Gomez Christopher
email: christopher.gomez@cnrs-bellevue.fr

Exposure to Volcanic Hazards, and Influence on Perception: A Case Study in Japan, Ten Years After the Unzen Fugendake Eruption

Keywords: Unzen, Nagasaki, Kyushu, Japan, Volcano, Eruption, Perception, Natural Hazards, Natural Hazards' exposure, pupils perception, trauma

This study aims at understanding the perception of volcanic hazards that adults and pupils have in the province of Nagasaki Kyushu, ten years after the volcanic eruption of the Unzen Fugendake. The Unzen Fugendake started to erupt in 1991 and produced several pyroclastic flows lahars and ashes rains. Shimabara city, located on the foot of the volcano was severely stroke by the eruption. Ten years after this eruption, a study was carried on in Shimabara city and in Saikai cityaway from any volcano -to compare the inhabitants' perceptions of volcanic hazards and the importance they have among other natural hazards. This study was carried on through questionnaires and interviews for the adults, as for the pupils we collected drawing, completed by informal interviews. From this survey, we can observe some signs of trauma among people who had been under direct exposure of the eruption, whereas indirectly exposed respondents did not show any sign of it. It seems that natural hazards' personal experience is always considered like the most important whenever the scale of the natural hazards that are compared. In Shimabara, majority of the respondents consider that the Unzen's eruption was the worst disaster in Japan for the twenty last years, whereas Saikai city's inhabitants hardly mentioned it, highlighting the "o-ame" event that triggered exceptional floods in Nagasaki city and in their own town. In the end, this study is also showing the difference between pupils and adults' perceptions. Indeed the first diverging point concerns the fact that pupils are highlighting the natural hazard itself, whereas adults preferentially highlighted the vulnerability of human beings and constructions. The second diverging point between adults and pupils is the fact that both cities children' perceptions do not vary as much as adults' ones do. Due to the absence of any direct exposure to volcanic eruption, the children' perception seems to tend towards some homogeneity, based on second-hand information.

Introduction

Japan is the third country for volcanic disasters, with 8% of the world fatalities due to volcanic hazards these last 4 centuries. Volcanic disasters can be defined as interactions between extreme volcanic events and vulnerable human populations (adapted from [START_REF] Blaikie | At Risk: Natural Hazards, people's vulnerability and Disasters[END_REF]. From 1991 to 1995, the town of Shimabara located on the foot of the Unzen volcano was stricken by its eruption, resulting in 43 fatalities and a reconstruction that cost almost US$2 billion (source: ministry of land and infrastructure). The eruption, its consequences and the authorities' actions were widely reported through the media all over Japan becoming of a national concern. Ten years after this disaster, this study aims at assessing the volcanic hazards' perception among inhabitants in Nagasaki Province, and more specifically to compare perception of population under the threat of a volcano and people not having any contact with a volcano in their everyday life.

The 1991-1995 eruption of Unzen volcano

The Unzen volcano -1486 m. a.s.l. at Heiseishinzan -is located in the Nagasaki province to the West of Shimabara city (fig. 1). The Unzen is well-known for Japan's greatest volcanic disaster (Mount Mayuyama): in 1792 around 15.000 people were killed by a massive landslide from Mount Maruyama which devastated Shimabara city, and entering in the sea generated a series of tsunamis [START_REF] Akagi | The tsunami height and damage area of tsunami occurred in Shimabara Peninsula[END_REF]Brantley and Scott, 1993;[START_REF] Furuya | A geomorphologic study of the collapse of Mayu-Yama in 1792[END_REF]. According to [START_REF] Katayama | Old records of natural phenomena concerning the "Shimabara Catastroph[END_REF], 14.528 died, 1518 people were injured, and 5972 houses were devastated around the Ariake Bay (tab. 1). Tab.1 : Damages after the Mayuyama disaster [START_REF] Katayama | Old records of natural phenomena concerning the "Shimabara Catastroph[END_REF] In 1990 the volcano awakened again [START_REF] Nakada | Overview of the 1990-1995 eruption at Unzen Volcano[END_REF]Ohta, 1991) for 5 years creating a new lava dome at the summit area of the Fugendake [START_REF] Hoshizumi | Geology and eruptive history of Unzen volcano, Shimabara Peninsula, Kyushu, SW Japan[END_REF]. This event started by phreatomagmatic eruptions in April-May 1991, and the lava dome began to grow in August 1991 to reach 1,2 Km long and 0.8 Km wide for a thickness ranging from 230m to 540m (Nakada et al.,1998). Domes' collapses generated pyroclastic flows, which 99% were of Merapi types. The first observed pyroclastic flows swept the Mizunashigawa valley in May 1991 and most of them occurred during the 1991-1993 periods that coincide with the exogenous dome growth (Tadahide et al., 1998). The Unzen Fugendake produced 9400 block-and-ash flows until 1995 (Miyabuchi, 1999) as frequently as 10 small pyroclastic flows a day and until 504 for the only month of September 1991 sweeping until 5,5 km from the dome (Brantley and Scott, 1993) killing 43 people -including French and American volcanologists. Tab. 3: Damages in Shimabara city due to the 1991-1995 eruption at Unzen volcano (Shimabara city hall, 2003).

According to Blong (1984), two of the factors that influence the social effects of an eruption on a community include (1) the nature of the volcanic hazards, (2) the deaths, injuries and damage incurred. In Shimabara fatalities remained little thanks to the population evacuation, since only 43 people died over this eruption (tab. 2). However building and infrastructure damages were important. Volcanic eruptions affect buildings through a variety of damage mechanisms. Indeed building may suffer collapse, burial, transport, foundation failure, undermining, racking, excessive wall or roof loads, impacts from ballistic projectiles, corrosion, or catch fire (Blong, 1984). The most destructive agent on Unzen volcano during the 1991-1995 eruption was lahars. From 1991 to 1995, lahars sweeping Unzen's valleys destroyed more than 1700 buildings. The second most destructive agent was pyroclastic flows. From 1991 to 1995, pyroclastic flows destroyed 808 buildings in Shimabara city (tab. 3).

The Unzen, a highly monitored volcano

The Unzen volcano has been systematically belted by a monitoring system including 13 automatic telemetered raingauges to survey the volume and intensity of rainfall pouring on the volcano, and that can potentially trigger lahars (fig. 1).

This system is completed by a rainfall's radar located in the village of Futsucho.

The raingauges and the radar are completed by sensors rigged on SABO dams to compute lahars' magnitudes. This system is wired to automatic video-camera recording the various lahars activities. This equipment is the first cell of a complete monitoring and population warning system (fig. 2).

surroundings against lahars. Sabo dams were created to stop, trap or slow down sediments carried by lahars (fig. 3).

Fig. 2: The post-eruption lahars' warning system on the Unzen volcano.

Fig. 3: The SABO dams on the Eastern flank of the Unzen volcano (Gomez C., 2004).

On the North-western flank, in the Nakaogawa, a series of 8 SABO dams were built between the altitude 270m and 350m in order to protect Shimabara city and upstream the Shinzan SABO dams 1 and 2. On the Northern flank, along 1 km; a series of 15 SABO DAMS oriented East-West between 300m and 450m a.s.l.

create a "staircase" meant to slow down the eventual lahars' runoff. Those "Oshigatani-shinzan" dams present a broad concrete structure 100m to 200m wide.

Downstream to the East, 3 major SABO dams cross the Mizunashigawa valley.

