Annie Vinter
email: annie.vinter@u-bourgogne.fr

Patrick Bonin

Pascal Morgan

The severity of the visual impairment and practice matter for drawing ability in children

Keywords:

Background and aims: Astonishing drawing capacities have been reported in children with early visual impairments. However, most of the evidence relies on single case studies. Hitherto, no study has systematically jointly investigated, in these children, the role of (1) the severity of the visual handicap, (2) age and (3) practice in drawing. The study aimed at revealing the specificities of the drawing in children deprived from vision, as compared to children with less severe visual handicap and to sighted children performing under haptic or usual visual control. Method: 148 children aged 6-14 years had to produce 12 drawings of familiar objects. 38 had a severe visual impairment, 41 suffered from low vision, and 69 were sighted children performing either under visual condition or blindfolded under haptic control. Results: Recognizability and other characteristics of the drawings were highly dependent on the child's degree of vision and level of drawing practice, and progressed with chronological age more clearly in the sighted children or those with low vision than in those deprived of vision.

Conclusion:

The study confirmed that all groups showed significant drawing ability, even the group totally deprived of visual experience. Furthermore, the specificities of the drawings produced by visually-impaired children appeared clearly related to their practice and the severity of their visual impairment. This should incite parents and professionals to encourage these children to practice drawing as early as possible.

improvement in performance of children deprived of vision between 6 and 8 years of age, whereas this improvement was noticeable when the 6-year-olds were compared to the children of 9-11 years of age. [START_REF] Bin | Examining explanations for differences in two-dimensional graphic spatial representation of cubes among totally blind subjects[END_REF] revealed a slow and less complex development in the drawing of cubes and cylinders by individuals without vision as compared to sighted persons. The former did not achieve the final stages of development observed in sighted children, except one who benefited from visual experience until 9 years of age and had prior experience with drawing.

The second factor that warrants attention is related to practice, and the role of this factor has been a concern in participants with visual impairments. Beyond differences in levels of performance, D' Angiulli and Maggi (2003) suggested that drawing in children with congenital visual impairments may develop, through practice, similarly to the way it does in sighted children. In their study, after 9 months of practice, 12-year-old visually-impaired children produced drawings comparable to those obtained from 4-to 8-yearold sighted children. In a follow-up analysis, D 'Angiulli, Miller and Callaghan (2008) demonstrated that these drawings displayed a greater structural complexity as practice increased, in much the same way as they do during the development of sighted children. Thus, the present study was aimed to tackle the main following questions: Can drawing ability be observed at a meaningful level when a quite large number of children with early visual impairments and no other associated disorders is studied? Is it possible to identify characteristics that are specific to the drawings produced by children deprived of vision? To what extent is drawing performance in these children impacted by their level of practice of this ability and by their chronological age? The study will establish that it is indeed mainly a function of the severity of their visual handicap and of their level of practice.

Method

Participants

A total of 148 children aged 6-14 years were included in the study, 79 of them were early visually impaired children (44 girls) and 69 were sighted (40 girls). They were generally from middle class families (74.3%). Thirty-eight children were totally deprived of vision (World Health Organization -WHOcategory 5, no light perception, n = 13) or had only minimal light perception (WHO category 4, best visual acuity 1/60, n = 25) from birth for 35 of them or early in infancy (n = 2 onset before 6 months of age, n = 1 onset at around 14 months of age). Forty-one children were suffering from low vision (WHO category 1, moderate low vision, best visual acuity 3/10, n = 19; WHO category 2, severe low vision, best visual acuity 1/10, n = 12; WHO category 3, profound vision loss, best visual acuity 1/20, n = 10), from birth for most of them (n = 38, and n = 3 with an onset before 12 months of age). These 79 visually impaired children were divided into four age groups (6-7 years, 8-9 years, 10-11 years and 12-14 years).

The children with visual impairments did not have any associated disorders of relevance for our study, in particular psychiatric or cognitive disorders. They were enrolled in the school grade that corresponded to their chronological age. Etiologies of the visual impairment were available for 67 out of the 79 children. These took the form of congenital cataracts (n = 10), retinopathy of prematurity (n = 10), Leber's congenital amaurosis (n = 9), congenital glaucoma (n = 9), severe myopia (n = 7), optic nerve atrophy (n = 6), amblyopia (n = 6), retinitis pigmentosa (n = 5), retinoblastoma (n = 5), microphthalmia (n = 5), and anophthalmia (n = 4). We asked the parents to code their child's current level of practice in drawing on a 5-point scale (level 1 = no practice at all, level 2 = rare practice, level 3 = 1 or 2 times a month, level 4 = 1 or 2 times a week, level 5 = almost daily practice). The control groups consisted of 69 typically developing sighted children aged 6-14 years, also divided into 4 age groups. Thirtytwo of them produced their drawings in the usual visual condition, using regular sheets of paper, while 37 children were blindfolded and drew in a haptic condition using the Swedish drawing kit. Their vision was normal or corrected to normal. They were enrolled in the school grade that corresponded to their chronological age. We checked that their level of drawing practice, coded by their parents, did not differ as a function of the drawing condition to which they were assigned, t(67) < 1, p > 0.50. The main characteristics of the different groups of children are summarized in Table 1.

Table 1 revealed two interesting points with regard to drawing practice. Gathering levels 1 and 2, the number of children without noticeable practice was much higher in the group without vision (12 out of 38) than in the two sighted groups (6 out of 69), χ 2 (1) = 9.2, p < 0.01. However, a high proportion of visually impaired children did practice drawing quite regularly (levels 3-5), enabling the investigation of the relationship between drawing performance and practice. Furthermore, gathering levels 3, 4, and 5, practice decreased with age in the two sighted groups (100% in the youngest, 81.25% in the oldest) and in the group with low vision (100% to 78.6%), while it remained stable (or even slightly increased with age) in the group with vision loss (66% to 75%). This difference may very well be due to specific educational interventions in the last group, encouraging these children to draw.

Whether visually impaired or not, the children were tested individually at home or at school. Informed written consent was obtained from the parents of each child. The experiment was conducted in accordance with the tenets of the World Medical Association Declaration of Helsinki on Ethical Principles for Medical Research Involving Human Subjects.

Material

The children either produced their drawings on white sheets of paper (A5 format, one drawing per sheet of paper) with black or colored pencils or used a Swedish raised-line drawing kit. The visually impaired children were free to choose the material they were most familiar with. Twenty-eight of them decided to draw with the Swedish raised-line drawing kit, including almost all (12 out of 13) children with no light perception, and half of those with minimal light perception (13 out of 25). This variable was thus largely redundant with the severity of the visual handicap. The sighted children produced their drawings with imposed material, either white sheets of paper (visual drawing condition) or the Swedish raised-line drawing kit (haptic drawing condition). In the latter case, the drawings were produced using a ballpoint pen on Mylar plastic sheets (21 cm × 14.7 cm), one drawing per plastic sheet placed on a rubberized board (Dycem support, 2 mm height). The pressure of the ballpoint pen on the plastic sheet produced a raised line (between approximately 0.5 and 0.7 mm height), thus providing haptic feedback during drawing execution.

3 left 3 right C4 = 4 C5 = 2 L2 = 2 L3 = 1 L4 = 2 L5 = 1 L1 = 2 L2 = 3 L3 = 1 L4 = 2 L5 = 3 L2 = 2 L4 = 5 L5 = 2 L1 =
= 5 L2 = 2 L3 = 5 L4 = 3 L5 = 1 L3 = 1 L4 = 6 L5 = 3 L3 = 2 L4 = 4 L5 = 1 L2 = 1 L3 = 3 L4 = 4 L5 = 2 L1 = 1 L2 = 2 L3 = 4 L4 = 6 L5 = 1 L4 = 6 L5 = 2 L3 = 3 L4 = 3 L5 = 2 L2 = 1 L3 = 2 L4 = 5 L2 = 1 L3 = 3 L4 = 4 L3 = 1 L4 = 5 L5 = 2 L3 = 3 L4 = 5 L5 = 3 L2 = 2 L3 = 3 L4 = 4 L5 = 1 L2 = 2 L3 = 3 L4 = 3 4

Procedure

The children were comfortably seated in front of a table and were instructed that they had to produce drawings of familiar objects. The experimenter told the children assigned to the haptic condition that they would wear a blindfold and draw in an unusual condition. They were shown how to use the raised-line drawing kit (both while seeing or while wearing the blindfold), and they were given some free time to practice in this condition. The duration of the familiarization period was variable, ranging from a few minutes to around 15 min.

