?

African multilingualism is well recognised (Wolff, 1999: 333;Obanya, 1999: 82;Trudell, 2008: 1) and multilingual and multicultural practices on the continent are real and current. Yet the policies which regulate language treatment and use as well as the linguistic attitudes are essentially exoglossic in nature because they accord primacy to and promote external languages (Ruiz, 1995:75) in a vast majority of countries. As offshoots of colonial perceptions and treatment given to African indigenous languages vis-à-vis foreign languages, these policies have tended to undervalue and so disregard the former. Over the postcolonial decades, therefore, colonial languages have continued to be perceived as the most acceptable instruments for nation construction. Coupled with this orientation has -nationapproach to language policy and planning. Consequently, except for a few countries that have defied generalised practices by adopting multilingual policies based on the realities of linguistic diversity, most of the countries have maintained the colonial legacy of the supremacy of foreign languages. In fact, even when the need for multilingual policies has been recognised, the putting in place of appropriate policies to cater for this need has not proven to be an easy task. While governments openly acknowledge the necessity for a shift in linguistic behavior, their actions on the ground reveal a completely different reality. Their actions demonstrate that they are either unconvinced of the need to undertake fundamental language policy reforms based on linguistic diversity or that they are simply unprepared to embrace a change in policy. Whatever the underlying motivations for this behavior, the impact of the maintenance of foreign language-based policies for Africa is not only increasingly being felt but also it has become the focus of debates in recent times.

Whenever the need for alternative language policy and planning has been expressed in society, education has been at the centre of the debate. The debates have been even fiercer in settings where a majority of the languages have minority statuses. In these contexts, demands and advocacy for more contextually-relevant 1 2 approaches to language policy and planning been observed to be greater. Long years of dependence on hegemonic languages, usually but not limited to foreign languages, have bred serious misconceptions about the value and functions of indigenous modes of communication. Faced with the challenge of innovation, both governments and the general population, probably due to limited understanding of the potential multi-dimensional gains of educational reforms based on indigenous African languages continue to resist change. A look at ongoing practices suggests that either Africa is uncertain of the benefits of any such innovations or that it simply wants to maintain the status quo by choosing the path of convenience rather than that of necessity. While the path of convenience has to do with the maintenance of foreign languages and the resources available in them for learning and other purposes, that of necessity entails radical overhauling of present policies and practices through obligatory integration of indigenous languages into all language policy and planning processes especially with respect to education. Obviously, the latter is more challenging because it involves changing long established linguistic attitudes. It entails the development and promotion in learning and other functions of languages hitherto considered as not good enough for these functions.

By opting to persist in the pursuance of foreign languagebased policies for a multilingual Africa, governments have failed to realise that effective development of nations depends greatly on the choices and options made with regard to educational planning and delivery. Clearly, on this continent, where multilingualism is the norm rather than an anomaly, these concerns are even greater. Regrettably, the linguistic choices that have guided educational promotion in Africa have not matched development requests founded on culturallysensitive and ideologically appropriate models of theorisation and application. The blame has not only been the subjection of educational planning and delivery to foreign language policies but also the determination by successive leaderships in postcolonial times to uphold their loyalties towards these policies, thereby, relegating to the background the huge linguistic resources of the continent. The puzzle that continues to burden many minds today, therefore, is why Africa cannot craft its own policies and so liberate itself from the linguistic colonisation and mental enslavement perpetuated by current policies and practices. How do we explain that unlike other continents that have suffered the pangs of colonisation, Africa seems to be the only continent that is least prepared to adopt alternative approaches to cultivating its development? For how long will this situation persist in the continent? Indeed, one wonders whether Africa is aware that no country has ever achieved development by despising and so discarding its languages and cultures in favor of alien ones. Put differently, one wonders people of the continent of Africa can break vibrant, promising and emerging economies today have defied the colonial yoke by ensuring that although they encourage the learning of foreign languages, their learning, the basis of modern socioeconomic and political advanced are founded fundamentally their languages and cultural values. The case of Asian countries that share the same colonial history with Africa is glaring. Questions like the ones asked here, signal the dangers of foreign language promotion perpetuity on a continent that is in dire need of linguistic and educational reforms.

Quite often when issues of language policy have been debated in Africa, the focus has been on the inability of governments to propose appropriate frameworks for effective language planning theorisation and actual application. Accordingly, failures in the development and promotion of education based on the language of instruction have been attributed to the absence of strong political will and unpreparedness to officially endorse learning in mediums other than languages inherited from colonisation. In recent years, though, the focus seems to be shifting fairly quickly to new dimensions of complication and challenges. This current shift in language policy and planning discourses has highlighted a rather paradoxical situation whereby, despite marked advancements registered at individual country levels, actual application has continued to suffer thus raising concerns as to whether there is really need for policy changes at all on the continent. The shift has thus has tended to draw attention to formulation inadequacies and implementation insufficiencies. On a general note, policies on the continent are either not well-formulated or they are well-designed but are not properly implemented. As the situation stands, this problem can be perceived from several angles involving but not status quo by ensuring that foreign languages continue to dominate learning at all levels, the unwillingness of parents and communities to adopt alternative educational models based on languages they consider as limited in the performance of functional roles in a society that is rapidly being transformed into a global village and in which language of limited diffusion have little or no space and the challenges involved in bringing indigenous African languages into the limelight of educational planning and delivery.

In this collection, we pool together a wide range of language policy and planning experiences from six countries in Africa. These are countries that have made some significant strides towards the establishment of African indigenous language-based policies. However, from experiences reported in the various contributions, it becomes clear that between language policy and its actual implementation, there exists a great variance (Prah, 1999: 543). In fact, what we gather from these experiences points to some kind of dilemma, indeed, a serious paradox. While in countries where appropriate multilingual policies have not yet been established, advocacy for such policies is rife, in those that have already adopted them the situation still leaves much to be desired they have fallen short of ensuring the effective implementation of the policies. In these countries, a critical look at implementation leaves one with more questions than answers. Despite the presence of attractive policies, the educational system as well as all written communication in the public arena is still largely dominated by foreign language use with or no official endeavor to usher in the required change. The very big question that we are tempted to ask here is: Why, despite all the expressed needs for contextually-relevant language policies and given the policy advancements observed, is effective language policy implementation so difficult to achieve? In other words, what is really holding us back from engaging in appropriate language policy and planning even when it is evident that governments are now more conscious of the benefits of alternative policies? The various contributions provide some clues to some of the intriguing peculiarities of language planning and policy at various countrylevels.

Gabriel Mba reviews language policy and planning in Cameroon. In Cameroon, coupled with a historical study of reflections that various stakeholders have had for its exploitation, he questions the different positions and actions regarding language policy and especially those with a say on the immense reservoir of mother tongues in the field of education in Cameroon. He observes that the linguistic complexity of the country has led stakeholders to engineer different language policy and management models based on their understanding of the situation, their expectations and felt needs. After briefly presenting language education enactments representing what is generally considered as Cameroonian language policy, he summarises the key policy orientations made by Cameroonian linguists. He thinks that the difficulties encountered in implementing present policy orientations.

Blasius A. Chiatoh examines language policy developments against the background of mother tongue-based education application. He argues that the failure to implement mother tongue education has generally been blamed on the absence of appropriate policies especially with respect to the specification of goals and the definition of strategies and mechanisms of implementation. He questions the appropriateness of decrees, legal and constitutional frameworks as language policy and submits that rather than constituting policy in themselves, they actually only constitute language policy-related legislation which in view can appropriately be referred to as insufficiencies of current legislation, it does provide some important window of opportunity for government and educational stakeholders to initiate transformative change through the introduction of mother tongue-based education in Cameroon. Drawing from Bagmbose (1999: 23-28), he highlights the essential qualities of an ideal language planning model for Cameroon. level of language planning and policy. Her contribution centres on the Mungaka experience of literacy promotion in Cameroon. She retraces the history of the development and promotion of Mungaka in education and evangelism notably the role of the Presbyterian Church in the colonial and postcolonial era (1800s-1970s). She argues that the successful spread and use of Mungaka as a language of education, literacy and evangelism provide evidence that a language policy that gives prominence to Cameroonian indigenous languages would enhance learning and subsequently accelerate development in the country.

In his chapter, Josephat Rugemalira argues that the Tanzanian experience in language policy and planning is not as unique and laudable as many scholars have been tempted to think. His submission is that the promotion of a single national language Swahili, as a replacement of the language of the colonial masters English is not a success story. In his opinion, the conflict observed in the three layers of the Tanzanian linguistic environment arises from two sources, namely, the purist tradition in linguistics and then the imbalanced economic relation between Tanzania and the Anglophone centre situated in Europe and America. He strongly argues that there are no guarantees in the long-term for Swahili as the national language to play its rightful role since the Tanzanian economy continue to be a negligible appendage of the powerful economies of the North.