The SABO dam located around 1700m a.s.l. is a slit dam which is meant to stop the largest blocks with its series of 16 seven meter high pillars. The two other dams are located downstream around 500m a.s.l. and 1100 m a.s.l., and are built like enormous reservoir for sediments. Then two series of 15 side protection dikes are canalizing the eventual flow towards the sea to avoid any flood towards the inhabited areas.

This SABO dam prevention system is completed by an active program that aims at stopping the rainfall triggered lahars with an artificial vegetation return. The

Regional & National Forestry Offices and the Shimabara Town-hall set up a program of pyroclastic deposits' stabilization on the Fugen-Dake Heisei

Objectives and hypothesis

This study aims at understanding the way residents of the Nagasaki province perceive, and evaluate the volcanic risks either if they are living near a volcanohere the Unzen -or more especially out of direct sight and influence of this one.

Risks' perception queries are a major stake for decision making processes and scientists, since people have their own perception of risks based on different criteria than scientists [START_REF] D'ercole | Volcanic risk perception: comparing public perception maps and scientific hazard maps[END_REF]. This fact was verified during the Unzen eruption in 1990. Indeed, despite evacuations ordered by the Japanese authorities, many peopled ignored the warnings because they did not have any knowledge of pyroclastic flows related risks, or thought pyroclastic flows would not reach their home [START_REF] Yoshii | Disaster Warnings and Social Response: The Eruption of Mt. Unzen in Japan[END_REF]. This perception has surely changed after the eruption, in Shimabara and elsewhere in Japan, since the Unzen eruption became of national concern, largely broadcasted on television. Therefore, ten years after the eruption, this paper tries to understand the influence of direct and indirect exposure to a volcanic hazard, and the perception of two populations, one under a direct threat of volcanic hazards and one other apparently without any volcanic hazards' concern.

This study is supported by a few hypothesis: (1) Firstly, in a country under the threat of numerous natural hazards, it is interesting to understand the place volcanic hazards have among all the natural hazards, and understand if the populations feel likely to be stricken by volcanic hazards rather than other hazards [START_REF] Johnston | Volcanic Hazard perceptions in Inglewood, New Zealand[END_REF].

(2) Volcanic hazards' perception is also certainly guided by the mean of information. We can observe direct exposure to the riskspersonal experience -and indirect exposure through media for example, and this should greatly change the populations' risks perception. For example, the data collected in may 1980 around St-Helens volcano which attests views exposed by the interviewed inhabitants were tinted by those expressed by volcanologists on the news [START_REF] Greene | The March 1980 Eruption of Mount St-Helens, Citizen Perceptions of Volcano Threat[END_REF][START_REF] Perry | Human Response to volcanic eruption: Mt St-Helens[END_REF]. The perception change then with the mean of information, which is itself controlled by the distance to the volcano (Blong, 1984[START_REF] Greene | The March 1980 Eruption of Mount St-Helens, Citizen Perceptions of Volcano Threat[END_REF][START_REF] Sorenson | Emergency response to Mount St-Helens' eruption: March 20 to[END_REF]. ( 3) Volcanic hazards perception is also influenced by the interviewed inhabitants' age [START_REF] Ronan | The effects of a "Benign" Disaster; Symptoms of Posttraumatic Stress in Children Following a Series of Volcanic Eruptions[END_REF][START_REF] Ronan | School Children's Risk Perceptions and Preparedness: A Hazards Education Survey[END_REF], therefore adults and pupils' perception must be slightly different

Methodology

The population sample and respondents' profile:

To carry out this study, the sample consisted in 400 participants recruited on a voluntary non paid basis. These participants can be divided into two groups depending on their reply to a questionnaire or answers through drawings. Both adults and children groups are composed of 200 participants each. The adult group is composed of respondents from the last grade of Junior High School to elderly, while children include students from eight to eleven years old -4 th grade to 6 th grade of elementary school. The children age repartition was pretty equal in between 4 th , 5 th and 6 th grade since querying was conducted in school classes. As for the adults' 5% in Saikai city and 21,5% in Shimabara-city were less than 20 years old, 12% and 18% were between 20 and 35 years old in respectively Saikaicity and Shimabara-city. The population comprised between 35 and 65 years old were for both town the most numerous with 35% and 40%, and population over 65 years old were 31% and 19,5% for respectively Saikai-city and Shimabara-city.

For Saikai and Shimabara cities 2,5% and 5% of respondents did not include the information about their age. In the end, the respondents' gender was represented by 60,4% by women and 39,6% by men.

Location choices

Queries were conducted in Shimabara city and Saikai city. Shimabara was choosen for this study because of its proximity to the volcano, since the city is located on the foot of the Unzen. Shimabara's counter part for this study is Saikaicity. This choice was driven by different reasons. Firstly Saikai city is 70km away from Shimabara and do not have any volcano in its surrounding. Indeed, if you are going eastward or southward from the Unzen you can find respectively the Aso volcanic complex, and the Sakurajima. Secondly, Shimabara and Saikai city are separated by Nagasaki city in between. Therefore Saikai city is partly polarized by Nagasaki city and do not entertain any obvious economical or cultural link with Shimabara city and the Unzen volcano. In the end, practical reasons drove the choice of Saikai-city: the author was living in this locality for more than a year, therefore personal and professional networks were of great help to conduct this study. Therefore Saikai city seemed a good counterpart for Shimabara city.

Procedure

For the questionnaire, each participant received one from the author or friends and acquaintance that helped for this study. Then, the respondents were invited to answer the questionnaire individually within a one month time period. They were encouraged to use a dictionary for difficult terms, and questionnaire given to the elderly were explicit with "furigana" of each kanji -phonetical pronunciation of each difficult Chinese sign used in Japanese language. Participants were also asked to circle terms they did not know or did not know the reading way. Once the questionnaire handed back to the author, a few interviews or questions were informally conducted, mostly on weekends at their home. For the drawings, they were conducted with the help of local teachers or parents, mostly at school during an hour school lesson. The students were undergoing a short talk on natural disasters with their teachers or parents, and then were asked to draw what they were thinking about when told about natural disaster. The exercise lasted for one hour the most. Some time later, students were approached in order to better understand what they have been drawing. These interviews were mostly conducted quite informally at respondents' home or on the school ground while playing during school breaks. The choice to keep the children in a familiar environment was motivated by the will of not creating too much stress to the children that could modify their answers.

Questionnaires' contents

Questionnaires were archived according to queried people characteristics that include name (not compulsory), age, gender, actual position, address, any change of address in the past, and the hometown information. The population sometimes omitted their age and names for privacy reasons but majority filled up all. Then, the query itself was divided into three parts: the first one was concerning natural hazards perception and an approach on the meaning of natural hazards. Indeed, Japanese language has inherent difficulties, and some terms are quite obscure, even for native speakers. This first part was also interested in personal experiences respondents have of natural hazards, with a classification of the worse natural disaster they can remember, and ones they experienced. The second part of the questionnaire concerns the queried people knowledge about volcanic activity, potential risks and their memory about the last Unzen eruption. In the end, they were asked to draw their perception of natural hazards.

Results

The volcanic hazards place among other natural hazards

Japan is stricken by a great variety of natural hazards; earthquakes, tsunamis, volcanic activity, landslides, mudflows, floods, etc. Therefore it was important at first to understand the place populations both in Shimabara city and Saikai city were giving to volcanic hazards. Considering pupils answers, over all the different drawings, a few elements are recurrent: volcanic activity, tsunami, rainfall, typhoon, lightning, earthquake and fire. We also observed irrelevant drawing

showing other subjects like pupils own home, families or school… According to some elementary school teachers we can attribute these mistakes to their young age, a mischief about the exercise and/or the stress of not producing anything compared to other comrades, even though they might not have understand the question. For Shimabara city, pupils most represented an erupting volcano.

