Households of couples (aged 20-59 years) sum up to 100% only with households of single-earner (female) and households without earners.

** Percentage points add to the respective number of dual-earner households only with households of women full-time/men part-time and women and men part-time; marginal part-time is defined with less than 15 hours per week; long working time is more than 40 hours a week; it is supposed to be 'high' if at least more than a third is used to work long hours; concentrated among men refers to significant differences in the share of long working hours between men and women.

This chapter looks into the details of female employment and public policies. In the first section, we have focused on the participation of women in Europe's labour market. We have described what are the main trends in each country. To get an idea of women's participation in the labour market, it is not sufficient to look at female employment rates. It is necessary to also consider actual working time. In some countries, while the activity rate is high, women tend to work essentially part-time. This implies different careers and a weaker investment of women in the labour force. Therefore, we have also described female working time in Europe.

In a next stage, we have explored the issue of gender discriminations on the labour market in terms of wages and career development. Finally, we have presented a variety of public policies that affect female employment, specifically focusing on taxation policy. Tax provisions may be such as to discourage the worker with the smallest wage in the couple (unfortunately, often the woman) to take on a paid job or to work more.

In the second section, we have analysed the more specific problem of female employment and family life. First, we have described, for Europe, the trend in mothers' employment according to the number of children. Second, we have studied the gender divide between paid and unpaid work within the household. Finally, we have presented a range of European family policies and their effects on mothers' employment, while trying to compare those policies from one country to another.

Women and labour market in Europe

In this section, we have first drawn the general picture of female employment in Europe from a quantitative point of view : What are the differences across the countries with respect to women's participation in the labour market ? Which European countries have the most active women? What is there to say about their working time?

In a second point, we have approached female employment from a qualitative perspective. As far as working women in Europe are concerned, we have tried to address the following questions: what kind of job do they have? What are their possibilities in terms of career development? Are discriminations between men and women on the labour market still relevant?

The third point introduces the topic of taxation policy and female employment. Which taxation system is most encouraging for female employment? 3.2.1. Female participation in the labour market in Europe

Female labour force

Since the 60's, there has been a strong and persistent growth in female activity in all European countries while male activity rates have stagnated or decreased. This persistent increase in the participation of women is fundamental. In fact, during recession periods, when women were expected to withdraw from the labour market and return home, female activity has kept increasing, and this all through the 80's and 90's. Women no longer constitute the so-called "workers reserve", to use a term of Margaret [START_REF] Maruani | Travail et emploi des femmes, Repères[END_REF], or in other words, precarious workers that are called upon during economic expansion periods and periods of lacking labour supply and removed from the labour market during crises and periods of unemployment. On average, 4 women on 10 were active in Europe in 1996. However, there are a lot of disparities among the different European countries. The following table shows the trend in the participation of women in the labour market across Europe. The activity rate is known to be the ratio of the female active population to the whole active population. .5 -* The source is not the same for the year 2000 which may explain the unusual differences. Source: Eurostat, Labour Force Survey; for the year 2000 : Moreno D., Escobedo A. and Moss P. (2002).

In terms of activity rate, France is part of the leading group composed mainly of Nordic countries: women represent almost half of the active population. Then come the UK, Germany, Austria, Belgium and the Netherlands with an activity rate around the EU average (42,5%). The Mediterranean countries (Italy, Spain and Greece), Ireland and Luxembourg form the last group with a rate below 40%.

Let us now look at the female employment rate, that is to say the ratio between the population of women in employment and the female working-age population. In line with [START_REF] Daly | Periodic Progress Report N°1 of the Thematic Network: "Working and Mothering: Social practices and Social Policies[END_REF], it is possible to distinguish between four clusters of developed countries as far as female employment is concerned. [START_REF] Daly | Periodic Progress Report N°1 of the Thematic Network: "Working and Mothering: Social practices and Social Policies[END_REF] established the following classification:

ü A group with a high female employment rate: Denmark, Finland, Sweden, France, Portugal, Norway, United States, Canada;

ü A group with medium to high activity: United-Kingdom, Australia;

ü A group with medium activity: Belgium, Germany, the Netherlands;

ü A low activity group: Ireland, Italy, Spain.

The following graph shows slight differences in the classification: Note: When comparing the above data with the results of Rubery et alii (2001), major differences were found for Ireland, Portugal and the Netherlands. For these three countries, Rubery et alii (2001) states total female employment rates of 53.24%, 60.36% and 63.39% respectively. These rates are between 2.9 and 6.3 percentage points lower than the rates provided by Care Work. On average, the Care Work statistics lie at about 0.46% higher than the employment rates calculated by Rubery et alii (2001).

Source: Moreno D., Escobedo A. and Moss P. (2002).

Amongst the European countries, labour force participation of working-age women ranged from a high of 72.5% in Denmark to a low of 39.7% in Italy in 2000. In other words, the 60%

Lisbon target for female employment was easily surpassed in Denmark. Besides Denmark, five other Member States met the target in 2000, the UK, Portugal, Finland, Sweden and the Netherlands. Austria was very close. 30 The second group's rate ranged between 50% and 60%

with Germany on top with a rate just below 60%, followed by France and then by Ireland, Belgium and Luxembourg, countries that will have to make a stronger effort in order to reach the European goal. The last group (Italy, Spain and Greece) lags behind with an employment rate of around 40%.

Extent of women's participation in the labour market

Participation rates per se or employment rates are relatively poor guides to the extent of women's involvement in and nature of their relationship to the labour market. How many hours they work each day is a more interesting measure of their actual attachment to the labour market. Only from this point of view is a European target of 60% for female employment relevant. Therefore, the European target, in the vague manner in which it is currently set, seems to be a way too poor indicator to produce information on female employment. Only by integrating a working time component can it be made really useful.

In real life, part-time work can cover a range of very different realities: part-time can be freely chosen; or, one can be forced to reduce one's working hours to part-time (in which case it is a form of unemployment); or, it can be shattered time; or, a form of flexible under-employment;

or, a way to adjust the working time. In the southern European countries, part-time work is not very developed (if women work, they tend to work full-time). In countries such as Germany, the Netherlands, and the UK, part-time work is a way for women to have access to the labour market, but their professional life is interrupted when they have children. On the contrary, in the Scandinavian countries (Sweden, Finland, Denmark), part-time has been very widespread since quite some time now, and it seems perfectly compatible with very high activity rates (close to those of men), and with a strong career continuity. France is a particular case. In France, part-time work developed in a period of economic recession to slow down the rise in unemployment, it was considered as a solution to the unemployment problem. In France, it also seems that part-time work in no way precludes a high activity rate among women and a strong continuity in their careers. Whilst before the economic crisis,

French women where massively entering the labour market and engaging in full-time employment, they now primarily opt for part-time activities. Eventually, in France, the development of "part-time jobs" has had the effect to remove women from the labour market, or, to put it more exactly, to make them reduce their working time.

In the following graph, total employment has been decomposed in full-time and part-time.

Graph 3.2:

Source: Moreno D. Escobedo A. and Moss P. (2002).

In the EU, part-time work has been central to any growth in employment, for both men and women. During the year 2000, 40% of net job growth for women was in part-time work 31 (although this is down on the 1999 trend when part-time jobs made up 47% of net job growth). For men, part-time work plays a weaker role. Just 10% of net job growth was in parttime work 32 .

30 Austria was at 59.64% in 2000 according to Rubery et alii (2001). 31 This corresponds to an increase of 0.7 million part-time jobs compared to 1.0 million full-time jobs. 32 This corresponds to an increase of 0.1 million part-time jobs compared to 1.2 million full-time jobs. Interestingly, when participation rates are standardised by the hours actually worked by women, Portugal and France emerge as leading countries levelling the Scandinavian countries. To clarify the information summarised in the table below (Table 3.2.), let us take

France as an example. For France, the total number of people employed was multiplied by a certain factor to obtain the corresponding number of full-time employed. This number is given in the first column of the table below. To put it in a very simplistic way, we could say that two women in part-time correspond to one FTE although actual computations are a little more subtle. To obtain the numbers in the second column, the same procedure is followed but for employed women. The first two columns are now adjusted for the number of hours actually worked. The third column presents the ratio of the previous two and, thus, corresponds to the actual share of female employment on the French labour market. For

France, this means that women's actual participation rate is 40%.

Quality of female employment in Europe and gender discrimination

Work/life cycle

Female employment, unlike male's, has a stop and start character with 'dips' in the years of early family formation. Nations are at their closest and labour force participation in general at its highest for the cohort of women in their mid-twenties. Viewed over the life cycle, the pattern of female employment varies a geat deal across countries. In France, Finland, and Sweden, the participation pattern of women in the labour market remains close to that of men over their whole life: it first increases and then decreases with age ( inverse U shape). In She also set forward six profiles for women's labour and therefore six corresponding clusters of countries :

Graph 3.6:

Source: Daly M. (1998.)

1 st cluster
Finland, Portugal, the USA and Canada are in the high/high part of the graph. Therefore, gender equality seems to be the greatest in these countries.

nd cluster

The Scandinavian pattern is the one displayed by Denmark, Norway and Sweden. Female participation rates are high but women's actual presence in the labour market is lower since many are engaged in part-time work. The public strategy in these countries is to encourage short full-times and long part-times.

France and Austria tend towards the Scandinavian pattern although they have far fewer women in part-time employment.

rd cluster

Australia, the UK and Germany fit yet another pattern of lower participation (largely due to a greater volume of part-time work) and a more interrupted career.

th cluster

Women in Ireland and Luxembourg participate rarely in the labour force and, if they do, their career path seems to be highly discontinuous, child births leading to a total drop-out among older women and to temporary career interruptions for the younger generations. Part-time employment is not very widespread.

Beside these trends, it is interesting to take a glance at the unemployment rates in the female labour force. Women are still more affected by unemployment than are men. Their employment seems to be more sensitive to the economic cycle and they usually have more precarious jobs than men which means that when the economy gets sluggy they are banned from the labour market quite easily. Table 3.4. shows that the unemployment rate of women aged between 15 and 64 years is higher than that of men in the same age category. The less educated people are, the higher is the level of unemployment, regardless of sex. But whatever the level of education, women have a higher and sometimes much higher unemployment rate (except for Austria and Ireland).

Gender discrimination and wage gap

Despite the fact that female employment opportunities have risen, 'informal' barriers still prevent their formal equality from transforming in substantive equality in the labour force.

Those 'informal' barriers are, for example, the lack of good-quality care services, glass ceilings, etc. As long as women remain pressured to scale back their economic activities and as long as care-giving remains predominantly women's work, there will continue to be a gender division of labour. However, the above discussion on part-time work has shown that gender differentiation is increasingly based on working time as women concentrate on parttime employment and domestic work and men engage in full-time jobs.