Linda C. Nkamigbo in her paper presents an overview of language policy recognises English and the three major languages Hausa, Igbo and Yoruba as official languages. As such, government encourages their use in vital public domains such as education and legislation. However, the policy fails to address the fate of Nigerian minority languages. As contained in the Federal Government National Policy on Education (NPE), indigenous mother tongues are to be promoted in the first three years of primary schooling with English used subsequently in later stages. However, she regrets that policy is not being implemented as English remains the medium of instruction in a vast majority of Nigerian schools right from the first year of the nursery. According to her, the failure to implement the policy comes from the lack of political will on the part of government. She then discusses the benefits of mother tongues in education particularly in early childhood and recommends a National Policy on Education based on the principle of biliteracy. Kathleen Heugh examines the South African experience in language policy and planning. She traces the history of language policy and planning over the last century. In her opinion, the marginalisation of Dutch (Afrikaans) speakers by British colonial policy after 1901 unavoidably had a huge impact on apartheid language policy during the second half of the 20 th century, a situation that led to implementation of mother tongue education that was perceived by speakers of African languages as a mechanism for the advancement of separate and unequal development for these different groups of people. In her argument, the post-apartheid language policy adopted for the country, although founded on multilingualism, has defaulted towards an English monolingual paradigm for three main reasons namely; the legacy of apartheid, the lack of political will and deliberate misinterpretation of language policy by key official advisors.

In her paper, Atikonda A. Mtenje reviews the Malawian experience in language planning. She examines the development of language policies in Malawi from the colonial era to postindependence (a period marked by the one-party state) to the era of democracy with a focus on the dominance of Chichewa over other local languages in the one-state period. She then goes further to explain the declaration of the Malawian Ministry of Education authorising the inclusion or integration of local languages into early primary school learning (standards 1-4) and posits that despite some major strides made in this direction, the declaration is yet to enjoy government approval and implementation. Among the main challenges affecting policy, she cites improper planning, lack of political will, negative attitude of the elite towards African languages and lack of sensitisation. At the end, she presents a number of outstanding lessons one could learn from the Malawian experience.

In the next chapter, Seyum Bekale attempts an overview of the Ethiopian experience in language policy and planning. Centering on the different historical epochs, he analyses the evolution of language policy in Ethiopia. The earliest historical period, the Pre-LP era, was the monarchs with other Ethiopian languages used in oral communication by native speakers until they were replaced by Amharic. As for the imperial and Derg periods, these saw the emergence of Amharic as the sole national official language within a one-nation-one-language state. This policy marginalised all the other languages of the country until recently with the official recognition of related to policy implementation, several Ethiopian languages now enjoy written development and are being used as languages of instruction and as official languages in decentralised localities as well as in the media. language policy and planning in Sudan. He addresses the problem of language policy in two different historical periods, namely, the colonial period and the postcolonial period. Basing his discussions on the language of education, arabicisation of education and the status of indigenous Sudanese languages, he explores the efforts made by different domains.

-makers in the North as well as in the South and concludes that so long as Arabic maintains its hegemony in the North with English gradually assuming a similar position in the South, indigenous languages will remain marginalised.

Blasius A. Chiatoh and W. Akumbu come in with a proposal for a national language policy for Cameroon. Observing that 50 years after reunification of the two Cameroons (1961), and despite its linguistic diversity, Cameroon still operates an exoglossic language policy based on the promotion of English and French in education and other domains. Moving from the position that such a policy is neither integrative nor representative and so cannot fully respond to the needs of Cameroonians, they strongly advocate for a total overhaul of present educational practices based on the language of instruction. In the guise of providing guidelines on the procedures and ingredients for adoption of an appropriate language policy, they briefly present language as having vital economic, educational, cultural, political and ideological value. They then propose general and specific principles to be considered in language policy elaboration, the qualities of an appropriate policy as well as the steps in policy formulation. They conclude their submission with a fervent appeal for the diagnosis and treatment of language issues in Cameroon to be informed by scientific and inclusive approaches.


Pius W Akumbu
email: akumbu_pius@yahoo.fr

TONE IN KEJOM (BABANKI) ASSOCIATIVE CONSTRUCTION &

This article presents the associative construction (AC) in Kejom (Babanki), while focusing on the behaviour of tone. Kejom is a Grassfields Bantu language spoken in the North West Region of Cameroon. Tonal processes that occur in the Kejom AC have been accounted for using Register Tier Theory. The analysis reveals that the tonal processes are sensitive to a lowtoned nasal that occupies the onset position of the noun roots involved. Furthermore, it has been demonstrated that some tones must be singly-linked, and not doubly-linked for the tone rules to apply. This study provides a simpler analysis of the AC in this language, making use of only four tone rules without much recourse to underlying floating tones.

Cet article est consacré à une étude des tons dans les constructions associatives en Kejom (Babanki), une langue bantoue du grassfield parlée dans la région nord ouest du Cameroun. La Théorie de Registre en Paliers est utilisée pour expliquer les changements tonaux observés. L'analyse révèle que ces processus tonaux sont sensibles à une nasale à ton bas qui occupe l'attaque de la racine des noms concernés. Elle révèle également que certains tons doivent être associés à une seule unité porteuse de ton (pas à deux) pour que les règles tonales s'appliquent. Cette étude propose une analyse plus simple de la construction associative en Kejom, car elle utilise seulement quatre règles tonales et ne fait pas appel à des tons flottants de structure profonde.

INTRODUCTION

This paper presents the associative construction (AC) in Babanki (henceforth Kejom, as the native-speakers prefer to call their language), while focusing on the behaviour of tone. Kejom is a Centre Ring Grassfields Bantu language spoken by approximately 25,000 people [START_REF] Akumbu | A phonological sketch of Kejom[END_REF] in two villages -Kejom Ketinguh (Babanki Tungo) and Kejom Keku (Big Babanki) in the North West Region of Cameroon. The AC in Kejom is of the form 'noun 1 (N1) of noun 2 (N2)', where the two nouns are separated by an associative marker (AM) which translates roughly as 'of'. In this type of construction, almost any noun can be juxtaposed with any noun. It has been demonstrated that in many African languages, the AC is conveyed by means of a tonal morpheme [START_REF] Akumbu | Njem tonology[END_REF][START_REF] Chumbow | Floating tones in Bangwa[END_REF][START_REF] Tamanji | Constituent structure of the associative construction in Grassfields Bantu[END_REF]. In Kejom, however, a segmental morpheme that depends on the noun class of the noun that occupies N1 position occurs as the AM. When nouns come together in such constructions tone behaves in very peculiar ways being constrained by a nasal when it occupies the onset position of noun roots. [START_REF] Hyman | Tonology of the Babanki noun[END_REF] proposes an account of ACs in this language, making heavy use of both floating L and H tones in underlying forms which differ markedly from their surface realizations. He shows what happened historically, retracing the steps that result to the present complex tonal system. I present a synchronic account within the framework of Register Tier Theory, making use of only a few tone rules, without much recourse to underlying floating tones. This paper is organized as follows: The basics of Register Tier Theory are presented in §1 followed by a presentation of the noun class system of the language in §2 since the discussion draws on nouns from different noun classes. In §3, I show the syllable structure of noun roots in the language because the behaviour of tone in these constructions is determined to a large extent by the structure of the syllables. Finally, §4 contains a discussion of the associative construction in Kejom.

REGISTER TIER THEORY

Register Tier Theory (RTT) [START_REF] Inkelas | Tone feature geometry[END_REF][START_REF] Inkelas | The phonology of intonation in Hausa[END_REF][START_REF] Snider | Towards the representation of tone: a three-dimensional approach[END_REF]Snider, , 1990Snider, , 1999) ) recognizes the following autosegmental features and tiers: the register features h and l on a REGISTER TIER, the tonal features H and L on a TONAL TIER, a TONAL ROOT NODE TIER (TRN), and a TONE-BEARING UNIT TIER (TBU). These tiers are geometrically arranged according to the configuration in (1) taken from Snider (1999:23). "Features on the Register tier and the Tonal tier are linked to structural nodes on the TRN. Geometrically, these tiers form a separate plane with respect to the TRN. Nodes on the TRN are, in turn, linked to moras (µ) on the TBU tier" (Snider 1999: 23).

The register features h and l are defined following Snider (1999:25) as "effect a register shift h=higher, and l=lower relative to the preceding register setting", and the tonal features H and L are defined as realize the "TBU at H=high pitch, and L=low pitch relative to the current register." This is shown in (2) (the dotted lines represent registers and the solid lines represent tones).

h --------- --------- H↑ L↓ L↓ l --------- L↓
The geometry in (1) and the features in (2) make it possible to specify up to four logically possible tonal distinctions, namely, a high tone on a high register, a high tone on a low register, a low tone on a high register, and a low tone on a low register. Notice firstly that the register feature of any given TBU is specified in relation to that of the preceding register. The register of the initial TBU for its part is construed to be higher than or lower than the reference point that native speakers usually have in mind when beginning an utterance. Secondly, the tonal feature associated to any given TBU specifies whether the tone is low or high in relation to the current register. RTT is used in this paper to insightfully explain the tonal processes, given that within this theoretical model features on each tier can behave independently of one another.

NOUN CLASS SYSTEM

Most Kejom nouns consist of a prefix1 +root with class ten nouns comprising a root+suffix. The noun class system includes Bantu classes 1, 2, 3, 5, 6, 6a, 7, 8, 9, 10, 13, and 19. Possible class pairings in Kejom are 1/2, 3/6, 3/13, 5/6, 5/13, 7/8, 7/6, 7/6a, 7/10, 9/6, 9/10, and 19/6a. The noun class system of this language is summarized in the following table. Apart from the glottal stop, all Kejom consonants can occupy the onset position. Most consonants in onset position can be labialized or palatalized, and prenasalized. At some point in history, the N in NC sequences might have been a prefix from class 1 or class 9, bearing a low tone, as analyzed by [START_REF] Hyman | Tonology of the Babanki noun[END_REF] but has now become part of the root. The coda of the syllable can be occupied by [e+r+m+l+oenqБ]. The way tones surface on the nouns in ACs depends to a large extent on whether or not a nasal occurs as part of the onset of the root. The roots with open syllables behave differently than those with closed syllables when they occur with the vocalic AM.