Indeed 40% of the children drew a volcano. The second natural hazard represented were typhoon and thunder with 17,7% of the drawings. Other natural hazards were represented between 7% and 8% each (fig. 6). In Saikai city, we observed about the same hierarchy with volcanic activity and typhoon/thunder that were represented each in 24% of the drawings. As for the other natural hazards, earthquakes and fires respectively represent 12% and 10%, others don't reach 8% (fig. 6). The place teenagers and adults give to volcanic hazards among all the other natural hazards represents only 10,4% in Saikai city and 21,5% in Shimabara city.

The different consequences of these hazards on human-beings are emphasizes rather than the events themselves. Indeed, the consequences on infrastructures and the human losses represent 48% and 41,1% of the answers in respectively Saikai city and Shimabara city (tab.4). This relation between natural hazards and consequences on infrastructures and inhabitants is also exemplified by the drawings teenagers and adults did (fig. 7). Thus, we can observe the importance given to personal experience when it comes to natural hazards perception, and the overvaluation of that experience in regards of other events, known by other mean. This observation was also the same when it comes to the Unzen eruption in Shimabara city.

Affective and cognitive perception of volcanic risks: a tribute to the information mean

The Unzen eruption's pith varies among Shimabara city and Saikai city inhabitants with a logical pre-eminence in Shimabara city. We supposed this difference due to the degree of direct exposure and therefore the mean that informed population. In Shimabara city, we can conjecture that direct experience has been far-reaching -as expressed previously -, as for Saikai city inhabitants the main information-mean was mass-media network and mainly television.

Indeed, the further away interviewed inhabitants are living from a volcano, the lesser importance they give to the eruption, and the greater the media influence [START_REF] Greene | The March 1980 Eruption of Mount St-Helens, Citizen Perceptions of Volcano Threat[END_REF]. Thus away from sight of the Unzen volcano, almost all Saikai city respondents have been informed about the eruption through massmedias -even if 6 respondents declared having news through their family in Shimabara, or through trips around the Unzen. This mass-media information has a direct consequence on the Unzen volcanic hazards perception. Firstly the 1990-1995 eruption perception is very poor in Saikai city, since less than 60% of the respondents knew that the major part of the activity was located towards the East.

The respondents in Saikai city also can't locate on a map the directions of erupted materials. Secondly, Saikai city inhabitants don't have an accurate perception of the volcanic risks in general. Damages due to ash falls are totally unknown, or forgotten, either on health or on individuals, properties and land. Moreover all the respondents are thinking that even in a case of a major eruption, there is absolutely no risk at about 70km away from the volcano. Saikai respondents are therefore showing a serious lack of knowledge about volcanic risks, even after the Unzen eruption.

On the foot of the volcano, in Shimabara city, the mean of knowledge is obviously different, at least for population who are old enough to have experienced that event. These respondents have direct references in their surrounding landscape, and the answers are accompanied by hands movements showing some places during the interviews. This attitude is not observed at all in Saikai city. The answers formulated in Saikai city are "dictionary" type of answers. The relations the respondents have with the volcanic hazard lies within the field of intellect and cognition, whereas in Shimabara city respondents are showing a relation which is cognitive and affective. Shimabara inhabitants are bonded in an affective relation with the volcano, and the old and new landscape.

The effect of direct exposure to the volcanic hazard is also symbolized in a "reexperiencing" process, even when it comes to project oneself in the future.

Indeed, when asked about future disasters that could still happened in Shimabara, lahars in the Mizunashigawa and Nakaogawa are appearing in more than 56% of the answers, a dome collapse in 58% of the answers. If the dome collapse's risk still exists, there is no real risk of dangerous lahar in the Mizunashigawa or the Nakaogawa. However inhabitants are projecting in the future their own past experience. This reaction was already observed by a query carried on by the Kyushu University in 1995 [START_REF] Takahashi | The Unzen eruption hazards plan for the disaster's recovery[END_REF]. At that time 68,3% of the respondents were afraid of lahars occurring in the Mizunashigawa and Nakaogawa, 35% of a dome collapse, 13,3% of ashfalls. Again, the fear were driven by past-even memory, more than a real estimation of the potential actual danger. The way populations are in contact with a volcanic hazard, through direct experience or through media is influencing the knowledge of the volcanic hazards, the perception of what is dangerous, and also the nature of the link between the human beings and the volcanic hazards: purely cognitive or "affectivo-cognitive".

Volcanic hazards perception among pupils

Volcanoes represented by children are always showing the same pattern of a regular mountain with a truncated summit where a vent takes place (fig. 8).

Regarding volcanic hazards developed by 40% of the children in Shimabara city and 24% in Saikai city -other treated of other natural hazards -, the drawings include mostly the following elements: Ashes and clouds coming out the volcanic main vent, lava, explosions, projections, thunder and earthquake linked to volcanic activity. In Shimabara city and in Saikai city, the most drawn phenomenon are the projections flying out of the main vent of the volcano. In Shimabara city 28% of the children drew projections, and in Saikai city 31% did.

The other most represented phenomenon are in Shimabara explosions and lava flow with respectively 21% and 14% of the drawings (fig. 9a). In Saikai city, the second and third positions are taken by these same phenomenons however in a different order, with 23% for the lava and a bit more than 15% for the explosions (fig. 9b). Then are coming earthquakes or "ground shake type" due to a volcanic eruption that was represented on a bit more than 18% of the drawings in Shimabara city and 15% in Saikai city. In the End, thunder linked to volcanic activity was represented in 9,3% of the drawings in Shimabara city and 3,8% in Saikai city.

For both locations, elements represented by the pupils are very similar when it comes to volcanic activity. The differences in perceptions observed within the adults don't seem to exist when it comes to pupils; however the information collection's mean was different for both adults and pupils groups, therefore it is really difficult to make any comparison. Nevertheless, Shimabara's pupils have no personal experience of the 1990-1995 eruption and their volcanic hazards perception is certainly not so different from the one Saikai-city students have.

Discussion and Conclusion

The population in Nagasaki province is showing through this study the different consequences of direct and indirect exposures to natural risks, and thus volcanic risks. From this degree of exposure, the type of relationship between the natural hazards and the human beings varies. A direct exposure is creating an affective bond between the human being and the hazard. This bond is due to the moral shock that endangered people's life, or created trauma or post-traumatic stress disorder. This kind of "pathological" relation seems to be expressed by the impossibility to forget the event and an over-emphasis of this one. It is also accompanied by body expressions. [START_REF] Ronan | The effects of a "Benign" Disaster; Symptoms of Posttraumatic Stress in Children Following a Series of Volcanic Eruptions[END_REF] has characterized a series of psychological disorders in his studies on "benign disasters" that struck children with 65% of reexperiencing, 24% of hyperarousal and 14% of psychic numbing cases. Population can also show clinic psychopatological disorders, but still seems to determine in a post-catastrophe population [START_REF] Keppel-Benson | Posttraumatic stress disorder in children and adolescents[END_REF][START_REF] Ronan | The effects of a "Benign" Disaster; Symptoms of Posttraumatic Stress in Children Following a Series of Volcanic Eruptions[END_REF] Such cases did not seem to appear during our study, certainly because the respondents were exposed to the eruption about 10 years ago, while the studies dealing with this subject are carried on just a few months after a catastrophe [START_REF] Eyre | More than PTSD: Proactive Responses Among Disaster Survivors[END_REF], and also because our respondents were for a greater part of them adults. It seems that the younger the respondents, the more vulnerable they seem to be to post-traumatic stress disorder and psychopathological disorders [START_REF] Atkins | Children's perspective of stress and coping: An integrative review[END_REF][START_REF] Belter | Impact of natural disasters on children and families[END_REF][START_REF] Gibbs | Factors in the victim that mediate between disaster and psychopathology : A review[END_REF]. However it is evident that the relationship between the hazards and the human beings under a direct exposure is totally different from the relationship with indirect exposure. Indeed, the kind speech patterns observed in Shimabara were totally absent from the respondents that have no direct experience of the risks. Thus the information mean linked with the degree of exposure is playing a very important role in risks' perception, whether it is through a media or through direct experience. Thus more than the distance itself, the degree of exposure to natural hazards seems to be the key of the psychological bound that links the hazards and the human beings, and then their response to a risk. However, this study can only be considered as a preliminary result on this point, and further studies are needed to be totally affirmative.

