Flora Thibaut

Tanguy Watrin

Fleur Méary

Sylvie Tricot

Virginie Legros

Pascal Pellen-Mussi

Dominique Chauvel-Lebret

EFFECTS OF ZOLEDRONIC ACID ON OSTEOBLASTS IN THREE-DIMENSIONAL CULTURE

Keywords: bisphosphonates (BPs), zoledronic acid (ZA), hFOB 1.19, multicellular spheroids (MCS), proliferation

Background/purpose: Bisphosphonates (BPs) are synthetic drugs with anti-tumour and bone antiresorptive activities. The use of BPs is suspected to favor the emergence of osteonecrosis of the jaw (ONJ), a putative side effect whose pathogenesis remains unclear. Thus, we aimed to get insights about the impact of BPs on osteoblasts functions, using a three-dimensional culture model, which is suggested to be more similar to the in vivo tissues than monolayers.

Materials and methods: Effects of low (0.1 µM) and high (10 µM) concentrations of zoledronic acid were investigated on osteoblasts (hFOB 1.19), cultured as multicellular spheroids (MCS).

Proliferation, apoptosis, spheroid growth kinetics and morphology were studied using MTT and APH assays, Caspase 3 western-blotting, phase contrast imaging and scanning electron microscopy.

Results: Proliferation, apoptosis and spheroid morphology showed that 10 µM ZA induced a significant reduction in the relative viable cell number, correlated with morphological alterations of spheroids and induction of apoptosis. On the opposite, lower ZA concentration (0.1 µM) promoted cell proliferation without affecting growth kinetics or spheroid morphology.

Conclusion: ZA sensitivity of osteoblasts depends on concentration and experimental models.

The dual dose-dependent effects of ZA on osteoblasts cultured as spheroids, thereby promoting or inhibiting cell proliferation, may provide opportunities in tissue engineering. At last, hFOB spheroid culture system represents a valuable model for the exploration of the molecular basis of BPs action on osteoblasts and for the development and evaluation of implantable biomaterials in bone site.

INTRODUCTION

Bisphosphonates (BPs) are synthetic compounds, both metabolically stable and structurally similar to inorganic pyrophosphate. They bind strongly to hydroxyapatite crystals and inhibit bone resorption by reducing physicochemical bone mineral dissolution and decreasing osteoclast activity. Due to their antiresorptive properties, these drugs are widely used in the clinical treatment of bone diseases with increased bone resorption such as post-menopausal osteoporosis, Paget's disease or lytic bone metastasis. [START_REF] Russell | Bisphosphonates: the first 40 years[END_REF] BPs can be divided into two groups depending on their molecular mechanism of action at the cellular level. (2) Simple BPs, such as clodronate and etidronate, are metabolized intracellularly to non-hydrolysable analogues of ATP. The accumulation of these metabolites in the cytosol of osteoclasts induces cell death, probably by inhibiting ATP-dependent intracellular enzymes. [START_REF] Frith | Antagonistic effects of different classes of bisphosphonates in osteoclasts and macrophages in vitro[END_REF][START_REF] Lehenkari | Further insight into mechanism of action of clodronate: inhibition of mitochondrial ADP/ATP translocase by a nonhydrolyzable, adenine-containing metabolite[END_REF] The more potent amino-BPs (e.g., zoledronic acid (ZA), alendronate and ibandronate), which are not metabolized, alter osteoclast function by inhibiting enzymes of the mevalonate biosynthetic pathway, notably the farnesyl diphosphate synthase (FPPS). Inhibition of FPPS leads to depletion of the metabolites FPP and geranylgeranyl diphosphate necessary for the post-translational modification (prenylation) of small GTPases, thereby affecting subcellular localization and function of these signaling proteins essential for osteoclast survival and activity. [START_REF] Coxon | The role of prenylated small GTP-binding proteins in the regulation of osteoclast function[END_REF] BPs use has been linked to osteonecrosis of the jaw (ONJ), particularly in patients after intravenous therapy in the setting of malignancy. BPs-Related ONJ (BRONJ) can be caused by oral trauma or dental extraction, but it can also occur spontaneously. [START_REF] Ruggiero | Bisphosphonate-related osteonecrosis of the jaw: an overview[END_REF][START_REF] Borromeo | A review of the clinical implications of bisphosphonates in dentistry[END_REF] Although causality between BPs exposure and ONJ is still controversial, a clinical staging system of BRONJ has been developed to more appropriately diagnose and clinically manage patients, ranging from stage 0 (no clinical evidence of necrotic bone, but non-specific clinical findings and symptoms) to stage 3 (exposed necrotic bone with pain, infection, pathologic fracture, extra oral fistula, oral antral/oral nasal communication…). ( 8 The objective of this in vitro study was to evaluate the effects of ZA -a commonly prescribed amino-BPs -on osteoblasts in a three-dimensional (3-D) culture system, namely multicellular spheroids (MCS). (21) The MCS model is based on the cell's propensity to self-aggregate when cultured in non-adhesive conditions. By restoring cell-cell and cell-matrix interactions, this 3-D system more closely mimics living tissue than monolayer cultures do, [START_REF] Lin | Recent advances in three-dimensional multicellular spheroid culture for biomedical research[END_REF][START_REF] Pampaloni | The third dimension bridges the gap between cell culture and live tissue[END_REF] making MCSbased assays more predictive of in vivo response to drugs. By examination of cell proliferation, cellular apoptosis and spheroid morphology after ZA treatment, we demonstrated that the dual dose-dependent effect of ZA already observed in monolayer culture is conserved in these osteoblastic microtissue-like cultures.

MATERIALS AND METHODS

Cell culture

The human fetal osteoblast cell line hFOB 1.19 (hFOB) was purchased from ATCC. hFOB cells were maintained in a 1:1 mixture of phenol red-free DMEM/Ham's F-12 medium, supplemented with 10% fetal bovine serum, 100 IU/mL penicillin, 100 µg/mL streptomycin, 20 mM HEPES buffer, 2 mM L-glutamine and 300 µg/mL G418. For cell expansion, hFOB cells were incubated at 33.5 °C and 5% CO 2 and fed twice a week. On reaching 80% confluency they were passaged using 0.05% trypsin-0.02% EDTA.

Generation and culture of MCS

Formation of hFOB spheroids was achieved using agarose-coated 96-well tissue culture plates (50 µL of 1.5% agarose in Phosphate-Buffered Saline (PBS). The seeding density was 2x10 3 cells per well. After 96 h of incubation, spheroid cultures were treated by replacing 50% (=100 µL) of supernatant with normal medium or medium supplemented with ZA (Zometa ® , Novartis, East Hanover, NJ, USA) diluted in PBS at 2X final concentration. Half of the supernatant was replaced every 2 or 3 days with drug-free or drug-supplemented medium to ensure constant culture conditions. hFOB MCS were incubated at 33.5 °C for all experiments.

MTT assay

The relative growth of hFOB cells in monolayer was evaluated using the MTT (methyltetrazolium salt; 3[4,5-dimethyl-thiazoyl-2yl] 2,5-diphenyl-tetrazolium bromide) assay as previously described.(24) Briefly, hFOB cells were seeded in 96-well tissue culture plates in 200 µL medium per well. After 24 h, the cells were incubated with different ZA concentrations for 3 and 10 days. hFOB cells were then exposed to MTT (1 mg/mL) for 3 h at 37 °C. After complete solubilization of formazan crystals in DMSO, optical density was measured on an ELISA plate reader at 570 nm.

APH assay

The relative cell proliferation of hFOB in 3-D culture was evaluated after 3 days and 10 days in absence or presence of 0.1 or 10 µM ZA using the acid phosphatase assay (APH) as previously described. ( 25) Briefly, 10 spheroids from the same culture conditions were pooled, washed with PBS and transferred into a well of a 96-well microplate. The supernatant was discarded to a final volume of 100 µL. Then, 100 µL of the assay buffer (Immunopure p-nitrophenyl phosphate 20 mg/10 mL, sodium acetate 0.1 M, Triton X100, 0.1%) was added to each well, and plates were incubated at 37 °C for 90 min. Following incubation, 10 µL of 1 N NaOH was supplemented to each well, and absorption at 405 nm was measured on an ELISA plate reader.

Microscopy analyses

Phase-contrast imaging of hFOB MCS was performed using an inverted microscope. Spheroids were assessed visually for differences in cell morphology from day 0 (i.e., 96 h after spheroid initiation) to day 14, in absence or presence of ZA. At least 48 MCS from each indicated time point were used to calculate the average diameter using Photoshop ® software. In addition, hFOB MCS cultivated for 3 days or 10 days in drug-free or drug-supplemented medium were examined by scanning electron microscopy (SEM). Spheroids were rinsed twice in PBS and fixed for 3 h with 2.5% glutaraldehyde in PBS. Then, samples were washed twice in PBS and dehydrated in graded alcohol (80°, 95° and 100°) and acetone. After critical drying point (CPD 010, Balzers Union, Balzers, Lichtenstein) using liquid carbon dioxide, samples were coated with a thin layer of gold-palladium and observed via SEM.