The fact that women mostly work part-time also explains existing gender differences in annual earnings even where part-timers are not discriminated against on the basis of their status. Of course, occupational sex segregation, and particularly the wage penalty associated to "women's jobs", also plays a major role in creating those gender differences in annual earnings.

Expected earnings are an important determinant of the decision to return to work. Women's average earnings are lower than men's in all European countries, and sometimes by a large amount, though the difference has been tending to decline slightly in most countries. Thanks to the OECD's earnings database, several interesting results with respect to the gender wage gap in developed countries and its evolution have been put forward (OECD, 2001, p.139).

The gender wage gap is defined as the ratio of difference between median male earnings and median female earnings, to median male earnings. The largest gap was observed in Portugal, Spain and the UK, and the smallest one was found in France, Belgium and Denmark. But it must be noted that the comparison does not take into account the differences in types of job performed by men and women. As it is stressed in the OECD study, the wage gap tends to be larger between the earnings of fathers and mothers of young children than between men and women in general. Mothers are more likely to hold part-time jobs in which wages are lower, whereas fathers of young children tend to work more than other men and earn higher wages.

Some employers may discriminate against mothers because they expect them to have a lower commitment to their jobs. Indeed, according to the OECD study (2001), in some Anglo-Saxon countries, the wages of mothers with children are found to be lower than those of women (non-mothers) working in similar jobs [START_REF] Harkness | The Family Gap in Pay : Evidence from seven Industrialized Countries[END_REF]. However, Datta, [START_REF] Gupta | Children and Career Interruptions: The Family Gap in Denmark[END_REF] found that this does not apply to Denmark. They have suggested the reason for this to be the generous parental/maternity leave benefits, taken up by virtually all Danish mothers. As a result, potential discrimination against mothers has been transferred to women in general.

We have seen that the proportion of part-time work is high in the Nordic countries (except Finland). The longer women work, the more they are paid, and the smaller the earnings gap tends to be. Therefore, occupational sex segregation is high since women remain the primary caregivers. In recent years, these countries have witnessed a decline in part-time as more and more young women engage in full-time employment but no change has occurred as to the high feminisation of the take-up rate of maternity and parental leaves.

According to the 2001 Employment Guidelines33 , the Council made a number of recommendations to individual member States for the year 2001. And once again Denmark, Finland, and Sweden were requested to consider ways of reducing levels of occupational and sector-level segregation. For example, in Denmark, the wage gap between men and women is still large: in 1999, this gap was 17% (EIRO, 2000). A study published by The National Institute for Social Research in May 2000 showed that of those 17%, 12% cannot be explained by differences in educational attainment, work experience, age or competence (EIRO, 2000). One of the most common explanations for this wage gap is the long and repeated interruptions of women's activity in the form of maternity and parental leaves.

By contrast, French women tend to have rather continuous employment patterns mainly by working long and well-protected part-times. While part-time employment is equally widespread in the rest of Continental Europe, women's labour continuity tends to be low and wage gaps high.

Portugal forms an exception to the Mediterranean group with a high female participation rate.

Part-time employment is not very popular and women enjoy continuous employment patterns.

In the other Mediterranean countries, female participation is much lower and thus the Council has called for improved female employment rates but it stays a fact that women either not work at all or in full-time jobs. The result is a small earnings gap.

British part-time jobs involve only short hours and are badly protected. Women are likely to drop out of the work force for childbearing and -rearing thus jeopardising the continuity of their work pattern. While the wage gap remains on the small side in the UK, women are exposed to relatively high occupational sex segregation. That is why the Council commissioned the UK (and Germany) once again to take action to reduce the gender pay gap and improve the quality of child-care provision.

While Ireland is part of the low activity group, part-time employment is more widespread than in the other countries of that category.Nevertheless, the wage gap is very high even among

Irish women working on a full-time basis.

In sum, each European country has still some way to go in strengthening and encouraging full-time female employment. Income taxation is often accused of being counter-productive in this sense.

Public policies and female employment in Europe

Of all public policies, taxation is one very important tool to either promote or discourage female employment (family policy is another means but it will be developed in the next section).

The general trend in taxation systems today is away from privileging the one-earner households towards individualised taxation of spouses/partners. Tax systems also tend to become more neutral towards employment on the part of the second spouse/partner.

Taxation of dual-earner couples may take various forms: separate, joint or quotient. The basic difference is whether the amount of taxes due is calculated on the basis of the sum of the two earned incomes or on the basis of the two incomes separately. In a system of joint taxation, regardless of its exact form, there is, in principle, a reduced incentive for the partner with the lower earnings (or lower potential income) to increase his/her earnings, since they will be subject to higher marginal tax rates, at least within a progressive taxation system. On the other hand, it is only by using a system of joint taxation that it is possible to achieve equality in the taxation of two couples with the same total income but a different distribution of that income.

However, the type of taxation is only part of the story. The effects of family-related tax reliefs and benefits can be of considerable importance in terms of incentives for the partners to work.

The right-hand side of the Table 3.5. illustrates this point.

In absence of any tax benefit system, the figures of the three last column of Table 3.5. would be respectively: 100, 140 and 200. In this sense, the closer to 140 the figure in the second column, the more the system is supportive of the second worker taking on a part-time job.

However, for European countries, this figure is usually low.

The figures on the right-hand side of Table 3.5. thus demonstrate that the type of taxation system does not necessarily determine the level of incentives in the sense used here: on average, the figures for countries where there is separate taxation are similar to those for countries with joint taxation.

Indeed, [START_REF] Dingeldey | European tax systems and their impact on family employment patterns[END_REF] has recently confirmed the non-existence of a clear "shaping effect" of tax systems on family employment patterns. She stressed the importance of other factors in determining these patterns: labour market regulation, family policy, childcare supply, families' preferences, and so on. The co-variation between the overall orientation of the tax system and female labour force participation is much weaker than the co-variation mentioned above between public caring policies and women's employment.

While [START_REF] Gustafsson | Tax regimes and labour market performance[END_REF] has confirmed that units of taxation defined in the various national systems have a major influence on the division of labour between marriage partners, it does not seem to be this simple. In splitting systems with a high rate of progressiveness, not only is one partner encouraged to withdraw from the labour market but the attractiveness of the parttime option relative to full-time employment is also enhanced.

The full-time/part-time combination as a family pattern of labour force participation attracts tax relief in all countries, particularly in those with splitting systems or systems with a transferable tax allowance with a high rate of progressiveness, so in those countries in which sole earners also benefit from considerable tax relief.

The majority of tax reliefs for the sole earner model are more or less in line with the principle of progressiveness. The 'family subsidy' increases with income. However, in countries with only 'formally individualised' systems, like Denmark, Belgium, and the Netherlands, sole earner households and households with a full-time/part-time combination and low income benefit most from transferable tax allowances.

Female employment and family life

Two main results concerning mothers' employment have derived from the labour supply theory. First, "having children" affects women's decision of whether or not to engage in paid work and, if yes, for how many hours. A child increases the value of time spent out of the labour market and affects women's preferences. The presence of a child also decreases the real wage women obtain in the market given the cost of child-care. Second, governments can take specific measures to decrease the pressure on female labour supply generated by "having children": child-care support, parental leaves, and so on. More recently, some studies have even analysed the relationship between school time (opening hours) and mothers' employment.

In this section, we specifically focus on mother's employment. We have first turned to the employment rate of women according to the number of children they have and we have tried to point out the differences that exist across European countries. Source: Eurostat, Labour Force Survey, 1996.

In France, for example, the majority of mothers does not interrupt their career when having children. Therefore, the activity rates for women with one or two children and for childless women are very close. The sociological gap seems to appear after the third child, but this obviously concerns only a minority of women. Nevertheless, since the beginning of the 80's, a growth in the activity rate of mothers' of three children and more has been observed. This trend seems to be set in all European countries but with different degrees and at different rhythms [START_REF] Maruani | Travail et emploi des femmes, Repères[END_REF].

The family pattern of paid and unpaid work in Europe

Dingeldey (2001) studied national profiles of family patterns of labour market behaviour in 1996 on the basis of the ELFS 34 . Rather than concentrating on women's and men's employment situation separately, she provided interesting information on the distribution of couples (with and without children) in dual and single earner situations. She also examined couples in terms of each partner's working hours. Table 3.7. summarises her results. [START_REF] Becker | A Theory of the Allocation of Time[END_REF] and [START_REF] Gronau | Leisure, Home Production and Work : the Theory of the Allocation of Time revisited[END_REF] explained this in terms of their assumed comparative advantages in both spheres. These explanations are now rejected by most economists [START_REF] Brusteel | La rationalité de la non-spécialisation dans les ménages: une rupture avec la théorie beckeérienne[END_REF]. In more recent models (bargaining models and so one), individual earnings are set as a factor determining within-household allocation of time. Lower labour market wages for women lead to lower incentives for women to engage in paid employment, which in turn leads to relatively high levels on unpaid work, and again, lower wages. Besides, employers tend to expect that mothers will invest relatively heavily in their child-care role.

This results in mothers and potential mothers having to meet tougher promotion standards than fathers, again a tendency towards confirming the traditional pattern of specialisation [START_REF] Lommerud | Mommy Tracks and Public Policy: On Selffulfilling Prophecies and Gender Gaps in promotion[END_REF].

A team of researchers at the University of Essex have made more comparable and harmonised a number of budget time surveys that cover 12 OECD countries [START_REF] Fisher | Technical details of Time Use Studies[END_REF]. The data shed information on the time use of individuals in households of a given type. The main results are the following: ü Full-time working mothers spend just over twice as much time, on average, compared to fathers, on child-care (housewives spend over three times as much); ü Full-time working mothers spend about twice as much time on other unpaid work compared to fathers (housewives spend around two and half times as much); ü On average, the total number of hours spent on paid and unpaid work is highest for women in full-time work, at around 10 hours per day, one hour more than the average for all men. Women working part-time have an average total of around nine and a half hours.

The time men spend on child-care has tended to increase in all the countries. However, these figures apply only to men in couples, and, thus, exaggerate the increase in the amount of child-care that is taken on by men. An increasing proportion of children are in lone-parent families headed by women. They often see little of their fathers (Dex, 1999).

International comparisons of time budget data must be made with considerable caution.

Nevertheless, it appears that Canadian and Swedish men contribute the most to unpaid household work, though still performing less than their spouses.