The second point brought by this study concerns our hypothesis on pupils' perception. Indeed the pupils' perception of natural risks can be differentiated from adults' perception by the points they are emphasizing. Pupils are emphasizing the natural hazards part of the risks whereas adults emphasized the human consequences. If we only consider volcanic risks pupils are representing the eruption itself -the hazards -whereas adults are representing the buildings destroyed, the inhabitants running away from the danger -the vulnerability. This orientation is the main difference between grown-up and pupils. This representation certainly comes from the absence of death in the elementary school level children; they are still living in an infinite time system where death does not have its place. The second point that differs from the adults is the fact that there is not such a difference between the answers in Shimabara city and Saikai city as it was observed for adults, certainly due to the absence of traumatic experience as we explained before. Even living by the volcano, the absence of direct experience of the eruption erased the differences observed among adults with direct and indirect experience. This is also showing the rapid resilience capacity of a population over natural disasters within only 2 generations, although memory is extremely important following this kind of event as suggested by Hasemi Yuuji (2001).

Thus, if there is no recent eruption, it seems that volcanic hazard perception tends to be quite similar regardless to the distance to the volcano, whereas the

Fig. 1 :

 1 Fig. 1: Unzen volcano, Shimabara and Saikai cities' locations in Nagasaki province -south

  Shinzan's flanks by accelerating the return of vegetation. It is not so much to protect population against lahars, but to avoid lahars triggering processes. Two methods have been used. For the upper part of the volcano, the most hazardous one, seeds have been spread out from an helicopter, and in the distal zones, the helicopter spread has been completed with trees plantations. The total surface regain by vegetation is around 1360 ha including 1344 in the main valleys touched by pyroclastic flows: Akamatsu, Mizunashigawa, Ashigatani, Nakaogawa and Yuegawa. 1.3 Inhabitants' characteristics: an aging population Both cities, Shimabara and Saikai are showing pretty similar inhabitants' evolution patterns, even though Shimabara city is showing some specificity due to the volcanic eruption. These last 20 years, Shimabara's population has been decreasing from 46.061 in 1986 to 38.959 in 2005. The number of inhabitants over 65 years old has been increasing whereas other classes have been decreasing. Indeed, inhabitants below 15 years old were 10436 in 1986 and only 5611 in 2005, 15 to 64 were 29394 in 1986 and 23192 in 2005. As for the over 65 years old, they pass from 6231 to 10156 from 1986 to 2005. The aging population process over this period has been accentuated by the volcanic eruption from 1991 to 1995 because young generations fled from the city (fig. 4). The population of Saikai city has been decreasing of -5,6% these last 20 years. The population that reached 9385 inhabitants in 1986 was only 8854 in 2005. As for Shimabara city, the number of over 65 years old inhabitants has been increasing from 1550 to 2555. Population below 65 years old has been decreasing. Below 15 years old were 1954 in 1986, and only 1364 in 2005, whereas 15 to 65 years old were 5878 in 1986, and 4935 in 2005 (fig. 5).

Fig. 4 :

 4 Fig.4: Population evolution in Shimabara city from 1986 to 2005.

Fig. 5 :

 5 Fig. 5: Population evolution in Saikai city from 1986 to 2005.

Fig. 6 :

 6 Fig. 6: Primary school most drawn natural hazards in Saikai city and Shimabara

  : consequences on infrastructures Tab. 4: Natural hazards importance for adults and teenagers in Shimabara city and Saikai city.The importance inhabitants give to natural hazards that drove to risks and sometimes to disaster vary with the age but also the exposure of the queried people. In both studied locations, respondents are emphasizing their own experience compared to other risks or disaster they did not experience themselves, but they got to know through Medias like television, etc. Indeed, in Shimabara city more than 38% of the respondents describe the Unzen eruption as the worse natural disaster of these last 20 years in Japan. Catastrophes

Fig. 7 :

 7 Fig. 7: Examples of natural hazards representation by adults and teenagers at the end of the questionnaires.

Fig. 8 :

 8 Fig. 8: Drawings examples of children in sixth grade of primary school.

Fig. 9 a

 9 Fig.9 a,b: Volcanic elements represented by primary school students in Shimabara city and in Saikai city.

Acknowledgments

The Shimabara Town-hall, and the Shimabara branch of the Japanese Ministry of Land and Infrastructures are thanked for their help and the different information provided to complete this study. Junko Yamada has also been of great help on various information collections and their understanding, therefore I would like to especially thank her for her all along help and support.

At last but not least, Gaillard J-C., Dibben C. and two anonymous reviewers are thanked for all the advices they provided and the constructive remarks they made to improve this manuscript.

differences could be more important just after an eruption due to the trauma and even psychopathology brought to the directly exposed population. Those trauma and psychopathology are absent from indirectly exposed populations.


B Bérard

F Cassez

S Haddad

Didier Lime
email: didier@cs.aau.dk

O H Roux

Comparison of Different Semantics for Time Petri Nets

Keywords: Time Petri Nets, Timed Bisimilarity, Expressiveness

In this paper we study the model of Time Petri Nets (TPNs) where a time interval is associated with the firing of a transition, but we extend it by considering general intervals rather than closed ones. A key feature of timed models is the memory policy, i.e. which timing informations are kept when a transition is fired. The original model selects an intermediate semantics where the transitions disabled after consuming the tokens, as well as the firing transition, are reinitialised. However this semantics is not appropriate for some applications. So we consider here two alternative semantics: the atomic and the persistent atomic ones. First we present relevant patterns of discrete event systems which show the interest of these semantics. Then we compare the expressiveness of the three semantics w.r.t. the weak time bisimilarity establishing inclusion results in the general case. Furthermore we show that some inclusions are strict with unrestricted intervals even when nets are bounded. Then we focus on bounded TPNs with upper-closed intervals and we prove that the semantics are equivalent. Finally taking into account both the practical and the theoretical issues, we conclude that persistent atomic semantics should be preferred.

Introduction

Since their introduction, Petri nets have been successfully applied for the design and analysis of discrete event systems. However with the development of critical systems, time has become a significant issue for their correctness. So numerous timed extensions have been proposed for Petri nets. These extensions can be roughly divided into three categories w.r.t. their application field:

-Timed Petri nets [START_REF] Ramchandani | Analysis of asynchronous concurrent systems by timed Petri nets[END_REF] include a duration associated with each transition in order to model for instance scheduling policies in production management; -Stochastic Petri nets [START_REF] Ajmone Marsan | Modelling with Generalized Stochastic Petri Nets[END_REF] include a probability distribution associated with each transition in order to evaluate the transient or steady-state behaviour of a system where duration of actions are obtained by statistical observations;

-Time Petri nets [START_REF] Merlin | A study of the recoverability of computing systems[END_REF] include a time interval associated with each transition (specifying the possible time elapsing before firing) in order to model systems which may evolve in a non deterministic way.