THE ASSOCIATIVE CONSTRUCTION

The ACs in this language can be divided into five sets based on the type of AM they take. The first set involves those for noun classes 2, 7, 13, and 19 that have a CV shape and occur on a high tone. The AM for class 10 nouns, which is also a high toned CV is in the second set. Recall that the nouns in class 10 take a suffix as their noun class marker unlike the other classes that take prefixes. The third set contains class 6a nouns whose AM has a CV structure that occurs on a low tone. The AM for classes 3, 5, 6, and 8 nouns is a high toned V and so they are placed in the fourth set. Finally the AMs for noun classes 1 and 9 nouns which have a low toned V shape form constitute the fifth set. They are grouped as in the following table. A comparison of tables 1 and 3 reveals a close correlation between the noun class affixes and the AMs for each class. Except for tone, the AM (high toned) and the noun class markers (low toned) for classes 2, 3, 5, 6, 7, 8, 13, and 19 are ordinarily identical. For classes 1, 6a, 9, and 10 both the AM and the noun class prefixes are identical.

Tonal processes that occur in the Kejom AC include h-spread, High Tone Spread (HTS), and Low Tone Spread (LTS). While h-spread is followed by the delinking of the l register feature, HTS is followed by the delinking of the low tone, and LTS by the delinking of the high tone from the target TBUs. These tonal changes are conditioned by the presence of a nasal in the onset position of the juxtaposed noun roots. For example, h-spread can only apply if no nasal is present. HTS from the AM to the prefix of N2 also occurs only if there is no nasal even though it does occur in the presence of a nasal if the N2 root is high-toned. All of these changes are discussed further below.

ACs INVOLVING CLASSES 2, 7, 13 AND 19 NOUNS

When nouns from noun classes 2, 7, 13 and 19 are juxtaposed, a high toned AM of the shape CV, identical to the noun class prefix of the first noun, occurs between them. Examples are given in the following data2 (In this paper, low tone raising is represented by ⍻ and downstep of high tone by ⍺) . It can be observed from these data that the low tone on the root of N1 is raised. In order to account for this raising, [START_REF] Hyman | Tonology of the Babanki noun[END_REF] proposes an underlying H-LḶ (where Ḷ represents a floating low tone) that becomes L-M through four different rules. First, the final floating low tone is absorbed into the preceding low tone. Next, H spreading occurs from the prefix to the root, followed by the prefix H lowering to L and then the HL on the root is simplified to M.

This procedure that results to the right output is rather complex and the proposal made in this paper is that the raising of the low tone is due to the presence of the following high tone of the AM. From a Register Tier Theory perspective, the low tone assimilates the high register feature of the high tone (through h-spread) and is realized on that higher register.

Another observation to be made from the above data is that the underlying L prefix of N2 surfaces as H while the H on the root surfaces as ⍺H. Hyman's account is that H spreading from the AM to the prefix gives an intermediate HL-H sequence that simplifies to H-⍺H (p. 166). The RTT account proposed here is that High Tone Spread (HTS) occurs from the AM to the prefix. This spreading is then followed by the delinking of the low tone since all the prefixes 3 of N2 nouns surface with high tones. As a consequence of the delinking, the low tone is left floating and as it happens in a good number of languages [START_REF] Akumbu | Njem tonology[END_REF]Snider, 1999;etc) this floating low tone causes the high tone on the root of N2 to be downstepped relative to the preceding high tone. The following rules are formulated to capture these changes. According to this rule, a high register feature spreads leftwards, and, in a subsequent process, delinks the low register feature. Notice that the target low tone is obligatorily preceded by another tone. For the h to spread, that preceding tone must be singly-linked 4 and not doubly-linked. In this regard, another, second, association line has been added to the node and an x put at the end of it to indicate that the tone must be singly-linked for the rule to apply. In other words if the tone is linked to another TBU, then h-spread does not apply. 3 An essential claim made in this paper is that all prefixes in Kejom are underlyingly low-toned (see [START_REF] Hyman | Tonology of the Babanki noun[END_REF] for a different view that most N1 prefixes are underlyingly high-toned). 4 Having rule application depend on single vs. multiple associations of nodes has some precedent in the literature. For example, Shein & Steriade (1986) states this constraint as the "Uniform Applicability Condition", [START_REF] Hayes | Inalterability in CV Phonology[END_REF] refers to it as the "Linking Constraint", and [START_REF] Goldsmith | Autosegmental and Metrical Phonology[END_REF] employs the "Conjunctivity Condition".

This rule states that a high tone spreads to the following low-toned bearing unit and, subsequently, delinks that low tone. Observe that the target low tone must also be followed by another tone. For the H to spread to the tonal root node of the L, that following tone must be singly-linked and not doubly-linked. Again, an x is put at the end of an additional association line to specify that the tone must be singly-linked. In order words, if the tone is linked to another TBU, then HTS does not apply. According to this rule, a floating low register feature spreads rightwards, and, in a subsequent process, delinks the following high register feature. The data in (1) can therefore be derived as follows: According to this derivation, HTS from the AM to the prefix of N2 results to the high tone of the final TBU being realized on a low register feature. The low tone on [j-∆ r] is realized on a higher register due to h-spread from the AM. Recall that the dotted lines represent register while the solid horizontal lines represent tone. The lower case h and l together with the arrows show how register changes. The phonetic representation above is therefore interpreted as follows: a low tone is realized on a low register followed by another low tone on a high register and a high tone on that same (high) register and then a high tone on a low register.

Figure 6 Input HTS l l h l h l l h l h L L H L H L L H L H o o o o o o o o o o = j«j-rj«j«en →j«j-rj«j«en→ Figure 7 Downstep h-spread l l h l h l l h l h = = L L H L H L L H L H o o o o o o o o o o j«j-rj«j«en → → → →j«j-rj«j«en→ → → →
However, h-spread fails to occur (and the N1 low tone is not raised) as in the following data. The fact that the low tone on low-toned roots in N1 is not raised can be attributed to the presence of a nasal in the onset position of such roots. One wonders why the nasal has the ability to regulate the behaviour of tone in ACs. [START_REF] Hyman | Tonology of the Babanki noun[END_REF] and [START_REF] Devisser | Tone in the noun system of Babanki[END_REF] propose that the N in NC sequences might have been a prefix from classes 1 or 9 bearing a low tone. When the low-toned nasal occurs, h-spread from the AM to the root of N1 fails to apply. As stated in (2) above, the rule can only apply if the low tone that precedes the target low tone is singly-linked. In (6), the nasal onset forms part of the root and bears the same low tone like the root vowel because of the OCP (Snider, 1999) that is enforced morpheme-internally in Kejom. Hence, the occurrence of the nasal leads to the violation of the condition that the tone that precedes the target low tone must be singly-linked for h-spread to occur. The data are therefore derived as follows: The failure of HTS to apply here is also due to the presence of the nasal onset. Both the nasal and root vowel are linked to the same low tone leading to the violation of the condition in (3) that the tone that follows the target low tone must be singlylinked for the H to spread. The data in (8) are therefore derived as follows: These data show that when a nasal occurs as part of the onset of high-toned N2 roots, the high tone is able to spread from the AM to the prefix of N2. This is allowed because the nasal onset bears a low tone while the root vowel bears a high tone. The tone that follows the target low tone is therefore singly-linked and the condition for HTS to apply is met.

Figure 9 Input HTS l l h l h l l h l h L L H L H L L H L H o o o o o o o o o o o o = u«,ms-mu«u«,k≠l →u«,ms-mu«u«,k≠l→
Figure 11 Input h-spread l l h l l l l h l l = L L H L L L L H L L o o o o o o o o o o o o u«,rtБu«u«,ms«Б → → → →u«,rtБu«u«,ms«Б→ → → →
HTS from the AM to the prefix of N2 is followed by the delinking of the low tone of the prefix. The delinked low tone merges with the low tone on the following nasal. The following derivation recapitulates these facts. While the sets of data in (6, 8, and 10) can be accounted for using h-spread and HTS, there are some nouns5 that behave differently, causing a high tone that follows them (e.g, that of the AM) to be downstepped. Consider the following data. In order to account for the downstep, Hyman proposes to have a floating L at the end of the N1 roots. Even though he eventually rejects such a possibility and posits a floating H for (12a) nouns, it is argued in this paper that the floating L tone analysis works well for both sets of data. The low register feature of this floating low tone spreads to and delinks the high register of the AM high tone causing the H tone to be realized on a low register. This can be seen in the following derivation.

