, similar to the intellectuals, they also thought it was the only means to legitimate the existence of the country in Europe, even if they thought the measure was also important for the internal development of Albanian society. The attitude of the political authorities, however, was less radical than that of the intellectuals, since it was at that time that King Zog decided to allow the reintroduction of religion courses in state schools in order to counter communist influences and to ensure the stability of the country. 40 40 AQSh, F. 152 (Ministry of Interior), Viti 1938, Dosja 688, fl. 1, letter of Zog to the Prime Minister, Tirana, 15/3/1937.

  Ankimet e popullit musliman dhe të Këshillit Naltë të Sheriatit mbi lanjen mas dore të mesimit besimit nga Qeverija revolucionare e Nolit", Zani i naltë, II/1, March 1925, p. 409-416 ; "Muslimanizmi dhe Kukuvajkat", Zani i naltë, II/2, April 1925, p. 438-442.22 Arkivi Qendror i Shtetit (Tirana), Fondi 152 (Minister of the Interior), viti 1924, dosja 594, leter of M. Tahir, H.I. Karakaçi and others to the Minister, Shkodër, 13/5/1924, fl. 2.

	20 Zani i naltë, I/4, January 1924, p. 125-126.
	21 "Të ndryshme për mbi moralin. Komunisza dhe feja", Zani i naltë, I/11, August 1924, p. 342-
	347; "Të ndryshme", Zani i naltë, I/12, September 1924, p. 381-383 ; "Të ndryshme", Zani i naltë,
	II/1, March 1925, p. 406-408 ; "

  On the other hand, Islam is much more present in Western Europe, because of the growing number of Muslim Diasporas. Europe often views the Islamic scene through the issues of fundamentalism and terrorism (especially after September 11). Neo-salafist trends have an important impact on young Muslims. Neo-brotherhood movements have developed in Turkey, where an "Islamist" political party democratically has come to power. The way of believing, of identifying oneself and of gaining knowledge is more and more individualized.In this context, the issue of Islam and Europe has an important place in the debates, but, similar to the time of inter-war Albania, it is today not the only concern for Albanian Muslims and Albanian non-Muslims. As far as Islam is concerned, the search for sainthood is very important, as shown by the strong impulse from below to recreate holy places and to restore the cult of saints, in order to ask for help and solace. This phenomenon has been accompanied by the (re)emergence of holy men or women acting as intermediaries between God and the people.41

	The post-communist context is characterized by a new freedom of movement and expression, by the rehabilitation of religion, banned in Albania during the previous 25 years. Dynamics in the international religious field are also quite different, compared to the inter-war period: neo-protestant and other sectarian groups are has other meanings than it had in the 1920s and 1930s. Today, it is no longer simply associated with civilization and progress, but with democracy, liberalism, now very active. Europe is already, for many people, synonymous with modernity. But Europe also capitalism, and the European Union.

Pierre-Jean Luizard, "Les effets du réformisme sur les transformations de l"autorité religieuse musulmane. Les cas des pays arabes du Moyen Orient, de la Turquie et de l"Iran", in Martine Cohen, Jean Joncheray et Pierre-Jean Luizard (eds.), Les transformations de l'autorité religieuse, Paris, L"Harmattan,

2004, p. 23-37.

See Nathalie Clayer, "Saints and Sufis in post-communist Albania", in Masatoshi Kisaichi (ed.), Popular Movements and Democratization in the Islamic World, London-New York, Routledge, 2006, p. 33-42.


Jean Dezert
email: jean.dezert@onera.fr

Albena Tchamova
email: tchamova@bas.bg

On the behavior of Dempster's rule of combination

Keywords: Data fusion, belief function, DSmT, DST

In this paper we present simple examples showing the insensitivity of Dempster's rule of combination proposed in Dempster-Shafer Theory (DST) of evidence with respect to the level of conflict between two sources of evidences. Aside famous Zadeh's example on the validity of Dempster's rule of combination, it is shown that for an infinite number of cases Dempster's rule does not respond adequately to combine sources of evidence even when the level of conflict between sources is low. For a comparison purpose, we present the solution obtained by the more efficient PCR5 fusion rule proposed originally in Dezert-Smarandache Theory (DSmT) framework.