The latter model has a lot of alternatives: for instance time constraints may be associated to places [START_REF] Khansa | P-Time Petri Nets for manufacturing systems[END_REF] or to arcs [START_REF] Abdulla | Timed Petri nets and BQOs[END_REF][START_REF] De Frutos Escrig | Decidability of properties of timed-arc Petri nets[END_REF], timed synchronisations may be added [START_REF] Diaz | Time stream Petri nets: a model for timed multimedia information[END_REF], etc. Here we stick to the original model for which significant theoretical developments [START_REF] Berthomieu | Modeling and verification of time dependent systems using time Petri nets[END_REF][START_REF] Popova | On time Petri nets[END_REF][START_REF] Lilius | Efficient state space search for time Petri nets[END_REF][START_REF] Berthomieu | State class constructions for branching analysis of time Petri nets[END_REF] have been obtained leading to efficient verification tools on models of large systems [START_REF] Berthomieu | The tool TINA -Construction of Abstract State Spaces for Petri Nets and Time Petri Nets[END_REF][START_REF] Gardey | Roméo: A tool for Time Petri Nets Analysis[END_REF]. Moreover when bounded this model can translated into timed automata [START_REF] Cassez | Structural Translation of Time Petri Nets into Timed Automata[END_REF] which are also extensively studied with successful applications [START_REF] Alur | A theory of timed automata[END_REF].

More precisely we consider a slight extension of TPNs where any kind of time interval can be associated with the firing of a transition instead of the closed ones in the original definition. Our work focus on the impact of the memory policy on the expressiveness of the model. The memory policy specifies which timing informations are kept when a transition is fired. The original model selects an intermediate semantics meaning that the transitions disabled after consuming the tokens, and the firing transition, are reinitialised. Here we propose two alternative semantics: the atomic and the persistent atomic ones. We first present significant examples where these semantics are more appropriate than the original one. Then we compare the expressiveness of the three semantics w.r.t. the weak time bisimilarity which is the standard equivalence relation used for such comparisons. First we establish inclusion results in the general case. Furthermore we show that some inclusions are strict with unrestricted intervals even when nets are bounded. The key point of the previous result is the presence of non upper-closed intervals. Then we focus on bounded TPNs with upper-closed intervals and we establish that the semantics are equivalent. All our translations are optimal since the size of the produced net is linear w.r.t. the size of the emulated net.

The paper is organised as follows. Section 2 introduces the syntax and semantics of TPNs, timed transition systems and timed bisimilarity. Section 3 is devoted to the comparison between the different semantics for TPNs with general intervals both in the bounded and the unbounded case. In section 4, we prove that the three semantics are equivalent for bounded TPNs with upper-closed intervals. Finally we conclude in section 5.

Time Petri Nets

Notations. Let Σ be a finite set (or alphabet). Σ * denotes the set of finite words on Σ. An interval I is a Q ≥0 -interval of R ≥0 iff its left endpoint belongs to Q ≥0 and its right endpoint belongs to Q ≥0 ∪ {∞}. We let I ↓ = {x | ∃y ∈ I ∧ y ≥ x} be the downward closure of I. We denote by I(Q ≥0 ) the set of Q ≥0 -intervals of R ≥0 .

If w = a 1 • • • a n , the length of w denoted |w|, is n. We also use Σ ε = Σ ∪ {ε} with ε ∈ Σ,

Timed Transition Systems (TTS) and weak time bisimulation

Timed transition systems describe systems which combine discrete and continuous evolutions. We present here a standard version.

Definition 1 (Timed Transition Systems). A timed transition system (TTS) over the set of actions Σ ε is a tuple S = (Q, q 0 , Σ ε , -→) where:

-Q is a set of states, -q 0 ∈ Q is the initial state, -Σ ε is a finite set of actions disjoint from R ≥0 , --→⊆ Q × (Σ ε ∪ R ≥0 ) × Q is a set of edges. If (q, e
, q ′ ) ∈-→, we also write q e --→ q ′ . For a transition q d --→ q ′ with d ∈ R ≥0 , the value d represents a relative time stamp.

We make the following common assumptions about TTS:

-Time-Determinism: if q d --→ q ′ and q d --→ q ′′ with d ∈ R ≥0 , then q ′ = q ′′ , -0-delay: q 0 --→ q, -Additivity: if q d --→ q ′ and q ′ d ′ --→ q ′′ with d, d ′ ∈ R ≥0 , then q d+d ′ ----→ q ′′ ,
-Continuity: if q d --→ q ′ , then for every d ′ and d ′′ in R ≥0 such that d = d ′ + d ′′ , there exists q ′′ such that q

d ′ --→ q ′′ d ′′ ---→ q ′ .
A run ρ of length n ≥ 0 is a finite sequence of transitions of the form

ρ = q 0 d0 --→ q ′ 0 a0 ---→ q 1 d1 --→ q ′ 1 a1 ---→ • • • q n dn ---→ q ′ n
where discrete actions alternate with (possibly null) durations. We also write this run as q d0a0...dn ------→ q ′ . Untimed(ρ) is the word of Σ * obtained by concatenation of labels a 0 , . . . , a n-1 (remember that ε is the empty word). Duration(ρ) = n i=0 d i . Furthermore, a run can be safely by inserting or deleting a null duration step and splitting a duration step or merging two consecutive duration steps as long as the total duration is the same.

It is well-known that the concept of weak timed bisimilarity is central among equivalence relations between systems since, for instance, two TTS are not distinguishable by formulas of most common timed arborescent temporal logics.

Let S = (Q, q 0 , Σ ε , -→) be a TTS. We define the relation

-→ >⊆ Q × (Σ ∪ R ≥0 ) × Q by: -q d -→ > q ′ iff
there is a run ρ = q σ -→ q ′ with Untimed(ρ) = ε and Duration(ρ) = d, -q a -→ > q ′ with a ∈ Σ iff there is a run ρ = q σ -→ q ′ with Untimed(ρ) = a and Duration(ρ) = 0, Definition 2 (Weak Timed Bisimilarity). Let S 1 = (Q 1 , q 1 0 , Σ ε , -→ 1 ) and S 2 = (Q 2 , q 2 0 , Σ ε , -→ 2 ) be two TTS and ≈ be a binary relation over Q 1 × Q 2 . We write q ≈ q ′ for (q, q ′ ) ∈ ≈. ≈ is a weak timed bisimulation relation between S 1 and S 2 if:

-q 1 0 ≈ q 2 0 ; -if q 1 a -→ > 1 q ′ 1 with a ∈ Σ ∪ R ≥0 and q 1 ≈ q 2 then ∃q 2 a -→ > 2 q ′ 2 such that q ′ 1 ≈ q ′ 2 ; conversely if q 2 a -→ > 2 q ′ 2 with a ∈ Σ ∪ R ≥0 and q 1 ≈ q 2 then ∃q 1 a -→ > 1 q ′ 1 such that q ′ 1 ≈ q ′ 2 .
Two TTS S 1 and S 2 are weakly timed bisimilar if there exists a weak timed bisimulation relation between S 1 and S 2 . We write S 1 ≈ W S 2 in this case.