Figure 15 Input

Downstep l h l h l l l h l h l l = L H L H L L L H L H L L o o o o o o o o o o o o j«,enj«j«,avÕm→ → → →j«,enj«j«,avÕm→ → → →

ACs INVOLVING CLASS 10 NOUNS

As mentioned previously, class 10 nouns have the structure root+suffix. When they are combined in an AC, [r«ae ] occurs between them as the AM. When this happens one expects to have the noun class suffix [-r«ae ] followed immediately by the AM [r«ae ]. However, only one of them surfaces, in order to avoid redundancy. Hyman (1980, pp. 228-229, 254) demonstrates that it is the class 10 suffix that is lost in favour of the AM. Consider the following data. Observe that downstep applies in these data between the high tone of the root and that of the suffix, where the H of [-r«ae ] is downstepped, relative to the H of the root. Downstep occurs when this class suffix is added to underlying low-toned nouns that do not have a nasal as part of the root (see 14a, b, and c). Hyman (1980, pp. 252-254) accounts for this downstep by providing historical evidence that there is a floating high tone that occurs before these noun roots. It docks onto the right (if there is no nasal onset) and delinks the low tone of the root. This low tone remains floating and causes the following high tone of the suffix to be downstepped relative to the high tone that now surfaces on the root, as shown in the following derivation. While almost all class 10 nouns whose singular forms do not take a nasal as part of the root in class 9 can be accounted for in this manner, two exceptional nouns have been found. These are [Aÿae ⍺r«ae ] /Aÿae ,r«à/ 'dogs' and [aiÕ◊ ⍺r«ae ] /aiÕ◊ ,r«ae / 'goats'. It is important to note that these are the only two high-toned nouns in class 9 whose roots do not have a nasal onset. All the nouns of this class without a nasal onset are underlyingly low-toned. Hyman (1980, p. 253) postulates a H-L stem for these roots in Kom (a Centre Ring Grassfields Bantu language), that is /Aÿae è and aiÕ◊ è/. While I adopt his analysis for these two words in Kejom, I further observe that the reason why they behave this way is precisely that the two are borrowed words from Kom which maintain their underlying structure from the donor language, making it possible for the floating low tone to cause downstep. These combinations confirm that the low tone of N1 roots is raised when followed by a high tone in case a nasal does not occur in onset position. HTS from the AM to the prefix of N2 also occurs only when there is no nasal in the onset of low toned nouns. When HTS does occur, it is followed by the delinking of the prefix low tone which remains floating and causes downstep of the following high tone. It should also be noted that the schwa is deleted when it occurs next to another vowel. Also observe that schwa deletion occurs after all tonal processes have applied. This explains why downstep of high tone still happens in [«∆ eÿae ⍺r≠◊ l] 'leaf of farm' from /«∆ , eÿae «ae «∆ ,r≠◊ l/.

Figure 16 Input HTS and r«à deletion h l h h h l h h l h h

Figure 19

Input The data also illustrate that when schwa deletion occurs, its stranded tone docks leftwards to form a falling contour tone on the preceding TBU. No other tonal processes occur in combinations involving classes 1 and 9 nouns.

HTS l l h l h l l h l h L L H L H L L H L H o o o o o o o o o o = «,s-Б«`,jvdm→«,s-Б«`,jvdm→ Figure 20 Downstep h-spread l l h l h l l h l h = = L L H L H L L H L H o o o o o o o o o o «,s-Б«`,jvdm →«,s-Б«`,jvdm→
Classes 1 and 9 nouns, like those of class 6a also cause N2 H tone nouns from other noun classes other than 1 and 9 to surface with L tones, as the following data show. This ability of classes 1, 6a, and 9 nouns, when found in N1, to cause Hs of N26 to become L can be attributed to the L of the AM of these classes. The L forces all tones that follow it to become L, on condition that there is no nasal in the onset of N2 roots. It should be noted that the L spreads first to the prefix and then to the root, deleting the prelinked tones. The following rule of low tone spread is formulated to account for the changes. According to this rule a low tone spreads to and delinks the tone of the following TBU. The following derivation is provided to illustrate the application of this rule.

CONCLUSION

The way tone behaves when nouns are brought together in the associative construction in Kejom has been discussed in this paper. The analysis has been done within the framework of Register Tier Theory. It has been demonstrated that tonal processes that occur in such constructions are sensitive to a low-toned nasal that occupies the onset position of the noun roots involved. For example, h-spread applies only if there is no nasal in the onset position of N1 nouns. HTS applies if there is no nasal in the onset position of N2 nouns or in the presence of a nasal if the N2 root is high-toned. A determining factor of the analysis presented in this paper is the constraint that some tones must be singly-linked, and not doubly-linked for the rules to apply. For example the tone that precedes the target low tone must be singly-linked for h-spread to apply whereas the tone that follows the target low tone must be singlylinked for HTS to apply. These kinds of rule application that depend on single vs. multiple associations of nodes find support in previously stated constraints such as the Uniform Applicability Condition (Shein & Steriade, 1986), the Linking Constraint [START_REF] Hayes | Inalterability in CV Phonology[END_REF], and the Conjunctivity Condition [START_REF] Goldsmith | Autosegmental and Metrical Phonology[END_REF].

Only four, as opposed to ten [START_REF] Hyman | Tonology of the Babanki noun[END_REF] tonological rules have been used to account for the realization of tone in ACs in this language. Some nouns have been shown not to follow the regular pattern and this has been attributed to either their noun class affiliation or to the fact that they are borrowed from Kom, a neighbouring language. The logic of the present analysis has been to make possible a simpler account over [START_REF] Hyman | Tonology of the Babanki noun[END_REF] analysis of ACs in Kejom.

Figure

  Figure 1 Geometry of tone

Figure

  Figure 2 Register features and tonal features _ _ H↑ H↑

  (1) a. u«∆ ,⍻rt∆ Бu«ae u«ae ,⍺sЀ«ae Б 'bottles of fon's aides' /u«∆ ,rt∆ Бu«ae u«∆ ,sЀ«ae Б/ c2-bottle AM c2-fon's aides b. u«∆ ,k≠◊ lu«ae u«ae ,⍺jvÕ◊ 'husbands of namesakes' /u«∆ ,k≠◊ lu«ae u«∆ ,jvÕ◊ / c2-husband AM c2-namesake c. j«∆ ,⍻j-∆ rj«ae j«ae ,⍺enae 'slave of thing' /j«∆ ,j-∆ rj«ae j«∆ ,enae / c7-slave AM c7-thing d. j«∆ ,ЀÕ◊ ◊ j«ae j«ae ,ЋvÕ 'place of spirit' /j«∆ ,ЀÕ◊ j«ae j«∆ ,ЋvÕ / c7-place AM c7-spirit e. s«∆ ,⍻ki«∆ oes«ae s«ae ,⍺r≠◊ l 'bamboos of farms' /s«∆ ,ki«∆ oes«ae s«∆ ,r≠◊ l/ c13-bamboo AM c13-farm f. s«∆ ,snae Бs«ae s«ae ,⍺fÕ◊ Б 'bushes of dew' /s«∆ ,snae Бs«ae s«∆ ,fÕ◊ Б/ c13-bush AM c13-dew g. e«∆ ,⍻j-∆ Бe«ae e«ae ,⍺sÕ◊ m 'tree of yard' /e«∆ ,j-∆ Бe«ae e«∆ ,sÕ◊ m/ c19-tree AM c19-yard h. e«∆ ,rdae re«ae e«ae ,k≠ l 'pepper of spice' /e«∆ ,rdae re«ae e«∆ ,k≠ l/ c19-pepper AM c19-spice

  Figure 3h-spread l h

Figure 4

 4 Figure 4 High Tone Spread (HTS) h l

  Figure 5 Downstep l h = H

  Figure 8Stray Erasure and Merger Phonetic Representation l h l

  (2) a. u«∆ ,m∆ s-∆ mu«ae u«ae ,⍺k≠◊ l 'pots of husbands' /u«∆ ,m∆ s-∆ mu«ae u«∆ ,k≠◊ l/ c2-pot AM c2-husband b. u«∆ ,oe∆ ftae Бu«ae u«ae ,⍺jvÕ◊ 'termites of namesakes' /u«∆ ,oe∆ ftae Бu«ae u«∆ ,jvÕ◊ / c2-termite AM c2-namesake c. j«∆ ,l-avÕ mj«ae j«ae ,⍺ЀÕ◊ 'thread of place' /j«∆ ,l-avÕ mj«ae j«∆ ,ЀÕù/ c7-thread AM c7-place d j«∆ ,m∆ r-ae mj«ae j«ae ,avÕ m 'frog of witchcraft' /j«∆ ,m∆ r-ae mj«ae j«∆ ,avÕ m/ c7-frog AM c7-witchcraft e. s«∆ ,oe∆ jv«∆ ms«ae s«ae ,⍺jnae 'tails of much money' /s«∆ ,oe∆ jv«∆ ms«ae s«∆ ,jnae / c13-tail AM c13-money f. s«∆ ,m∆ stae s«ae s«ae ,⍺v`ae r 'shoots of vegetable' /s«∆ ,m∆ stae s«ae s«∆ ,v`ae r/ c13-shoot AM c13-vegetable g. e«∆ ,m∆ ЀÕ Бe«ae e«ae ,k`∆ l 'beetle of net' /e«∆ ,m∆ ЀÕ Бe«ae e«∆ ,k`∆ l/ c19-beetle AM c19-net h. e«∆ ,oe∆ ft¿ e«ae e«ae ,sЀn 'hook of squirrel' /e«∆ ,oe∆ ft¿ e«ae e«∆ ,sЀn / c19-hook AM c19-squirrel

  Figure 10 Downstep, Merger and Stray Erasure Phonetic Representation l h l h

  Figure 12 Stray Erasure and Merger Phonetic Representation l h l H↑ L H L L↓ l9 h: L↓ L↓ o o o o o o