Protein extraction and Western blot analysis

After 3 or 10 days of ZA treatment, hFOB MCS were collected, rinsed twice in cold PBS and homogenized in cold RIPA protein extraction buffer (50 mM Tris-HCl pH 7.4, 150 mM NaCl, 1% NP-40, 0.5% Na-deoxycholate, 0.1% SDS, 2 mM EDTA). Homogenates were then sonicated, -7 -incubated on ice for 30 min and centrifuged at 14,000 g for 15 min at 4 °C. Supernatants (total protein) and pellets (cellular debris) were collected and stored at -80 °C until use. Proteins were quantified with the Micro BCA™ Protein Assay Kit (Thermo Scientific, Waltham, MA USA), according to the manufacturer's instructions. 20 µg of total proteins were separated by electrophoresis in a 10% polyacrylamide SDS gel and transferred onto a PVDF membrane (Millipore, Merck KGaA, Darmstadt, Germany). After saturation, membranes were incubated with rabbit monoclonal antibodies raised against cleaved caspase-3 or alpha-tubulin. HRPconjugated secondary antibodies were used subsequently before ECL signal detection. All antibodies were purchased from Cell Signaling Technology.

Statistical analysis

The data are expressed as means ± standard deviation (s.d.). All experiments were conducted in triplicate. Comparisons across multiple experimental groups were performed using a one-way analysis of variance (ANOVA) followed by PLSD Fisher test using StatView v5.0 software. The observed differences relative to a probability of P < 0.05 were considered significant.

RESULTS

Effect of ZA on hFOB cell proliferation in monolayer and 3-D cultures

To test whether the culture model influences ZA sensitivity of hFOB cells, a comparative study of relative cell growth between monolayer and spheroid cultures was carried out.

Analysis of cell proliferation in hFOB monolayer culture was realized using MTT assay after 3 days or 10 days culture with different ZA concentrations (Figure 1). Results are documented relative to the respective untreated control (set to 100%). Exposure of hFOB to 0.1 µM ZA did not alter cell proliferation when compared with control cells (101.7% ± 7.2% at day 3, 102.5% ± 5.5% at day 10). In contrast, relative growth of hFOB cells in monolayer culture was significantly decreased after 3 days and 10 days in presence of 10 µM ZA, respectively to 87.4% ± 7.4% (P < 0.0001) and 40.4% ± 9.7% (P < 0.0001) of untreated cells' growth.

To explore the sensitivity of hFOB cells to ZA treatment in 3-D cultures, the relative number of viable cells in hFOB MCS incubated in absence or presence of 0.1 or 10 µM ZA was determined using the APH assay (Figure 2) in comparison with respective untreated control (set to 100%).

No significant difference was found between experimental conditions at day 3 (104.3% ± 9.9% for 0.1 µM, 94.2% ± 8.4% for 10 µM), whereas a lower relative viable cell number was observed after 10 days incubation in 10 µM ZA (80.4% ± 12.3%; P < 0.0001), in agreement with MTT assay results although growth inhibition appeared much more pronounced in monolayer cultures.

On the other hand, 10 days treatment with 0.1 µM ZA seemed to promote slightly hFOB cell proliferation (114.5% ± 9.0%; P < 0.0001) only when cultured as MCS.

Impact of ZA on hFOB MCS growth and morphology

Monitoring of ZA effects on hFOB MCS growth and morphology was achieved through a combination of phase-contrast imaging and SEM analysis. Spheroid growth kinetics as determined through MCS diameter measurement (Figure 3, Figure 4) revealed a very gradual and linear increase of mean spheroid size when cultured in absence of ZA, from 280 µm on day 0 to 405 µm on day 14. A nearly identical growth curve was obtained with spheroids incubated with 0.1 µM ZA, without any statistically significant difference in spheroid size at any time point. On the opposite, growth inhibition was evident for hFOB MCS treated with 10 µM ZA. Indeed, the diameter increase was almost totally abrogated as soon as 3 days culture, the difference in diameter being highly significant at day 7 (P < 0.0001) when compared with untreated MCS.

Interestingly, spheroid size reached a stable plateau (around 290 µm) maintained for at least 14 days of culture. To test whether ZA-induced growth inhibition of hFOB spheroid was correlated with morphological alterations, SEM analysis was carried out at 3 days or 10 days culture in drug-free or drug-supplemented medium (Figure 5). Microscopic examination of untreated spheroids after 3 or 10 days showed a compact spherical architecture without any distinguishable osteoblast -except for the round-shaped detaching ones -indicating an extremely dense cellular network. At high magnification (x3000), the surface of the MCS displayed filamentous structures, suggestive of cell membrane protrusions. SEM micrographs of spheroids treated with 0.1 µM ZA for 3 or 10 days, or with 10 µM ZA for 3 days did not reveal any morphological difference with control spheroids. In contrast, the structural integrity of hFOB MCS incubated for 10 days with 10 µM ZA appeared dramatically affected. Indeed, although spheroids conserved a spherical aspect, they were characterized by a highly irregular and grainy surface.

Immunodetection of the active form of caspase-3 during ZA treatment

Because exposure to the highest concentration of ZA compromised hFOB growth kinetics and morphology, we further examined whether the caspase cascade was triggered in response to ZA by performing Western blot analysis of cleaved caspase-3 (Figure 6). Immunodetection on total protein extracts from treated and untreated hFOB MCS revealed the presence of the active form of caspase-3 after 3 days incubation with 10 µM ZA. Moreover, no electrophoretic band was detected at day 10, indicating a transient activation of the apoptotic cascade.

DISCUSSION

While BPs are widely used in clinical practice for their bone antiresorptive action, accumulating evidence demonstrates that osteoclasts are not the only direct cellular targets of BPs and that these drugs affect as well the activity of bone-forming osteoblastic cells. The findings that BPs modulate expression of RANKL (receptor activator of NF-kappaB ligand) and osteoprotegerin in osteoblasts that may in turn indirectly interfere with differentiation and survival of osteoclasts highlight the complexity of BPs action on bone cells. [START_REF] Pan | The nitrogen-containing bisphosphonate, zoledronic acid, influences RANKL expression in human osteoblast-like cells by activating TNF-alpha converting enzyme (TACE)[END_REF][START_REF] Viereck | Bisphosphonates pamidronate and zoledronic acid stimulate osteoprotegerin production by primary human osteoblasts[END_REF] In this context, the clinical study of Recker and colleagues showing an increase of the mineral apposition rate in patients intravenously treated with ZA exemplifies the current view that BPs modulate both arms of bone remodelling. ( 28) However, the molecular mechanisms by which BPs regulate the activity of osteoblast is not completely understood.

As exposed above, literature data indicate that high (micromolar) concentrations of BPs compromise viability and function of osteoblastic cells whereas lower concentrations stimulate osteoblast proliferation/differentiation and inhibit osteoblast/osteocyte apoptosis. In line with these observations, we demonstrated that ZA treatment of hFOB MCS induces positive or negative effects on osteoblasts in a dose-dependent manner. The highest concentration (10 µM) blocks cell proliferation, alters MCS morphology and induces apoptosis that might be partly responsible for the development of BRONJ, whereas the lowest (0.1 µM) promotes slightly but significantly cell growth.

Our findings that 10 µM ZA inhibits the increase of MCS diameter after three days of incubation, and that the size of MCS remains constant the following days, suggest ZA exerts a cytostatic effect on osteoblasts, as already shown in other cell types. [START_REF] Cornish | Bone-bound bisphosphonate inhibits growth of adjacent non-bone cells[END_REF][START_REF] Ohnuki | Zoledronic acid induces S-phase arrest via a DNA damage response in normal human oral keratinocytes[END_REF] The transient activation of caspase cascade may signify that ZA has a pro-apoptotic effect as well but only on some sensitive cells. To test whether ZA-induced growth inhibition was correlated with induction of bone-forming activities, von Kossa staining was performed but did not reveal any mineralized nodule (data not shown). We conclude that the non-proliferative state of hFOB cells is not associated with the differentiation process, in contrast with previous results obtained with hFOB monolayer cultures. [START_REF] Reinholz | Bisphosphonates directly regulate cell proliferation, differentiation, and gene expression in human osteoblasts[END_REF][START_REF] Reinholz | Distinct mechanisms of bisphosphonate action between osteoblasts and breast cancer cells: identity of a potent new bisphosphonate analogue[END_REF] Further supporting the idea that 3-D MCS model presents specific properties conditioning the effect of ZA on osteoblastic cells, our results from MTT and APH assays showed that the anti-proliferative effect of high concentration of ZA is much more acute in monolayer than in MCS, suggesting a protective effect of the 3-D experimental condition.