Let C and C ′ be two classes of TTS. The next definition formalises the relative expressiveness of C and C ′ .

Definition 3 (Expressiveness w.r.t. Weak Timed Bisimilarity). The class C is more expressive than C ′ w.r.t. weak timed bisimilarity if for all B ′ ∈ C ′ there is a B ∈ C s.t. B ≈ W B ′ . We write C ′ ≤ W C in this case. If moreover there is a B ∈ C s.t. there is no B ′ ∈ C ′ with B ≈ W B ′ , then C ′ < W C. If both C ′ ≤ W C
and C ≤ W C ′ then C and C ′ are equally expressive w.r.t. weak timed bisimilarity, and we write C ≈ W C ′ .

Time Petri Nets

Time Petri Nets (TPN) were introduced in [START_REF] Merlin | A study of the recoverability of computing systems[END_REF] and extend Petri Nets with timing constraints on the firings of transitions. In a TPN, a time interval is associated with each transition. An implicit clock can then be associated with each enabled transition, and gives the elapsed time since it was last enabled. An enabled transition can be fired if its clock value belongs to the interval of the transition. Furthermore, time cannot progress if time elapsing would result in leaving the interval of a transition. The following definitions formalise these principles.

Definition 4 (Labeled Time Petri Net). A Labeled Time Petri Net N is a tuple (P, T, Σ ε , • (.), (.) • , M 0 , Λ, I) where:

-P is a finite set of places; -T is a finite set of transitions with P ∩ T = ∅; -• (.) ∈ (N P ) T is the backward incidence mapping; (.) • ∈ (N P ) T is the forward incidence mapping; -M 0 ∈ N P is the initial marking;

We also use

• t (resp. t • ) to denote the set of places • t = {p ∈ P | • t(p) > 0} (resp. t • = {p ∈ P | t • (p) > 0}) as it is common is the literature 4 .
The net of figure 1 illustrates the graphical representation of a TPN. Each transition is represented with its label and its interval. For instance transition t 1 has label λ 1 and interval [1, ∞[. Semantics of Time Petri Nets. The semantics of TPNs is given in terms of Timed Transition Systems. A marking M of a TPN is a mapping in N P and M (p) is the number of tokens in place p. A transition t is enabled in a marking M iff M ≥ • t. We denote by En(M ) the set of enabled transitions in M . To decide whether a transition t can be fired, we need to know for how long it has been enabled: if this amount of time lies within the interval I(t), t can actually be fired, otherwise it cannot. On the other hand time can progress only if the enabling duration still belongs to the downward closure of the interval associated with an enabled transition. Let ν ∈ (R ≥0 ) En(M) be a valuation such that each value ν(t) is the time elapsed since transition t was last enabled. A configuration of the TPN N is a pair (M, ν). An admissible configuration of a TPN is a configuration (M, ν) s.t. ∀t ∈ En(M ), ν(t) ∈ I(t) ↓ . We let ADM(N ) be the set of admissible markings.

• • • p1 p2 t1, λ1, [1, +∞[ t2, λ2, [1, 1]
When defining the semantics of a TPN, three kinds of policies must be fixed.

The choice policy concerns the choice of the next event to be fired (scheduled). For TPNs (and also timed automata), this choice is non deterministic (possible alternatives use priorities, probabilities, etc.). The service policy concerns the possibility of simultaneous instances of a same event to occur. In the context of Petri nets, this is formalised by the enabling degree of a transition. Here we adopt the single-server policy (at most one instance of a firing per transition in every state). Our results could be extended to the multiple server at the price of intricate notations.

The memory policy concerns the updating of timing informations when a discrete step occurs. The key issue in the semantics is to define when to reset the clock measuring the time since a transition was last enabled. This can only happen when a transition is fired. We let ↑enabled(t ′ , M, t) ∈ B be true if t ′ is newly enabled by the firing of transition t from marking M , and false otherwise.

Let M be a marking and t ∈ En(M ). The firing of t leads to a new marking

M ′ = M -• t + t • .
↑enabled I (t ′ , M, t) = (t ′ ∈ En(M -• t + t • ) ∧ t ′ ∈ En(M -• t) ∨ (t = t ′ ) (1)
A: The atomic semantics considers that the firing of a transition is obtained by an atomic step. The corresponding mapping ↑ enabled A (t ′ , M, t) is defined by:

↑enabled A (t ′ , M, t) = (t ′ ∈ En(M -• t+t • ))∧ t ′ ∈ En(M )∨(t = t ′ ) (2)
P A: The persistent atomic semantics considers that the firing of a transition is also obtained by an atomic step. The difference with the A semantics in only on the value of ↑enabled A (t ′ , M, t) when t = t ′ . The fired transition here is handled as any other one:

↑enabled P A (t ′ , M, t) = t ′ ∈ En(M -• t + t • ) ∧ (t ′ ∈ En(M )) (3)
Note that we have the relation: (3) ⇒ (2) ⇒ (1) but as we shall see on the example this does not imply any inclusion relation between the different behaviours.

We now define the semantics of a TPN, which is parameterised by the choice of the ↑enabled predicate.

Definition 5 (Semantics of TPN). Let s ∈ {I, A, PA}. The s-semantics of a TPN N = (P, T, Σ ε , • (.), (.)

• , M 0 , Λ, I) is a timed transition system S N = (Q, q 0 , T, →) where: Q = ADM(N ), q 0 = (M 0 , 0), and -→∈ Q × (Σ ε ∪ R ≥0 ) × Q consists of the discrete and continuous transition relations:

-∀(M, ν) ∈ ADM(N ), ∀t ∈ En(M ), s.t. ν(t) ∈ I(t)
the discrete transition relation is defined by:

(M, ν) Λ(t) ---→ (M -• t + t • , ν ′ ) where ∀t ∈ En(M -• t + t • ), ν ′ (t) = 0 if ↑enabled s (t ′ , M, t), ν(t) otherwise. -∀(M, ν) ∈ ADM(N ), ∀d ∈ R ≥0 , s.t. ∀t ∈ En(M ), ν(t) + d ∈ I(t) ↓ , the con- tinuous transition relation is defined by: (M, ν) d --→ (M, ν + d)
We simply write (M, ν) w -→ to emphasise that a sequence of transitions w can be fired in S N from (M, ν). If Duration(w) = 0 we say that w is an instantaneous firing sequence. The set of reachable markings of N is Reach

(N ) = {M ∈ N P | ∃(M, ν) | (M 0 , 0) w -→ (M, ν)}. A net is bounded iff there exists an integer B such that ∀M ∈ Reach(N ), ∀p ∈ P, M (p) ≤ B.
We illustrate the different semantics on the example of fig. 1. (M 0 , 0) 1t1t1 ---→ is a run for the PA semantics but not for the other ones since the second firing of t 1 should imply a delay of at least 1 time unit. (M 0 , 0) 1t1t2 ---→ is a run for the PA and A semantics but not for the I semantics since as t 1 consumes (and produces again) the token in place p 2 this should imply a delay of at least 1 time unit before the firing of t 2 . (M 0 , 0) 1t11t2 ----→ is a run for for the I semantics but not for the other ones since after t 1 fires, t 2 is not newly enabled and time cannot elapse before it is either fired or disabled.