  (5) a. e«∆ ,ْ∆ ْÕ◊ m⍺e«ae e«ae ,oe∆ jtae Б 'bird of slave' /e«∆ ,ْ∆ ْÕ◊ me«ae e«∆ ,oe∆ jtae Б/ c19-bird AM c19-slave b. j«∆ ,enae ⍺j«ae j«ae ,avÕ m 'thing of witchcraft' /j«∆ ,enae j«ae j«∆ ,avÕ m/ c7-thing AM c7-witchcraft

  Figure 16 Merger Phonetic Representation l h l l

  (6) a. a`ae Б,⍺r«ae Ћÿae ,⍺r«ae 'cliffs of bees' /a`∆ Б,r«ae r«ae Ћÿ∆ ,r«ae / cliff-c10.AM bee-c10 b. Ѐtae ,⍺r«ae a`ae Б,⍺r«ae 'fishes of cliffs' /Ѐt∆ ,r«ae r«ae a`∆ Б,r«ae / fish-c10.AM fish-c10 c. cyøae l,⍺r«ae l-a`∆ m,r«ae 'backs of fences' /cyø∆ l,r«ae r«ae l-a`∆ m,r«ae ae / back-c10.AM fence-c10 d. m∆ s-∆ m,r«ae eih◊ ,⍺r«ae 'pots of kidneys' /m∆ s-∆ m,r«ae r«ae eiÕ ,r«ae / pot-c10.AM kidney-c10 e. l-oeÿ∆ ,r«ae l-a`∆ m,r«ae 'ropes of fences' /l-oeÿ∆ ,r«ae r«ae l-a`∆ m,r«ae ae / rope-c10.AM fence-c10

5. 3

 3 ACs INVOLVING CLASS 6a NOUNS When nouns from this class combine in ACs they are separated by a low-toned AM [l«è], as the following examples show. (7) a. l«∆ ,rn∆ l«∆ l«∆ ,l-eÿae 'boils of medicines' /l«∆ ,rn∆ l«∆ l«∆ ,l-eÿae / c6a-boil AM c6a-medicine b. l«∆ ,m∆ Ѐ≠◊ ◊ l«∆ l«∆ ,m∆ kit∆ Б 'tears of wine' /l«∆ ,m∆ Ѐ≠◊ ◊ l«∆ l«∆ ,m∆ kit∆ Б/ c6a-tear AM c6a-wine c. l«∆ ,rdae rl«∆ l«∆ ,m∆ ЋÕ Б 'peppers of oil' /l«∆ ,rdae rl«∆ l«∆ ,m∆ ЋÕ◊ Б/ c6a-pepper AM c6a-oil d. l«∆ ,sÿae Бl«∆ l«∆ ,l-eÿae 'calves of medicine' /l«∆ ,sÿae Бl«∆ l«∆ ,l-eÿae / c6a-calf AM c6a-medicine These data show that none of the processes described above occurs when nouns from this class combine in ACs. However, some H tones surface as L when found in N2 position, as the following data show. (8) a. l«∆ ,m∆ kit∆ Бl«∆ j«∆ ,eÿ∆ 'wine of medicine' /l«∆ ,m∆ kit∆ Бl«∆ j«∆ ,eÿae / c6a-wine AM c7-medicine b. l«∆ ,rdae rl`∆ r`∆ oe 'peppers of maize' /l«∆ ,rdae rl«∆ `∆ ,r`ae oe/ c6a-pepper AM. c6 maize c. l«∆ ,sÿae Бl«∆ j«∆ ,f«∆ oe 'calves of grass' /l«∆ ,sÿae Бl«∆ j«∆ ,f«ae oe/ c6a-calf AM c7-grassIt should be mentioned that the majority of high toned nouns from classes 5, 7 and 19 whose high tones surface as low tones when found in N2 position are those that have their plural forms in either classes 6 or 6a. Iterative low tone spread formulated in (23 below) can be used to account for the data in (17).

  Figure 18Input Low Tone Spread l h l l h l h l l h

Figure

  Figure 21 Schwa deletion Stray Erasure and Merger Phonetic Representation l h l

(

  11) a. ْ∆ ْ`∆ l«∆ j«∆ ,j≠ l 'animal of crab' .ْ∆ ْ`∆ l«∆ j«∆ ,j≠◊ l/ c9.animal AM c7-crab b. r`∆ oe«∆ s«∆ ,r≠ l 'month of farming' /r`∆ oe«∆ s«∆ ,r≠◊ l/ c1.month AM c13.farm c. Ћt∆ u«∆ ,sr-∆ oe 'snake of thieves' .Ћt∆ «∆ u«∆ ,sr-ae oe/ c9.snake.AM c2.thief d. vÕ Б «∆ e«∆ ,sÕ m 'person of yard' / vÕ Б«∆ e«∆ ,sÕ◊ m/ c1.person AM c19.yard

Figure

  Figure 22 Low Tone Spread (iterative) l

  Figure 23Input Low Tone Spread l l l h l l l h

Table 1 :

 1 Kejom Noun Classes

	Noun	Prefix (Suffix,	Example		Gloss
	Class		Class 10)
	1		÷∆ ,		r`∆ oe+rt∆ Б+vt∆ k≠◊ l		'month', 'bottle', 'man'
	2		u«∆ ,+÷∆ ,	u«∆ ,r`∆ oe+u«∆ ,rt∆ Б+kÿae ⍺l«ae	'months', 'bottles', 'men'
	3		«∆ ,		«∆ ,evÕ◊ m+«∆ ,vnae oe		'leg', 'market'
	5		«∆ ,		«∆ ,Ѐ≠◊ +«∆ ,֙nae oe		'eye', 'spear'
	6		`∆ ,
	19		e«∆ ,		e«∆ ,ْ∆ ْÕ◊ m		'bird'
				3. SYLLABLE STRUCTURE OF NOUN ROOTS
	A majority of Kejom noun roots are monosyllabic and this study is limited to
	such nouns. Monosyllabic noun roots have the structure CV, CGV, CVC, CGVC,
	NCV, NCGV, NCVC, NCGVC, where G represents a glide as illustrated in table 2
	with the most common tonal melodies, L and H.
	Table 2: Syllable Structure of Noun Roots
		CV		CGV	CVC	CGVC	NCV		NCGV NCVC	NCGVC
	L	`∆ ,ad∆	ai≠	cy«∆ l	j«∆ ,aiÕ Б	j«∆ ,m∆ cÿ∆		m∆ ci«∆	j«∆ ,l-e≠◊ e	oe∆ fv`∆ Б
		'liver'	'pit'	'back'	'dust'	'whip'		'thing'	'blind s.o'	'seed'
	H «∆ ,eÿae	«∆ ,kvÕ◊	s«∆ ,j`ae oe	«∆ ,evÕ◊ m	j«∆ ,l-anae	oe∆ fvdae	oe∆ j≠◊ l	l-ai«ae Б
		'leaf'	'nose'	'skies'	'leg'	'madness'	'name'	'basket'	'shoulder'

`∆ ,evÕ◊ m+`∆ ,Ѐ≠◊ +`∆ ,֙ÿae 'legs', 'eyes', 'feet' 6a l«∆ , l«∆ ,ْ∆ ْÕ◊ m+l«∆ ,eÿae 'birds', 'medicines' 7 j«∆ , j«∆ ,l-an∆ +j«∆ ,֙ÿae +j«∆ ,enae 'bag', 'foot', 'thing' 8 «∆ , «∆ ,l-an∆ 'bags' 9 ÷∆ , l-a`∆ Б+Aÿae 'insult (N)', 'dog' 10 ,r«ae l-a`∆ Б,r«ae +Aÿae ,⍺r«ae 'insults (N)', 'dogs' 13 s«∆ , s«∆ ,vnae oe+s«∆ ,֙nae oe 'markets', 'spears'

Table 3 :

 3 Kejom Associative Markers

	Noun Class	Association Marker
	2, 7, 13, and 19	High toned CV (u«ae

+j«ae +s«ae +ande«ae )

	10	High toned CV (r«ae )
	6a	Low toned CV (l«∆ )
	3, 5, 6, and 8	High toned V («ae )
	1, and 9	Low toned V («∆ )

ae Б,⍺r«ae Ћÿae ,⍺r«ae

													h l h
	H L H	H H L H	H L H H H L H
	o	o	o	o	o	o	o	o	o	o	o	o	o	o
									=				=
	a`Бr«r«Ћÿr« → → → →a`Бr«'r«(Ћÿr«→ → → →
	Figure 17		Downstep and Stray Erasure			Phonetic Representation
			h l h		h l h
			=		=
		H L H	H L H
										H↑		H↑
										l9		l9
		o	o	o	o	o	o				H↑		H↑
	a`Бr«Ћÿr«	a`

& This paper was first presented in 2009 at the WOCAL6 conference in Cologne (Germany). I thank the audience at this conference for a stimulating discussion and the anonymous JWAL reviewer for very helpful comments. I also appreciate very useful comments from Prof. Ngessimo Mutaka.

An initial vowel may or may not occur before the prefix or root (i.e. it is optional). The initial vowel can occur in all the noun classes. It can surface before all consonants. When found before vowels, it is deleted following a schwa deletion process that applies in the language, deleting the schwa when it is adjacent to another vowel.

As a native-speaker of Kejom I have provided most of the data used in this paper. I have also made substantial reference to the data found in[START_REF] Hyman | Tonology of the Babanki noun[END_REF]. Unlike Hyman, I present data according to noun classes especially because the ACs are divided into sets based on the noun class of N1. In order to be quite sure of the different phonetic tone levels I asked Hyginus Abufon, who was born and has lived in Kejom Ketingu for more than 20 years to say some of the constructions.