Emergence of a hypoxia-related mechanism of resistance inside the MCS may provide an explanation for this phenomenon, hypoxic conditions preventing the anti-proliferative and proapoptotic effects of amino-BPs on osteoblast-like cells. (32) Alternatively, limited penetration and diffusion of ZA within the MCS may confine the pharmacological action of ZA to the cells located in the outermost layer of the spheroid, reducing consequently the proportion of cells exposed to ZA in comparison with monolayer cultures. Further studies would greatly benefit from fluorescently labeled conjugates of amino-BPs, which may provide useful information on the gradient distribution of these drugs within MCS as well as on the mechanisms of cellular uptake and potential transcytosis. [START_REF] Coxon | Visualizing mineral binding and uptake of bisphosphonate by osteoclasts and non-resorbing cells[END_REF][START_REF] Kashemirov | Fluorescently labeled risedronate and related analogues: "magic linker" synthesis[END_REF]) Conversely, we have described that 0.1 µM ZA moderately promotes cell growth of MCS. This anabolic effect on osteoblasts has been demonstrated for other amino-BPs and is thought to involve binding to and inhibition of protein tyrosine phosphatases. (35) However, why this proliferative effect is restricted to hFOB cells cultured as MCS remains to be solved.

Our observation that the biological response to ZA not only depends on the concentration but also on the experimental model, as already proposed, [START_REF] Cvikl | The response of dental pulp-derived cells to zoledronate depends on the experimental model[END_REF] emphasizes the great potential of MCS model as microtissue-like model for drug evaluation. (37) Indeed, the restoration of cell-cell and cell-matrix interactions confers to the MCS model structural and functional properties physiologically more relevant than monolayer cultures, giving to MCS-based assays a better predictive value of in vivo drug efficacy. For these reasons, in cancer research, the MCS represents an incomparable system for drug screening and metastasis analysis, as an avascular tumor model reproducing the in vivo pathophysiology of tumor tissue. [START_REF] Friedrich | Spheroid-based drug screen: considerations and practical approach[END_REF][START_REF] Vinci | Advances in establishment and analysis of threedimensional tumor spheroid-based functional assays for target validation and drug evaluation[END_REF] Also, the MCS constitutes a highly valuable research tool for tissue engineering with multiple applications in regenerative medicine. For example, Kelm and colleagues have generated vascularized macrotissues by assembling individual MCS coated with endothelial cells, and observed that these macrotissues were connected to the host's vascular system after in vivo implantation, illustrating the promising potential of heterotypic MCS as minimal building block for ex vivo generation of macrotissues. ( 39

  ) Various hypotheses have been put forward to explain BRONJ pathophysiology, which is likely to be multifactorial. It is postulated that BRONJ localizes on the jaws because of the heavy bone turnover in this area, causing BPs to accumulate preferentially in jaws at cytotoxic concentrations. Pathophysiology may also involve suppression of bone turnover and angiogenesis, altered function of oral mucosal cells, microbial flora, antiinflammatory effects and genetic predispositions. (9) Beyond well-described direct effects of BPs on osteoclastic activity, significant clinical and experimental evidence indicates that BPs also influence osteoblast functions. (10,11) In vitro and in vivo studies demonstrated dose-dependent effects of BPs on cell proliferation, cell differentiation, apoptosis and matrix mineralization. High concentrations (> 10 µM) generally produce adverse effects on these processes that may account for the occurrence of BRONJ. (12-17) On the opposite, lower BPs concentrations act positively on osteoblastic function by promoting survival, proliferation and differentiation of bone-forming cells. (17-20) While this dual effect of BPs on osteoblasts is now well established, uncertainty remains concerning the threshold of cytotoxic and clinically relevant concentrations of BPs, probably because of multiple experimental models used for investigations.

)

  Concerning the specific field of bone tissue engineering, multiple therapeutic strategies based on direct (coating, incorporation) or indirect (adjunct therapy) association of BPs with scaffolds (bioactive glasses, hydroxyapatite scaffolds, implants) have been developed. (40) These applications rely on the mechanical and putative osteoconductive and osteoinductive properties of bone substitutes, in conjunction with the local or systemic delivery of BPs, to promote bone regeneration. In this context, the hFOB spheroid culture system represents a convenient in vitro model for studying the effects of BPs on osteoblasts, and may greatly contribute to our understanding of their mechanisms of action on bone cells in the context of both tissueengineered bone regeneration and physiopathology of BRONJ. Based on the hFOB MCS model, this study shows that ZA exerts dual dose-dependent effects on osteoblast functions in 3-D cultures, inducing MCS growth arrest and apoptosis at micromolar concentration, but promoting cell proliferation at a lower level. Besides constituting a model of choice for elucidating the mechanisms of action of BPs on osteoblast, the hFOB MCS culture

FIGURE LEGENDS Figure 1 .

 LEGENDS1 FIGURE LEGENDS

Figure 2 .

 2 Figure 2. Relative cell growth of hFOB cells in 3-D culture. Relative cell growth in hFOB

Figure 3 .

 3 Figure 3. Monitoring of hFOB MCS growth. Graph visualizes spheroid diameter evolution

Figure 4 .

 4 Figure 4. Phase-contrast imaging of hFOB MCS. Representative phase-contrast images of

ACKNOWLEDGMENTS

The authors thank Joseph Le Lannic (CMEBA, University of Rennes 1), the members of the H 2 P 2 core facility (H 2 P 2 -histopathology core facility IFR 140, Biogenouest, University of Rennes 1, Inserm), Céline Allaire (Faculty of Dental Surgery, University of Rennes 1) and Isabelle Renard-Derrien (SCELVA, University of Rennes 1) whose help and cooperation aided in the completion of this study.

system may also be very useful in assessing the effects of new bone substitutes on bone-forming cells and, more generally, in screening potential therapeutic molecules. Nevertheless, further studies are needed to improve the physiological relevance of the MCS model, for example through the development of heterotypic spheroids mixing osteoblasts and endothelial cells. Such experimental models may not only contribute to in vitro molecular and cellular studies but also have great applications in tissue engineering, especially as building blocks for tissue reconstruction with autologous cells.


domain: math info shs

text:
Christophe Fouqueré

Myriam Quatrini

Myriam Quatrini Divergence In Dialogues

Divergence in Dialogues *

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

During these last decades, dialogue has become again the subject of intensive studies. This is the case inside Logics. The relevance of the dialogical frame for Logics is well known since Aristotle and it continues to be an important ingredient for logical studies [START_REF] Keiff | La dialectique entre logique et réthorique[END_REF][START_REF] Castelnérac | Arguing for inconsistency : Dialectic games in the academy[END_REF]. The study of dialogues is also essential in argumentation theory, in which debates and controversies are consubstantial to argumentation. The study of natural language makes also use of dialogical situations for grasping various aspects of natural languages, semantical but also syntactical ones [START_REF] Winterstein | Ludics and presupposition projection[END_REF][START_REF] Asher | Logics of conversation[END_REF]. In this perspective, works of J. Ginzburg are noticeable as they consist in developing directly a Theory of Dialogue (KoS, [START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF]). Our aim is very close to his approach as we also intend to define a theoretical framework able for studying linguistic phenomenas starting from their manifestations in dialogues. Our proposal shares with KoS numerous methodological postulates: -Meaning should not be studied on decontextualized sentences, by adding phonetical and pragmatical modules to syntactical and semantical ones. Instead, all features have to be grasped together.

-For studying and describing a communication, it is relevant to take seriously into account its successful transmission of contents as well as its failures (e.g., misunderstanding) and its repair attempts.