Family policies and female employment

As the last section has shown, the presence of children constitutes a slow-down factor for female activity. The reconciliation of family and professional life is and remains a women's problem. Family policy is a means to help mothers to keep working while they have children, even it is not sufficient on its own: a growing participation of men in unpaid work would be a much more efficient way to support female employment.

Family policies have many different components, all directed at two main objectives. The first one is to cover the cost of having children in order to maintain the living standard of families with children relative to those without. With respect to this objective, the most relevant component of family policy are cash transfers such as family allowances. We have chosen not to develop this aspect of family policy here because it does not directly concern mothers' employment. Transfers in kind, such as child-care provision, constitute a more important aspect of family policy from our point of view. These transfers in kind are likely to have a significant impact on mothers' employment, especially since the 'burden' of children is still mainly carried by women. First, we have compared the different European child-care systems in terms of convenience, generosity and quality. Second, we have focused on maternity and parental leaves that are fundamental in shaping mothers' attachment to the labour market.

However, there remain a lot of different aspects of family policy that we have not approached in this chapter, either because they do not directly affect female employment or because it is just too onerous to develop each and every public measure implemented in each of the countries.

Child-care provisions

Female labour is highly influenced by public services. Indeed, the effects of child-care can be analysed in two ways. First, the presence of children affects mothers' preferences with respect to non-market time and market time. The more satisfactory the provisions for childcare, the more the preferences of mothers for time spent at home relative to time spent at work areweak, ceteris paribus [START_REF] Blau | The Economics of Women and Men, and Work[END_REF]. Second, children affect the budget constraints of mothers [START_REF] Connelly | The Effect of Child Care Costs on Married Women's Labor Force Participation[END_REF]). The cost of child-care can be considered as a tax on mothers' hourly wage. A high child-care cost will have the same effect as a decrease in the net wage and will, therefore, result in a decrease of mothers' employment and working time.

It follows that, in theory, every improvement in child-care conditions and prices will favour female employment. [START_REF] Schmidt | Gendered labour Force Participation : Political and Social Determinants of Female and Male rates of Labor Force Participation in 18 OECD countries[END_REF] has shown that the increase in child-care possibilities from 1961 to 1980 is positively correlated to the increase in employment rates of women in 18 OECD countries.

The attitude of governments towards families varies across countries and thus, public support appears to be quite different across the European arena. For example, in the UK, the choice of having children is considered a private one, one which has to be made by all parents, in general. As a result, family support is mainly targeted at poor or single parents. In other countries, the role of the government in providing support to families is, traditionally, much larger. When children are considered a public matter, public policies are supposed to cover the cost of children regardless of family income. In France, the "Duty of the Nation" towards families is embedded in the French Constitution. Gornick et alii (1997) compare the family policies of 14 OECD countries in order to construct an aggregated indicator. The high diversity across countries implies that it is difficult to conclude on the intensity of public support regarding mothers' employment. A composite indicator is, nevertheless, a means to measure each country's effort.

The provision of public services reflects the "dual-earner" gender ideology. Nordic countries are the most evident adherents to this ideology. Their official gender-equality policies led them to support dual-earner families and female employment. In practice, this ideology has been translated into extensive public care services. Possibly, an exception can be made for France but in the rest of Continental Europe, official policies, while mentioning gender equality, fail to increase the level and quality of female employment. Child care is largely left to the private sector to provide. Nonetheless, general support to families with children tends to be high.

Child-care arrangements can take one of four main forms: ü Group care in child-care centres (nursery, kindergarden, play-school) ü Residential care, including specialist services such as care for disabled children ü Childminders, based in their own home, looking after one or more children.

ü Care provided by a carer who is not a family-member but frequently lives in with the family.

Generally, countries that have a high level of public funding, spend a large part of it to the first two types of child-care. Most governments provide special support for children who are considered to be at risk of abuse or neglect and for children in lone-parent families. Some governments offer arrangements for child-care in others ways. For example, Austria and France require home-based childminders to be registered. In France, the Allocation de Garde d'Enfant à Domicile (AGED) supports parents who employ a person at home to take care of their children (50% to 75% of the employer's charges can be deducted from taxable income). Table 3.8. shows the proportion of children under three who benefit from one of the four formal child-care arrangements described above. Nordic countries (Denmark and Sweden) have the highest proportion (above 40%). A much lower proportion of children benefit from formal provisions in Southern European countries. For children above three years old, the coverage tends to be much higher, reaching 90% and more in several countries. The share of above 3s in formal arrangements is also more uniformly distributed across countries. Nordic countries and other European countries mainly rely on publicly financed centres for this age group, whilst in Ireland, the UK, and the Netherlands formal child-care is primarily provided by the private or commercial sector.

OECD studies tend to explain the lower provision of child-care services for children under 3 in terms of of the higher cost involved in caring for very young children. Adema (2002) shows that the average proportion of children covered in countries relying mainly on public expenditure is only slightly higher than in those countries that rely on private expenditure (some non-European countries, Canada and the US, with mainly private funding also have a high rate of coverage).

Finally, a number of countries (Denmark, France, Finland...) provide subsidies to parents looking after their own children at home. The application rules differ a lot from one country to another. These systems are closer to parental leaves and are, therefore, also developed in the following section. However, unlike parental leave provisions, systems relying on such subsidies do not necessarily guarantee the right to return to one's previous job.

Firms' contribution to the reconciliation of work and family life is also crucial. We have not developed this point but it is important to draw attention to it. Family-friendly arrangements by firms can take several forms: leave from work for family reasons, changes in the work arrangement for family reasons, practical help with child-care and provision of training and information. * For some countries, the covergae was not given by a number of hours but by "full time" or "part time", then we have considered that a full time day was 8 hours a day and a part time day 4 hours.

Source: OECD, Employment Outlook, 2001;[START_REF] Adema | An Overview of Benefits that foster the reconciliation of Work and Family life in OECD countries[END_REF] In conclusion, three groups of countries can be distinguished according to Gornick et alii (1997). France, the Nordic countries (Finland, Denmark and Sweden), Belgium and Italy offer the highest support to mothers' employment. The second group is composed of Luxembourg, Canada, Germany, the Netherlands and Norway, followed by the third group that is mainly made up of Anglo-Saxon countries (UK, US and Australia).

women's employment; but if the leaves are extended, they are associated with stereotypical roles for women and with reductions in their relative wages.

The cases of France and Denmark are illustrations of how parental leave can have a negative effect on female employment. In France, the parental leave, associated with a benefit called the Allocation Parentale d'Education (APE), has had a strong negative effect on the employment of women in precarious situations. It strongly encourages them to leave the labour market because the expected low wage of these women is very close to the benefit they would receive if they went on leave (APE). If they opt for the APE, they are allowed to stay inactive for 5 years. Women who were at the fringe of the labour market before taking up the allowance or whose job was a precarious one thus got stuck in an inactivity trap which it was impossible to get out of by themselves in order to re-enter the labour market. Viewed from such an angle, the full-rate APE can be considered as a kind of mother's wage but with a temporary character since it only applies until the youngest child reaches the age of 3. The APE has had positive consequences for most women who have taken it up at a partial rate or who have chosen the full-rate but could fall back on a secure job. Usually, these women are skilled. However, it has had the strong perverse effect of removing from the labour market those women who, generally speaking, were unskilled. Once the three years of entitlement to the APE had expired, these women were generally no longer able to find a job because of the training opportunities foregone and their inactivity during too long a period. In conclusion, it has encouraged unskilled women to return home and has strengthened the disparity and inequality, that were already strong to start with, between skilled and unskilled women.

We have already described the situation in Denmark. A very recent law, adopted in March 2002, has increased the parental leave from 32 weeks to 52 weeks. This seems to be a good measure in order to facilitate the reconciliation of family and working life. However, in fact, this measure could result in an increase in gender discrimination. Indeed, the law contains two specifications that could have huge consequences: 14 weeks of the 20 extra weeks must be taken up by the woman and of the 32 weeks of parental leave that already existed, fathers are no longer obliged to take almost 4 weeks, but instead, only 2. Therefore, all the law has really done is increase the maternity leave. This often implies a long withdrawal of mothers from the labour market, a situation that could translate into a penalty for their future career and working life. This is a women's problem since most often it is women who take up the parental leave rather than men : in 2001, according to the Danish Statistics Institute, only None but with job protection during the leave. 35 Working parents have the right to take an additional paid parental leave (family leave) to care for an ill child under age 16 at home. For one-child families, this leave corresponds to 6 days a year, families with 2 children are entitled to 8 days and in case there are three or more children 10 days can be taken. Furthermore, working parents have the right to take 4 days a year to visit the child's school. In other words, 100% when included in taxable income. 37 However, Irish parents are entitled to 3 days of paid family or emergency leave. 38 In Sweden, parents are entitled to take 60 days of paid leave a year to care for an ill child or in case the usual caregiver of the child is ill.

Conclusion

Female employment is still too low in many European countries. Having children and caring for them remains a major problem for women, far more than for men. Reconciling work and family life has also kept a predominantly female character.

The international perspective has led to a number of findings of policy relevance. The first is that, in countries with relatively well-developed systems of work/family reconciliation policies, women tend to have higher employment rates in their thirties. Both formal child-care coverage of young children and paid maternity leave policies appear important from this viewpoint. For all countries, it is an issue of major importance to increase the possibilities for child-care and to improve their quality in order to generate a strong effect on mothers' employment. A lot of European countries still have a long way to go if they want to reach the EU target by 2010. However, even if they succeed, there will still remain a lot to be done.

Indeed, the EU target lacks sense in that it is not sufficient to increase female employment rates if women keep having mainly part-time, less paid, or less protected jobs. If such remains the scenario, then gender equality will not be achieved. A target such as the European one must, therefore, be specified more precisely by integrating a range of "job quality" measures, related to working time, wage, promotion, etc.

Finally, sociological and psychological factors have a great role to play: as long as women are considered the most competent to fulfil domestic tasks, caring for children included, and as long as women consider themselves to be so, it will be difficult for them to conquer a place equal to that of men on the labour market. Nevertheless, public policies can significantly improve their situation and their independence in many ways.

  Germany and the Netherlands, the pattern of female participation shows a peak at the left: female employment is high until they are 25 years of age and then decreases progressively. In the UK, the female employment pattern has a U shape with an increase in employment up to the age of 25 approximately, and then it falls during the childrearing period to increase again, once children are older. The last two patterns are very different from men's. Female behaviour on the labour market varies in reaction to the presence of children. Across Europe, Daly (1998) has distinguished three patterns of age-specific female labour force activity : a) Continuous participation rates over the life-cycle (inverted U-curve, typical male pattern): Scandinavian countries; b) More or less permanent exit from the labour market during child-bearing years (lefthanded peak): Ireland, Luxembourg, Spain; c) High but declining participation for the older age-groups: France, Portugal, UK and Germany (with a sharper decline in the latter two countries), Greece and Italy (stability over the life course but a lower overall rate and a faster decline from the mid-40s onwards).