All the children were asked to produce 12 drawings of animate entities (man drawing, animals: dog, fish and bird) and inanimate objects (objects that can be held in the hand: toothbrush, glass and banana; tangible objects that cannot be held in the hand: tree, house and bed; non-manipulable objects, sources of strong tactile sensations: sun and rain2 ). These objects were familiar to the children, and they did not present a complex geometrical structure that would make drawing difficult. They corresponded to a range of tactile experiences due to their diversity of shapes, sizes and textures. Finally, they had a high imageability value according to available databases (however restricted to sighted people, as no database existed for visually-impaired people). All of them rated above 4.5 on the Bonin et al., 5-point scale (2003) for instance.

The order in which the 12 drawings were requested was randomized for each child. For each drawing, the experimenter asked the child: "Can you draw a….?" None of the children refused to draw and no feedback was given to them on their performance.

Data coding

Two raters unaware of the and visual status of the children who produced the drawings worked independently and coded the variables defined below (see Fig. 1 for illustrations). They were trained together to apply the criteria before they started to code the drawings with the small sample of drawings gathered in Fig. 1, and were then tested with different series of 10 randomly extracted drawings until they reached 80% of agreement. Inter-rater reliability for the coding of each variable was examined using Cohen's Kappa coefficient, which varied from 0.74 to 0.85 (ps < 0.05). This indicated a high level of agreement in all cases [START_REF] Landis | The measurement of observer agreement for categorical data[END_REF]. Disagreements were settled before data analysis.

*the degree of recognizability of the drawing (low = 0the coder was unable to tell what was represented in the drawing; medium = 0.5-the coder identified some parts of the drawing and could only guess its identity; high = 1). The raters knew the complete list of the 12 drawings produced, but not the specific title of each one. The percentage of inter-rater agreement was 87%. Illustrations in Fig. 1A; *the presence of positioning errors of elements relative to each other or to the overall shape (e.g., arms anchored in the head; percentage of inter-rater agreement: 85%). This variable was not coded for the banana and the rain. Illustrations in Fig. 1B; *whether the drawing was made up of juxtaposed elements (each mark depicting an element of the represented object) instead of a single integrated shape (percentage of inter-rater agreement: 89%). This variable was not coded for the rain and the banana. Illustrations in Fig. 1C; *whether the correctly positioned elements were disconnected or not (disconnected = 1; mixture = 0.5; connected = 0): the shape was integrated but the segments were not connected to one another (percentage of inter-rater agreement: 81%). This variable was not coded for the rain, the sun and the banana. Illustrations in Fig. 1D; *whether volumes were reproduced with sticks instead of closed curves standing for surfaces (mainly sticks = 1; mixture = 0.5; mainly surfaces = 0, percentage of inter-rater agreement: 80%). This variable was not coded for the rain and the sun. Illustrations in Fig. 1E; *whether contour lines were used to draw the overall shape or profile of the object (percentage of inter-rater agreement: 79%). This variable was not coded for the rain and the sun. Illustrations in Fig. 1F.

Statistical analysis

Insofar as we were not interested in the effect of object category, we averaged the coded variables across the 12 drawings for each child and obtained various scores of mean occurrences ranging from 0 to 100% (or 0 and 1). Nonparametric tests were used because homoscedasticity did not apply in most cases. The Kruskal-Wallis test (H value) was employed to test overall differences due to Visual Status (no vision: no or minimal light perception, low vision, sighted in visual condition, blindfolded sighted in haptic condition), Age (6-7 years, 8-9 years, 10-11 years and 12-14 years) and WHO categories (5 levels). The Mann-Whitney test (U value) enabled performance to be compared when 2 levels were of interest. Correlations between drawing practice and different criteria of drawing performance were computed with the Kendall test (t value).

Results

Degree of recognizability and positioning errors

First of all, it is worth reporting that all groups produced recognizable drawings, including the group of children with no vision who, regardless of age, obtained a mean score of recognizable drawings of 0. As shown in Fig. 2, the children without vision (or at best minimal light perception) produced the drawings that were the most difficult to recognize. The sighted children in the visual condition outperformed those who drew blindfolded under haptic control (Mann-Whitney test, U = 336, N1 = 32, N2 = 37, p < 0.01), with the latter producing drawings that were no easier to recognize than those of the children with low vision (Mann-Whitney test, p > 0.10). The recognizability of the drawings differed significantly as a function of age, regardless of visual status, moving from 0.57 at 6-7 years of age to 0.79 at 12-14 years of age. However, the developmental trend was much less pronounced in the group with no vision. In this latter group, the difference between the levels observed in the two youngest age groups (M = 0.36, SD = 0.28) and that recorded in the two oldest (M = 0.45, SD = 0.30) failed to reach significance (Mann-Whitney test, p > 0.20).

A more fine-grained analysis of the role of the severity of the visual handicap was carried out. A Kruskal-Wallis test run with the WHO category factor revealed significant differences between categories, H(5, N = 148) = 36.4, p < 0.001, and Fig. 3A showed that the greater the visual handicap, the less recognizable the drawings were. Even the difference between the drawings made by the children in WHO category 4 (minimal light perception, M = 0.50, SD = 0.27) and those in WHO category 5 (no light perception, M = 0.23, SD = 0.24) was noticeable, U = 70, N1 = 25, N2 = 13, p < 0.01, whereas the Mann-Whiney test did not indicate any significant differences between the first 3 categories, ps > 0.20.

The correlation between recognizability and drawing practice reached significance in the 38 children with vision loss, Kendall's t = 0.31, p = .005, but failed to yield significance in the 41 children with low vision, t = 0.18, p = .10, as well as in the 69 sighted children, t = 0.12, p = .12.

The production of element positioning errors was a function of Visual Status, H(3, N = 148) = 64.7, p < 0.001, but not of Age, H (3, N = 148) = 5.6, p = .13. As Fig. 4A shows, positioning errors were produced almost solely by the children deprived of vision and the blindfolded sighted children assigned to the haptic task, with no significant difference between these two groups, U = 64, N1 = 38, N2 = 25, p = .56.

As illustrated in Fig. 3B, the occurrence of positioning errors increased as the level of vision decreased, H(5, N = 148) = 41.85, p < 0.001, although the children with moderate and severe low vision made very few positioning errors (M = 2.9%, SD = 1.1%), no more than the sighted children drawing in the visual condition (M = 3.6%, SD = 1.4%). Children who did not perceive light (WHO category 5), introduced significantly more positioning errors in their drawings than the children who could perceive light (WHO category 4), U = 78, N1 = 13, N2 = 25, p < 0.01. Furthermore, these errors just failed to be significantly negatively related to their level of drawing practice, t= -0.18, p = .09.