I. INTRODUCTION

The variety of the fusion rules dealing with imperfect and uncertain information are based on different mathematical models and on different methods for transferring the conflicting mass of belief to specific hypotheses of the frame of the problem under consideration. DST [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] has been the first mathematical theory of evidence for combining uncertain information expressed as basic belief assignments. The fusion rule proposed by Shafer in DST is Dempster's rule of combination, i.e. the normalized conjunctive rule of combination. The legitimacy and the justification of Dempster's rule for combining sources of evidence has been a source of open endless strong debates in the community of users of belief functions since the publication in 1979 of Zadeh's famous example in [START_REF] Zadeh | On the validity of Dempster's rule of combination[END_REF], [START_REF] Zadeh | Book review: A mathematical theory of evidence[END_REF]. In this paper, we present new interesting examples to show another potential problem, or at least questionable behavior, of Dempster's rule of combination. Clearly, we show that in some case Dempster's rule is unable to respond adequately to the fusion of different basic belief assignments (bba's) whatever the level of conflict is. The problem is not due to the level of conflict between sources (contrary to Zadeh's example) but it is due to the inadequate normalization step done in Dempster's rule and the way the conflicting mass is redistributed back to focal elements through this normalization step. Such very simple examples reinforce the justification for using more efficient rules of combination to circumvent such unsatisfactory behavior of Dempster's rule, or at least to use Dempster's rule with extreme caution in applications.

Dezert-Smarandache Theory (DSmT) [START_REF] Smarandache | Advances and Applications of DSmT for Information Fusion[END_REF] has been developed to overcome the limitations of DST and circumvent the problems of inconsistency of Dempster's rule in possibly highly conflicting fusion problems and to go beyond the limitations of Shafer's model. DSmT distinguishes three possible classes of underlying models for the frame of discernment on which the basic belief assignment is defined depending on the nature of the problem: 1) Shafer's model (as in DST) considers the frame of discernment Θ = {θ 1 , . . . , θ n } as a finite set of n exhaustive and exclusive hypotheses according to the possible solutions of the problem under consideration; 2) A free-DSm model, is an opposite to Shafer's model and consists in assuming that all elements θ i , i = 1, 2, ..., n of Θ are not exclusive. The only requirement is for their exhaustivity. Shafer's model can be considered as the most constrained model; 3) Between Shafer's and free models, there exists a set of fusion problems represented in term of DSm hybrid models where the frame of discernment could involve both fuzzy continuous and discrete hypotheses, in accordance with the exact nature of the problem. In DST the bba's m(.) are defined on the power-set 2 Θ whereas in DSmT, and when working with and underlying hybrid model for the frame, the bba's are defined on the hyperpower set D Θ that corresponds to Dedekind's lattice. The mathematical definition of D Θ with many detailed examples can be found in [START_REF] Smarandache | Advances and Applications of DSmT for Information Fusion[END_REF], Vol. 1 and is out of the scope of this paper.

The main purpose of this paper being the analysis of Dempster's rule of combination, we will work in the classical DST framework where Shafer's model is assumed valid for the frame Θ and therefore we don't need to work with hyperpower set D Θ . More precisely, one has D Θ = 2 Θ when Shafer's model holds. From a frame of discernment Θ, a basic belief assignment (bba) is defined [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] as a mapping m(.) : 2 Θ → [0, 1] associated to a given source of evidence:

m(∅) = 0 and X∈2 Θ m(X) = 1 (1)
The elements of the power set having a strict positive mass of belief are called focal elements of m(.). The set of all focal elements is called the core of m(.) and is denoted K. The measures of credibility and plausibility of any proposition X ∈ 2 Θ are defined from m(.) by Bel(X)

Y ⊆X Y ∈2 Θ m(Y ) (2)
Pl(X)

Y ∩X =∅ Y ∈2 Θ m(Y ) (3)

A. Dempster's fusion rule

Dempster's rule of combination, also called Dempster-Shafer's (DS) rule since it was proposed by Shafer in his mathematical theory of evidence [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF], is a normalized conjunctive operation. Based on Shafer's model of the frame, Dempster's rule for two sources is defined by m DS (∅) = 0, and ∀X ∈ 2 Θ \ {∅} by

m DS (X) = m 12 (X) 1 -m 12 (∅) (4)
where

m 12 (X) X1,X2∈2 Θ X1∩X2=X m 1 (X 1 )m 2 (X 2 ) (5)
corresponds to the conjunctive consensus on X between the two sources. The degree of conflict between the sources is defined by