  The Netherlands and Belgium are characterised by moderate participation rates and relatively high discontinuity. The volume and concentration of part-time work among parents in the Netherlands distinguishes this country from any other.5 th clusterSpain, Greece and Italy have a profile of low but relatively continuous employment. The female employment pattern in those countries is bipolar: continuous full-time employment versus inactivity. Part-time work is rare.

Full

  includes both domestic chores and care work. Traditionally, mothers tend to spend more time than fathers on child-care and unpaid household work.

Table 3 .1: Evolution of the proportion of women in the active population between 1983 and 1996

 3

				(in %)
		1983	1987	1991	1992	1993	1996	2000*
	Germany	39.1	39.5	40.9	42.6	42.5	43.1	43.8
	Austria	-	-	-	-	-	43.4	44
	Belgium	36.9	38.5	40.0	40.6	41.2	41.5	43.4
	Denmark	45.5	46.0	46.6	46.8	46.9	45.7	46.6
	Spain	-	32.9	35.3	36.1	36.6	38.5	37.3
	Finland	-	-	-	-	-	47.8	47.4
	France	41.9	43.3	44.3	44.6	45.0	45.5	44.8
	Greece	34.1	35.8	36.8	37.0	37.3	38.7	37.7
	Ireland	31.1	32.8	34.1	35.4	36.6	38.4	40.7
	Italy	34.1	35.6	37.1	36.8	36.7	37.8	36.8
	Luxembourg	33.5	35.2	35.6	37.2	36.4	37.0	39.4
	Netherlands	33.8	37.6	39.5	40.4	40.6	41.8	42.8
	Portugal		41.8	43.8	44.4	44.7	45.2	45.1
	Sweden	-	-	-	-	-	47.7	47.5

Graph 3.1: Employment rate of working age women (2000)

	90 80 70 60 50	52,2 70,3	72,5 81,4	73,6 73,6 58,2 42,1 40,6 70,2 69,1 55	77,7 53,9	68,8 39,7	75,4 50,3	63,7 82,8	77 60,2	63,8 65,5 81 71,8	70,5 74,3	65,4 79,4	women
	40													men
	30
	20
	10
	0
		B DK D EL E	F IRL I	L NL A P FIN S UK

Decomposition of total employment in full-time and part-time jobs (2000)

	The Netherlands springs out with 41.2% of all jobs being part-time in 2000. This trend seems
	to affect particularly women. In 2000, the Netherlands witnessed a growth of female part-time
	work of 144,000 and a drop in the number of full-time jobs by 12,000. To quote Plantenga
	"this trend suggests that the one-and-a-half earner model has become firmly established into
	Dutch society and there are no reasons to expect that this will change very rapidly" (Rubery
	et alii, 2001). Furthermore, Scandinavian countries (except Finland) tend to have a high
	proportion of part-time work in contrast to Mediterranean countries with small part-time
	sectors. Liberal nations, Ireland excepted, tend towards the Scandinavian model with more
	than one fifth of all employment being organised on a part-time basis. Continental Europe is
	somewhere in between although the proportion of part-time is on the rise there.
	The graph below shows how female employment is decomposed in full-time and part-time in
	Europe in 2000:
	Graph 3.3:
	120
	80 100	79,3 78,3 80,6	95,4 91,8	83,1 83,2	91,2 88,7		83	89,3 87,8	77,2 75,1
	60											58,8				full-time
	40											41,2				part-time
	20	20,7 21,7 19,4	4,6	8,2	16,9 16,8	8,8	11,3		17	10,7 12,2	22,8 24,9
	0
		B	DK	D		EL	E	F	IRL	I	L	NL	A	P	FIN	S	UK
											Source: Moreno D. Escobedo A. and Moss P. (2002).
	Part-time work is a very important form and component of female employment. The sex
	imbalance among part-time workers is the largest in Continental Europe. A first phase of
	rapid and steep growth in part-time work among women occurred between the 1950s and the
	1970s. The second phase was one of more modest growth (apart from the Netherlands) and
	began in the late 1970s. Finally, some parts of Europe (Denmark, Norway and Sweden) have
	experienced a decline in part-time recently while in others the growth in part-time jobs has
	remained steady or has increased even further. However, Southern European countries, as

Decomposition of total female employment in full- time and part-time jobs (2000)

	well as Finland have not witnessed the general growth in part-time employment since the
	1970s. Denmark, Sweden, the Netherlands, and the UK are high part-time work countries.
	Beneath these variations lies a further set of differences in the hours worked (long, medium or
	short part-time). Information on the number of hours worked is presented in the following
	graph:
	Graph 3.4:
	100		92,1	82,8	82,6		83,6 83,1 Source: Moreno D., Escobedo A. and Moss P. (2002).
	80 60 40	60,1 39,9	64,8 62,1 35,2 37,9	69 69,3 31 30,7		74 26	70,6 29,4	67 33	63,7 36,3	55,5 44,5	full-time part-time
	20		7,9	17,2	17,4			16,4 16,9
	0
		B DK D EL E F IRL I	L NL A P FIN S UK

Average usual weekly hours of work of full-time and part-time employees (2000)

	50														43,6
	40	38,5 39,3 40,1 40,9 40,6 38,9 39,9 38,6 39,8 39	40,1 40,3 39,3 40
	30 20	22,5	19,9	18	21,4	18	23,1	18,6	24	21,3	18,8	22	20 20,8	22,8	18,4	full-time part-time
	10
	0
		B DK D EL E F IRL I	L NL A P FIN S UK

Table 3 .2: Total employment FTE* (a) Female employment FTE* (b) (b)/(a)

 3

	Belgium	3444,2	1299,6	38%
	Denmark	2066,5	871,3	42%
	Germany	32539,7	12077,5	37%
	Greece	3983,9	1436,2	36%
	Spain	13169,3	4574	35%
	France	20920,1	8412,1	40%
	Ireland	1546,7	538,7	35%
	Italy	20272,3	6968,3	34%
	Luxembourg	171,5	61,1	36%
	Netherlands	5705,7	1848,3	32%
	Austria	3366,8	1284,8	38%
	Portugal	4531,2	1952,8	43%
	Finland	1929,7	849,8	44%
	Sweden	3338,2	1420	43%
	UK	20567,4	7970,1	39%
	* in thousands of people		Source: Moreno D., Escobedo A. and Moss P. (2002).
	The graph below shows the full-time equivalent female employment rates. With this measure,
	only three EU member states meet the 60% target: Denmark, Finland and Sweden.

Graph 3.5: Female FTE Employment Rates

	70
	60
	50
	40
	%
	30
	20
	10
	0	AU	BE	GE	DK	SP	FI	FR	GR	IRL	I	LUX	NL	PT	SW	UK	EU15
	FTE1997 51,3	40,5	45,2	59,7	30,3	55,5	46,1	37,8	39,3	34,3	41,3	36,6	53,1	56,7	48	42,6
	FTE1998	51	40,9	45	59,8	31,5	56,4	46,6	38,6	41	35	41,2	38,3	54,5	56,4	48,3	43,1
	FTE1999	51	43,25 45,8	62,1	33,8	60,2	47,1	38,9	43,6	35,7	43,5	40	56,1	58,5	49,2	44,3
	FTE2000	0	44,79 46,1	62,2	36,6	60,5	48,7	40	45,2	36,7	44,6	40,1	57,1	60,2	49,7	45,3

Source: Rubery J. Grimshaw D.,

[START_REF] Rubery | Gender Equality and the European Employment Strategy: An Evaluation of the National Action Plans for Employment[END_REF]

.

Table 3 .

 3 3.(Rubery et alii, 2001) groups the EU member states according to divergent trends in headcount employment rates. When the headcount measure is used instead of the FTE measure, then four countries have reached or even surpassed both European targets, the target of 70% for total employment and that of 60% for female employment. These countries are Denmark, the Netherlands, Sweden and the United Kingdom. Two more are quickly moving closer to the targets, Spain and Ireland. While Finland has no more catching up to do in terms of female employment, efforts are still to be made in order for total employment to rise to the targeted level. In a similar position to that of Finland, but lagging further behind in terms of the general employment target, are Austria and Portugal. The remaining countries have not yet reached either one of both European targets. Belgium and Luxembourg are getting there rather quickly as far as the target for female employment is concerned but are moving at a

much slower pace towards meeting the second target. Germany, France, Greece, and Italy still have a long way to go before reaching either one of the targets.

Table 3 .3: Divergent paths in meeting Lisbon targets (headcount measure)

 3

	Female 60% target
	At or above	Quickly closing	Slowly closing
	target	gap	gap

Source:

[START_REF] Rubery | Gender Equality and the European Employment Strategy: An Evaluation of the National Action Plans for Employment[END_REF]

.

Table 3 .4: Unemployment rate in Europe by sex in 2000 (people aged 15-64) and by level of education in 1999 (people aged 25-64)

 3

	Country			Men			Women
		all	0	1	2	All	0	1	2
	Austria	4.8	8.0	3.4	1.8	4.6	6.0	4.0	2.2
	Belgium	5.3	10.0 4.6	2.4	8.3	15.6	8.3	3.9
	Denmark	4.0	6.8	3.3	2.6	5.0	7.2	5.1	3.5
	Finland	9.2	12.0 9.3	3.3	10.6	14.4	9.8	5.9
	France	8.5	14.1 7.2	5.3	12.0	16.7	12.0	7.1
	Germany	7.7	17.7 8.4	4.4	8.7	14.1	9.4	5.8
	Greece	7.5	5.5	6.6	5.3	16.9	13.7	17.3	10.3

Source: OECD, Employment

Outlook, 2001.