Juxtapositions, disconnections, stick drawings and contour lines

As clear in Fig. 4B, the children deprived of vision drew the different parts of the depicted entity juxtaposed with one another significantly more often than the other groups did, H(3, N = 148) = 71.2, p < 0.001. This feature appeared to be characteristic of the former group. The percentage of juxtapositions in the blindfolded sighted group (M = 12.6%, SD = 16.2%) was similar to that obtained for the children with profound vision loss (WHO category 3, M = 11.25%, SD = 31.4%), but was much lower than that of children with no or minimal light perception (WHO categories 4 and 5, respectively M = 25.7%, SD = 24.2% and M = 62.4%, SD = 31.1%). Age did not yield significant differences, p = .64. Fig. 3C shows that the degree of severity of the visual handicap induced significant differences, H(5, N = 148) = 62, p < 0.001, with no cases of juxtaposition being reported in the drawings made by the children with severe or moderate low vision, and significantly less occurrences in the children who could perceive light than in those totally deprived of light perception, U = 61.5, N1 = 25, N2 = 13, p = .002. Among these 38 children deprived of vision, those with the greatest drawing practice were those who produced the fewest drawings with juxtaposed elements, t = -0.35, p < 0.001. The presence of disconnections between drawn segments was significantly influenced by the child's visual status, H(3, N = 148) = 86.5, p < 0.001, as indicated in Fig. 4C. This was frequent in the participants with no or minimal light perception, and rare in those with low vision and in sighted children, except when they drew blindfolded in the haptic condition. The WHO visual category factor induced significant differences, H(5, N = 148) = 41.8, p < 0.001, as Fig. 3D shows. The correlation between the level of drawing practice and the frequency of disconnections reached significance in the children without vision, t = -0.45, p < 0.001.

One feature that appeared to be quite characteristic of these latter participants was the occurrence of stick drawings, H(3, N = 148) = 66, p < 0.001. Half of their drawings contained stick representations, while this was the case in only around 15% of the productions of the blindfolded sighted children in the haptic condition (Fig. 4D). This behavior was negatively correlated with their level of drawing practice, t = -0.33, p < 0.01. Fig. 3E illustrates the significant differences related to the WHO visual category factor, H(5, N = 148) = 65, p < 0.001. These children with no or minimal light perception produced significantly more stick drawings than the children with low vision, U = 203, N1 = 38, N2 = 41, p < 0.001. Age did not yield significance, H(3, N = 148) = 1.1, p = .78.

These results are consistent with the fact that the children deprived of vision did not often draw using contour lines, as confirmed by a significant Visual Status effect, H(3, N = 148) = 49.8, p < 0.001, which was complemented by a significant Age effect, H(3, N = 148) = 27, p < 0.001. Fig. 5 presents these findings.

Contour lines increased with age in all the groups, apart from the group deprived of vision where age did not induce significant differences, H(3, N = 48) = 3.6, p = .30. Though the blindfolded haptic group used contour lines more often than the former group, U = 187, N1 = 37, N2 = 38, p < 0.001, they employed this drawing strategy significantly less often than the sighted children in the visual condition, U = 392.5, N1 = 37, N2 = 32, p < 0.05. Moreover, a fall-off in the use of this strategy at 12-14 years of age was observed in the sighted children drawing under haptic control. Fig. 3F shows the significant differences induced by the WHO visual category factor, H(5, N = 148) = 48.2, p < 0.001, and highlights the scarcity of contour lines in the children with no or minimal light perception (categories 4 and 5), in comparison to the children with low vision (categories 1-3), U = 256, N1 = 38, N2 = 41, p < 0.001. p < 0.05. No other significant effects were observed.

Discussion

The literature provides numerous convincing pieces of evidence that testify to unexpected drawing capacities in children with visual impairments (e.g., [START_REF] Kennedy | Drawing and the blind. Pictures to Touch[END_REF]. Though [START_REF] Kennedy | Drawing and the blind. Pictures to Touch[END_REF] tested children who were unpracticed in drawing, the extent to which these clear findings are representative of the general capacities of visually-impaired individuals or, on the contrary, of those who are highly motivated to draw and do so regularly, remains unclear. With a few exceptions (e.g. [START_REF] Millar | Translation rules or visual experience?: Drawing the human figure by blind and sighted children[END_REF], there have been no drawing studies that have enlisted a significant number of visually-impaired participants with different levels of visual handicap or of drawing practice. The aim of the present study was to provide a more analytical overview of drawing abilities in children who had a total or partial lack of vision, in order to relate specific drawing characteristics to various levels of degraded visual information. Furthermore, increasing the number of participants enabled us to study the impact of their level of practice in this skill as well as the chronological age-related development of different drawing criteria in both children deprived of vision and children with low vision. As suggested by Kennedy and collaborators, our study confirmed that vision is not a pre-requisite for drawing. When asked to draw 12 familiar objects, the children who had no light perception and those benefitting from minimal light perception produced drawings that were recognizable, by sighted persons, at a level of 40% on average, given knowledge of 12 options. This percentage is similar to that reported by D' Angiulli and Maggi (2003). As illustrated in Fig. 1, these children used lines to roughly copy the shapes of the objects, corroborating other findings that demonstrate that visual imagery is not necessary in order to activate the spatial representations that can guide drawing (e.g. D 'Angiulli & Maggi, 2003;[START_REF] Heller | Perspective taking, pictures: and the blind[END_REF][START_REF] Vinter | Exploratory procedures of tactile images in visually impaired and blindfolded sighted children: how they relate to their consequent performance in drawing[END_REF]. The spatial understanding about objects that is needed to drive drawing from memory (as required in our task) derives from tactile, auditory or conceptual information, as well as from degraded visual information for children with low vision [START_REF] Cornoldi | Mental imagery in blind people: the role of passive and active visuo-spatial processes[END_REF]. In line with this, the language used by children deprived of vision to describe similar familiar objects appears to refer largely to tactile and auditory attributes [START_REF] Vinter | Verbal definitions of familiar objects in blind children reflect their peculiar perceptual experience[END_REF]. Following [START_REF] Cattaneo | Blind vision[END_REF], it can be argued that the most economical account of such results is to consider that individuals with severe visual impairments mentally consider the shapes of the objects as they progress in their drawing, merging all these sources of information together. However, our experiment also showed that their drawings were less recognizable, for sighted persons, than those produced by the other children. What factors are involved in this issue?

Both visual status and drawing practice are evident in our findings. We will discuss the effect due to the severity of the visual impairment first. To our knowledge, this is the first study to show that as visual acuity falls, drawings as those required in our study become less recognizable, include fewer elements, contain more element positioning errors and disconnections between drawn segments, are more likely to be made up of juxtaposed parts, themselves represented as stick configurations, and use fewer contour lines. Some of these findings have already been reported when visually impaired children were compared to sighted ones, in particular with regard to positioning errors and disconnections [START_REF] Kennedy | Drawing and the blind. Pictures to Touch[END_REF][START_REF] Millar | Translation rules or visual experience?: Drawing the human figure by blind and sighted children[END_REF], but not as a function of a large diversity of degrees of severity of the visual impairment. In most cases, the data reported in Figs. 2 and4 support the hypothesis of a huge gap between drawings of individuals deprived of vision (WHO categories 4 and 5) on the one hand and of individuals with low vision (categories 1 and 2 particularly) on the other hand, of whom drawings did not differ, for most criteria, from those of sighted children. Although the loss of vision did not prevent drawing abilities, the level of severity of the visual impairment determined the degree of recognizability of the produced drawings. This finding is similar to others reporting that participants with low vision outperformed those deprived of vision in spatial tasks [START_REF] Blanco | Haptic exploration and mental estimation of distances on a fictitious island: from mind's eye to mind's hand[END_REF][START_REF] Cattaneo | Blind vision[END_REF]. Even significantly degraded visual information can "make the difference" and, when associated with haptic information, can enable individuals with a visual handicap but residual vision to perform spatial tasks adequately.

The comparison with the blindfolded sighted group that performed the haptic task permits a better understanding of some of the characteristics of the drawings produced by the group with severe visual impairments. Drawing under haptic feedback led the drawers to reduce the number of elements included in their productions, which, moreover, contained disconnected segments and element positioning errors (see also [START_REF] Millar | Translation rules or visual experience?: Drawing the human figure by blind and sighted children[END_REF]. All these features are presumably induced by the lower spatial resolution of tactual compared to visual information (e.g., [START_REF] Dassonville | Haptic localization and the internal representation of the hand in space[END_REF], by the narrow tactile perceptual field even when multiple fingers are used [START_REF] Loomis | Similarity of tactual and visual picture recognition with limited field of view[END_REF]. It may also be due to the sequential nature of this mode of perception, which imposes a considerable load on working memory [START_REF] Revesz | Psychology and art of the blind[END_REF] and affects the way information is mentally reconstructed in individuals with vision loss [START_REF] Cattaneo | Imagery and spatial processes in blindness and visual impairment[END_REF]. Following this line of analysis, the decrease in the use of contour lines at 12-14 years of age in the blindfolded haptic group probably indicates a strategic adaptation to the task constraints, preventing the overloading of working memory as a result of a continuous drawing movement performed under haptic control. Note however that this use of stick drawings could likely decrease if the blindfolded sighted children drawing with the Swedish kit would be extensively trained.