-Dynamics of language cannot be ignored. The meaning of an utterance is elaborated in a conversational process. The attempts for making explicit, for repairing, for updating parts of meaning have an important role. Nevertheless, our approach has to be distinguished from KoS with respect to the theoretical framework on which it relies. Indeed, we place ourselves in the perspective of recent philosophical positions, as for example Pinker's claim: "Dialogue is at the origin of language" [START_REF] Pinker | The stuff of thought, language as a window into the human nature[END_REF], or Brandom's position with his inferentialist semantics based on asking and giving reasons [START_REF] Brandom | L'articulation des raisons. du cerf[END_REF]. According to these approaches, interaction is ontologically the primitive fact of language. This is why we choose Ludics as our formal framework. Ludics is a logical theory developed by J.Y. Girard [START_REF] Girard | Locus solum: From the rules of logic to the logic of rules[END_REF] as an achievement of both proof theoretical and computational theoretical considerations. Above all, it is a theory of interaction in the sense that interaction is ontologically its primitive concept. Preliminary works have shown the interest of this logical framework for the modelling of dialogues (e.g., [START_REF] Lecomte | Ludics and its applications to natural language semantics[END_REF][START_REF] Lecomte | Figures of Dialogue: a View from Ludics[END_REF][START_REF] Fouqueré | Ludics and natural language[END_REF]). Ludics being a logical theory, our framework is relevant for grasping logical dimensions of language, especially for studying argumentation [START_REF] Fouqueré | Argumentation and inference: A unified approach[END_REF][START_REF] Fouqueré | Un cadre formel issu de de la théorie de la démonstration pour la théorie de l'argumentation[END_REF] but also divergences in dialogue as we show in this paper. In particular complementary aspects of argumentation and divergence in dialogues are modelled in a uniform way: from logics and pragmatics to semantics and syntax. Furthermore we account for several features of dialogues directly at the formal level. Therefore, we may expect a fine-grained representation of various dialogical phenomenas. More precisely we illustrate this last point by describing how different divergences in dialogue may be distinguished in our modelling.

A Frame for a Dialogue Theory

A few Facts on the Theoretical Framework

Ludics is a theory of interaction: its main object is the interaction itself and its dynamics. Designs are the support of interaction. Interaction between designs is allowed when these designs are (at least partially) localized at the same address (i.e., a cut in terms of logics). Such a situation triggers the dynamical process of interaction that abstracts the logical process of cut-elimination. We refer the reader to [START_REF] Girard | Locus solum: From the rules of logic to the logic of rules[END_REF] for a thorough presentation of Ludics, we just sketch below elementary aspects.

The elementary step of an interaction is an action (a move in Game Semantics), which is represented as a triple: (i) a polarity, that is positive (resp. negative) with respect to one of two interacting designs, when the action is produced (resp. received) by this design; (ii) the current address of the interactive process; (iii) a finite set of new addresses that the action creates (positive polarity) or expects (negative polarity) and on which the interaction may continue. In addition, there is a special positive action called daimon and denoted by which enables to terminate an interaction.

A base is a finite set of addresses that localizes the design: interaction may begin on such addresses. Formally it is a sequent Γ ∆ where Γ contains at most one address, which is said in negative position, and ∆ contains a finite set of addresses, said in positive position. A design based on Γ ∆ is a set of coherent alternate sequences of actions starting on the address in Γ if it exists, in ∆ otherwise.

An interaction between two designs occurs when their bases share an address in dual position. The process of interaction is a step by step travel through dual alternate sequences of actions. This process may diverge if these two (dual) sequences yield actions which are not dual each other, it may converge when one of these two (dual) sequences yields , it may also continue indefinitely.

Two modes of interaction are available: either a closed mode when the base contains only addresses shared by the two designs, or an open mode. Interaction may then be interpreted in two ways. The closed mode focuses on the process itself: it tests the possibility that interaction may go on and converge (to daimon), without paying attention to any context. The open mode focuses on the transformation that this process induces on the context (i.e., the addresses that are not shared by the interacting designs). In terms of Logics, the first case is similar to a cut-elimination between dual formulas and it enables to decompose them with respect to their structure, the second case corresponds to a modus ponens: from a proof of A and a proof of A ⇒ B, we get a proof of B, where B is the context. In terms of natural language and reasoning, the first case helps modelling interacting processes, e.g., dialogue processes, while the second one may be used for modelling explicitation processes, e.g., calculi, deductions or context updating.

A few Facts on the Model

Our model of dialogue is organized in two levels, that we may intuitively compare to the two modes of interaction in Ludics. The first level deals with the surface of dialogues, i.e., the flow of interventions and their role inside the process. At the second level, we take into account the content of these interventions.

The surface of dialogues

With respect to the first level, that considers the dynamics of a dialogue, a dialogue is seen only as an alternate sequence of interventions among which we may distinguish the one which initiates the exchange and the one which eventually ends the exchange. Interventions are only considered according to (i) their role in the flow of interventions: one intervention is anchored on a previous one and opens possible continuations of the dialogue, (ii) the fact that they are produced by one locutor while they are in the same time received by the other locutor.

We interpret interventions by means of dialogue acts. Formally, a dialogue act is an action of Ludics anchored in an address L opening addresses {L 1 , . . . , L n } together with the expression e that reveals the dialogue act in the intervention: that is a quadruple (+/-, L, {L 1 , . . . , L n }, e). It is worth noticing that an address created by a dialogue act (with positive polarity) may represent what Ginzburg and Fernandez call a question under discussion (QUD) [START_REF] Ginzburg | Computational models of dialogue[END_REF]. The dialogue act is the minimal unit of communication, whose role is to fuel the dynamics and determine the shape of the dialogue. At this level, the dialogue is reduced to a process. A dialogue act expresses an entitlement or a decision of the speaker, and also its acknowledgment by the addressee. In some sense, it is quite close to a speech act. However, dialogue acts are more elementary than speech acts. Let us consider, for example, the speech acts of argumentation, as presented in [START_REF] Fouqueré | Argumentation and inference: A unified approach[END_REF]. While the fact to argue, to ask or to negate are accounted by a single dialogue act, the concession is accounted by a sequence of two dialogue acts: the first one, denoted κ is positive with respect to the locutor who makes the concession: κ enables to precise which assertion is conceded, it is anchored on the pending address created by the assertion which is conceded; moreover, κ creates a single new address L on which the next dialogue act is anchored. The action with address L is then negative with respect to the same locutor. This latter dialogue act creates no new addresses, therefore pending addresses relative to the conceded utterance disappear and cannot be used in the following of the dialogue.

An intervention, or turn of speech, is a sequence of dialogue acts. From the viewpoint of the locutor who produces it, the intervention begins and ends with a positive dialogue act. When there is no divergence or rupture of the dialogue, the sequence of dual actions is recorded by the locutor who receives the intervention. A turn of speech is then twofold: producing an utterance to her interlocutor and expecting an utterance from her interlocutor.

A dialogue defines two sequences, called paths: each of them being the dialogue seen from the viewpoint of one locutor. Finally each path defines a design, hence a dialogue may be considered as the trace of the interaction between two designs, one for each locutor. This first level of modelling captures the surface of a dialogue. It is already enough to take into account important aspects of dialogues, namely the pragmatical and rethorical ones [START_REF] Fouqueré | Ludics and natural language[END_REF]. More fundamentally, this level enables to pay attention to the notion of convergence/divergence in the dialogue. Viewing the dialogue as a trace of an interaction between two designs makes explicit the fact that there are two points of view. Therefore it allows to observe the success or the failure of a communication between these two points of view, i.e., the fact that the dialogue may fail or end with a drop. Dialogues may badly end because of misunderstandings, disagreements, lacks of informations . . . and it is necessary to be able to represent such situations. At each step in Ludics, an interaction may continue if a negative action exists in one design that is dual to the positive action 'played' by the other design. Transposed in our modelling, a dialogue may continue without divergence provided that the current intervention produced by a locutor (a positive action) is correctly received/expected by her addressee (the dual negative action exists).

Example 1. A person P conducting a survey gives to a native the following informations: "All the Kpelle cultivate rice. Mister Smith does not cultivate rice". The person P asks the following question to the native N: -P: "Is Mister Smith a Kpelle?" -N: "I do not know Mister Smith, I have never seen him." At that level, we consider only the question and its answer that we represent each by a single dialogue act: • (+/-, L 0 , {L 1 }, Is Mister Smith a Kpelle?): positive for the person P that produces the question, negative for the native N that may receive it.

• (+/-, L 1 , {L 2 }, I do not know...): positive for the native N that produces the answer, negative for the person P that should expect it. But the person P expects a logically correct answer, e.g., "no", that we denote (+/-, L 2 , {L 3 }, No). He plans also to receive an incorrect answer, e.g., "yes" or "it may be the case", that we denote

(+/-, L 2 , {L 4 }, Yes). L 3 L 4 L 1 L 0 The viewpoint of P L 2 L 1 L 0
The viewpoint of N

The interaction between the two designs associated resp. to the dialogue seen from the viewpoint of P and of N diverges: the design corresponding to the viewpoint of P has no negative action dual to (+, L 1 , {L 2 }, I don't know...).

The dialogue cannot continue.

A full account of dialogues

We can refine this first approach in order to take care of other dimensions of dialogue, typically phonetical, lexical, syntactical, semantical, pragmatical and dialectical ones. In particular, we set a cognitive base for each locutor. A cognitive base contains, roughly speaking, the various knowledges and abilities1 used for building utterances as well as for receiving and recording them. Such a cognitive base is twofold: (1) sets of statements, that are assumed by each locutor;

(2) the linguistic elements used and received during the process of dialogue. While the level of surface of dialogues makes only use of the closed mode of interaction, the management of cognitive bases, their updating and their use in the dialogue in progress make use of the open mode of interaction2 .