K 12 m 12 (∅) = X1,X2∈2 Θ X1∩X2=∅ m 1 (X 1 )m 2 (X 2 ) (6)
The mass of conflict is distributed to all meaningful propositions (i.e. the non-empty elements belonging to the intersection of the cores of m 1 (.) and m 2 (.)) of the power set through a simple normalization procedure 1 (with the division by 1m 12 (∅), assuming m 12 (∅) < 1). As already pointed out by Zadeh's [START_REF] Zadeh | On the validity of Dempster's rule of combination[END_REF], this rule has a very questionable behavior when K 12 → 1 because Dempster's rule can reflect the minority of opinions and moreover it is insensitive to inputs values as shown in [START_REF] Smarandache | Advances and Applications of DSmT for Information Fusion[END_REF], p. 114. That is why we have proposed to use the PCR5 fusion rule developed originally in DSmT framework [START_REF] Smarandache | Advances and Applications of DSmT for Information Fusion[END_REF]. It has been proved that PCR5 does not suffer of such unexpected behavior even in highly conflicting situations, but at the price of higher complexity for its implementation with respect to the complexity of Dempster's rule.

B. PCR5 fusion rule

The idea behind the Proportional Conflict Redistribution rule no. 5 [START_REF] Smarandache | Advances and Applications of DSmT for Information Fusion[END_REF] (Vol. 2) is to transfer conflicting masses (total or partial) proportionally to non-empty sets involved in the model according to all integrity constraints. The general principle of PCR rules is then to : 1 ) calculate the conjunctive rule of the belief masses of sources; 2 ) calculate the total or partial conflicting masses; 3 ) redistribute the conflicting mass (total or partial) proportionally on non-empty sets involved in the model according to all integrity constraints. Under Shafer's model 2 of the frame Θ, the PCR5 combination rule for only two sources of information is defined as: m P CR5 (∅) = 0 and 1 When the sources are in total contradiction, i.e. when m 12 (∅) = 1, then Dempster's rule cannot be used since one has a division by zero in (4). 2 We consider only Shafer's model here to make the comparison with Dempster's rule results.

∀X ∈ 2 Θ \ {∅} m P CR5 (X) = m 12 (X)+ Y ∈2 Θ \{X} X∩Y =∅ [ m 1 (X) 2 m 2 (Y ) m 1 (X) + m 2 (Y ) + m 2 (X) 2 m 1 (Y ) m 2 (X) + m 1 (Y ) ] (7)
where all denominators are different from zero. All sets involved in the formula are in canonical form. All denominators are different from zero. If a denominator is zero, that fraction is discarded. No matter how big or small is the conflicting mass, PCR5 mathematically does a better redistribution of the conflicting mass than Dempster's rule since PCR5 goes backwards on the tracks of the conjunctive rule and redistributes the partial conflicting masses only to the sets involved in the conflict and proportionally to their masses put in the conflict, considering the conjunctive normal form of the partial conflict. PCR5 is quasi-associative and preserves the neutral impact of the vacuous belief assignment.

II. A COUNTER-EXAMPLE TO DEMPSTER'S RULE

Here we point out a new (counter) example showing the inadequate behavior of Dempster's rule. This example is not related with the level of conflict between sources. In this example the level of conflict can be chosen at any low value and Dempster's rule is not responding adequatetly to the combination of different bba's since it provides always same results which is not a good expected behavior for a good fusion rule for applications. Stated otherwise, whatever the (strictly positive) level of conflict is, Dempster's rule gives always same result which is very surprising and counter-intuitive.