The intermediate semantics I, based on [START_REF] Aura | A causal semantics for time Petri nets[END_REF][START_REF] Berthomieu | Modeling and verification of time dependent systems using time Petri nets[END_REF] is the most common one. However, we provide two significant patterns (among other ones) of discrete event systems where the other semantics are more appropriate. Consider the net of figure 2 which models a component whose state is checked by an observer in order to react (by firing transition t). Let us emphasise that this modeling by a loop is standard in the untimed context. This observation does not interfere with the behaviour of the component using A and P A semantics while the I semantics renitialises the delay of transitions t 1 and t 2 . The subnet of fig. 3 models clients waiting for some information to be produced by a source (in 1 time unit) and then sent to every client (in a negligible time). The two firings of t are performed at the same time only with the P A semantics.

The comparison of expressive powers of the three semantics has not been investigated and is the topic of our work.

• • • clients diffusion source t, λ, [1, 1]
Fig. 3. A case where P A is more convenient than A and I: instantaneous multicast 3 Inclusion results for the semantics I, A and P A

Large inclusion results

We now establish inclusion relations between the three semantics for TPNs.

In order to alleviate the figures, transitions are filled in black when their firing interval is [0, 0] and we omit their label when equal to ε. Proposition 1. Let N be a time Petri net with intermediate semantics. There exists a TPN N with atomic semantics which is weakly timed bisimilar to N . The size of N is linear w.r.t. the size of N . Furthermore if N is bounded then N is bounded.

Proof. The construction is quite easy. Let T ′ ⊆ T be the subset of transitions which have at least one input place. The set of places of N is obtained by adding to the set of places of N a new place for each transition t from N : -for all p ∈ P , M (p) = M (p) + Σ t∈T ′ t • (p).M (p t ) -for all t ∈ T \ T ′ , ν(t) = ν(t) and for all t ∈ T ′ ∩ En(M ), ν(t) = ν(t -) if t -is enabled in M and 0 otherwise. The latter case corresponds in N to a newly enabled transition.

P = P ∪ {p t | t ∈ T ′ }.
To prove that R is a bisimulation, we first note that, with the definition above for markings, from any configuration (M , ν), we can reach instantaneously a configuration (M 1 , ν 1 ) such that M 1 (p t ) = 0 for all t ∈ T ′ , with the firing of a (possibly empty) sequence of transitions in T + . Moreover, the relation between valuations implies that (M 1 , ν 1 ) is still equivalent to (M, ν).

Consider now a pair ((M, ν), (M , ν)) ∈ R.

-if (M, ν) t -→ (M ′ , ν ′ ) with t ∈ T ′ , then from the remark above, we first fire a sequence from M to empty all places p t ′ , leading instantaneously to (M 1 , ν 1 ), which is equivalent to (M, ν). Then transition t -can be fired from (M 1 , ν 1 ), immediately followed by t + , leading to (M ′ , ν ′ ), where all places p t are empty again. Moreover, the transitions newly enabled by t + in N are exactly those which were newly enabled by t in N , so that (M ′ , ν ′ )R(M ′ , ν ′ ).

-Conversely, suppose that a transition is fired from (M , ν) in N . If the transition is some t + , then the new configuration (M 1 , ν 1 ) is still equivalent to (M, ν) (as above), thus no move at all is necessary in N .

If (M , ν)

t - -→ (M ′ , ν ′ ), then t can be fired from (M, ν) and the resulting marking is, (M ′ , ν ′ ), equivalent to (M ′ , ν ′ ).

-if (M, ν)

d -→ (M, ν + d),
for some delay d, then again we have to apply the emptying sequence from (M , ν), to reach a configuration (M 1 , ν 1 ) still equivalent to (M, ν), where time can elapse. The relation between ν and ν 1 implies that this is possible, leading to (M 1 , ν 1 + d).

-Conversely, if (M , ν) d -→ (M , ν + d), then all places p t are empty in M , so that the move (M, ν)

d -→ (M, ν + d) is also possible in N .
The cases for a transition t ∈ T \ T ′ are straightforward. Thus R is a bisimulation. Assuming that N is bounded, the boundedness of N is mainly due to the following observation: if a place p t is unbounded in N then any input place of t is unbounded in N .

⊓ ⊔ Proposition 2. Let N be a time Petri net with atomic semantics. There exists a TPN N with persistent atomic semantics which is weakly timed bisimilar to N . The size of N is linear w.r.t. the size of N . Furthermore if N is bounded then N is bounded.

Proof. Here again, the construction is simple. Note that the only difference between the two semantics concerns the question wether a transition t can newly enable itself. With atomic semantics, this is the case as soon as t is enabled in the new marking while with persistent atomic semantics, this is never possible.

In order to ensure that a transition t will be newly enabled if it is enabled in the new marking, we add an input place En + t and an output place En - t to the transition, with an instantaneous loop b t leading back to En + t , once the transition has been fired. The construction is represented in Figure 5, again from left to right. We consider the equivalence relation R which contains all pairs ((M, ν), (M , ν)) such that:

-M (p) = M (p) for all places p in P , and -for a transition t ∈ En(M ), ν(t) = ν(t) if t is enabled in M and 0 otherwise.

Again the latter case corresponds in N to a newly enabled transition.

Like in the previous proposition, the proof is mostly based on the fact that from any configuration (M , ν), we can reach instantaneously a configuration (M 1 , ν 1 ) such that M 1 (En + t ) = 1 for all t, with the firing of a (possibly empty) sequence of transitions b t , and again (M 1 , ν 1 ) still equivalent to (M, ν). Furthermore the claim about boundedness is mainly due to the following invariants M (En + t ) + M (En - t ) = 1. ⊓ ⊔

Strict inclusion

The next proposition shows that the expressive power of TPNs depends on the chosen semantics even in the bounded case.

Proposition 3. There exists a bounded TPN N with persistent atomic semantics such that no TPN (even unbounded) with atomic semantics is bisimilar to N .

Proof. Consider the following bounded TPN with P A semantics composed by a single transition t labeled by ε:

t, ε, [0, 1[
The (observable) behaviour of this net is simply to let the time elapse without reaching 1. Suppose that there is a TPN N with atomic semantics bisimilar to this TPN and let d min be the minimum of the non null upper bounds occuring in the intervals associated with the transitions of N and 0.5 (in fact any value less than 1 would be convenient).

There must be a sequence (M 0 , ν 0 )

d0t1...t k d k --------→ (M, ν) with Σ k i=0 d i = 1 - d min /2 and (M, ν) bisimilar to (∅, 1 -d min /2).
From (M, ν), we fire or disable the transitions enabled at this configuration, which leads to a new configuration (M ′ , ν ′ ) bisimilar to some (∅, 1δ ′ ) with 0 < δ ′ ≤ d min /2. Now since (M ′ , ν ′ ) is bimilar to (∅, 1δ ′ ) there must be a sequence (M ′ , ν ′ ) The double arrow at the end indicates that the place end t is both an input and an output place for the corresponding transition: time cannot progress. Clearly this part of the construction does not apply when the interval is not upper-closed. Now, using the fact that the TPN is bounded, we consider its upper bound B and we associate with each place p a complementary place p such that for any reachable marking M , M (p) = B -M (p). Figure 7 represents a part of the subnet (on the right) for transition t (on the left), where test 1 is the beginning of the test step for what timing updates are required by the firing of t, and M utex ensures that the updates are done (instantaneously) before anything else, as explained further. Fig. 6. Time subnet for interval

d ′ 0 t ′ 1 ...t ′ k ′ d ′ k ′ ---------→ with 0 < Σ k ′ i=0 d ′ i < δ ′ .
I(t) = [a, b] of transition t • • • • • • • p1 p2 t, Λ(t), [a, b] t, Λ(t), [0, +∞[ p1 p2 p2 p1 true [a,b]
M utex test1 Fig. 7. From a transition in N to its translation in N