Three different [L-H] tone classes can be distinguished for Kejom[START_REF] Hyman | Tonology of the Babanki noun[END_REF]. Examples include the following: L-H class 1: e«∆ ,ْْÕ◊ m 'bird' L-H class 2: j«∆ enae 'thing' L-H class 3: j«∆ j≠◊ l 'crab' These three L-H tone classes act differently in two ways: (i) L-H classes 1 and 2 cause a following H tone to become downstepped. (ii) L-H classes 1 and 3 become L-L after classes 1,

6a, and 9 low tone associative markers.

Such nouns were classified by[START_REF] Hyman | Tonology of the Babanki noun[END_REF] in Class II (see footnote 4 above).


Hassane Alla

Optimal Supervisory Control Synthesis

Keywords: controller, Place invariant, Petri net, forbidden states, linear constraints contrôleur, invariant de marquage, réseau de Petri, états interdits, contrainte linéaire

The place invariant method is well known as an elegant way to construct a Petri net controller. It is possible to use the constraint for preventing forbidden states. But in general case, the number forbidden states can be very large giving a great number of control places. In this paper is presented a systematic method to reduce the size and the number of constraints. This method is applicable for safe and conservative Petri nets giving a maximally permissive controller.

Introduction

Petri Nets (PN) is an appropriate and useful tool for the study of Discreet-Event Systems (DES) because of their modeling power and their mathematical properties. One of the advantages of PN models in DES control synthesis is the use of a PN marking as a distributed representation of the system state. For control goals represented as state avoidance problems, this often allows a decomposition of the control synthesis problem into a series of smaller problems which are more efficiently solved. In the last decade, the research in the field of controller synthesis of DES became one of the most active domains [1],[2],[3], [4].

In this paper we present an efficient method for synthesizing a controller to solve forbidden state problems for a particular class of DES's. We consider control specification which can be stated in the PN model. Synchronous composition between model of system and model of specification by existence the uncontrollable transitions, produce the forbidden states.

The forbidden state problem for a DES was introduced in a paper by Ramadge and Wonhom in which they demonstrated several fundamental properties of the solution [?]. The main limitation of such approach is the lack of structure in controlled automata and the large number of states of the related state transition structures.

Li and Wonham [??] have presented an algorithm which calculates the optimal solution for the systems modeled by Petri nets whose uncontrollable sub nets are loop-free. The controller has to solve on-line at each step linear integer programs. The approach will be difficult to apply to the grand systems because of computational complexity. A method for the computation of a maximally permissive controller using theory of region is presented in Ghaffari and Xie []. The main idea is to reduce the reachability graph to the set of admissible markings, i.e., markings that respect the specifications. Control places are added to the original network, so that the closed loop plant respecting the specifications. The theory of region is used on the reachability graph for the computation of the control places. For the computation of control places, a system of linear inequalities must be resolved. The solution of the system must be an integer positive solution. The advantage of this method is that gives a maximally permissive solution of the control problem, but the computation method is difficult and does not guarantee a solution. In this method, for each forbidden state, we must compute a control place. Then there is many control places and it increases the complexity of system. There is many of research for solving the problem of forbidden states [],[],[],[]. Each of them proposed the method for solving this problem. Most relevant to our work are [5], [6]. In [5], author presented a method for construct a optimal controller for the system modeled by Grafcet. The method is based on the equivalence between the set of forbidden states and the set of linear constraints deducted from it. The use of the Yamalidou technique in [5],[6] allows constructing a set of control places which represent the optimal controller. However the number of control places is equal to the number of constraints and can become very important. Intuitively, we realized that the PN solution is not unique from the point of view of the structure, but the reachable space (graph of the markings) is naturally unique. The problem comes from all the linear constraints which can be simplified by holding structural properties of the PN. In this paper is proposed a systematic method of reduction of the number of constraints. The minimal solution is not unique in the general case, but a method to choose the best solution is presented. This approach is based on the hypothesis that the PN models are safe and conservative. This paper is organized as below:

In the second section will be presented the fundamental definitions then how we can construct the linear constraints from forbidden states. Later, in the fourth section, the method to reduce the number and the size of the constraints will be presented. At the end, this method will be explained through an example.

Fundamental definitions

In this paper it is supposed that reader knows the bases of the Petri net paradigm and we present only the notions used in this paper. For more details, see the book in [START_REF] David | Discrete, Continuous, and Hybrid Petri Nets[END_REF]. A PN is presented by a 4-uplet N = {P, T, Pre, Post} where P is the set of places, T the set of transitions, Pre: P×T →N is the pre-incidence function that defines weighted arcs from places to transitions. Post: T×P →N is the post-incidence function that defines weighted arcs from transitions to places. C= Post -Pre is the incidence matrix.

The reachability graph is an automata = {M, Σ, δ, m 0 } where M is the states set, Σ is the events set, δ : M x Σ → M the evolution function, m 0 is the initial state. This graph corresponds to all the possible evolutions of the PN. The reachability graph consists of nodes which correspond to the accessible markings, and of arcs to the firing of transitions. In this paper, we use the word state instead of marking, represented by all the marked places.

Note 1: The corresponding state to the marking M i will be noted as following:

M i = (P i1 …P ij …P in ) │ ∀j ∈ {1, …, k} m(P ij ) =1 & m(P l ) = 0 for other places
For example the marking M 1 = [0, 1, 1, 0, 0, 1, 0, 0, 0] will be presented as state M 1 = P 2 P 3 P 6 .

In the reachability graph, there are some of states: the authorized state, the forbidden state and the not reachable state. Among the forbidden states a particular and important subset is constituted by the border forbidden states, it is denoted by the set M B . These states are such that all the input transitions are controllable. In this paper we use the definition that presented in (Toma 2005).

Definition 1: The set of forbidden states is noted M F and is expressed:

M F = {m∈M | ∃ w∈L R \ L Rd and m =δ(m 0 ,w)} i.e.,
M D = {m∈M | ∃ w∈Σ u * and δ(m, w) ∈M F } i.e.
, a dangerous state is a reachable marking from which there is at least a sequence of uncontrollable transitions who leads to a forbidden state. We will consider, also, that a forbidden state is a dangerous state.

Definition 3:

The set M A of admissible states is the greatest set of reachable states so as:

• M A ∩ M D = ∅
• If m∈M A and δ(m, e) ∈M F than e∈Σ c , i.e., the passage from an admissible state to a dangerous state is made by the firing of a controllable transition. Furthermore, M A is the most permissive behavior respecting the specifications.

Definition 4:

The set of border states is:

M B = {m∈ M D | ∃ e∈Σ c ∧∃ m a ∈M A , s.t. δ(m a ,e)=m}
i.e., the set of dangerous states which are reached by the firing of a controllable transition from an admissible state.

Definition 5: A controller is maximally permissive (MPC) if all the admissible markings of M A are reachable under supervision and all the firings of a transition, which cause the evolution of the plant from an admissible state to a non-admissible one, are inhibited.

In this paper, an important definition will be also used; it concerns the notion of over-state. Definition 6: Let be M i = P i1 P i2 …P in an admissible state. M j = P j1 P j2 …P jm is an over -state of M i if:

1) ∀ P jk ∈ M j ∃ P ik ∈ M i such as P jk = P ik = 1 2) ∀ P jl , l∉ [1,…, m] M j (P jl ) = 0 or 1 For example state M 1 = P 1 P 4 P 7 is an over -state of M 2 = P 1 P 4 P 7 P 10 .

From Forbidden States to Linear Constraints

The modelling approach presented in this paper, is the classic approach developed by Ramadge andWonham (1987a, b, 1988). The model of the process is synchronized with the specifications giving the closed loop desired functioning. The existence of uncontrollable transition often leads to the existence of forbidden states including the border forbidden states. They are systematically determined by using the Kumar algorithm [START_REF] Kumar | Supervisory control of deterministic Petri nets with regular specification languages[END_REF]. We consider here that the PNs models of both process and specifications are safe, but this does not mean that the marking of the control places is Boolean. Let f i be a border forbidden state from F and {P i1 P i2 P i3 … P in } be all the marked places which correspond to it. From the forbidden states1 , linear constraints can be constructed [START_REF] Kattan | Synthèse structurelle d'un contrôleur basée sur le Grafce[END_REF][START_REF] Caramihai | ″A DES-Supported Agent-Based Control Architecture for Flexible Manufacturing Systems″[END_REF]. The linear constraint for the forbidden state f i is given by Equation (1).

∑ = n k 1 m ik ≤ n -1 (1)
Where n is the number of marked place of f i , and m ik is the Boolean marking of place P ik in state f i . For example: If the state (P 2 P 5 P 7 ) is a forbidden state, it can be avoided by using Equation ( 2)

m 2 + m 5 + m 7 ≤ 2
(2) In this paper the forbidden state and its constraint is presented as below:

f i = {P i1 P i2 P i3 … P in } c i = {P i1 P i2 P i3 … P in , n-1}
(3)

The last term in c i corresponds to the number of marked places in the forbidden state minus 1, it is called the bound. This is true only at the beginning.

Reduction of the Size and Number of Constraints

In real systems, the number of forbidden states can be very large. Knowing that for each forbidden state, one control place is added, the complexity of the controller can become unmanageable. For this reason, in this paper we propose a systematic method to reduce the size and the number of the constraints.