Example 1: Let's consider the following frame Θ = {A, B, C} with Shafer's model. We consider the following two bba's associated with two distinct bodies of evidence as follows with 0 < a < 1 and 0 < b < 1:

Focal elem. \ bba's m 1 (.) m 2 (.) A a 0 A ∪ B 1 -a 0 C 0 b A ∪ B ∪ C 0 1 -b Table I INPUT BBA'S m 1 (.) AND m 2 (.)
Using the conjunctive consensus operator, one gets:

m 12 (A) = a(1 -b) m 12 (A ∪ B) = (1 -a)(1 -b) K 12 = m 12 (∅) = m 1 (A)m 2 (C) + m 1 (A ∪ B)m 2 (C) = b Applying Dempster's rule by normalizing by 1 -K 12 = 1 -b, one gets m DS (A) = a(1 -b)/(1 -b) = a m DS (A ∪ B) = (1 -a)(1 -b)/(1 -b) = 1 -a
Clearly, one sees in this example that: 1) m 2 (.) plays here the same role as the vacuous belief assignment represented by m v (A ∪ B ∪ C) = 1 since one gets finally m DS (.) = m 1 (.). 2) Dempster's rule is not sensitive (i.e. adequately responding) to the values of the bba m 2 (.) because the result is independent of the input parameter b. This is not very satisfactory because Dempster's rule doesn't capture efficiently the impact of the level of conflict between two sources due to the inadequate normalization procedure. If now we apply PCR5 rule of combination in this example, we obtain the following result:

m P CR5 (A) = a(1 -b) + a 2 b a + b m P CR5 (A ∪ B) = (1 -a)(1 -b) + (1 -a) 2 b 1 -a + b m P CR5 (C) = ab 2 a + b + (1 -a)b 2 1 -a + b
It can be easily verified that m P CR5 (.) is normalized. We see clearly that PCR5 does react more efficiently to the variations of the inputs bba's contrary to Dempster's rule. When b tends to zero, m P CR5 (.) tends to m 1 (.) which is normal since m 2 (.) coincides with the vacuous belief assignment. When b = 1, then one combines the bba's of Table II Focal elem. \ bba's

m 1 (.) m 2 (.) A a 0 A ∪ B 1 -a 0 C 0 1 Table II INPUT BBA'S m 1 (.) AND m 2 (.)
Because of the principle of proportional redistribution of the masses of partial conflicts to the focal elements involved into them, one will obtain in this special case

m P CR5 (A) = a 2 1 + a m P CR5 (A ∪ B) = (1 -a) 2 2 -a m P CR5 (C) = a 1 + a + 1 -a 2 -a
The figures 1-3 show the evolution of fusion results. The figure 1 shows the mass of belief committed to A, the figure 2

III. INFINITE CLASS OF COUNTER-EXAMPLES

The previous example showing the inadequate behavior of Dempster's rule is not unique and actually there exists an infinity of cases where this non-responding behavior occurs with Dempster's rule as we prove in this section. Example 2: Let's modify a little bit the previous example by still considering Θ = {A, B, C} with Shafer's model and by taking the two non-Bayesian bba's given in Table III where

a ∈ [0, 1] and b 1 , b 2 > 0 such that b 1 + b 2 ∈ [0, 1]. Focal elem. \ bba's m 1 (.) m 2 (.) A a 0 A ∪ B 1 -a b 1 C 0 1 -b 1 -b 2 A ∪ B ∪ C 0 b 2 Table III INPUT BBA'S m 1 (.) AND m 2 (.)
Using the conjunctive operator, one gets:

m 12 (A) = a(b 1 + b 2 ) m 12 (A ∪ B) = (1 -a)(b 1 + b 2 ) K 12 = m 12 (∅) = 1 -b 1 -b 2
which yields after the normalization by 1 -

K 12 = b 1 + b 2 , m DS (A) = m 12 (A)/(1 -K 12 ) = a m DS (A ∪ B) = m 12 (A ∪ B)/(1 -K 12 ) = 1 -a
Clearly, Dempster's rule is still insensitive to m 2 (.) for this new example because whatever the different values of input paremeters b 1 and b 2 are, the rule provides exactly the same result, i.e. m DS (.) ≡ m 1 (.) even if m 2 (.) doesn't correspond to the vacuous belief basic assignment. It can be easily verified in this second example (and for any cases actually) that PCR5 does respond efficiently for combining these two bba's.