Concretely, we make use of the following elements for modelling a dialogue between two protagonists A and B: -The dialogue is a sequence of triples 3

(C i A , C i B , [[D A , D B ]] i
), each triple indexed by i models what is at stake in the i th intervention. -C i A and C i B are the cognitive bases of respectively locutors A and B when the i th intervention occurs.

-X being one of the locutors A or B, the context C X is managed in two parts: the set of utterances publicly assumed by X, denoted by E X and a record R X of the linguistic elements used for building these utterances. All these elements (utterances assumed by locutors and linguistic elements) are represented by designs. Interactions between these designs enable to account for their (de)constructions. In this paper, we do not describe the part R X of linguistic elements and we only focus on the part E X of commitments. Furthermore we do not describe the correspondence between an utterance, seen as a knowledge, and its logical form. Therefore, in this paper, an "utterance" is an element of E X . -Designs D A and D B are the interacting designs representing the surface of the dialogue as presented in the previous subsection. The result of the interaction at step i is noted

[[D A , D B ]] i .
This result is the state of the dialogue after the i th intervention.

We illustrate these elements of modelling with the following dialogue between A and B:

Example 2.
-A: Who wrote to Paul? -B: Pierre.

We consider three steps in this dialogue: the first one corresponds to the question asked by A; the second one is the answer given by B. The last one is not expressed explicitly in this dialogue. Nevertheless, A may consider that the answer given by B is sufficient and ends properly the dialogue: the dialogue is convergent.

1. The first intervention is produced by A. Interpreting the surface of this dialogue, it is accounted by a unique dialogue act. This latter is positive according to A, its address is initial (the dialogue is starting), it creates just a new address on which the answer of this wh-question is expected. Finally, its phonetical manisfestation is Who wrote to Paul?.

We just account the first part of the cognitive base of A. The set of commitments of A, denoted by E 1 A , contains two utterances: -Someone wrote to Paul. (P 1 ).

-Someone is known by the two locutors A and B; his name is Paul (P 2 ).

As soon as B correctly received the intervention of

A, the answer given by B is taken into account by the dialogue act which is positive according to B, which is anchored on the address created by the previous dialogue act (that B correctly received). It creates a single address, the one corresponding to the information Pierre wrote to Paul. This information is expressed by the single word Pierre.

Since B receives correctly the wh-question and answers it, we may consider not only that the utterances P 1 and P 2 are available in E 2 B and may be activated, but also that there are other propositions in E 2 B : B knows who wrote to Paul, therefore E 2 B = {P 1 , P 2 , P 3 , P 4 } where P 3 corresponds to the utterance Pierre wrote to Paul and P 4 corresponds to the utterance Someone is known by us (A and B), his name is Pierre.

The third step is supposed to be expressed implicitly by A, as an acknowledgment of

B's answer. Therefore, the last dialogue act is produced by A and is the daimon, which terminates the dialogue on a convergence. Finally, the set of utterances publicly assumed by A contains after this third step the proposition P 3 (the answer given by B) and also the proposition P 4 (the fact that Pierre is known by A and B). Besides these two utterances, utterances P 1 and P 2 are also present in E 3 A . The formal representation of the dialogue is summed up in the following tables:4

• Dialogue surface (seen from A's viewpoint):

Steps

Dialogue acts of A Dialogue acts of B 1 (+, λ, {wh}, W ho wrote to P aul) 2 (-, λ, {wh}, W ho wrote to P aul) (+, λ.wh, {R}, P ierre) (-, λ.wh, {R}, P ierre) 3 ( , )

• Contexts (cognitive bases):

Steps C A C B 1 E 1 A = {P 1 , P 2 } E 1 B = ∅ 2 E 2 A = {P 1 , P 2 } E 2 B = {P 1 , P 2 , P 3 , P 4 } 3 E 3 A = {P 1 , P 2 , P 3 , P 4 } E 3 B = {P 1 , P 2 , P 3 , P 4 }

More on Divergence

Let us consider some alternative dialogues to the previous one given in example 2.

Example 3.

-A: Who wrote to Paul? -B: Did Paul receive a letter?

Example 4.

-A: Who wrote to Paul? -B: Who is Paul?

In both cases, the first step is formalized exactly as previously. After that step, in dialogue 3 as in dialogue 4, the locutor B does not receive as expected this starting intervention. In dialogue 3, B cannot use the knowledge P 1 : Someone wrote to Paul for her answer since this latter is not in her cognitive base. In dialogue 4, proposition P 2 : The name of someone that we both know is Paul. is not in E B . In both cases, the reaction of B is to try repairing this "miscorrespondance of contexts". Nevertheless the parallel between the two situations stops here.

• In dialogue 3, the locutor B may receive easily the intervention from A, by using the design associated to Paul in her cognitive base E 2 B , and by using elementary syntactical and semantical abilities. Afterwards, she may extend her context which becomes E 2 B = {P 1 , P 2 }. The question Did Paul receive a letter?, by means of which B initiates a new sub-dialogue, anchored on a new address γ, is mainly the expression that B recorded this information, new for her. The dialogue may end on a convergence, A answering or not.

Namely we have:

-Contexts in dialogue 3: The interaction between the designs corresponding to the (surface of) dialogue seen from the viewpoints of respectively A and B converges.

Steps C A C B 1 E 1 A = {P 1 , P 2 } E 1 B = ∅ 2 E 2 A = {P 1 , P 2 } E 2 B = {P 1 P 2 } 3 E 3 A = {P 1 , P 2 } E 3 B =
• In dialogue 4, B cannot receive as expected the intervention from A. In her cognitive base, B lacks utterances P 1 and P 2 that are needed and the relative linguistic elements.

Once the intervention is done by A, the base of B is provided with two new utterances: Someone received a letter, his name is Paul. (P 3 ) and B does not know who is this Paul. (P 4 ). B has to initiate a new dialogue (anchored on δ) in order to complete her cognitive base and to share with A informations present in A's cognitive base.

-Contexts in dialogue 4: This situation is different from the two previous ones. An intervention of a locutor is misunderstood by her addressee. Hence the addressee needs to complete her cognitive base (knowledges and linguistic elements) in order to be able to continue the dialogue. If the dialogue stops here, it is divergent. The interaction between the designs corresponding to the (surface of) dialogue seen from the viewpoints of respectively A and B, does not converge, if it stops now.

Steps C A C B 1 E 1 A = {P 1 , P 2 } E 1 B = ∅ 2 E 2 A = {P 1 , P 2 } E 2 B = {P 3 , P 4 } 3 E 3 A = {P 1 , P

Conclusion

In our theory of dialogues, elements of knowledge and ability are considered individually: a cognitive base is modelled as a set of designs, each design representing a knowledge or an ability. Such a cognitive base is the set of facts that are mobilized by each locutor during the dialogue. It enables to account ruptures, failures and repair attempts occurring in dialogues. J. Ginzburg [START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF] makes use of a similar approach, by using context updating as the dialogue unfolds. We may outline some of the features of our formal framework which are relevant for such a methodology. First of all, Ludics provides us with a theoretical frame that integrates interaction as a primitive and fundamental notion. Note that the same formal process is used for interpreting all levels of communication. In particular, there are two modes of interaction in Ludics. We make use of one or the other depending on which property of dialogue has to be considered. A first aspect of dialogue is modelled as a closed interaction. It integrates the necessity of keeping communication convergent. This imperative determines the surface of dialogue. A second aspect of dialogue, modelled as an open interaction, concerns the exchange of datas between locutors, the record of new informations, the update of other ones. This dimension is accounted by cognitive bases. In some sense, these two levels provide us with a more or less fine-grained account of the interaction at work in dialogues. We may either observe the flow of interaction, or focus on what is interacting (and therefore we fix, step after step, the interacting supports).

The complementarity of these two levels is illustrated by the analysis of dialogues 3 and 4. At first sight, these dialogues seem to be formally identical, both ending on a question instead of an expected answer. However a sharper study, that our modelling enables, reveals differences. In this analysis, the two levels of our formal framework are important: the model of the surface of the dialogue enables to remark that one is divergent while the other is convergent, the model of the cognitive bases enables to explain this difference: commitments sets are not equal.