The PNs models are assumed to be safe and conservative; then it is impossible for two places belonging to the same place invariant, to be marked simultaneously. This basic idea will be used for the simplification of the constraints.

Property 1: Let be {(P 1 P i1 … P i(n-1) ),...,(P r P i1 … P i(n-1) )} r forbidden states in M F .

m 1 + m 2 +…+ m r = 1 (4)
The r constraints are equivalent to one constraint as below:

m 1 + m i1 +…+m i(n-1) ≤ n-1 m 2 + m i1 +…+m i(n-1) ≤ n-1 … m r + m i1 +…+m i(n-1) ≤ n-1 ⇔ m i1 +…+m i(n-1) ≤ n-2
Where n is the number of marked place.

Proof: necessary Condition:

The sum of all constraints gives the constraint follow: (m 1 +m i1 +…+m i(n-1) ) + (m 2 +m i1 +…+m i(n-1) ) + … + (m r +m i1 +…+m i(n-1) ) ≤ r(n-1)

(5) (4), ( 5

) ⇒ 1+r(m i1 +…+m i(n-1) ) ≤ r(n-1) ⇒ m i1 +…+m i(n-1) ≤n-1-1/r (m i : Integer number) ⇒ m i1 +…+m i(n-1) ≤n-2 Sufficient Condition: m i1 +…+m i(n-1) ≤ n-2 ∀ k ∈ {1,…,r} m k ≤ 1 ⇒ m k + m i1 +…+m i(n-1) ≤ n-1
Let be F = {(P 1 P 4 P 7 ), (P 1 P 4 P 8 ), (P 1 P 4 P 9 )} set of a forbidden states and m 7 +m 8 +m 9 = 1, thus the over-state (P 1 P 4 ) is forbidden state, without attention to marked place of the set {P 7 , P 8 , P 9 }, or it is possible to use instead of three constraints only one constraint as below:

m 1 +m 4 +m 7 ≤ 2 m 1 +m 4 +m 8 ≤ 2 ⎯ ⎯ ⎯ ⎯ ⎯ → ← = + + 1) m m (m 9 8 7 m 1 +m 4 ≤ 1 m 1 +m 4 +m 9 ≤ 2
NOTE: it is possible to use property 1 more than one time.

For example, if the set F is as below: F = {(P 1 P 4 P 7 P 10 ) , (P 1 P 4 P 7 P 11 ) , (P 1 P 4 P 8 P 10 ) , (P 1 P 4 P 8 P 11 ), (P 1 P 4 P 9 P 10 ), (P 1 P 4 P 9 P 11 )} and m 10 + m 11 = 1 m 7 +m 8 +m 9 = 1 After the first use of Property 1 for the invariant between P 10 and P 11 the result is: {(P 1 P 4 P 7 ), (P 1 P 4 P 8 ), (P 1 P 4 P 9 )}. Now it is possible to use Property 1 for the other invariant. The final result will be the over-state (P 1 P 4 ).

Let us consider another set of forbidden states as below;

F ' = {(P 1 P 4 P 7 P 10 ), (P 1 P 4 P 7 P 11 ), (P 1 P 4 P 8 P 10 ), (P 1 P 4 P 8 P 11 )} At the end of first simplification the result is: {(P 1 P 4 P 7 ), (P 1 P 4 P 8 )} and it is not possible to use Property 1, but is it possible to reduce the number of constraint? The answer is yes. Because of the invariant between P 7 , P 8 and P 9 , the states P 7 and P 8 cannot be marked in the same time. Thus both constraints can decrease in a constraint as below:

m 1 +m 4 +m 7 +m 8 ≤ 2 (6)
The property discussed above, is formalized in property 2:

Property 2: Let be C = {(P 1 P i1 …P i(n-1) , k), (P 2 P i1 … P i(n-1) , k), ..., (P r P i1 … P i(n-1) , k) } the equivalent constraints to forbidden states of marking graph and

m 1 +m 2 +…+m r ≤ 1 (7)
The r constraints are equivalent to one constraint as below:

m 1 +m i1 +…+m i(n-1) ≤ k m 2 + m i1 +…+m i(n-1) ≤ k … m r + m i1 +…+m i(n-1) ≤ k ⇔ m 1 + m 2 + …+ m r + m i1 +…+m i(n-1) ≤ k
Where n is the number of marked place.

Proof: necessary Condition:

∀ j ∈ {1,…, r} m j + m i1 +…+ m i(n-1) ≤ k (8) and

m 1 + m 2 +…+ m r ≤ 1 ⇒ (m 1 + m 2 + …+ m r )+m i1 +…+m i(n-1) ≤ k+1
We are going to show that limit k+1 is never reached. If not, it is necessary that m 1 + m 2 +…+ m r = 1 and m i1 +…+m i(n-1) = k. But if m i1 +…+m i(n-1) = k, thus because of the constraints (8) it is necessary that for ∀ j ∈ {1,…, r} m j = 0. Then m 1 + m 2 + …+ m r = 0. Thus limit k+1 never is reached.

Sufficient Condition:

(

m 1 + m 2 + …+ m r ) + m i1 +…+m i(n-1) ≤ k and ∀ i ∈ {1,…,r} m i ≥ 0 Then ∀ i ∈ {1,…,r} m i + m i1 +…+m i(n-1) ≤ k NOTE:
While exploiting this property, constraints are used instead of states since the bound may change according to the simplifications.

Unreachable states

The objective of this approach is reduction the size and the numbers of the constraints. The final result is simpler if Property 1 is used. For this reason, our idea is to add the unreachable states in order to increase the space of forbidden states.

Definition 7: The unreachable states are the states that are not accessible from the initial state of the PN model or are reachable from border forbidden states.

If these states help to reduce constraints, it is possible to use these states as forbidden states. But at the end, it is not necessary to choose a constraint which does not covered any forbidden state. This will be explained later in Section 5.

Minimum of constraints

In order to have the minimum of constraints, it is necessary to choose the simplest constraints containing all the forbidden states. To reach this goal it is necessary to write all the final constraints with the forbidden states. A method similar with the one proposed by Mac Cluskey for the reduction of the logical expressions is used for determining the best choice(Morris [START_REF] Mano | Digital Design[END_REF]. Firstly, we must choose the simplified constraints which contain the forbidden states that are not in other constraints. Secondly for the forbidden states which are in two or several simplified constraints, it is needed to choose the constraint containing the greatest number of forbidden states which were not chosen, or the constraint which is simpler. In generally, there is not only one answer. This will be more explained in the following section.

EXAMPLE

To illustrate the ideas developed in this paper, let us consider the example which was presented in [START_REF] Kattan | Synthèse structurelle d'un contrôleur basée sur le Grafce[END_REF]). A manufacturing system is composed of two independent machines M1 and M2, two transfer robots of the parts and one test bench where the final products are tested (Fig. 1).

Fig. 1. Production line

Each machine has the following operating cycle: By occurrence of the event c i , the machine starts working. When the work is finished (occurrence of event f i ) the produced part is transferred by the robot on the test bench, and a new cycle can be started again by occurrence of event t i . There are two types of events in this system: the controllable events and uncontrollable events, only events c 1 and c 2 are controllable:

∑ c = {c 1 , c 2 } et, ∑ u = {f 1 , t 1 , f 2 , t 2 }.
The model PN for this system is given in Fig. 2. The model of the specification of this system is presented in the Fig. 3. It is asked that firstly, the robot transfers the product of machine M 1 and then the product of machine M 2 .

Desired functioning in closed loop

The determination of the forbidden states is made by the study of the controllability of the language of specification with regard to the language generated by the model of desired functioning in closed loop. In this example only transitions c 1 and c 2 are controllable and the other transitions are not controllable. The closed loop model of this system is illustrated in Fig. 4 and will be denoted as R d . In this paper this model of PN is called Quasi-PN defined below. Definition 8: A Quasi-PN is a PN which respects the following rules of firing: 1) A controllable transition is firable in the same way as in a classical PN.

2) An uncontrollable transition is firable if all its input places belonging to the process are marked

The difference of reachable markings between these two models (PN and Quasi-PN) indicates the forbidden states.

Definition 9: If in from a marking, an uncontrollable transition is firable for the Quasi-PN model, but not for the real PN model, then this state is forbidden.

To determine the forbidden states by this method, it is necessary to construct the reachability graph for both models.

Reachability graph of the functioning closed loop

The reachability graph is an identification tool of the behaviour of the PN. A node in the reachability Graph represents the reachable markings, and arcs characterize the possible passages between these markings. A marking corresponds to a combination of marked places. There are 2 N possible states for a safe PN with N places. But for a conservative PN, the number of states is smaller. Thanks to the method presented in [START_REF] David | Discrete, Continuous, and Hybrid Petri Nets[END_REF] , the set of possible states can be calculated. The number of state is determined as below:

N= ∏ = m i 1 (n i ) (9)
Where n i is the number of places in each marking invariant and m is the number of invariant.

From the reachability graph, the theory developed by Ramadge and Wonham allows to obtain the maximal permissive supervisor. If the language generated by the reachability graph is controllable, then the problem is resolved and the PN model R d constitutes the controller. But if it is not controllable, it is necessary to determine all the forbidden states.