What then remained typical of the drawings of children with no or minimal light perception? Only they produced more than half of their drawings as stick representations of juxtaposed elements, each depicting a particular part of an object (e.g. legs, arms, wings), as if they had in mind a list of elements to include in their drawings [START_REF] Millar | Translation rules or visual experience?: Drawing the human figure by blind and sighted children[END_REF]. As a consequence, the spontaneous use of contour lines was extremely rare. They tended to draw the object's elements piece by piece, which were, nevertheless, sufficiently well combined relative to one another for the resulting overall configuration to be recognized in 40% of cases on average. Note that prior teaching strategy or exposure effects are not likely to account for this result. We are ignorant of tactile illustrated books that contain tactile pictures depicted as stick configurations, and drawing in this way was not taught in the centers our visually impaired children attended.

However, our findings also revealed that these characteristics of their drawings were dependent on their level of practice. The production of visually impaired children with current regular drawing practice were more recognizable, contained more elements, were less piecemeal, with fewer disconnections and stick configurations than those who had rare or null practice. The impact of practicing spatial skills has already been pointed out (e.g., D 'Angiulli & Maggi, 2003;[START_REF] Dulin | The effects of visual experience and training in raised-line materials on the mental spatial imagery of blind persons[END_REF][START_REF] Ittyerah | Hand ability and practice in congenitally blind children[END_REF]. [START_REF] Likova | Drawing enhances cross-modal memory plasticity in the human brain: a case study in a totally blind adult[END_REF] provided impressive illustrations of improvements in the drawings of an adult with congenital visual impairment after training, and performed an fRMI analysis to demonstrate that the primary visual cortex of this participant exhibited rapid changes during learning, becoming massively activated as drawing from tactile memory was enhanced (see also [START_REF] Amedi | Neural and behavioral correlates of drawing in an early blind painter: a case study[END_REF]. This high cross-modal plasticity of the brain after specific stimulations or experience is evidently involved in the astonishing drawing competencies sometimes exhibited in individuals deprived of vision who intensively practice this skill. It is likely that the low level of contour lines use in children deprived from vision in our study is a consequence of both limited drawing practice and reduced exposure to pictures during their childhood: would they have been given more opportunity, their performance would have been more close to that of sighted children, as claimed by [START_REF] Warren | Blindness and early childhood development[END_REF] with regard to different perceptuo-motor, cognitive and social skills. In this vein, guessing that children with vision loss will have, in the future, major exposure to pictures and increased practice in drawing, we can wonder if their drawings will still be different from those made by sighted individuals. Our data incite us to think that it will no longer be the case.

Finally, while our results showed an age-related development on a number of drawing criteria in the sighted as well as in the children with low vision, the developmental trends in the children with the most severe visual impairments were tenuous. Their number with a high level of drawing practice in each age group was unfortunately too low in our experiment to permit an assessment of drawing development in these selected children. Despite this, the age-related development of recognizability tended to suggest that the developmental trends were similar in the sighted and visually impaired children, although undoubtedly slower in the latter, especially in those with vision loss. Be that as it may, we can also ask whether it might not be somewhat questionable to expect drawing behavior to develop in similar ways in sighted children and children with the most severe visual impairments. If children build representations of objects that are anchored in their specific perceptual experience, children deprived of vision should tend to integrate features inaccessible through their own perceptual experience in their drawings only in response to explicit instructions (see, for instance, [START_REF] Heller | Production and interpretation of pictures of houses by blind people[END_REF], for a demonstration of the influence of explicit instructions in haptic drawing recognition). Alternatively, it may be that children deprived of vision feel the continuity, proportion and limb-positions of objects veridically, and that the presence of disconnections or juxtapositions in drawings has to do with drawing practice, not with specific perception of objects.

In conclusion, the present study demonstrated that "the less one sees, the less well one draws" (from sighted persons' point of view), though drawing is a spatial skill that individuals without vision can definitely develop. Furthermore, drawing practice can help them develop drawing skills. This important finding could encourage parents, caregivers and teachers to increase drawing opportunities for children deprived of vision, as producing drawings is one important media of communication between one self and the others, and contributes to cognitive development (e.g. [START_REF] Freeman | Strategies of representation in young children: analysis of spatial skills and drawing processes[END_REF][START_REF] Willats | Making sense of children's drawings[END_REF]. In further studies, it could be interesting to study if severely visually-impaired children are able to identify their own drawings, whether these drawings are judged as recognizable or not by sighted persons. If the errors in drawings that made them not recognizable are mainly due to insufficient practice, they should not recognize them better than sighted persons would.

  Abbreviations: WHO categories: C1 = moderate low vision; C2 = severe low vision; C3 = profound vision loss; C4 = participants with minimal light perception; C5 = participants without light perception. Drawing Practice: L1 = no practice at all; L2 = rare practice; L3 = 1 or 2 times a month; L4 = 1 or 2 times a week; L5 = almost daily practice.

Fig. 1 .

 1 Fig. 1. Illustrations of the different criteria with drawings made by children with a visual handicap or not (C1 = moderate low vision; C2 = severe low vision; C3 = profound vision loss; C4 = participants with minimal light perception; C5 = participants with no light perception; Si-H = blindfolded sighted participants in the haptic condition; Si-V = sighted participants in the visual condition).

  41, indicating an ability to draw. In comparison, the blindfolded sighted group performing in the haptic task reached a 0.70 score of recognizability. The Kruskal-Wallis test revealed significant Visual Status, H(3, N = 148) = 20.6, p < 0.001, and Age differences, H(3, N = 148) = 34 p < 0.001.

Fig. 2 .

 2 Fig. 2. Evolution with age of the degree of recognizability of the drawings made by the four child groups. Values are means with standard error bars.

Fig. 3 .

 3 Fig. 3. Frequency of occurrence of different drawing criteria as a function of the WHO (World Health Organization) visual categories. Values are means with standard error bars.

Fig. 4 .

 4 Fig. 4. Frequency of occurrences of different drawing criteria in the four child groups. Values are means with standard error bars.

Fig. 5 .

 5 Fig. 5. Frequency of use of contour lines in the children's drawings as a function of age and visual status. Values are means with standard error bars.

Table 1

 1 Main characteristics of the four children's groups involved in the experiment.

		Age group Number Sex	Mean age (SD)	Hand used for	WHO visual	Drawing	Use of Mylar
					yrs,mths	drawing	category (N)	Practice (N)	material or paper
	Group	6-7 yrs	6	4 girls	M = 7,1
				2 boys	(0,59)

Early: means an onset at the time of birth or shortly after, in the course of the first year of life.

Of course, the sun and the rain can also be perceived visually, like the other objects, and their shapes can be described verbally. We qualified the inanimate objects according to the type of direct hand-object relationships they enable.

Acknowledgments

The authors are profoundly grateful to the following people for their help in data collection and/or quotation: V. Fernandes (whom PhD included part of these data), P. Claudet, R. Champton, M. Jousset, M-C. Pichon, D. Schloesing, R. Panetta, C. Jeannin, M. Dolange and A.Vernier. We also sincerely thank T. Pownall for his careful correction of the English of the ms. This ms has also benefited of very thoughtful comments of two anonymous reviewers that we want to thank.

This research was supported by the French National Agency for Research (ANR).


Maria Boritchev
email: maria.boritchev@loria.fr

Philippe De Groote
email: philippe.degroote@loria.fr

Maxime Amblard
email: maxime.amblard@loria.fr

An inquisitive account of wh-questions through event semantics

Keywords: Questions, Logic, Semantics, Inquisitive Semantics, Events

. Our model results in an operationalized view of wh-questions and inquisitive existentials which we illustrate on several examples of wh-questions including who, which and where.