Finally, it is worth noticing that interaction, as defined in Ludics and as we use it in our framework, enables to update the cognitive bases, either from one interacting side or from the other one. Therefore instead of relying on composition, our framework relies on interaction. As Ludics is a general-purpose logical and computational framework, the same formalization may be used for dealing with linguistic aspects of utterances as we do for commitments. As a long term issue, we intend to investigate in which extent this slight shift of viewpoint might renew semantical and syntactical studies. Even if computational applications are not immediately expected, a roadmap may be sketched. To start, two questions have to be investigated. The first one concerns argumentative dialogues, the second one general dialogues. In [START_REF] Fouqueré | Argumentation and inference: A unified approach[END_REF], we succeeded in characterizing speech acts of argumentation as invariant sequences of dialogue acts. We have to complete this formalization by providing other invariant sequences of dialogue acts for argumentative markers, e.g. the ones studied by Winterstein [START_REF] Winterstein | La dimension probabiliste des marqueurs de discours[END_REF]: but, nevertheless, otherwise, etc. In parallel, we intend to study dialogue ends. In particular, we should study what happens with respect to knowledge bases (a step towards such an issue is done in this paper). We should also be able to classify dialogue ends knowing that addresses that are created are closed either in the following of the dialogue or implicitly with the last interventions. In a next step we should test our formalization on a corpus of dialogues in order to refine and precise it.

  {P 1 , P 2 } -Surface of dialogue 3:

	Steps		Dialogue acts of A	Dialogue acts of B
	1	(+, λ, {wh}, W ho wrote to P aul)
	2			(-, λ, {wh}, W ho wrote to P aul)
				(+, γ, {Q}, Did P aul receive a letter?)
		(-, γ, {Q}, Did P aul receive a letter?)
	3		( , (yes))
			γ.Q λ.wh
	λ.wh	γ.Q	λ.wh, γ
	λ	γ	λ γ
	A's viewpoint	B's viewpoint

  2 , P 4 } E 3 B = {P 3 , P 4 } -Surface of dialogue 4:

	Steps	Dialogue acts of A	Dialogue acts of B
	1	(+, λ, {wh}, W ho wrote to P aul)
	2		(-, λ, {wh}, W ho wrote to P aul)
			(+, δ, {wh}, W ho is P aul ?)
	3	(-, δ, {wh}, W ho is P aul ?)

Once again, we intend to consider all different dimensions of language, hence knowledges and abilities concern linguistic ones (phonetical, lexical, syntactical, semantical, pragmatical and dialectical ones) as well as logical or psycho-social ones.

We retrieve the logical dimension of a dialogue only at this level.

This notion is very close to the notion of Dialogue Gameboard used byGinzburg.

is the empty production.

* This work has been partially funded by the French ANR projet blanc "Locativity and Interaction in Logic, Linguistics and Computer Sciences" LOCI ANR-10-BLAN-0212 03.


domain: info math shs

text:
Christophe Fouqueré
email: christophe.fouquere@lipn.univ-paris13.fr

Myriam Quatrini
email: quatrini@iml.univ-mrs.fr

Inferences and Dialogues in Ludics

We propose to use Ludics as a unified framework for the analysis of dialogue and as a reasoning system. Not only Ludics gives a denotational semantics for Linear Logic, but it uses interaction as a primitive notion. We first sketch a model for pragmatical and rhetorical aspects of dialogue after a brief review of the way the interactive aspect of dialogue may be represented in Ludics. Then we show how taking into account inferences that occur during a dialogue, with respect to a ISU -like model of dialogue. Through various examples we give an analysis of deductive inferences as well as processes making facts explicit that take place during knowledge updating.

Introduction

Inference and Dialogue: Interactive Viewpoint

Ludics [START_REF] Girard | Locus solum: From the rules of logic to the logic of rules[END_REF] is a reconstruction of logic with interaction as a primitive notion, in the sense that the primary logical concepts are no more formulas and proofs but cut-elimination interpreted as an interaction between objects called designs. The sole purpose of a design is that it carries on the interaction: a design is nothing else but a set of potential paths where interaction may take place.

At first, let us recall that both the proof-theoretical notions of cut and of cut-elimination are closely related to the (more philosophical) notion of inference: -A cut occurs between two proofs when one formula is the conclusion of the first proof, whereas the same formula is a hypothesis of the second one. Called modus ponens, it is the main ingredient of reasoning and it is the paradigm of deductive inference.

-Cut-elimination is one of the main results of proof theory: it states that it is possible (in Linear Logic as in Classical Logic) to normalize a proof, i.e. to transform a proof with cuts to an equivalent proof without cuts. As emphasized by Girard in [START_REF] Girard | Towards a Geometry of Interaction[END_REF], the cut-elimination process has also an inferential reading: it makes proofs explicit.

Secondly, let us observe that, in Ludics, we consider two modes of interaction: either we focus on the travel itself and we are interested in the possibility of the continuation, step after step, of a travel between two designs interacting each other, without paying attention to any context, or we focus on the transformation that this travel induces on some context or environment. In terms of Logic, the first case corresponds to a close situation: the interaction between proofs of dual formulas 1 enables to decompose such formulas with respect to their structure2 , the second case corresponds to applying a modus ponens: from a proof of A and a proof of A 'implies' B, we get a proof of B. In terms of natural language and reasoning, the first case helps modelling interacting processes, e.g. dialogues, while the second one may be used for modelling explicitation processes, e.g. calculus or deduction.

Modelling Dialogues and Inferences in Ludics

Ludics has already been used in a series of papers for representing several aspects of Natural Language (e.g. [START_REF] Lecomte | Pour une étude du langage via l'interaction : dialogues et sémantique en Ludique[END_REF][START_REF] Lecomte | Figures of Dialogue: a View from Ludics[END_REF][START_REF] Fouqueré | Ludics and Natural Language: First Approaches[END_REF]). They are all based on the idea that interaction is fundamental in Natural Language, dialogue being archetypical in that way. Remark that Ludics has also been used in the same spirit for formalizing web processes [START_REF] Fouqueré | Ludics and web: Another reading of standard operations[END_REF]. These modellings use the "primitive" objects of the interaction that Ludics set, upstream of a reconstruction of logic. In fact, Ludics is primarily a theory of interaction and its basic elements, actions and paths, are enough to grasp the rudimentary aspects of dialogical interactions. The formalisation of dialogue we obtain that way has to be related to the one obtained by means of Game Theory, see for example the study of dialogue done by Lascarides and Asher [START_REF] Lascarides | Agreement, Disputes and Commitments in Dialogue[END_REF] or other works for the more specific case of argumentation dialogues [START_REF] Walton | New directions in the logic of dialogue[END_REF][START_REF] Mackenzie | Four dialogue systems[END_REF][START_REF] Poesio | The predictive power of game structure in dialogue act recognition: experimental results using maximum entropy estimation[END_REF][START_REF] Prescott | Process and policy: Resource-bounded nondemonstrative reasoning[END_REF][START_REF] Prakken | A formal model of adjudication dialogues[END_REF]. However, note that several aspects of Ludics allow to go beyond what is done in Game Theory for modelling Natural Language: among them, an action has a more precise description than a move in Game Theory, and a specific action, the daimon, specifies the end of an interaction.

We must remark that the fact that Ludics is also a logical theory has not fully been taken into account in these previous works that consider Ludics for formalizing Natural Language. Let us shortly recall how Logic is retrieved from Ludics. The concrete basic steps of interaction (the actions) correspond to the basic steps of cut elimination. Hence a path, that is a sequence of actions used in an interaction, may correspond to the exploration of a formula through its main connective to its successive subformulas inside a proof. Finally, a design, that is a set of pairwise coherent paths, may correspond to a proof of a formula. However, a design abstracts from the notion of formula: a design has only loci, i.e. addresses, where interaction goes through. When a set of designs is given, the space where interaction with these designs may take place is also given. This space is defined by all the counter-designs of each design of this set. This gives an external viewpoint on designs, from which one may observe regularities. Formulas may then be retrieved as sets of designs closed relatively to these counter-designs. Indeed, exploring a set of designs reduces to exploring each design of this set. When it is closed, a set of designs describes all the ways to explore the object it represents, until its undecomposable elements. This has to be related to the concept of a formula, defined inductively by its connectives and subformulas that compose it until obtaining propositional variables or constants. Moreover, and this is an essential property of Ludics, some designs associated with a formula may denote proofs (of this formula): those designs that satisfy suitable properties, among which, precisely, the fact that the exploration can always continue until a suitable term. So, truthness of a formula requires the existence of some proof belonging to the set of designs associated to this formula.