If we apply the PCR5 fusion in this example, the proportional redistribution of the mass of the partial conflict

m 1 (A)m 2 (C) = a(1 -b 1 -b 2 ) is done by x A m 1 (A) = x C m 2 (C) = m 1 (A)m 2 (C) m 1 (A) + m 2 (C) = a(1 -b 1 -b 2 ) a + 1 -b 1 -b 2 whence x A = a 2 (1-b1-b2)
a+1-b1-b2 and x C = a(1-b1-b2) 2 a+1-b1-b2 ; and similarly the proportional redistribution of the mass of the partial conflict m

1 (A ∪ B)m 2 (C) = (1 -a)(1 -b 1 -b 2 ) is done by y A∪B m 1 (A ∪ B) = y C m 2 (C) = m 1 (A ∪ B)m 2 (C) m 1 (A ∪ B) + m 2 (C) whence y A∪B = (1-a) 2 (1-b1-b2) 1-a+1-b1-b2 and y C = (1-a)(1-b1-b2) 2 1-a+1-b1-b2 .
Therefore with PCR5, one gets a fusion result that does react efficiently to the values of all the masses of focal elements of each source since one has

m P CR5 (A) = m 12 (A) + x A = a(b 1 + b 2 ) + a 2 (1 -b 1 -b 2 ) a + 1 -b 1 -b 2 m P CR5 (A ∪ B) = m 12 (A ∪ B) + y A∪B = (1 -a)(b 1 + b 2 ) + (1 -a) 2 (1 -b 1 -b 2 ) 2 -a -b 1 -b 2 m P CR5 (C) = x C + y C = a(1 -b 1 -b 2 ) 2 a + 1 -b 1 -b 2 + (1 -a)(1 -b 1 -b 2 ) 2 2 -a -b 1 -b 2
Example 3: Let's modify again a little bit the previous example by still considering Θ = {A, B, C} with Shafer's model and by taking the two non-Bayesian bba's given in Table IV where

a 1 , a 2 > 0 such that a 1 + a 2 ∈ [0, 1] and b 1 , b 2 > 0 such that b 1 + b 2 ∈ [0, 1]. Focal elem. \ bba's m 1 (.) m 2 (.) A a 1 0 B a 2 0 A ∪ B 1 -a 1 -a 2 b 1 C 0 1 -b 1 -b 2 A ∪ B ∪ C 0 b 2 Table IV INPUT BBA'S m 1 (.) AND m 2 (.)
Using conjunctive operator, one gets:

m 12 (A) = a 1 (b 1 + b 2 ) m 12 (B) = a 2 (b 1 + b 2 ) m 12 (A ∪ B) = (1 -a 1 -a 2 )(b 1 + b 2 ) K 12 = m 12 (∅) = 1 -b 1 -b 2
which yields after the normalization by 1 -

K 12 = b 1 + b 2 , m DS (A) = m 12 (A)/(1 -K 12 ) = a 1 m DS (B) = m 12 (B)/(1 -K 12 ) = a 2 m DS (A ∪ B) = m 12 (A ∪ B)/(1 -K 12 ) = 1 -a 1 -a 2
One sees that also in such very simple example 3, Dempster's rule is insensitive to the input parameters b 1 and b 2 of m 2 (.) because they are automatically simplified through the normalization procedure used in Dempster's formula. More generally, an infinite class of counter-examples based on a generalization of this type of examples can be easily constructed and therefore the following theorem holds.

It this example, in can be easily verified that PCR5 fusion provides the following result:

m P CR5 (A) = a 1 (b 1 + b 2 ) + a 1 2 (1 -b 1 -b 2 ) a 1 + 1 -b 1 -b 2 m P CR5 (B) = a 2 (b 1 + b 2 ) + a 2 2 (1 -b 1 -b 2 ) a 2 + 1 -b 1 -b 2 m P CR5 (C) = a 1 (1 -b 1 -b 2 ) 2 a 1 + 1 -b 1 -b 2 + a 2 (1 -b 1 -b 2 ) 2 a 2 + 1 -b 1 -b 2 + (1 -a 1 -a 2 )(1 -b 1 -b 2 ) 2 2 -a 1 -a 2 -b 1 -b 2 m P CR5 (A ∪ B) = (1 -a 1 -a 2 )(b 1 + b 2 ) + (1 -a 1 -a 2 ) 2 (1 -b 1 -b 2 ) 2 -a 1 -a 2 -b 1 -b 2
This result shows that PCR5 does react efficiently to the masses of all the focal elements of m 2 (.). It can be verified also that m P CR5 (.) is normalized of course.