Introduction

Modeling the meaning of sentences in natural languages is a task that can be approached from different perspectives, starting from distributional semantics [START_REF] Harris | Analyse du discours[END_REF] and up to formal semantics [START_REF] Bos | A survey of computational semantics: Representation, inference and knowledge in wide-coverage text understanding[END_REF]. The study presented in this paper is conducted from the point of view of and in the scope of formal semantics.

There are multiple formal accounts of declarative sentences, starting from Richard Montague's [START_REF] Montague | Universal grammar[END_REF]. Montagovian approaches are compositional and truthconditional. They assign a truth value to each declarative sentence by computing it using the sentence's constituents. In this work, our object of study is interrogative sentences, and for this type of sentence, truth-conditional approaches do not hold, as it is not clear what assigning a truth value to a question would mean [START_REF] Groenendijk | Studies on the Semantics of Questions and the Pragmatics of Answers[END_REF]. Is a yes/no-question true or false? What is a negative answer to a yes/no-question? What does it mean for a wh-question to be true or false?

To keep the compositional approach but be able to account for interrogative sentences, a possible method is to use a logic that treats interrogative and declarative sentences without differentiation. It is the case of Inquisitive semantics [START_REF] Ciardelli | Inquisitive semantics[END_REF]. Inquisitive semantics is not directly compositional, but studies such as [START_REF] Ciardelli | Composing alternatives[END_REF] show that the two approaches are compatible.

Following classic literature on formal semantics of interrogative sentences such as [START_REF] Hamblin | Questions in montague english[END_REF], [START_REF] Hintikka | New foundations for a theory of questions and answers[END_REF], [START_REF] Groenendijk | Studies on the Semantics of Questions and the Pragmatics of Answers[END_REF], we investigate questions through the semantic type of queries they raise. This leads us to envision a compositional approach that makes extensive use of thematic roles [START_REF] Fillmore | Case grammar. Universals in Linguistic Theory[END_REF][START_REF] Gruber | Studies in lexical relations[END_REF]: Neo-Davidsonian Event Semantics [START_REF] Parsons | Events in the Semantics of English[END_REF].

The main idea behind our construction is that it is possible to see a whextraction as an existential quantification on a variable inhabiting the thematic role corresponding to the wh-word used in the wh-question. To do so, we define a procedural calculus that produces such a logical formula. Our idea is that of a parallel between wh-extractions and quantification, as in both cases, the scope is central to the interpretation. We achieve a technical solution that presents a similar approach for wh-extractions and quantification through the use of Inquisitive semantics, which translates the wh-extractions by existential quantifiers.

In this article, we present a compositional method for logically computing the semantic representations of wh-questions. We use compositional Neo-Davidsonian Event Semantics as presented in [START_REF] Champollion | Parts of a whole: Distributivity as a bridge between aspect and measurement[END_REF] joined with Inquisitive semantics [START_REF] Ciardelli | Inquisitive semantics[END_REF]. We implement our encoding using the Abstract Categorial Grammars framework [START_REF] De Groote | Towards abstract categorial grammars[END_REF]. This paper unravels as following: first, we present the preliminary notions (in Section 2) needed to introduce our model in Section 3. Then, we make a focus on a crucial part of our method that is responsible for the control of the derivations in Section 4 and we discuss our proposition in Section 5. Finally we compare our work with other approaches in Section 6 and conclude in Section 7.

Linguistic & mathematical preliminaries

Our approach is rooted in classic formal accounts of interrogative sentences such as [START_REF] Hamblin | Questions in montague english[END_REF], [START_REF] Hintikka | New foundations for a theory of questions and answers[END_REF], [START_REF] Groenendijk | Studies on the Semantics of Questions and the Pragmatics of Answers[END_REF], [START_REF] Ginzburg | The interactive stance[END_REF]. We base our method on several existing and well-studied frameworks, presented in the following sections, which we combine in a systematic and operational way. We start with semantic considerations, through the presentation of thematic roles and Neo-Davidsonian Event Semantics, and turn to more technical aspects with Inquisitive semantics and Abstract Categorial Grammars. [START_REF] Qian | Event in compositional dynamic semantics[END_REF] and [START_REF] Qian | Modal subordination in type theoretic dynamic logic[END_REF] have shown how these different theories can be used in a shared perspective.

Wh-questions and thematic roles

As defined in [START_REF] Ginzburg | Interrogative investigations[END_REF], wh-questions, in English, are questions that give rise to answers whose semantic types match those of the wh-phrase contained in the interrogative. A wh-phrase is introduced by a wh-word: what, when, where, who, whom, which, whose, why, how [START_REF] Aarts | The Oxford dictionary of English grammar[END_REF].

For this definition to be operational, it is necessary to systematically define the semantic types. A way to do so is by using thematic roles, as inspired by [START_REF] Dowty | Thematic roles and semantics[END_REF][START_REF] Kratzer | Semantics in generative grammar[END_REF]. This raises many discussions related to the interpretation and definition of thematic roles. To tighten up, we use a list of thematic roles inspired by Fillmore's and Gruber's works [START_REF] Fillmore | Case grammar. Universals in Linguistic Theory[END_REF][START_REF] Gruber | Studies in lexical relations[END_REF], adapted from [START_REF] Bonial | A hierarchical unification of lirics and verbnet semantic roles[END_REF] and [START_REF] Bunt | The dit++ taxonomy for functional dialogue markup[END_REF]: participant, actor, cause, agent, undergoer, instrument, theme, pivot, patient, attribute, location, see the definitions in Table 1.

Table 1: Adapted from [START_REF] Bonial | A hierarchical unification of lirics and verbnet semantic roles[END_REF], without thematic roles specific to events with symmetrical participants, events of perception, or events of communication, and with the addition of the Location role from [START_REF] Bunt | The dit++ taxonomy for functional dialogue markup[END_REF].

Thematic role Definition

Participant Entity involved in a state or event Actor Participant that is the instigator of an event Cause

Actor (animate or inanimate) in an event, that initiates the event, without intentionality or consciousness, existing independently of the event Agent

Actor in an event who initiates and carries out the event intentionally or consciously, existing independently of the event Undergoer

Participant in a state or event that is not an instigator of the event or state Instrument Undergoer that is central to an event or state that is not an instigator of the event or state Theme Undergoer that is central to an event or state that does not have control over the way the event occurs, is not structurally changed by the event, and/or is characterized as being in a certain position or condition throughout the state Pivot Theme that participates in an event with another theme unequally but is central to the event Patient

Undergoer in an event that experiences a change of state, location, or condition, that is causally involved or directly affected by other participants, and exists independently of the event Attribute

Undergoer that is a property of an entity or entities, as opposed to the entity itself Location Place where an event occurs or a state is true

We note that a more detailed list of thematic roles is presented in the DIT++ schema [START_REF] Bunt | The dit++ taxonomy for functional dialogue markup[END_REF], a semantically based framework for the analysis and annotation of dialogue. Following [START_REF] Bender | Linguistic fundamentals for natural language processing ii: 100 essentials from semantics and pragmatics[END_REF]'s statement that no fixed list of thematic roles can be established (nor crosslinguistic, nor for English only), we choose to showcase our method on a shorter list for the sake of readability. Our model can be easily tailored to a different list of thematic roles.

Once the thematic roles are set, asking a wh-question corresponds to interrogating the content of a thematic role. Therefore, for a compositional computation of a formal representation of an interrogative sentence, we need a formalism that gives access to variables corresponding to the content of the thematic roles. Neo-Davidsonian Event Semantics [START_REF] Parsons | Events in the Semantics of English[END_REF] is such a formalism.