We may then enrich the framework of our modelling, using still Ludics, to grasp the part of dialogue that is relative to inferences. This is particularly interesting when we consider argumentation dialogue as the authors did in [START_REF] Fouqueré | Un cadre formel issu de la théorie de la démonstration pour la théorie de l'argumentation[END_REF]. As remarked previously, this latter dimension of dialogues takes sense according to an environment or a context. We propose in this paper to integrate to our framework the Information State Update approach (ISU ) developed by various authors [START_REF] Larsson | Information state and dialogue management in the trindi dialogue move engine toolkit[END_REF][START_REF] Asher | Logics of Conversation[END_REF][START_REF] Fernández | Information state update: Semantics or pragmatics?[END_REF][START_REF] Lascarides | Agreement, Disputes and Commitments in Dialogue[END_REF], according to which inferences are managed in a context of dialogue. Of course, the logical dimension of Ludics is fully relevant for considering correctness (or not) of deductive inferences.

Our contribution

As mentioned above, previous works, e.g. [START_REF] Lecomte | Pour une étude du langage via l'interaction : dialogues et sémantique en Ludique[END_REF][START_REF] Lecomte | Figures of Dialogue: a View from Ludics[END_REF][START_REF] Fouqueré | Ludics and Natural Language: First Approaches[END_REF], were devoted to model the surface of dialogues, which concerns both the intra-locutor justification relation and the inter-locutors connection relation, i.e. relations that may exist between dialogical interventions. We assume that this interactive level is central in order to grasp and understand other dimensions of dialogues: semantics (as started in [START_REF] Lecomte | Ludics and its applications to natural language semantics[END_REF]), syntax and even lexical aspects. In this paper we are interested in the pragmatical and rhetorical dimensions. Our aim is to make more explicit the continuity between the interactive structure of dialogues and the eristic or dialectic role of interventions occurring in such dialogues. Our approach integrates two complementary aspects: one concerning pragmatical and rhetorical dimensions of dialogue, and another one concerning inferences that may take place inside an argumentation as well as the process of making facts explicit in informative dialogues. The content of this paper is organized as follows: -in section 2, we sketch a model for pragmatical and rhetorical aspects of dialogue after a brief review of the modelling in Ludics of the interactive aspect of dialogues; -in section 3, we show how taking into account inferences that occur during a dialogue, with respect to a ISU -like model of dialogue. Through various examples we give an analysis of deductive inferences as well as calculus/'explicitation' processes that take place during knowledge updating. Because of lack of space, we do not recall formal aspects of Ludics. However the reader may find complete presentations in [START_REF] Girard | Locus solum: From the rules of logic to the logic of rules[END_REF][START_REF] Curien | Introduction to linear logic and ludics, part i[END_REF][START_REF] Curien | Introduction to linear logic and ludics, part ii[END_REF].

Modelling Dialogues

The modelling of dialogues in Ludics may be considered as a refinement of what is done with Game Theory [START_REF] Lascarides | Agreement, Disputes and Commitments in Dialogue[END_REF]. Roughly speaking, each locutor is a player in a game, the dialogical interventions are represented by moves and a dialogue itself is represented by means of an alternate sequence of moves, that is a play. Nevertheless, in Ludics, the duality, expressed by alternating positive and negative actions, does not correspond to a question/answer duality as in game semantics but to a production/reception duality in dialogue modelling. A positive action models the production of an utterance whereas a negative action is an expectation of utterances. A turn of speech is then twofold: producing an utterance to her interlocutor and expecting an utterance from her interlocutor. A dialogue defines then two sequences, called paths: each of them being the dialogue seen from the viewpoint of one locutor. Finally each path defines a design, hence the dialogue may be considered as the trace of the interaction between two designs, one for each locutor. We show in this section that the framework of Ludics, in particular the use of designs, is useful to give an account of various aspects of dialogues, namely presuppositions, divergence, argumentation, . . . The starting point of the formalisation is only based on the interactive feature of dialogues. This level is already enough to take into account some important elements of dialogues. We refine then this first approach in order to take care of other dimensions of dialogue. We focus on the pragmatical and rhetorical levels.

Interactive level

At this level, we only consider the interactive surface of dialogues: a model for the intra-locutor justification relation and for the inter-locutor connection relation. With respect to this interactive level, dialogues are only alternate sequences of interventions among which we may distinguish the one which initiates the exchange and the one which eventually ends the exchange. Interventions are only considered according to: -their role in the flow of interventions: one intervention is anchored on a previous one and opens possible continuations of the dialogue (justification relation), -the fact that they are produced by one locutor while they are in the same time received by the other locutor (connection relation).

An alternate sequence of interventions may then be represented by an alternate sequence of actions in Ludics, as illustrated in the following example:

Example 1 (C. Dickens, David Copperfield) The following dialogue is extracted from the novel "David Copperfield" of C. Dickens. The dialogue takes place between a coachman (C) and David (D), the coachman brings David to London: The dumplings is uncommon fine down there . . .

C
As a first approximation, an action κ 1 , . . . , κ 9 is associated to each utterance I 1 , . . . , I 9 . The justification between actions is given Fig. 1, on right. The first action κ 1 is initial, indeed, with this intervention C initiates the dialogue. Actions κ 3 , κ 7 and κ 9 are justified by the second intervention κ 2 : in the three cases, these interventions refer to 'Suffolk', entity present in the second intervention that may introduce topics concerning hunting, native soil, gastronomy . . . Each other intervention is justified by the action that precedes it immediately where the focus is given.

Ludics allows for rebuilding the interaction that produces this alternate sequence of interventions as a trace. The alternate sequence of interactions κ 1 , . . . , κ 9 may be represented choosing the positive polarity for the interventions of C: in that way, the dialogical interaction is represented with the point of view of C. Reversing the polarities, one gets the point of view of D. The trace of interaction between the two designs is the dialogue. In left part of Fig. 1, the design on the left is the point of view of C, and the design on the right is the point of view of D, circled actions are the positive ones, black edges account for the justification relation, the flow of red arrows is the interaction between the two designs.

Viewing the dialogue as a trace makes explicit the fact that there are two points of view. Moreover it allows to observe the success or the failure of these two points of view, i.e. the fact that the dialogue may fail or end with a drop. Dialogues may badly end because of misunderstandings, disagreements, . . . and it is necessary to be able to represent such situations. Ludics distinguishes

κ 9 κ 9 κ 8 κ 8 κ 7 κ 6 κ 6 κ 5 κ 5 κ 4 κ 4 κ 3 κ 3 κ 7 κ 2 κ 2 κ 1 κ 1 Viewpoint of C Viewpoint of D κ 1 κ 2 κ 3 κ 4 κ 5 κ 6 κ 7 κ 8 κ 9
Trace of the interaction with justification between actions.

Figure 1: The interaction between two designs/viewpoints two cases. An interaction is convergent when it ends with a daimon3 , it is divergent when a positive focus has no dual counterpart. These two cases allow for interpreting two standard final situations in dialogues: either the dialogue finishes well or there is a misunderstanding between the two locutors.

The following example is given by M. Chemillier [START_REF] Chemillier | Eléments pour une ethnomathématique de l'awélé[END_REF] to illustrate the difficulties that arise when one wants to isolate the logical part of a dialogue.

Example 2 A person P conducting a survey gives to a native the following informations: All the Kpelle cultivate rice. Mister Smith does not cultivate rice, the person P asks the following question to the native N:

-P: "Is Mister Smith a Kpelle ?" (κ 1 ) -N: "I do not know Mister Smith, I have never seen him." (κ 2 )

Here we consider only the question and its answer that we represent each by a single action (resp. κ 1 and κ 2 ). The person P expects a logically correct answer, e.g. "no", we note κ 3 . He plans also to receive an incorrect answer, e.g. "yes" or "it may be the case", we note κ 4 . Hence the design for the person P may be either κ 1 followed by κ 3 , or κ 1 followed by κ 4 . The interaction in Ludics is given Fig. 2: it is divergent as there is no negative action dual to κ 2 in what is expected in the design of P. The dialogue cannot continue.

Pragmatical and Rhetorical Levels

In order to take into account other dimensions of dialogues, the formalisation is refined. A first step is to replace the notion of action by the one of dialogue act. The notion of dialogue act is defined by Landragin [START_REF] Landragin | Vers l'identification et le traitement des actes de dialogue composites[END_REF] as "the minimal unit of communication in a dialogical context". A dialogue act is a communicational fact whose role is to fuel the dynamic and determine the shape of the dialogue. It may be explicit or implicit, verbal or not (e.g. an acknowledgment given as a gesture). It may appear

κ 3 κ 4 κ 2 κ 1 κ 1 ?
Viewpoint of the person P conducting the survey Viewpoint of the native N as one or more propositions, but also as part of a proposition (word, adverb, . . . ). It expresses an entitlement or a decision of the speaker, and also its acknowledgment by the addressee. In some sense, it is quite close to a speech act. However a speech act may correspond to several dialogue acts as shown in following examples. Dialogue acts are indeed more elementary than speech acts. They can be seen as the basic blocks from which one builds interpretation for dialogical interventions or even utterances. Formally, a dialogue act may be defined using the complete definition of action in Ludics: a polarity, an address and a finite set of new addresses which may be used as adresses of following actions. A dialogue act is such an action together with the expression that reveals the dialogue act in the intervention. Such an expression may be a proposition, a word (e.g. a single adverb, a noun), a prosodic feature, a non verbal sign (a nod, a shake, a slap, . . . ). In trivial cases, an intervention is a unique dialogue act. Otherwise a turn of speech has to be decomposed into a sequence of dialogue acts, hence may give rise to a complex design. Note that the representation of an utterance in terms of dialogue acts is dependent of the context of the dialogue, and in particular of past interventions that occurred.