Theorem 1: Let's consider the frame of discernment Θ = {θ 1 , θ 2 , . . . , θ n } with n ≥ 3 and satisfying Shafer's model and a given element, say θ i of Θ. If m 1 (.) is a dogmatic bba3 having the core K 1 = {θ j for some indexes j = i, X} where X denotes the disjunction of all θ j (i.e. a partial ignorance), and if m 2 (.) is a non-dogmatic bba having the core K 2 = {θ i , X, I t } where I t = θ 1 ∪θ 2 ∪. . .∪θ i ∪θ j ∪. . .∪θ n is the total ignorance, then Dempster's rule doesn't respond adequately to the fusion of sources since it is insensitive to m 2 (.), and one gets m DS (.) = m 1 (.) whatever the masses of focal elements of m 2 (.) are.

Proof:

We need to compute Dempster's fusion result for the combination of the following two normalized bba's satisfying theorem's conditions:

Focal elem. \ bba's m 1 (.) m 2 (.) θ j , j = i > 0 0 X j > 0 > 0 θ i 0 > 0 It 0 > 0 Table V INPUT BBA'S m 1 (.) AND m 2 (.)
Applying directly the conjunctive operator yields

m 12 (θ j ) = m 1 (θ j )[m 2 (X) + m 2 (I t )] m 12 (X) = m 1 (X)[m 2 (X) + m 2 (I t )] K 12 = m 12 (∅) = m 2 (θ i )[ j m 1 (θ j ) + m 1 (X)] = m 2 (θ i ) = 1 -m 2 (X) -m 2 (I t )
After the normalization by 1 -K 12 = m 2 (X) + m 2 (I t ) one gets:

m DS (θ j ) = m 12 (θ j )/(1 -K 12 ) = m 1 (θ j ) m DS (X) = m 12 (X)/(1 -K 12 ) = m 1 (X)
Hence m DS (.) = m 1 (.) which completes the proof.

Note: Of course, the previous theorem can be a little bit relaxed by not forcing X to be a focal element of m 1 (.) and of m 2 (.). Actually, even if m 1 (X) = 0 or/and m 2 (X) = 0 are chosen in the previous proof, then one still gets m DS (.) = m 1 (.). This corresponds to the case of Example 1.

Generalization of Theorem 1

Theorem 1 can also be easily extended by considering for m 2 (.), not only the focal elements I t , X and θ i , but also I t , X and any θ i1 , θ i2 , . . . , θ i k and their possible partial ignorances which yields to Theorem 2 stated as follows:

Theorem 2 (Generalization of Theorem 1): Let's consider: 1) the frame of discernment Θ = {θ 1 , θ 2 , . . . , θ n } with n ≥ 3 and satisfying Shafer's model; 2) a given non-empty (proper) subset of Θ denoted Θ ′ = {θ i k , i k ∈ {1, 2, . . . , n}}. If m 1 (.) is a dogmatic bba having the core K 1 = {θ j ∈ {Θ \ Θ ′ }, X} where X denotes the disjunction of all θ j (i.e. a partial ignorance), and if m 2 (.) is a non-dogmatic bba having the core K 2 = {I t , X, 2 Θ ′ \ {∅}}, then Dempster's rule doesn't respond adequately to the fusion of sources since it is insensitive to m 2 (.), and one gets m DS (.) = m 1 (.) whatever the masses of focal elements of m 2 (.) are.

Proof: Applying directly the conjunctive operator yields

m 12 (θ j ) = m 1 (θ j )[m 2 (X) + m 2 (I t )] m 12 (X) = m 1 (X)[m 2 (X) + m 2 (I t )] K 12 = m 12 (∅) = [ Y ∈2 Θ ′ m 2 (Y )][ j m 1 (θ j ) + m 1 (X)] = Y ∈2 Θ ′ m 2 (Y ) = 1 -m 2 (X) -m 2 (I t ) After normalization by 1 -K 12 = m 2 (X) + m 2 (I t ) one gets: m DS (θ j ) = m 12 (θ j )/(1 -K 12 ) = m 1 (θ j ) m DS (X) = m 12 (X)/(1 -K 12 ) = m 1 (X)
Hence m DS (.) = m 1 (.) which completes the proof.