Neo-Davidsonian Event Semantics

Neo-Davidsonian Event Semantics (NDES) is a formalism in which every sentence is considered in terms of occurring events and ways the sentence's semantic constituents relate to this event. Recent updates such as [START_REF] Champollion | Parts of a whole: Distributivity as a bridge between aspect and measurement[END_REF] present a compositional version of NDES, called quantificational event semantics. Representations of declarative sentences can then be built in a way that flattens down the sentence's syntactic structure and makes thematic roles accessible for further work [START_REF] Amblard | Sémantic representations and generative grammar: the Minimalist Categorial Grammars[END_REF]. NDES follows the Montagovian truth-conditional tradition presented in the introduction, so in order to use it for both declarative and interrogative sentences, we need a different type of logical approach.

More formally, quantificational event semantics is based on three atomic types:

e : individual/entity t : truth value v : event e and t are inherited from Montague Semantics [START_REF] Montague | Universal grammar[END_REF]; v is specific for event semantics. [START_REF] Champollion | Quantification and negation in event semantics[END_REF] presents a simple example for a declarative sentence:

(1) [John kissed Mary] = ∃e.kiss(e) ∧ agent(e, john) ∧ patient(e, mary)

In ( 1), the type of the predicate of the verb is restricted to v → t. The link with the variables of the other elements used in the sentence is made with the event variable e of type v in thematic roles predicates. One of the main arguments in support of NDES is its flexibility for the representation with regard to the optional arguments.

The semantic representation of a declarative sentence S (of type t) is then built through a process that assumes an existing definition of a syntax-semantic interface:

1. get the syntactic parsing of S ; 2. building of the deep syntactic tree of S (also called logical form of S) ; 3. translate the elements of the tree to their semantic representations through the syntax-semantics interface (e.g. λ-terms); 4. composition of the elements of the tree following the types constraints of the semantic representations and syntactic tree structure.

Inquisitive Semantics

Most frameworks for semantic representation do not focus on question processing. An exception is Inquisitive semantics which introduces different levels of interpretation. In Inquisitive semantics, all propositions are considered to have both an informative and an inquisitive content, which allows for a uniform treatment of declarative and interrogative sentences, and also to manage interactions with quantifiers, modals, or discourse structures. In the view of dynamic semantics, [START_REF] Heim | The semantics of definite and indefinite noun phrases[END_REF], in Inquisitive semantics, an utterance is considered to be a context update function. While declaratives introduce only one possibility of update, questions open up alternatives.

The logic features two projection operators, ! and ?. The first one transforms any proposition into a purely informative one by canceling its inquisitive content, which corresponds to a declarative sentence. Conversely, the second one transforms any proposition into a purely inquisitive one by canceling its informative content. Figure 1 shows the decomposition of a proposition in its non-informative and non-inquisitive projections.

Non-informative propositions

Non-inquisitive propositions ?P P =!P ∩?P !P Fig. 1: Propositions with informative and inquisitive content, projection operators, [START_REF] Ciardelli | Inquisitive semantics[END_REF].

More formally, inquisitive semantics interprets a proposition as a set of sets of possible worlds. Intuitively, an inquisitive proposition may therefore be seen as a set of classical (modal) propositions. Then, if s is the type of possible worlds and t the type of truth values, the type of inquisitive propositions p is defined as follows:

p = (s → t) → t
Which gives the following definition of first-order inquisitive logic, with ϕ and ψ of type p and e the type of individuals/entities:

R i t 1 . . . t n := P(R t 1 . . . t n ) where P a = λb s→t . ∀w s . (b w) → (a w) ϕ ∧ i ψ := λa s→t . (ϕ a) ∧ (ψ a) ϕ ∨ i ψ := λa s→t . (ϕ a) ∨ (ψ a) ϕ → i ψ := λa s→t . ∀b s→t . (∀w s . (b w) → (a w)) → ((ϕ b) → (ψ b)) ¬ i ϕ := λa s→t . ∀w s . (a w) → ¬(∃b s→t . (ϕ b) ∧ (b w)) ∀ i x e . ϕ x := λa s→t . ∀x e . ϕ x a ∃ i x e . ϕ x := λa s→t . ∃x e . ϕ x a !ϕ := ¬ i ¬ i ϕ ?ϕ := ϕ ∨ i ¬ i ϕ
In Hintikka's tradition of possible worlds, Inquisitive semantic includes connectives (e.g., and, or, not), quantifiers, and modal operators (e.g., may, should).

The aim is to bring together propositional, predicate, and modal logics while allowing a more detailed study of the relations between the elements involved. This sheds new light on various technical and linguistic aspects, but also brings to light new questions, thanks to a more refined analysis methodology.

One important remark for our purpose is that the interpretation of quantifiers is not the usual one. A universal quantifier cannot be the one of a question because the properties will hold for all the instantiation of the variable. An existential quantifier binds a variable that can be both inquisitive and informative. Then, using an existential also corresponds to a missing piece of information (which is not the variable). The use of ! projects the content of the formula into a purely informative content, thus without any missing element.

Abstract Categorial Grammars

The Abstract Categorial Grammars (ACGs) are a type-theoretic grammatical formalism for natural languages, based on the implicative fragment of multiplicative linear logic. It was introduced in [START_REF] De Groote | Towards abstract categorial grammars[END_REF], based on the Lambek calculus [START_REF] Lambek | The mathematics of sentence structure[END_REF] and semantic recipes expressed as type λ-calculus. The core principle of ACGs is to use the same type theory for both syntax and semantics. By doing so, ACGs define concrete and abstract signatures, linked by a lexicon.

An ACG is defined on a set of atomic types (A). The set T (A) of linear implicative types built upon A is inductively defined as follows:

if a ∈ A, then a ∈ T ; if α, β ∈ A , then (α → β) ∈ T (A).
Then a higher-order linear signature is defined as a triple Σ = A, C, τ where A is a finite set of atomic types, C a finite set of constants and τ : C → T (A) a function that assigns to each constant in C a linear implicative type in T (A). It is then possible to use a set of linear λ-terms build upon these signatures.

Given two higher order linear signatures, named vocabularies,

Σ 1 = A 1 , C 1 , τ 1 and Σ 2 = A 2 , C 2 , τ 2 , a lexicon L from Σ 1 to Σ 2 is defined as a pair L = F, G such that:
1. F : A 1 → A is a function that interprets the atomic types of Σ 1 as linear implicative types built upon A 2 ; 2. G : C 1 → Λ(Σ 2 ) is a function that interprets the constants of Σ 1 as linear λ-terms built upon Σ 2 ; 3. the interpretation functions are compatible with the typing relation.

The structure of these relations is summarized in Figure 2. In the following, we define signatures and lexicons in such a way that we produce question representations that integrate thematic roles, NDES, and Inquisitive Semantics.

A categorial formalisation of the syntax and semantics of wh-interrogatives

It is usual in the categorial grammar tradition to distinguish between the quantified noun phrases and the mere noun phrases, at the syntactic level. While the latter are assigned a simple atomic type np, the former are assigned the functional type (np → s) → s, which reflects the fact that a quantified expression takes a scope. This allows for a smooth treatment of scope ambiguities.

Let us illustrate this approach by considering the following sentence:

(2) every farmer fed a donkey the two possible readings of which are captured by the two syntactic structures of Figure 3. In order to give an abstract categorial account of sentence (2), one may declare abstract constants of the following types: This categorial treatment of scope ambiguities, which directly derives from Montague [START_REF] Montague | The proper treatment of quantification in ordinary english[END_REF], might be problematic when the targeted semantic formalism is Davidson's event semantics. It has indeed been argued that Montague's treatment of quantification does not combine smoothly with event semantics. The problem is that, in event semantics, a declarative sentence that is ultimately interpreted as a truth value (t) is first interpreted as a set of events (v → t). Then, switching from the latter interpretation of a sentence to the former necessitates an existential-closure operator, which may badly interact with the quantifiers that occur in the interpretation of the sentence. Fortunately, the literature provides at least two solutions to this problem. A first one is due to Champollion [START_REF] Champollion | The interaction of compositional semantics and event semantics[END_REF], and a second one to Winter and Zwarts [START_REF] Winter | Event semantics and Abstract Categorial Grammar[END_REF]. We follow this second solution since it is in line with the categorial tradition that we are advocating.