Employment rates of women with and without young children

  Then we have attempted to analyse the gender share within households of paid and unpaid work. The gendered division of work might explain the weak employment rate of mothers because of the double charge

	Graph 3.8:
	Graph 3.7:
	%	EU Employment Rates With and Without Young Children, 1992-2000
		100
		90
		0
		80 AU BE GE	SP	FR GR	IT	LU NL	PT UK EU
	No Child 1999 80,8 74,6 79,6 53,5 73,6 53,4 57,1 74,6 81,2 75,3 83 73,4
	70 No Child 2000 79,8 76,5 80,8 56,8 75 53,6 58,4 74,8 81,9 76,1 72,2 72,6
	Child 0-6 1999 67,3 69,1 52,3 42,5 57,5 49,1 46,2 49,4 61,4 72,4 56,4 54,1
	60 Child 0-6 2000 68,2 68,9 53,7 45,3 59,1 49,9 46,6 56	64	75	0	55
	Graph 3.9:	50
		40
			1992	1993	1994	1995	1996	1997		1998	1999	2000
	men without children	87,32	85,87 84,94	85,19	85,1	85,13 85,85	86,36 87,72
	men with young children	91,48	89,84	89,3	90,27 89,89	90,39 90,98	91,68 92,91
	women without children	70,18	69,87 70,05	70,56 71,12	71,59 72,51	73,49 72,61
	women with young children 48,06	47,96 48,15	49,65 50,36	51,27 52,26	54,17 55,05
	all without children	78,43	77,57 77,28	77,69 77,93	78,22 79,07	79,82 79,96
	all with young children	69	68,11 67,92	69,11	69,3	69,98	70,7	72,07 73,18
			Source: Rubery J. Grimshaw D., Smith M. and Figueiredo H. (2001).

(work and family) they have to face. In the third part of this last section, we have presented the family policies in European countries in order to show how they can help or discourage mothers to work.

3.3.1. Mothers' employment in Europe

De-segregation of workforce groups by parenthood reveals important disparities in the working patterns of men and women. Across the EU, men with children are more likely to work than men without children, but the reverse is true for women, leading to very wide gender gaps among parents of young children. The impact of children's presence on female employment rates is high in Germany, Luxembourg and the Netherlands, but smaller in Portugal and Greece, despite these two countries displaying widely different patterns of female employment rates.

Employment Gender Gap of women with and without young children %

		AU	BE	GE	SP	FR	GR	IT	LU	NL	PT	UK	EU
	No Child	8,96	13,12	5	35,73 10,01 33,08 33,84 19,29	8,22	17,22	2,67	12,86
	No Child	9,66	12,21	4,95	33,12	9,23	32,27 32,18 21,47	9,2	15,3	16,54 15,11
	Child 0-6 27,39 23,96 38,89 49,89 33,04 46,77 46,16 46,62 33,18 22,56 33,35 37,51
	Child 0-6 26,54 25,55 37,75 48,11 32,66 46,41	46,8	40,82 30,53 20,79	0	37,86

Source: Rubery J. Grimshaw D.,

[START_REF] Rubery | Gender Equality and the European Employment Strategy: An Evaluation of the National Action Plans for Employment[END_REF]

.

Table 3 .

 3 6. shows the activity rate of mothers when the number of children is taken into account.

Table 3 .6: Activity rates of women between 25 and 49 years old according to the number of children under 15 years old in the EU, in 1996 in %*

 3

	Country	0	1 child	2 children	3 children and more
	Germany	84.5	74.1	60.9	42.6
	Austria	81.9	76.5	66.6	57.8
	Belgium	74.7	75.5	77.0	55.6
	Spain	58.5	56.1	52.2	43.8
	Finland	87.3	84.1	82.1	65.8
	France	85.6	82.8	75.0	50.1
	Greece	58.9	57.9	55.6	46.8
	Ireland	72.6	56.6	52.2	40.0
	Italy	59.2	56.6	48.2	36.9
	Luxembourg	70.1	55.6	42.9	29.1
	Netherlands	79.2	65.5	61.3	48.6
	Portugal	78.2	80.2	77.0	59.9
	UK	86.5	73.1	67.0	48.8

* Data are not available for Sweden and Denmark.

Table 3 .7: National profiles of family patterns of labour market behaviour in 1996 Types Dominance of the dual-earner model Tendency towards egalitarian patterns of labour market behaviour Dominance of the male breadwinner model Continuation in modernised form

 3

	Countries	DK	SWE	P	D	F	A	GB	B	NL	D
		('94)			(East)						(West)

E

Household

types (as % of all households of couples/of all households of couples with children (younger than 15 years)* Dual-earner Single earner (male) Dual-earner with children

	68.9	74.3	60.4	60.0	58.6	59.5	67.2	56.0	56.4	55.8	31.1
	-	21.1	21.1	23.4	28.5	28.2	19.6	31.9	32.3	31.4	52.2
	-	71.1	67.1	64.0	57.3	60.6	61.0	61.6	52.4	50.8	32.7

Table 3 .8: Summary indicators of formal child-care coverage Proportion of young children using formal child-care arrangements Daily Hours of child-care coverage* Country

 3

		Year Aged under 3	Aged 3 to	0 to 3 years	3 to 7 years
				mandatory
				school age
	Denmark		64	91	10 to 12	8
	Finland		22	66	8 to 10	4 to 10
	Sweden		48	80	5	6
	Greece		3	46	na	na
	Italy		6	95	9 to 11	8
	Portugal		12	75	4 to 11	8
	Spain		5	84	na	na
	Ireland		38	56	na	na
	UK		34	60	na	5 to 8
	Austria		4	68	8	8
	Germany		10	78	9	8 to 9
	Netherlands		6	98	8	5 to 6
	Belgium		30	97	8	4
	France		29	99	8	8
	Luxembourg	-	-	-	8	4 to 6
	USA		54	70	na	na
	Note : Data were provided by national authorities.

Table 3 .9: Leave arrangements in Europe

 3

	Countries	Maternity leave	Replacement rate	Paternity leave	Parental leave	Parental leave allowance
		(weeks)	(in % of earnings)
	Austria	16 (8+8)*	100%	None	Per family: 24 months	Flat-rate (13.48 € per day), paid until child is 18
						months if only mother takes leave, until 24th month if
						father takes some leave as well.
	Finland	18 (6+12)	70%	6-12 days (paid as for	Per family: 26 weeks + 10	Paid as for maternity leave; low flat-rate if child not in
			(except for high earners)	maternity leave)	weeks for multiple births;	publicly funded child care (252 €/month + 84€/month
					followed by care leave (child	for each child under 3 + FIM 50.5 €/month for each
					rearing leave) until child is 3	child over 3 + supplement by local authorities of 168
					years	€/month on average)
	Greece	16 (8+8)	70% if the woman	1 days for men in the	3.5 months/parent	none 35
			worked more than 200	private sector
			days in the 2 years
			preceding birth (in fact
			this two-year period
			ends 3 months before
			birth)
			means-tested cash
			benefit if not
	Luxembourg	16 (8+8)**	100%	None	6 months/parent	One parent can opt for a flat-rate benefit (412.5
						€/month for 22 months) without guarantee of re-
						employment or for a flat-rate benefit (1.5 €/month net
						of tax for 6 months) with guarantee of re-employment
	Spain	16 (6+6)***	100% (up to a	2 days (paid at 100%	Per family: 36 months by
			maximum)	of earnings)	mother OR father

  Per family: benefit at 80% of earnings (100% for civil servants) during 360 days and flat-rate benefit of SEK 6,5 €/day for 90 days; an upper limit exists around 72 €/day or SEK 290315 €/year ; the last 3 months are unpaid but job-protected.38

	Belgium	15 (1+8)	82% first month, then	Private sector: 3 days	3 months/parent (in addition,	Flat-rate benefit of 496 € on average (during the
			75% (up to a maximum)	(paid at 100% of	career breaks of 3 to 12	career break: flat-rate benefit of 297 €/month on
				earnings); public	months up to a total of 5	average)
				sector: 4 days (paid	years can be taken)
				at 100% of earnings)
	France	16-26	84% (not taxed) 36	3 days	Per family: 36 months by	None if parents only have one child, flat-rate income-
		(6+10/20)****			mother OR father	tested benefit of FF 3,024/month if they have two or
						more children
	Ireland	14 ( + 4)***** 70% (not taxed)	None	14 weeks/parent	None 37
	Netherlands	16 (4-6 + 10-12) 100% (up to a	None	6 months/parent only if	None
			maximum)		parents continue to work
					part-time (20 hours)
			lower benefit for
			unemployed mothers
	Sweden	14 (6+8) see: note Sweden	10 working days	18 months/parent plus an
				(paid at 80% of	extra 6 months per child in
				earnings)	the event of multiple births;
					an extra 30 days must be
					taken by father or else they
					are lost
	Denmark	18 (4+14)	Flat-rate benefit =	10 days (paid as for	Per family: 10 weeks	Parental leave: (paid as for maternity leave); Child-
			unemployment benefit	maternity leave)	Parental leave + 2 weeks for	Care leave: flat rate benefit of 60% of unemployment
			(about DKK 2,846/week		the father; 13 weeks Child-	benefit (about 230 €/week in 2000) + possible
			in 2000)		Care leave for each child	supplement by local authorities up to maximum of
					under the age of 8 extended	4712 €/year******
					to 26 weeks if child is under
					1 or recently adopted;
	Germany	14 (6+8)*	100%	None	per family: 36 months by	Flat-rate income-tested benefit of 307 €00/month until
					mother OR father	child is 2 if household income is under a certain level;
						some Länder pay benefits for at least 6 months during
	36

an extra four weeks after birth for multiple or premature births ** plus an extra four weeks for multiple births *** plus an extra 2 weeks for multiple births, 4 weeks of the leave are transferable to the father **** 16 weeks for first and second order births and 26 weeks for third or later order birth; first child: plus an extra 2 weeks for twins, plus an extra 12 weeks for triplets; second child: plus an extra 12 weeks for multiple births, plus another 2 weeks for multiple births if mother already has two or more children. ***** plus an extra 4 weeks if the mother requests but not paid ****** children under 3 cannot at the same time be in publicly funded child care, children aged 3 to 8 may attend on a part-time basis. note Sweden: 60 days leave before birth if woman cannot continue ordinary job and cannot be transferred to alternative duties. Maternity allowance covers 50 days and parental leave allowance the other 10 days. Alternatively, women can take up to 60 days of parental leave before birth paid at 80% of earnings.

  30% of earnings for the 6 months of parental leave that can be taken before the child reaches 3 years of age. The rest of the leave has to be taken before the child's eighth birthday and is unpaid except for lowincome families.39

						the child's third year.
	Italy	5 months (2+3) 80%	None	10 months if only mother
					takes leave; 11 months if
					father at least takes 3 months
					(more than 2 months in a
					row)
	Portugal	6 months ( ?+60	100%	None	6 months/parent; in the case	None (a benefit is paid during the adoption leave) 40
		days)	(not taxable)		of the birth of a third or
					higher order child, 2 years
					can be taken when there are
					two children and 3 years
					when there are 3; 60 days
					adoption leave until the child
					is 3
	UK	18 weeks if not 1	Flat-rate payment; 90%	None	13 weeks/parent/child but	None
		year with same	earnings for 6 weeks and		max. 4 weeks/year
		employer; 40 weeks	flat-rate payment for up
		(11+29) if 1 year	to a further 12 weeks
		with same
		employer

* plus

More specifically, in the context of the guidelines relative to Pillar 4 "Strengthening equal opportunities for women and men".