Note: If we want, we can also relax a little this theorem by not forcing X to be a focal element of m 1 (.) and of m 2 (.), and also not to force all the elements of 2 Θ ′ \ {∅} to be focal elements of m 2 (.) as shown in the next example. VI, where at least one a i > 0 and one b i > 0. Using conjunctive operator, one gets:

Focal elem. \ bba's m1(.) m2(.) A a1 0 B a2 0 A ∪ B 1 -a1 -a2 b1 C 0 b2 D 0 b3 D ∪ F 0 b4 C ∪ E ∪ F 0 b5 A ∪ B ∪ C ∪ D ∪ E ∪ F 0 1 - P 5 i=1 bi
m 12 (A) = a 1 (b 1 + (1 - 5 i=1 b i )) = a 1 (1 - 5 i=2 b i ) m 12 (B) = a 2 (b 1 + (1 - 5 i=1 b i )) = a 2 (1 - 5 i=2 b i ) m 12 (A ∪ B) = (1 -a 1 -a 2 )(b 1 + (1 - 5 i=1 b i )) = (1 -a 1 -a 2 )(1 - 5 i=2 b i ) K 12 = m 12 (∅) = 1 -b 1 -(1 - 5 i=1 b i ) = 5 i=2 b i
which yields after the normalization by 1-K 12 = 1-

5 i=2 b i , m DS (A) = m 12 (A)/(1 -K 12 ) = a 1 m DS (B) = m 12 (B)/(1 -K 12 ) = a 2 m DS (A ∪ B) = m 12 (A ∪ B)/(1 -K 12 ) = 1 -a 1 -a 2
One sees that Dempster's rule is insensitive to the input parameters of m 2 (.) because they are automatically simplified through the normalization procedure used in Dempster's formula.

If we apply PCR5, one will get for this example (the verification is left to the reader): One can see again that PCR5 rule does react efficiently to the masses of all focal elements of the sources in the combination whereas Dempster's is unable to take properly into account the mass of focal elements of m 2 (.) since it works as if m 2 (.) was equal to the vacuous belief assignment m v (Θ) = 1. This is clearly what we consider as a very not satisfactory behavior of Dempster's rule.

IV. CONCLUSION

In this paper, we have introduced interesting counterexamples that show what we consider as a very unsatisfactory behavior of Dempster's rule of combination because Dempster's rule is insensitive to the second source, even if this second source is different from the vacuous belief assignment. The problem of Dempster's rule pointed out here doesn't come from the level of conflict between sources to combine (as in famous original Zadeh's example), but it comes only from the intrinsic structure of focal elements of sources and the inadequate normalization procedure. These examples are as important as Zadeh's example and show clearly that Dempster's rule can be non responding adequately to different bba's inputs in some cases and not necessarily only when sources are highly conflicting. Therefore, we consider that Dempster's rule must always be used with extreme caution in applications and we recommend to replace Dempster's rule by a more efficient rule of combination whenever possible. The PCR5 fusion rule appears to be a very good candidate for the combination of bodies of evidence because it has a more rational behavior and it also reduces the uncertainty of the combined basic belief assignment thanks to the powerful proportional conflict redistribution principle.

  the mass of A ∪ B and the figure 3 the mass of C obtained with Dempster's rule and PCR5 rules for couples (a, b) of input parameters varying in [0, 1]. One sees clearly the nonresponding of Dempster's rule to the variation of the input parameter b involved in m 2 (.).

Figure 1 .

 1 Figure 1. m DS (A) and m P CR5 (A) for a, b ∈ [0, 1].

Figure 2 .Figure 3 .

 23 Figure 2. m DS (A ∪ B) and m P CR5 (A ∪ B) for a, b ∈ [0, 1].

Example 4 :

 4 Let's take Θ = {A, B, C, D, E, F } satisfying Shafer's model and the two normalized non-Bayesian bba's given in Table

a 1 + b 5 + a 2 b 2 5 a 2 + b 5 + ( 1 -a 1 -a 2 )b 2 5 1

 5525115 a 1a 2 + b 5

Table VI INPUT

 VI BBA'S m 1 (.) AND m 2 (.)

By definition[START_REF] Smets | The combination of evidence in the transferable belief model[END_REF], a dogmatic bba doesn't admit Θ as a focal element, i.e. the total ignorance has a zero mass of belief.