The remaining part of N is devoted to the test of the other transitions from the original TPN, including t itself. Consider a given transition (say t i ), with again two input places p i 1 and p i 2 . The corresponding subnet consists of 4 modules, one for each case, depending on wether t i can be fired or not before and after t. For this, two additional places are associated with t i : E ti , which contains a token if t i was enabled before the firing of t and N E ti its complementary place. If t i is initially enabled then E ti is initially marked otherwise N E ti is marked. This group of 4 modules has a common input place test i and a common output place test i+1 , which means that the tests are to be executed sequentially (and instantaneously), except for the last one where all outgoing transitions are linked to M utex. These places are not shown in the following figures. Case 1: transition t i is enabled both before and after firing t. To test this case, we use the simple module on the left of Figure 8, where E ti (test before t) and p i 1 and p i 2 (test after t) are input and output places. Case 2: t i is not enabled before but enabled after firing t. The module is very similar to the previous one and is on the right of Figure 8. Note that, in this case only, because of the PA semantics, there must be a reset on the valuation of the transition, which explains why the initial place start ti of the time module for t i is an output place.

Case 3: t i is enabled neither before nor after firing t. To test this, we must find an input place of t i , where the current number of tokens disable t i . Here is the point where the boundedness hypothesis is required. In order to perform this test, we check whether B -• t i (p) + 1 tokens can be removed from a complementary place p. testing subnet is empty, with (M 1 , ν 1 ) still equivalent to (M, ν). The details are omitted.

⊓ ⊔

We conclude this section with the following table summarizing our results.

(

Conclusion

Since the introduction of TPNs, numerous works have lead to verification algorithms, tool development and applications to real-time systems. In this paper, we investigated some questions relative to the expressiveness of three different semantics for TPNs. First, we presented some standard patterns of discrete event systems where the original semantics (i.e. the intermediate one) is not the most appropriate, thus showing that atomic and persistent atomics semantics could be interesting alternatives.

Then we undertook a theoretical analysis of the three semantics w.r.t. weak time bisimilarity. We established a hierarchy between these semantics and proved that the P A semantics is strictly more expressive than the two other ones. Since the last result is due to non upper-closed intervals and is still valid for bounded nets, we focused our study on bounded nets with upper-closed intervals. In this last case, the three semantics are equivalent.

However, even in the restricted case we feel that P A semantics should be preferred to the other ones since the emulation of the other semantics by the P A semantics is natural whereas the reverse emulation is rather involved and yields a TPN whose readability is doubtful. Furthermore most verification techniques are based on the class graph construction which applies with the same complexity for these three semantics.

  where ε is the empty word. B A stands for the set of mappings from A to B. If A is finite and |A| = n, an element of B A is also a vector in B n . The usual operators +, -, < and = are used on vectors of A n with A = N, Q, R and are the point-wise extensions of their counterparts in A. The set B denotes the boolean values {tt, ff} and R ≥0 denotes the set of non negative reals. A valuation ν over a set of variables X is an element of R X ≥0 . For ν ∈ R X ≥0 and d ∈ R ≥0 , ν + d denotes the valuation defined by (ν + d)(x) = ν(x) + d. 0 denotes the valuation s.t. ∀x ∈ X, ν(x) = 0.

Fig. 1 .

 1 Fig. 1. An example of TPN

Fig. 2 .

 2 Fig. 2. A case where P A or A are more convenient than I: observation of a component

Fig. 4 .

 4 Fig. 4. From I to A

Fig. 5 .

 5 Fig. 5. From A to P A

2 Fig. 10 .

 210 Fig. 10. Testing transition ti: case 4 (names of transitions are omitted)

  Three semantics are possible: I: The intermediate semantics (I) considers that the firing of a transition is performed in two steps: consuming the input tokens in • t, and then producing output tokens in t • . The fact that a transition t ′ is newly enabled on the firing of a transition t = t ′ is determined w.r.t. the intermediate marking M -• t. When a transition t is fired it is newly enabled whatever the intermediate marking. We denote by ↑enabled I (t ′ , M, t) the newly enabled predicate in this case. This mapping is defined by:

  Choose the first d ′ i > 0 and let (M * , ν * ) be the state reached before the duration d ′ i . Since time may elapse in this state, all enabled transitions have non null upper bound for their interval, hence these bounds are greater than or equal to d min . Since the transitions have been enabled at or after configuration (M ′ , ν ′ ),we have ∀t, ν * (t) ≤ d min /2δ ′ < d min /2, thus (M * , ν * ) But (M * , ν * ) is bisimilar to (∅, 1δ ′ )which cannot let time elapse for a duration of d min /2. Due to the strict inclusion established by proposition 3, we now restrict our study to bounded TPNs, with upper-closed intervals, i.e. with intervals [a, b], [a, ∞[, ]a, b] or ]a, ∞[. Proposition 4. Let N be a bounded TPN with upper-closed intervals and with persistent atomic semantics. There exists a bounded TPN N with intermediate semantics which is weakly timed bisimilar to N . The size of N is linear w.r.t. the size of N and the logarithm of the bound.Proof. In this case, the construction of N is more involved. Like above, we show how to simulate a transition t equipped with interval [a, b], for a ≤ b (the other cases are similar). We first build a time subnet for t (Figure6below), to simulate time elapsing from the last time t was enabled, until reaching (and staying inside) interval [a, b]. The token is in place start t if the transition is enabled in the initial marking.

	dmin/2 -----→. This is a contradiction.
	A similar proof could be developed with any interval ]a, b[ or [a, b[ instead of
	[0, 1[.
	⊓ ⊔
	4 Equivalence result for bounded TPNs with
	upper-closed intervals

Whether • t (resp. t • ) stands for a vector of (N P ) T or a subset of P will be unambiguously defined by the context.

Fig. 8. Testing transition ti: cases 1 and 2

Fig. 9. Testing transition ti: case 3

Case 4: t i is enabled before but not after firing t. In this case, we have a module (see Figure 10) similar to the one above, except that we must also test for all the different configurations of the time subnet corresponding to t i , to disable the transitions by removing the tokens.

It can be seen in Figure 10 that there is a transition for each pair (p, state), where p is an input place of t i and state may be either the place start ti , the pair of places (in ti , true [ai,bi] ) or the pair (end ti , true [ai,bi] ). Like above, an edge from p to a transition must be labeled with B -• t i (p) + 1 (which is omitted in the figure). Note that the number of states (here 3) depends on the kind of intervals but is bounded by a constant. Thus this construction is still linear w.r.t. the number of input arcs of t.

We consider the equivalence relation R containing all pairs ((M, ν), (M , ν)) such that -M in N is obtained by projection: M (p) = M (p) for each place p ∈ P , -for a transition t in T enabled by M : ν(t) = 0 if the time subnet of t is empty, ν(t) = ν(l t ) if the place start t contains a token, ν(t) = a + ν(u t ) if the place in t contains a token and ν(t) = b if the place end t contains a token. Note that in both latter cases, true [a,b] also contains a token and the transition t can be fired in N .

Also note that if M (start t ) = 1 and ν(u t ) = a, then with instantaneously firing u t , transition t can also be fired. By a development similar to the previous ones, we can show that R is a bisimulation relation. More precisely, the proof is mainly based on emptying sequences from a configuration (M , ν) of N : it is always possible to reach instantaneously a configuration (M 1 , ν 1 ) such that the