In Italy, parents can take unlimited unpaid job-protected sick leave to care for an ill child under 3.

The most generous countries specifically target families with pre-school children. Conversely, Anglo-Saxon countries are least generous for these families. Their systems inadequately protect mothers' employment during the maternity period, and during the first months after birth, they offer only limited access to public child-care.

Within each group, a classification of countries can be made. France offers the largest support to families with preschool-aged children thanks to a rule that says children can enter school (école maternelle) at 3 years of age. However, French support for infants (children under 3 years of age) is still limited. In Finland, the situation is inverse: support goes mainly to infants, not to preschool-aged children.

Maternity and parental leave policies

A European Union directive mandating a paid 14-week maternity leave was adopted as a health and safety measure in 1992 and a directive mandating a three-month parental leave was enacted in 1998. The critical cross-national difference in policy is philosophical, and has to do with the extent to which policy is designed • To support family work and child-rearing and to create an incentive for women to leave the labour force when children are very young; • To facilitate women's work outside the home and help reconcile work and family life, by protecting and promoting the well-being of children while their parent(s) are in the labour force.

In sum, maternity and parental leave policies provide for almost a year of fully job-protected leave and are targeted on parents with strong prior labour force attachment. Payments received during this leave are close to full wage replacements. These policies also guarantee a place for children in good-quality, affordable, out-of-home care. Therefore, they appear to be supportive of both "parental choice" and child well-being. On the other hand, policies providing more extended leaves, but associated with low-level benefits, limited job protection, and with insufficient places for toddlers in care, are likely to create an incentive for women (in particular, wives) to leave the workforce and remain at home. It follows that the duration of the leave and the level of the benefit received during the leave are both key issues. Research suggests that there have been no negative consequences of leave policies for women or for employers where short and intermediate leaves are concerned; but there may be negative consequences for women who take extended leaves (e.g. three years) in particular, if they go on multiple sequential breaks. Parental leaves are usually associated with increases in 5000 men were on a parental leave against 44 000 women. On average, in 1999, men took up only 2.2 weeks of parental leave (4% of their right) against 44.8 weeks taken up by women (including the maternity leave). It follows that such generalisation of parental leave among women reinforces employers' beliefs that women constitute an unstable labour force. Gornick et alii (1997) have observed some changes in the relative score of two countries: Italy and Norway. In Norway, by the end of the 80's, maternity leaves were generous but were associated with weak child-care support, poor public investments to the benefit of preschoolaged children, and a high age for compulsory school attendance to begin at. Italy offered a different mix of support to families : weak support to families with very young children but more generous support for those with preschool-aged children. From the end the maternity leave until the child reaches the age of 3, Italian mothers face strong barriers to enter the labour market. Table 3.9. summarises the European members' maternity, paternity and parental leave schemes at the end of the 1990s.


domain: spi math info phys

text:
Xavier Pucel

Stéphanie Roussel

Louise Travé-Massuyès
email: louise@laas.fr

Valentin Bouziat

Meta-diagnosis via preference relaxation for state trackability

In autonomous systems, planning and decision making rely on the estimation of the system state across time, i.e. state tracking. In this work, a preference model is used to provide non ambiguous estimates at each time point. However, this strategy can lead to dead-ends. Our goal is to anticipate dead-ends at design time and to blame root cause preferences, so that these preferences can be revised. To do so, we present the preference-based state estimation approach and we apply a consistency-based meta-diagnosis strategy based on preference relaxation. We evaluate our approach on a robotic functional architecture benchmark.

Introduction

For autonomous systems, state tracking is a critical task because it strongly influences decision making, which is essential to the life of the system. It provides the means to diagnose faults and to react to the various hazards that can affect the system.

In this paper, we focus on discrete event systems [START_REF] Zaytoon | Overview of fault diagnosis methods for discrete event systems[END_REF] in which states are Boolean variable assignments and transitions are propositional logic formulae. When the system state is partially observable, the number of candidate state estimates may quickly become too large to be usable. In embedded or distributed systems, memory and communication limitations may also become a problem, even with symbolic representation techniques such as in [START_REF] Torta | An on-line approach to the computation and presentation of preferred diagnoses for dynamic systems[END_REF]. These limitations lead us to propose a single-state estimation strategy that retains only one state out of the set of candidate estimates at each time step, as in [START_REF] Bouziat | Preferential discrete model-based diagnosis for intermittent and permanent faults[END_REF]. Although this can be seen as an extreme strategy, it is efficient to feed the decision system with a clear input and it is consistent with main stream works that select a limited number of best candidates according to some preference criterion, for example probabilities [START_REF] Williams | A model-based approach to reactive self-configuring systems[END_REF][START_REF] Kurien | Back to the future for consistency-based trajectory tracking[END_REF]. However, the more we limit the number of estimates, the more we may be confronted with the problem of dead-ends, i.e. observations proving that previous estimates were wrong and evidencing no continuation of the estimated trajectory. Because backtracking [START_REF] Kurien | Back to the future for consistency-based trajectory tracking[END_REF] is not a viable solution in regard to real time constraints, some approaches ( [START_REF] Bouziat | Single state trackability of discrete event systems[END_REF][START_REF] Roussel | Model-based synthesis of incremental and correct estimators for discrete event systems[END_REF]) build single-state estimators that are guaranteed to avoid dead-ends, making the system single-state trackable. However, this is not always possible. In this case, there are two options: to refer to multi-estimator strategies as proposed in [START_REF] Coquand | Dead-end free single state multiestimators for DES -the 2-estimator case[END_REF] or to detect and diagnose dead-ends at design time and to identify which part(s) of the estimator should be modified to circumvent the dead-end.

In this work, we focus on the second option that can be qualified as a meta-diagnosis strategy. Following the approach of [START_REF] Pralet | Diagnosis of intermittent faults with conditional preferences[END_REF][START_REF] Pucel | Intermittent fault diagnosis as discrete signal estimation: Trackability analysis[END_REF][START_REF] Bouziat | Preferential discrete model-based diagnosis for intermittent and permanent faults[END_REF], a single-state estimator is defined by a totally ordered conditional preference model [START_REF] Boutilier | Preference-based constrained optimization with CP-nets[END_REF], which allows to describe conditions for estimating truth values of state variables. Diagnosing a dead-end means identifying a set of preferences that, if modified appropriately, circumvent the dead-end. The approach implements a consistency-based diagnosis approach based on preference relaxation. This paper resumes the work of a short two-pages paper presented as a poster in [START_REF] Bouziat | Preference-based fault estimation in autonomous robots: Incompleteness and meta-diagnosis[END_REF]. It provides the details of the meta-diagnosis approach and the results of a set of experiments used in the validation phase.

The paper is structured as follows. In Section 2, we formally define the process of incremental single-state estimation and dead-ends. In Section 3, we define the semantic for relaxing conditional preferences and describe a meta-diagnosis strategy to circumvent dead-ends based on relaxing preferences. In Section 4, we present a robotic functional architecture used to test our approach and provide the results in Section 5.

We then conclude and discuss future work in Section 6.

Incremental Single State Estimation

Preliminary notations. For a variable set V, an assignment over V is a function that assigns a truth value (true, false) to each variable of V. An assignment over V is classically extended to assign a value to formulae whose scope is in V.

For a variable set V and a variable v ∈ V, v (resp. v ) denotes the assignment in which v is assigned true (resp. false). For two assignments x and y on the variable sets X and Y, x .y is the assignment on X ∪ Y such that ∀x ∈ X, xy(x) = x (x) and ∀y ∈ Y, xy(y) = y(y).

For two sets of variables X and Y such that Y ⊆ X, and an assignment x on X, the projection of x on Y is denoted x ↓Y such that ∀y ∈ Y, x ↓Y (y) = y(y).

System Definition

In the following, we adapt definitions of previous works [START_REF] Pucel | Intermittent fault diagnosis as discrete signal estimation: Trackability analysis[END_REF][START_REF] Bouziat | Preferential discrete model-based diagnosis for intermittent and permanent faults[END_REF]. We suppose that the system is governed by discrete dynamics where each time step lasts the same duration.

Definition 1 (System).

A system M is a tuple (O, E, S, s 0 , ∆) where:

-0 and E are two finite sets of propositional symbols that represent respectively observable and estimated features of the system; -S is a subset of assignments on 0 ∪ E and represents the states of the system; s 0 ∈ S is the initial state of the system; -∆ ⊆ S × S is the set of transitions of the system.

Without loss of generality, we consider that all the states of the system are reachable from the initial state s 0 . In order to avoid an explicit representation of transitions, ∆ is represented by a set of propositional logic formulae ∆ p that can relate to both the variables of 0 ∪ E, here denoted P and the same variables but referring to the previous time step, denoted P pre . Formally, we define a bijection pre : P → P pre such that for all p ∈ P, pre(p) refers to the variable p of the state at the previous time step. For a state s ∈ S, s pre is the assignment on P pre such that ∀p ∈ P, s pre (pre(p)) = s(p). The set of transitions ∆ is the set {(s, t) ∈ S 2 |s pre .t(∆ p ) = true}.

An observation is a projection of a state on observable symbols O. The set of observations of a system is denoted O.

We define the function cands : S × O → 2 S such that for all states s in S, for all o in O, cands(s, o) represents the set of successors of s that have observation o. Formally, ∀s ∈ S, ∀o ∈ O, cands(s, o) = {t ∈ S|(s, t) ∈ ∆ and t ↓O = o}.

We also define nextObs the function S → 2 O such that nextObs(s) is the set of observations that can be observed just after the system is in state s. Formally, ∀s ∈ S, nextObs(s) = {o ∈ O|cands(s, o) = ∅}.

A state sequence seq is a list (s 0 , s 1 , . . . , s n-1 ) where each s i is a state in S; |seq| = n is the length of the sequence and seq[i] = s i is the ith state in the sequence; last(seq) designates the last state of seq; if s is a state, seq • s is the sequence of length |seq| + 1 that begins with seq and ends with s.

Definition 2 (Language, observation language). The language associated with a system M = (O, E, S, s 0 , ∆) is the set of state sequences accepted by the system and starting with s 0 . Formally

L(M ) = {seq ∈ S + |seq[0] = s 0 and ∀i ∈ [1, |seq| - 1], (seq[i -1], seq[i]) ∈ ∆}.
The observation language is the language accepted by the system projected on the observations. Formally, L obs (M ) = {seq ↓O |seq ∈ L(M )}.