Winter and Zwarts' solution consists in assigning two different syntactic types to the sentences. On the one hand, a first type (s 0 ) is used for the "open" sentences, i.e., the sentences that are semantically interpreted as sets of events, and, on the other hand, a second syntactic type (s) is used for the sentences that are interpreted as truth values. Then, the existential closure operator allows values of type s 0 to be coerced into values of type s. Accordingly, the abstract signature we have sketched above is transformed as follows: farmer, donkey : n a, every : n → (np → s) → s fed : np → np → s 0 e-clos : s 0 → s This ensures that the existential closure operator will always take a narrower scope with respect to the other quantifiers. Now, it is well known that there is a strong analogy between quantifier raising and wh-extraction. Following this analogy suggests that we should assign to a wh-noun phrase the type that we assign to quantified noun phrases. Typically:

who : (np → s) → s
Similarly, a wh-determiner must be assigned the same type as a quantificational determiner:

which : n → (np → s) → s
Finally, wh-adverbs must be assigned the same type as the quantified adverbial modifiers, which are assigned ((s 0 → s 0 ) → s) → s [START_REF] De Groote | A type-logical account of quantification in event semantics[END_REF]. Accordingly, we have:

where : (s 0 → s 0 ) → s) → s
Putting everything together, we end up with an abstract syntax specified by the signature given in Table 2, where q is a syntactically silent operator that will allow a declarative proposition to be turned into a polar question. In abstract categorial grammars, the language generated by an abstract signature (such as the one given in Table 2) acts as a pivot language between surface forms and semantic interpretations. Figure 5 is typically the way an abstract categorial grammar models the syntax-semantics interface:

Consequently, in order to complete the picture, it remains to give the syntactic and the semantic translations of the abstract syntax specified by the signature of Table 2. These are given in Table 3 andTable 4, respectively.

We may now illustrate the overall approach by treating the following example:

(3) where did every farmer feed a donkey Its two possible readings are captured by the two following λ-terms built upon the signature given in Table 2:

4 Controlling wh-extraction and quantifier raising

The grammar we have sketched in the previous section is quite simple, and has the advantage of highlighting the parallel that exists between declarative and interrogative sentences. In particular, it is based on a uniform treatment of quantification raising and wh-extraction. This simplicity, however, is not without its drawbacks. These are threefold. Firstly, our grammar assigns the same syntactic categories to both the declarative and the interrogative forms (for instance, every and which are both assigned n → (np → s) → s). This gives rise to a grammar that generates ungrammatical surface forms such as the following one:

(7) * every farmer fed which donkey in which meadow

Secondly, allowing the quantifiers to take any possible scope results in spurious ambiguities. For instance, a sentence such as:

(8) a farmer fed a donkey in a meadow will give rise to six different abstract syntactic structures the semantic interpretations of which are all logically equivalent. Finally, the interactions between wh-extraction and quantifier raising must obey some constraints. For instance, in a wh-question, the wh-quantifier must always take the wider scope. Consider again example (3). The signature given in Table 2 allows one to built syntactic structures, such as the following one, that do not respect the wh-quantifier wider scope constraint:

(9) every farmer (λx. where (λf. a donkey (λy. e-clos (f (did-feed y x)))))

Consequently, our grammar might allow nonsensical semantic interpretations to be derived. The three kinds of defects that our grammar presents are all the consequence of a same fact: the signature of Table 2 allows too many abstract syntactic structures to be derived. We should therefore be able to select among the λterms that can be built upon the signature of Table 2 the ones that correspond to legitimate abstract syntactic structures. A modular and efficient solution to this problem consists in using another abstract categorial grammar in order to control the Abstract Syntax signature, which corresponds to the grammatical architecture shown in Figure 6.

Control Abstract Syntax v v ( (

Surface Forms Semantics

Fig. 6: Grammatical architecture of our model.

We have developed such a control grammar, which is unfortunately too large to be presented here 1 . Just to give a flavor of it, by way of illustration, we give the excerpt that allows example (3) to be dealt with. where := λp. q (where (λf. p (λs. e-clos (f s)))) SQ4 := λpqf. p (λx. q x f ) did-feed2 := λxyf. f (did-feed x y)

QRq13 := λpqf. p (λx. q x f )

Discussion

In the previous sections, we have presented a categorial formalisation of the syntax and semantics of wh-interrogatives. In its current state, our model doesn't give a direct solution for the representation of wh-questions that contain the whwords "what", "whose", "why", and "how".

"what" appears to be close in behavior either to "who" and "whom", or to "which". The difference between "what" and "which" seems to come from pragmatic considerations: the interpretation of "what" hugely depends on the context in which this interrogative word is used, while "which" is restrained in its interpretation by the definition of the set from which the choice of the response is made. "what did the farmer do" is an example that we cannot account for in the current state of our model: the issue raised by this question concerns the action, therefore the event, in our Neo-Davidsonian view. In order to be able to model this example, we would need a quantification on the predicates, so a higher-order logic.

Regarding "how", the difficulty is quite close to the one encountered with "what": the meaning of the wh-word "how" depends on the expression "how" is paired with. Consider the difference between "how old" and "how far": in the first case, the interrogated thematic role could be something time or age-related; in the second, the thematic role is location.

"whose" raises the problem of modeling the possessive relation. The semantics of possessives is a subject on its own, see for example [START_REF] Peters | The semantics of possessives[END_REF] for investigations on the semantics of English possessives. In our case, modeling "whose" would require an extension of our grammar with the addition of the preposition "of" used as a possessive, therefore as a modifier of a noun.

Finally, the difficulty with "why?" comes from its very large scope. Indeed, "why" can take a whole sentence as its argument (ex: "why did a farmer feed every donkey?"). In particular, it means that "why" introduces a quantification on the event predicate, which requires a higher-order logic treatment.

Related work

We compare our model to existing works by discussing related approaches, in particular related to the linguistic and mathematical tools that we use.

In [START_REF] Champollion | Quantification and negation in event semantics[END_REF], Lucas Champollion's presents Quantificational event semantics, an original take on Neo-Davidsonian event semantics. Quantificational event semantics is compositional and natively computational. Our method takes advantage of Champollion's compositional presentation of event semantics while following a different path in terms of sentence closure, to be able to account for interrogative sentences.

[15] introduces typed Inquisitive Semantics, a compositional take on Inquisitive Semantics, that addresses quantification, negation, some wh-questions, notions of entailment and coordination. [START_REF] Champollion | Some questions in typed inquisitive semantics[END_REF] uses typed Inquisitive Semantics to model questions and presuppositions. Our approach can be viewed as very similar to a combination of typed Inquisitive Semantics and quantification event semantics, which, to our knowledge, had not yet been done.

[24] presents a dynamic approach to discourse modeling. Based on Montague semantics, it is compositional and computational through the usage of simple type theory [START_REF] Church | A formulation of the simple theory of types[END_REF] and λ-calculus. It is operationalized through the ACG toolkit [START_REF] De Groote | Towards abstract categorial grammars[END_REF][START_REF] Pogodalla | ACGTK: un outil de développement et de test pour les grammaires catégorielles abstraites[END_REF].

Conclusion & Future work

We present a compositional method for the computation of logical representations of wh-questions through the use of thematic roles, based on Neo-Davidonian Event Semantics and Inquisitive Semantics and implemented through the Abstract Categorial Grammars framework. This paper shows that our approach is operational and computationally sound. We showcase our method on multiple examples and present several thoughts on future extensions.

In particular, the approach presented in this paper aims at staying as general as possible with regard to language specificities. We focused the presentation on English because of the simplicity of wh-words-based presentation, but we aim to produce a system that would be easily extendable to other languages such as French and Russian, using for example insights from [START_REF] Zamaraeva | Assembling Syntax: Modeling Constituent Questions in a Grammar Engineering Framework[END_REF].