The reachability graph of the real -PN and quasi -PN are presented in Fig. 5 and6. The comparison of the two graphs, gives the forbidden states. E = {(P 7 P 6 P 1 ), (P 7 P 6 P 2 ), (P 7 P 6 P 3 ), (P 8 P 4 P 3 ), (P 8 P 5 P 3 ), ( P 8 P 6 P 3 )} ( 10)

From these states, weakly forbidden states are deduced. By the technique presented in [START_REF] Kattan | Synthèse structurelle d'un contrôleur basée sur le Grafce[END_REF], it is possible to find all the forbidden and weakly forbidden states that are called dangerous states. To prevent from reaching the dangerous states, it is sufficient to forbid the border states. All the input transitions of these states are controllable.

For the example, the border states are: F = {(P 7 P 5 P 1 ), (P 7 P 5 P 2 ), (P 7 P 5 P 3 ), (P 8 P 4 P 2 ), (P 8 P 5 P 2 ), (P 8 P 6 P 2 )} (11)

From these 6 forbidden states, 6 constraints are deduced. We can now use the presented properties to reduce the size and the number of these constraints. For this, it is necessary to construct a set containing all the possible states and to indicate the type of each one.

In this example the marking invariants are obvious. For this example the number of possible states is calculated as below:

N= ∏ m i (n i )= 3*3*

Use of Properties 1 and 2

In this example there are three invariants and Property 1 is used for all.

I: m 1 +m 2 +m 3 =1: Finding this invariant in Fig. 7 is easy, because in this case, all of the states are in one row. At the end it is obtained two over-states as below:

(P 5 P 7 ), it covers the states: {P 7 P 5 P 1 , P 7 P 5 P 2 , P 7 P 5 P 3 } (P 6 P 7 ), it covers the states: {P 7 P 6 P 1 , P 7 P 6 P 2 , P 7 P 6 P 3 } (13)

IІ. m 4 +m 5 +m 6 =1: By applying Property 1 for this invariant, the result is as below:

(P 8 P 2 ), covering {P 8 P 4 P 2 , P 8 P 5 P 2 , P 8 P 6 P 2 } (P 8 P 3 ), covering {P 8 P 4 P 3 , P 8 P 5 P 3 , P 8 P 6 P 3 } ( 14)

IIІ. m 7 +m 8 =1 (P 5 P 2 ), covering {P 7 P 5 P 2 , P 8 P 5 P 2 } (P 6 P 2 ), covering {P 7 P 6 P 2 , P 8 P 6 P 2 } (P 5 P 3 ) , covering {P 7 P 5 P 3 , P 8 P 5 P 3 } (P 6 P 3 ), covering {P 7 P 6 P 3 , P 8 P 6 P 3 } (15)

The following table contains all the over states and indicates all the forbidden states which are covered by them. If may happen that a state is not covered, then it must be added in the final selection. It is not the case in this example (Fig. 8). Here, Property 1 cannot be used longer, but in general case, it is possible to examine the table for new simplifications. Property 2 can now be used.

From now, we use the constraints instead of states or over-states since the bound may change according to the simplifications. There is two over-states or two constraint {(P 5 P 7 , 1), (P 6 P 7 , 1)} which can be simplified by using Property 2. It is possible to use instead of two constraints only one constraint as bellow:

{(P 5 P 7 , 1), (P 6 P 7 , 1)} ⇒ (P 5 P 6 P 7 , 1)

The result after using Property 2 for all the constraints of Fig. 8 is illustrated in Fig. 9.

Constraint

The states which are covered S P 5 P 6 P 7 , 1 P 7 P 5 P 1 , P 7 P 5 P 2 , P 7 P 5 P 3 , P 7 P 6 P 1 ,

P

Minimum of constraint for this example

At the end of this level, it is necessary to choose the minimum and simplest constraints, containing all the forbidden states. In general case, it is possible to have some choice. The difference between two choices is the number of arc of the control places. The best choice is determined after the calculation of the control places. At the end, the best solution contains two linear constraints(Fig. 10).

m 5 +m 6 +m 7 ≤ 1 m 2 +m 3 +m 8 ≤ 1 (17)

Calculation of control places

To calculate control places for every linear constraint, there is a systematic method given in [START_REF] Yamalidou | Feedback control of Petri Nets based on place invariants[END_REF]. The incidence matrix of the PN R d , corresponding to the model of functioning in closed loop is W p . The synthesis approach consists in adding to the incidence matrix W p a matrix W c corresponding to the control places.

The incidence matrix of controlled system W will be constructed by adding a line to the matrix of incidence of controlled process W p for every control place.

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = c p W W W (18)
It is then possible to calculate the matrix W c and its initial marking as bellow:

W c = -L.W P (19) And M cinit = b -L.M pinit (20)
The controller is known by the determination of his matrix of incidence W c and his initial marking M cinit . By applying this result to the example, the result is: In the large system, the number of possible states is enormous and by this method we need to large memory and long time for calculating the simpler controller. By example for a system AVG by two inputs and two outputs and 4 workstations and 4 critical zones, we have 30,965,760 states possible and computing the controller is very complicated. But it is obvious that the situation of output system don't effect on the situation of Wagon In zone 1. Therefore we can compute the controller for the zone 1 without attention to mark in the model of output system. The Petri Net model for controlled AVG is shown in Fig. 13.

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - - = 0 0 1 1 0 0 1 0 0 0 0 1 Wc
In this example, supervisor is presented by the constraints. By Yamalidou's method, the supervisor is computed. But because of uncontrollable transition this supervisor doesn't work correctly. For resolve the problem of first zone we can neglect the model of workstation 1, 2, 3 and the Wagons that move out of zone 1. This model is presented in Fig. 14. Now, we can compute the forbidden states and border forbidden states for this system and compute the controller by the method that presented in this paper. Remark: The controller isn't always simpler controller, because some times we can construct one control place for two or more critical zone.

Remark: In supervisory idea the critical zone produced when we have an uncontrollable transition common between model of system and model of supervisor.

We must find all of this type of transitions and dependence invariants. Finding the invariants that depend to each critical zone is possible intuitively for the simple system, but in the large system we need to an algorithm to computing the dependence invariant for each critical zone. The result for this example is shown in Fig. 15. In this example we can apply the constraints for supervisory as the forbidden states in constructing the controller. This result is presented in Fig 16.

Conclusion

In this paper we have presented a systematic method to reduce the number of forbidden states or the number of equivalent linear constraints. The basic idea is to use the markings invariants of the PN to simplify these constraints. This is realized by using the not reachable states.

We obtain a structural solution easy to implement. The solution obtained by our approach gives the optimal controller. This optimality comes from the equivalence between all the admissible markings and all the linear constraints. The method that is presented in this paper is applicable for the systems modelled by Grafcet if the place of control is safe.

Our future work consists in: 1) establishing complete algorithms and, 2) developing this approach for the system that is modelled by safe PN model not necessarily conservative.

Fig. 2 .

 2 Fig. 2. PN process model.Fig. 3. PN specification model

Fig. 3 .

 3 Fig. 2. PN process model.Fig. 3. PN specification model

Fig. 4 .

 4 Fig. 4. Petri net model of closed loop system

Fig. 5 .

 5 Fig. 5. Reachability graph of real -PN Fig. 6. Reachability graph of quasi -PN By using Definition 3 it is possible to determine the set of forbidden states as below:

Fig. 7 .

 7 Fig. 7. All possible states

Fig. 11 .

 11 Fig. 11. The control matrix W c The initial marking of the control place is calculated: m c1 = 1; m c2 = 0; The PN of the final controller is represented in Fig.12.

Fig. 12 .

 12 Fig. 12. Petri net model of closed loop system with control places

Fig. 13 .

 13 Fig. 13. The controlled Automated Guided Vehicle

Fig. 14 .

 14 Fig. 14. The dependence invariants to zone 1

Fig. 15 .

 15 Fig. 15. The controller for zone 1

Fig. 16 .

 16 Fig. 16. The controlled system without supervisor for zone 1

a state is forbidden if it is a reachable state and if it violates the specifications. Definition 2: The set of dangerous states is:

  P 5 P 1 , P 7 P 5 P 2 , P 7 P 5 P 3 0 P 6 P 7 P 7 P 6 P 1 , P 7 P 6 P 2 , P 7 P 6 P 3 Ф P 8 P 2 P 8 P 4 P 2 , P 8 P 5 P 2 , P 8 P 6 P 2 0 P 8 P 3 P 8 P 4 P 3 , P 8 P 5 P 3 , P 8 P 6 P 3 Ф

	states
	P 5 P 7 P 7 P 5 P 2	P 7 P 5 P 2 , P 8 P 5 P 2	0
	P 6 P 2		P 7 P 6 P 2 , P 8 P 6 P 2	0
	P 5 P 3		P 7 P 5 P 3 , P 8 P 5 P 3	0
	P 6 P 3		P 7 P 6 P 3 , P 8 P 6 P 3	Ф
	.
	Fig. 8. All states after first simplification
	Over-	The states which are covered	S

  7 P 6 P 2 , P 7 P 6 P 3 0 P 8 P 2 P 3 ,1 P 8 P 4 P 2 , P 8 P 5 P 2 , P 8 P 6 P 2 , P 8 P 4 P 3 , P 8 P 5 P 3 , P 8 P 6 P 3 0 P 5 P 6 P 2 P 3 ,1 P 7 P 5 P 2 , P 8 P 5 P 2 , P 7 P 6 P 2 , P 8 P 6 P 2 , P 7 P 5 P 3 , P 8 P 5 P 3 , P 7 P 6 P 3 , P 8 P 6 P 3

	0
	Fig. 9. All states after the last simplification

Where there is no ambiguity, the word border will be omitted.