We illustrate this modelling approach on a small model that is a simplified version of the experimental benchmark of Section 4.

Example 1 (System). We consider a simple robot functional architecture with three functions: movement, communication and power supply. The health status of each function is represented by the three respective variables h mv , h com and h pow . Two alarms al mv and al com can be raised when movement and communication fail respectively. We estimate the value of health variables from the sequence of alarms, i.e. O = {al mv , al com }, and

E = {h mv , h com , h pow }. The initial state is s 0 = al mv .al com .h mv .h com .h pow .
∆ is the conjunction of the following formulae :

¬pre(h pow ) → ¬h pow (δ 1 ) ¬h mv → al mv (δ 3 ) ¬h pow → (al mv ∧ al com ) (δ 2 )
¬h com → al com (δ 4 ) It expresses that the fault in the power supply is permanent (δ 1 ), and causes both alarms to be raised (δ 2 ), movement faults cause movement alarms (δ 3 ) and communication faults cause communication alarms (δ 4 ). Note that alarms can also occur without any fault being present (false positives, external perturbations, etc.), and that movement and communication faults are intermittent, i.e. they are independent of their previous values.

At time step 1, let us assume we receive observation o 1 = al mv .al com . We have cands(s 0 , o 1 ) = {al mv .al com .h mv .h com .h pow , al mv .al com .h mv .h com .h pow }.

In the first candidate state, the al mv alarm is explained as a false alarm or caused by some unknown event, while it is explained by a fault in the move module in the second candidate.

Preference-based estimation strategy

We now focus on the estimation part for systems defined above. Since we consider nondeterministic, partially observable systems, there may be several state sequences that explain a given observation sequence. We adopt an incremental approach to select a unique explanation, called an estimation strategy [START_REF] Roussel | Model-based synthesis of incremental and correct estimators for discrete event systems[END_REF].

Definition 3 (Estimation strategy

). An incremental single-state estimation strategy for a system M is a function estim : S × O → S such that for all s in S, for all o in nextObs(s), estim(s, o) represents the estimated state of the system at time step k if it was estimated in state s at time step k -1 and if o is observed at time step k. We impose the estimation strategy to be consistent both across time (i.e. estim is a function) and with the system behaviour (i.e. estim(s, o) belongs to cands(s, o)).

Following [START_REF] Pucel | Intermittent fault diagnosis as discrete signal estimation: Trackability analysis[END_REF][START_REF] Bouziat | Preferential discrete model-based diagnosis for intermittent and permanent faults[END_REF], the estimation strategy is based on a sequence of conditional preferences.

Definition 4 (Conditional preference). Let M be a system. A conditional preference γ on a variable e of E is defined by cond : e ≺ e , where cond is a propositional formula on P ∪ P pre . cond is called γ's condition and e is γ's target.

Informally, cond : e ≺ e expresses the fact that we prefer to estimate e as true if and only if cond is true. Note that it is equivalent to ¬cond : e ≺ e .

From now on, we consider that the set of estimated variables E contains n variables and that these variables are ordered, from e 1 to e n . This order can be seen as the order used to estimate the state at every time step.

Definition 5 (Conditional preference model).

A conditional preference model Γ for a system M is an ordered sequence of preferences (γ 1 , γ 2 , . . . , γ n ) such that γ i = cond i : e i ≺ e i is a conditional preference on e i and that its condition cond i only uses variables from P pre ∪ O ∪ {e j |1 ≤ j < i}.

Informally, if γ i appears before γ j in Γ , then the condition of γ j can depend on the outcome of γ i , but the reverse is forbidden.

A conditional preference model Γ allows to define an estimation strategy estim as presented in Algorithm 1. Let s ∈ S be the state of the system at time step k -1 and o ∈ nextObs(s) the observation received at time step k. We initialise the sets of preferred candidates for estim(s k -1 , o k ) with cands(s k -1 , o k ) (line 1) and the general idea is to remove non-preferred candidates from this set until it is a singleton. To do so, for each preference γ i , we compute the value val of cond i with the assignment s k -1

pre .o k .estTargets where estTargets is the assignment containing truth values for variables e j with j < i (line 3). If there exists a state t in preferredCands such that t(e i ) = val , we apply the preference and consider that estim(s k -1 , ok) = val (line 5) and we remove from preferredCands all states that do not have value val for e i (line 6). If such a t does not exist, it means there is no choice for this preference, i.e. only the negation of val is possible for e i in preferred states (line 7). Note that such an algorithm corresponds to the best transition with respect to a partial order on transitions formally defined in [START_REF] Bouziat | Preferential discrete model-based diagnosis for intermittent and permanent faults[END_REF].

Algorithm 1: PreferredEstimation(M , s k -1 , o k ) Returns a singleton containing the preferred state estim(s k -1 , o k ) preferredCands ← cands(s k -1 , o k ); estTargets ← true 1 for i ← 1 : n do 2 val ← s k -1 pre .o k .estTargets(condi)
Example 2 (Preference model). We consider the system of Example 1 and now define the conditional preference model:

¬pre(al mv ) ∧ al mv ∧ ¬pre(al com ) ∧ al com : h pow ≺ h pow (γ 1 ) al mv ∧ h pow : h mv ≺ h mv (γ 2 ) pre(al com ) ∧ al com ∧ h pow : h com ≺ h com (γ 3 )
If both alarms are raised simultaneously, we blame their common cause, i.e. the power supply (γ 1 ). Otherwise we blame the respective functions (γ 2 and γ 3 ). Moreover, for the communication function, we dismiss the first alarm as noise, and only diagnose a communication fault when the alarm persists during several time step (γ 3 ). Thus, we have estim(s 0 , o 1 ) = al mv .al com .h mv .h com .h pow .

Definition 6 (Estimated sequence). Let M be a system, seq obs be an observation sequence in L obs (M ) and estim an estimation strategy for M based on a preference model Γ . The estimated sequence for seq obs is the state sequence seq ∈ L(M ) such that seq[0] = s 0 and for all

i in [1, | seq| -1], seq[i] = estim( seq[i -1], seq obs [i]).

Dead-end

At some point, the estimator may choose a state sequence different from the one actually taken by the system. This may be an issue when the following conditions happen: a) the system is in state s, the estimator estimates that it is in state ŝ with ŝ = s; b) the system moves to state t and produces observation t ↓O ; c) the set of candidates cands(ŝ, t ↓O ) is empty.

Definition 7 (Dead-end). Let M be a system and estim an estimation strategy based on a preference model Γ . A sequence of observations seq obs • o in L obs (M ) is a deadend if there exists an estimated sequence seq for seq obs and cands(last( seq), o) = ∅.

Example 3 (Dead-end). In the system and the estimation strategy presented in Examples 1 and 2, the sequence of observations (al mv .al com , al mv .al com , al mv .al com ) is a dead-end. In fact, at time step 1, two faults occur simultaneously in the movement and communication functions in the system, causing both alarms to activate. The estimator explains it with a fault in the battery, due to preference (γ 1 ), which constitutes a divergence between the real system and the diagnosis. At time step 2, the faults disappear in the real system, causing both alarms to stop and the estimator cannot explain this observation as the fault in the battery is permanent.

Meta-diagnosis via consistency-based diagnosis of preferences

While dead-ends can be eliminated by modifying the system, we aim at eliminating them by modifying the estimation strategy defined by the the preference model. Indeed, some dead-ends can be circumvented by modifying the preference conditions in Γ . We address this problem with a consistency-based diagnosis approach [START_REF] Hamscher | Readings in model-based diagnosis[END_REF]. More precisely, given a dead-end and a preference model, we want to know whether the observation sequence associated with the dead-end could be accepted by the estimator if some preferences that we aim to identify were "relaxed". When this is the case, we can indicate to the designer that the dead-end can be avoided by modifying the conditions for the identified subset of preferences. Correcting the preference conditions is left for future work.

Relaxed preference model

In this subsection, we generalise the notion of preference by introducing relaxed preferences. Then, we define the notion of (general) preference, allowing us to define a relaxed preference model. A relaxed preference targeting the variable e in E declares that the valuations of e are incomparable, i.e. there is no preference on the valuations of e in any context.

Definition 8 (Relaxed preference model).

A relaxed preference for e in E has the form e e . A (general) preference for e ∈ E, denoted ϕ, is either a conditional preference cond : e ≺ e or a relaxed preference e e . Given a system M , a conditional preference model Γ = (γ 1 , γ 2 , . . . , γ n ) and a subset of preferences

Ω ⊆ Γ , a relaxed preference model Γ Ω is a sequence of preferences (ϕ 1 , ϕ 2 , . . . , ϕ n ) such that ϕ i = γ i if γ i /
∈ Ω, and ϕ i = e i e i otherwise.

In comparison with the conditional preference model presented previously, a relaxed preference model may result in more than one preferred state in the set of candidates at each time step. Preferred states only differ with respect to variables that are target of relaxed preferences. In Algorithm 1, two lines should be modified to compute preferred states in that case. First, at line 2, the condition of the loop should be "for all i ∈ [1, n] such that φ i is a conditional preference". This implies that candidates states can only be removed by conditional preferences. Then, as variables e j that are target of relaxed preferences are not assigned a truth value in estTargets, line 3 should be replaced by "val ← isSAT(s k -1

pre .o k .estTargets, cond i )", where for an assignment x over variables X, and form a formula whose scope is F, isSAT(x, form) returns true if and only there exists an assignment y over variables X ∪ F such that y ↓X = x and y(form) is true. Therefore, val is true if and only if it is possible to assign targets of relaxed preferences to make condition cond i true. Finally, the returned preferred candidates may be several as some steps are skipped in the loop. Note that, following [START_REF] Bouziat | Preferential discrete model-based diagnosis for intermittent and permanent faults[END_REF], it would be possible to formally define the new order on transitions defined by a relaxed preference model but we do not present it here for conciseness purposes.

Relaxed estimation process

We now generalise the notions of estimation strategy, estimation sequence and deadend. Informally, a relaxed estimation strategy returns a set of preferred candidates instead of a unique preferred candidate. A relaxed estimated sequence is a sequence of states in which any successive states s and t in the sequence are such that t belongs to the set of preferred states of the relaxed estimation strategy for state s. A relaxed deadend is a sequence of observations for which the last observation cannot be explained by any relaxed estimation sequence.