The grammar presented in this article is a toy grammar for a fragment of English. It can easily be extended, for example in order to include syntactic tags such as the ones from the Stanford parser2 [START_REF] Klein | Fast exact inference with a factored model for natural language parsing[END_REF], currently available for 6 languages: Arabic, Chinese, English, French, German, and Spanish, or following other exploration on question and answer relations [START_REF] Cruz-Blandón | Toward dialogue modeling: A semantic annotation scheme for questions and answers[END_REF].

One of the future development directions for our work would be to add dynamicity to our model. A way to achieve this goal would be by integrating [START_REF] De Groote | Towards a montagovian account of dynamics[END_REF]'s framework along with Inquisitive Dynamic Epistemic Logic [START_REF] Ciardelli | Inquisitive dynamic epistemic logic[END_REF].

The next step for our work will be in addressing the wh-questions that are out of the scope of our current model, such as "why", see Section 5. Then, we would like to integrate this work in the perspective of a dialogue model. In such a model, it is necessary to be able to represent declarative sentences, interrogative sentences, but also non-sentential utterances (for partial answers or clarification questions). A dialogue model also needs to have context-related parts. We would like to extend a model such as the one presented in [START_REF] Boritchev | On dialogue modeling: a dynamic epistemic inquisitive approach[END_REF] using the method from this article.

Fig. 2 :

 2 Fig. 2: Schematic view of an ACG.

Fig. 3 :

 3 Fig. 3: Syntactic structures of two possible readings for sentence (2).

Fig. 4 :

 4 Fig.4: λ-terms corresponding to the syntactic structures presented in Figure3.

Table 4 :

 4 Semantic Interpretation Lexicon Semantic Interpretation farmer := λx. farmer x donkey := λx. donkey x meadow := λx. meadow x the := λp. the (λx. p x) a, some := λpq. !(∃x. (p x) ∧ (q x)) every := λpq. ∀x. (p x) → (q x) which := λpq. ∃x. (p x) ∧ (q x) in := λxp. λe. (p e) ∧ (location e x) fed, did-feed := λxy. λe. (fed e) ∧ (patient e x) ∧ (agent e y) who := λp. ∃x. p x where := λp. ∃x. p (λq. λe. (q e) ∧ (location e x)) e-clos := λp. !(∃e. p e) q := λx. ?x ?∃x.!(∃y.(donkey y) ∧ (∀z.(farmer z) →!((∃e.(fed e) ∧ (patient e y) ∧ (agent e z) ∧ (location e x)))))

1

  a : n → npe every : n → npu Inpe : npe → np Inpu : npu → np farmer, donkey : n did-feed : np → vpq where : sq 0 → s SQ4 : np → vpq → sq 0 did-feed2 : vpq np QRq13 : npe → sq 0npe → sq 0 It is about three times larger than the grammar presented in Section 3.a := a every := every Inpe, Inpu := λx. x farmer := farmer donkey := donkey did-feed := λpxf. p (λy. f (did-feed y x))

Table 2 :

 2 Abstract Syntax Signature

Abstract Syntax farmer, donkey, meadow : n the : n → np a, some, every, which : n → (np → s) → s in : np → s0 → s0 fed, did-feed : np → np → s0 who : (np → s) → s where : (s0 → s0 ) → s) → s e-clos : s0 → s q : s → s

https://stanfordnlp.github.io/CoreNLP/

Abstract Syntax

in := λxy. y + in + x fed := λxy. y + fed + x did-feed := λxy. did + y + feed + x who := λx. who + (x ) where := λq. who + (q (λx. x)) e-clos := λx. x q := λx. x (4) q (where (λf. every farmer (λx. a donkey (λy. e-clos (f (did-feed y x))))))

(5) q (where (λf. a donkey (λx. every farmer (λy. e-clos (f (did-feed x y))))))

Indeed, both ( 4) and ( 5) yield the same result when they are applied the translation given in Table 3:

(6) where + did + every + farmer + feed + a + donkey By contrast, when applied the translation given in Table 4, (4) and ( 5) yield the two expected different readings: (agent e y) ∧ (location e x))))) (1)


Denture by Jems Mayor; Teeth Gan Khoon

Adrien Coquet

Gym by supalerk laipawat, Fruit by Eucalyp, Milk Sandra Steak

Molecule Tomas Knopp

Protein by Turkkub, polyphenols Ben Davis

Francesca Ameglio

Steak

Sandra

Equilibrated proteins but also with specific amino acids

Et alors?

  Khoon, Denture by Jems Mayor; Teeth by Adrien Coquet; Steak by Sandra, Gym by supalerk laipawat, Fruit by Eucalyp, Milk by Tomas Knopp, Molecule by Ben Davis, Protein by Turkkub, polyphenols by Francesca Ameglio from the Noun Project Dardevet et al. Proc Nutr Soc. Khoon, Denture by Jems Mayor; Teeth by Adrien Coquet; Steak by Sandra, Gym by supalerk laipawat, Fruit by Eucalyp, Milk by Tomas Knopp, Molecule by Ben Davis, Protein by Turkkub, polyphenols by Francesca Ameglio from the Khoon, Denture by Jems Mayor; Teeth by Adrien Coquet; Steak by Sandra, Gym by supalerk laipawat, Fruit by Eucalyp, Milk by Tomas Knopp, Molecule by Ben Davis, Protein by Turkkub, polyphenols by Francesca Ameglio from the

  key and more than just their amino acid composition has to be taken into account The determinants associated with all the dimentions of protein quality have to be taken into account much more with the vegetarianization of dietary proteins However, « greening » significantly dietary proteins in older adults is possible but with some cautions and it should be supervised and followedIn protein nutrition, an ally to optimize and constrain the increase of protein intake could be a program of adapted physical activity

40 Animal 24 63 88 70 58 99 51 16 68 Equilibrated proteins but also with specific amino acids Plant Proteins in Older Adults?

	Amino Acid Amino Acid	HIS HIS	ILEU ILEU	LEU LEU	LYS LYS	CYS + MET CYS + MET	TYR + PHE TYR + PHE	THR THR	TRP TRP	VAL VAL
	mg/g of dietary protein mg/g of dietary protein	16	30 30	61 61	48 48	23 23	41 41	25 25	6.6 6.6	40
	Animal		63	88	70	58	99	51	16	68
			43	68	75	19	73	41	-	47
			37 125 27	35	87	36	7	48

Equilibrated proteins but also with specific amino acids In general, plant proteins are not optimal in their EAA composition RDA RDA 0.83 1.00 1.00 1.20 1.47 1.77 Plant Proteins in Older Adults?

	Amino Acid	HIS	ILEU	LEU	LYS	CYS + MET	TYR + PHE	THR	TRP	VAL
	mg/g of dietary protein		30	61	48	23	41	25	6.6	40
	Animal		63	88	70	58	99	51	16	68
			43	68	75	19	73	41	-	47
			37 125 27	35	87	36	7	48

Equilibrated proteins but also with specific amino acids In general, plant proteins are not optimal in their EAA composition Cereal 60g * Basal Whey 35g * Older Adult Anabolic effect Cereal Gorissen SH J Nutr. 2016 Plant Proteins in Older Adults?

	Amino Acid	HIS	ILEU	LEU	LYS	CYS + MET	TYR + PHE	THR	TRP	VAL
	mg/g of dietary protein		30	61	48	23	41	25	6.6	40
	Animal		63	88	70	58	99	51	16	68
			40	96	51 27,5 80	38	7	47,5

Equilibrated proteins but also with specific amino acids Solution is to combine pulse and cereal protein sources 50% 50% Older Adult Anabolic effect Basal Whey Wheat/ Wheat/ Wheat/ Bean Lentil Pea There are other limiting factors associated with plant protein sources Plant Proteins in Older Adults?

Couple by Gan Khoon, Denture by Jems Mayor; Teeth by Adrien Coquet; Steak by Sandra, Gym by supalerk laipawat, Fruit by Eucalyp, Milk by Tomas Knopp, Molecule by Ben Davis, Protein by Turkkub, polyphenols by Francesca Ameglio from the Noun Project