Definition 9 (Relaxed estimation strategy). A relaxed estimation strategy for a system M and a relaxed preference model Γ Ω is a function estim Ω Γ : S × O → 2 S such that for all s in S, for all o in nextObs(s), estim Ω Example 4. Let us consider the dead-end from Example 3: seq obs = (al mv .al com , al mv .al com , al mv .al com ), and let us relax two preferences Ω = {γ 1 , γ 2 }. TOTOTO TOTOTOTO TOTOTOTO The sequence of states (s 0 , s 1 , s 2 ) with s 1 = al mv .al com . h pow .h mv .h com and s 2 = al mv .al com .h pow .h mv .h com is a relaxed estimation sequence for seq obs , which means that seq obs is not a relaxed dead-end for estim Ω Γ .

Consistency-based preference diagnosis

Checking whether a given observation sequence is a relaxed dead-end can be considered as a form of consistency check (due to Proposition 1). Then, searching for the smallest set(s) of preferences that circumvent a dead-end can be done with a classical consistency-based diagnosis algorithm [START_REF] Reiter | A theory of diagnosis from first principles[END_REF].

Definition 12 (Preference Meta-Diagnosis). Let M be a system, estim an estimation strategy based on a conditional preference model Γ , and seq obs a dead-end for this estimation strategy. A set of preferences

Ω ⊆ Γ is a preference meta-diagnosis if seq obs is not a relaxed dead-end for estim Ω Γ . A meta-diagnosis Ω is a minimal meta-diagnosis if and only if there is no meta- diagnosis Ω ⊆ Γ such that Ω ⊂ Ω.
A meta-diagnosis Ω is interpreted as follows: it is possible to modify the conditions for the preferences in Ω so that the diagnoser does not dead-end on the associated observation sequence. It does not guarantee anything with respect to other potential dead-ends. Proposition 1 ensures that if Ω is a meta diagnosis, then all supersets of Ω are meta-diagnoses as well.

Example 5 (Minimal meta-diagnosis). In example 4, we have seen that seq obs was not a relaxed dead-end for estim Ω Γ . This means that Ω = {γ 1 , γ 2 } is a meta-diagnosis for this dead-end. The minimal meta-diagnosis for seq obs is in fact Ω 2 = {γ 1 }. Thus, modifying the condition of γ 1 , can eliminate dead-end seq obs , for example by replacing γ 1 with γ 1 = : h pow ≺ h pow . However it implements a different fault management strategy, that must be validated against the robotic mission requirements.

To check if a sequence of observations seq obs is a relaxed dead-end for a given set of relaxed preferences, it is possible to compute at each time step the set of preferred candidates. To do so, we follow the modification of Algorithm 1 described previously in this section. Then, starting from the initial state, it is possible to compute all relaxed estimation sequences and therefore compute whether seq obs is a relaxed dead-end. Note that this approach is combinatorial in both the number of relaxed preferences and the length of seq obs .

By testing the meta-diagnosis candidates, one can find all the minimal preferences meta-diagnoses. Approaches such as the FASTDIAG algorithm [START_REF] Felfernig | An efficient diagnosis algorithm for inconsistent constraint sets[END_REF] can be used to efficiently browse the meta-diagnosis candidate space.

Experiments

We have experimented our approach on a functional robotic architecture along with a complex dynamic and complex preference model. We consider a system with three functions: movement, communication and power supply. It can raise two alarms al mv and al com (observations of the system) if they are performing poorly. We model the trust we have in each function with variables t mv , t com and t pow , in the sense that as we receive alarms, we lose trust in the system's operational capacity. Variables f mv and f com model external perturbations that impede movement and communication (obstacles, slippery terrain, distance to antenna, etc). f pow represents a loss of voltage in the power supply.

The estimator receives alarms as input, and estimates if each function can be trusted for autonomous operation. Figure 1 details the model and illustrates this architecture.

∆ represents that we trust the movement and communication functions only if we trust the power supply function as well (δ 1 ). Movement (resp. communication) perturbations or low voltage cause a movement (resp. communication) alarm (δ 2 , δ 3 ). Once we have lost trust in the power supply, we never trust it again (δ 4 ). For communication, the alarm is a perfect indicator of our trust in the function (δ 5 ). In the preferences associated with this model, we use the temporal logic operators from PtLTL to express formula compactly. We rely on [START_REF] Havelund | Synthesizing monitors for safety properties[END_REF] to efficiently translate PtLTL formulae into propositional logic formulae. The formula up(f) is true when the formula f was false at the previous time step and is now true. The formula H κ (f) (with κ > 0) is true when the formula f has been true for the last κ time steps, including now. The formula O(f) is true when f has been true at least once in the experiment, including now.

(tmv ∨ tcom) → tpow (δ1) (fmv ∨ fpow) → almv (δ2) (fcom ∨ fpow) → alcom (δ3
Γ is the ordered sequence (γ 1 , γ 2 , γ 3 , γ 4 , γ 5 , γ 6 ) and implements the following strategy. We blame low voltage if and only if both alarms fire simultaneously (γ 1 ), or if low voltage was already blamed at the previous time step. After a continuous period of size κ with low voltage, we lose trust in the power supply 1 (γ 2 ). When a movement alarm not explained by low voltage is on for two time steps, we blame the associated environmental perturbations (γ 3 ). After 4 time steps without alarm we trust the movement function (γ 4 ). Communication alarms not explained by low voltage are blamed on environmental perturbations (γ 5 ). In doubt, we trust the communication function (γ 6 ).

This model has a dead-end when both movement and communication alarms are raised simultaneously, stay on for κ time steps, then al com turns off. We can control the minimal length of the dead-ends with the κ parameter in (γ 2 ), as the shortest dead-end has κ+2 time steps. There are two minimal meta-diagnoses for this dead-end: {γ 1 } and {γ 2 }. If we replace γ 1 by γ 1 = : f pow ≺ f pow or γ 2 by γ 2 = : t pow ≺ t pow , the dead-end is circumvented, although it changes the estimation strategy.

Results

We use Sat4j [START_REF] Le Berre | The Sat4j library, release 2.2[END_REF] as a SAT solver to directly compute if there exists a candidate with a particular variable value (see Algorithm 1 line 5), and to implement the isSAT function described in Section 3.1. Experiments have been conducted on an Intel(R) Xeon(R) CPU E5-2660 v3 @ 2.60GHz processor with 62GiB of RAM, although only around 2Gib were used.

The results depicted in Table 1 show that the computation is very fast for short deadends, but grows exponentially with dead-end length. This is due to the fact that at each time step, for each relaxed preference, there may be two outcomes, which means that in the worst case we need to explore an exponential number of paths to check if a path is a relaxed dead-end.

We consider the performance satisfactory for two reasons. First, memory usage is not a limiting factor, and time is not a constraint during design. Second, long dead-ends are difficult to find, but also difficult to interpret and debug by the designer. When an estimation model becomes too large for maintenance, architectural responses may help dividing it in smaller decentralized models.

Conclusion

In this paper, we present an approach for blaming a dead-end on a set of preferences at design time. It follows a consistency-based meta-diagnosis strategy based on relaxing conditional preferences. We have defined and implemented algorithms with satisfactory performances. For large benchmarks, several approaches can be explored to improve the associated computation time. For instance, we could parallelize the algorithms by dividing the space search among several computation cores. We could also define an intelligent heuristic for finding relevant scenarios faster.

During our experiments we noted that many dead-ends reproduce the same pattern. A perspective is to identify dead-end patterns to represent them more compactly. Another perspective is to find relaxations that circumvent several dead-ends at once. A final perspective is the correction of conditions of preferences belonging to a meta-diagnosis to avoid a dead-end or a set of dead-ends.

3 if

 3 ∃t ∈ preferredCands such that t(ei) = val then 4 estTargets ← estTargets.[ei ← val ] 5 preferredCands ← preferredCands -{t|t(ei) = val } 6 else estTargets ← estTargets.[ei ← val ]

Fig. 1 :

 1 Fig. 1: ∆, Γ and schema of the simple architecture model. Variables labelled with + are observable,estimated. Arrows represent functional dependency.

Table 1 :

 1 Meta-diagnosis computation time (in seconds) against dead-end length.

	κ Dead-end len. Comp. time (s)	κ Dead-end len. Comp. time (s)
	3	5	0.56	8	10	11.57
	4	6	0.84	9	11	23.48
	5	7	1.52	10	12	47.72
	6	8	3.03	11	13	94.26
	7	9	6.07	12	14	186.93

Γ (s, o) represents the set of preferred estimated states of the system at time step k if it was estimated in state s at time step k -1 and if o is observed at time step k.Definition 10 (Relaxed estimation sequence). Let M be a system, seq obs be an observation sequence in L obs (M ) and estim Ω Γ a relaxed estimation strategy for M . A relaxed estimation sequence for seq obs is a state sequence seq ∈ L(M ) such that seq[0] = s 0 and for alli in [1, | seq| -1], seq[i] ∈ estim Ω Γ ( seq[i -1], seq obs [i]). Definition 11 (Relaxed dead-end). Let M be a system and estim Ω Γ a relaxed estimation strategy. A relaxed dead-end is a sequence of observations seq obs • o in L obs (M ) such that for all relaxed estimated sequences seq for seq obs , cands(last( seq), o) = ∅.Intuitively, relaxing preferences allows to increase the set of estimated sequences and therefore might reduce the number of relaxed dead-ends. This is expressed through the following proposition.Proposition 1 (Relaxed dead-end inclusion). Let M be a system and estim Ω1 Γ and estim Ω2 Γ two relaxed estimation strategies such that Ω 1 ⊆ Ω 2 . If seq obs is a relaxed dead-end for estim Ω2 Γ then it is also a relaxed dead-end for estim Ω1 Γ . Proof. We first show that for a state s in S and an observation o in nextObs(s), estim Ω1 Γ (s, o) ⊆ estim Ω2 Γ (s, o), i.e. preferred states are still preferred after relaxing preferences. Then, we can show that a relaxed estimation sequence for seq obs in estim Ω1Γ is also a relaxed estimation sequence for seq obs in estim Ω2 Γ . It follows that if

Given constraint (δ4), preference (γ2) could equivalently be written as Hκ(fpow)) : tpow ≺ tpow . However, we claim that Γ should represent the estimation strategy independently from the system model, for modularity purposes.

Acknowledgements

This project has been supported by ANITI, the "Artificial and Natural Intelligence Toulouse Institute", through the French "Investing for the Future -PIA3" program under the Grant agreement ANR-19-PI3A-0004.

all relaxed estimation sequences for seq obs in estim Ω2 Γ cannot be followed by o, so it is for all relaxed estimation sequences for seq obs in estim Ω1 Γ .