Definition 1 (Dialogue Act) A dialogue act κ is:

• either a proper dialogue act, that is to say a tuple ( , ξ, I, e) where the finite sequence of integers ξ is the focus of κ: the location of the act with respect to the dialogical interaction one considers, -I is the ramification of κ: the openings created by the dialogue act on which new dialogue acts may be produced, e is the expression of the dialogue act, that is to say the language or communicational fact by means of which the dialogue act manifests itself,

the polarity of the act may be positive (+) or negative (-). The act is positive for the locutor that produces it except when this act is a constraint from its interlocutor (see below presupposition case). These acts have a dual polarity when received by the interlocutor.

• or a particular positive dialogue act, still called daimon anot noted ( †, e), that registers the end of an interaction that went well. In that case, the expression e of the dialogue act may often be empty.

Our modelling of dialogues in Ludics, together with the notion of dialogue act, allows for analysing different pragmatical aspects as well as rhetorical features. These two dimensions appear as being closely linked: both concern primitively dialogues, both are central in argumentation dialogues, both are at the frontier of linguistics and philosophy and both are closely related to Logic. For example, the pragmatical notion of speech acts is systematically used for describing argumentation dialogues by means of an almost exhaustive list of speech acts of argumentation. The notion of fallacies seems being at first relevant for rhetorical issue but the presuppositions are one of the current study objects of Pragmatics. Without trying to radically separate these two dimensions, the sole interactive level of dialogues, enriched by the notion of dialogue act enables to characterize different fallacies as well as speech acts of argumentation, according to their behaviour with respect to interaction (see also [START_REF] Lecomte | Pour une étude du langage via l'interaction : dialogues et sémantique en Ludique[END_REF][START_REF] Fouqueré | Un cadre formel issu de la théorie de la démonstration pour la théorie de l'argumentation[END_REF]). We illustrate below the modelling of fallacies in Ludics on an example of presupposition.

Presupposition in Dialogues

A presupposition is an implicit assertion concerning the world, whose validity is accepted in the dialogue.

Example 3 Let us consider this well known example due to Aristotle; a judge asks a young delinquent this question: "Have you stopped beating your father?".

Answering this question by yes or no supposes that the answer of the implicit question is yes: -Q 1 : "Did you beat your father?" -R: "Yes." -Q 2 : "Have you stopped beating him?" This may be modelled by the following interaction between two designs:

κ 3 κ 2 κ 2 κ 1 κ 1
Point of view of the judge Point of view of the young adult

where the dialogue acts κ 1 , κ 2 and κ 3 refer respectively to interventions : "Did you beat your father?"; "Yes"; "Have you stopped beating him?".

The question asked by the judge, "Have you stopped beating your father?", is a sequence of three dialogue acts:

(κ 1 , Q 1 )(κ 2 , R)(κ 3 , Q 2 ).
If the young adult continues the dialogue, he implicitly assumes that the two (partial) designs do not diverge. He accepts to justify the intervention he will have on the negative dialogue act dual to (κ 3 , Q 2 ), this dialogue act being anchored on the positive dialogue act dual to (κ 2 , R): he assumes he would have answered yes to the implicit question.

Argumentation Dialogues

Argumentative dialogues show some peculiarities. In particular speech acts that compose them are limited in number: assertions, arguments, denial, concessions, . . . In addition, a notion of winnings appears, which seems specific to argumentative dialogues. In a dialogue whose aim is to exchange information, to share knowledge or a feeling, the question of whether one of the speakers wins is quite irrelevant. But to know who is right after a controversy is essential. An argumentative dialogue is distinguished from (common) dialogues in the sense that two arguments are opposed, each party having his thesis to defend or advance. The dialogue is then mainly an exchange of arguments and counter-arguments with the sole purpose that the dialogue ends when a thesis is considered winning.

In order to take care of argumentative dialogues, we give below an account of speech acts of argumentation by means of dialogue acts. It is necessary to precise what is a "gain". One of the locutors becomes a "loser" in the following situation: he abandons the dispute conceding the last arguments of his addressee (who becomes the winner): the winner has the last word. We can precise the situations when a daimon is used: either the set of addresses where a dialogue act may take place is empty, hence the daimon is the only possible dialogue act and the opponent had really "the last word", or there exist still locations where actions may take place and the locutor prefers to quit the dialogue. Note that this notion of winner is linked to a specific dialogue: it may be the case that the current loser may be a winner with another opponent. We must mention that the notion of "winning" design exists in Ludics: a design is winning if, in particular, it does not contain the daimon. Hence every interaction with such a design is either divergent or ends with a daimon in the opposite design: such a design wins every time.

The speech acts of argumentation:

• To claim a thesis (to affirm, to assert). This corresponds to posing a proposition on which the interlocutor may continue the dialogue by negating it or by conceding it or by asking more explanations. Such a proposition may constitute the initial thesis of a dialogue, it may also be an additional element for an on-going discussion. In our Ludics framework, this is represented by a proper and positive dialogue act (+, ξ, I, e), that may be initial or justified by one of the previous dialogue acts that introduces one of the subjects of the dialogue.

• To argue. This corresponds to posing propositions that serve as argumentative premises and in the same time the sound structuring between these premises and the conclusion,i.e. the fact on which the locutor argues, that should be a location open to discussion. For example, the locutor may affirm a thesis A and justify it with B and B ⇒ A, hence propositions that are posed are actions corresponding to the two elements B and B ⇒ A. This speech act is represented by a unique positive proper dialogue act (+, ξ, I, e). Contrarily to the previous item, the ramification I, hence the locations opened by this act, cannot be a singleton: it contains at least all the premises and the relation between the premises and the conclusion.

• To negate (to refuse). We consider this speech act as a kind of thesis, a claim of the locutor. However such a claim cannot be an initial act of the dialogue: it is a refusal of a previous claim 4 . In a general setting, in such a turn of speech, the locutor may negate and pose a counter-argumentation by giving new facts. It is represented as a proper positive dialogue act (+, ξ, I, e), necessarily justified by a previous act that introduces the proposition that is negated.

• To ask, to request some justification. With respect to the structure of a dialogue, there is no difference with the previous item except the linguistic form it has.

• To concede. This consists in accepting one assertion that the interlocutor claimed. Hence the dialogue will not continue on this element. As such, a concession is only a part of a turn of speech: either the dialogue ends with this (last) concession and the locutor abandons, or the locutor continues its intervention on another element still under discussion. A sequence of two dialogue acts is associated with the concession: (+, ξ, {0}, e 0 ), (-, ξ.0, ∅, e 1 ) where -the first one, positive, enables to express what is conceded, its focus is the one created by the affirmation that one concedes and its ramification is a singleton.

-The second dialogue act is a negative one, it is focused on ξ.0: the locus that the immediate previous act created and it has an empty ramification. The effect is that the conceded affirmation disappears.

• To abandon. This act ends an argumentation dialogue in a convergent way. The locutor does not mention any misunderstanding but recognizes he has nothing more to add. The positive dialogue act ( †, e) fits exactly to this situation. Its use finishes the interaction.

In order to illustrate the effect of an intervention during an argumentation dialogue, we give below an example due to Prakken [START_REF] Prakken | A formal model of adjudication dialogues[END_REF] and that we analyzed more exhaustively in [START_REF] Fouqueré | Argumentation modeling in Ludics[END_REF].

Example 4 A juridical controversy between a plaintiff P and a defendant D starts as follows:

I 1 Plaintiff:
I claim that defendant owes me 500 euro. I 2 Defendant: I dispute plaintiff 's claim. I 3 Plaintiff:

Defendant owes me 500 euro by r 1 since we conclude a valid sales contract, I delivered but defendant did not pay. I 4 Defendant: I concede that plaintiff delivered and I did not pay, but I dispute that we have valid contract.

To start, we focus on the argumentative dialogue. According to our modelling, the two first interventions I 1 and I 2 create each only a new locus, respectively L 1 and L 2 . On the contrary, the intervention L 3 is represented by a dialogue act which creates four new loci. Indeed the utterance I 3 'Defendant owes me 500 euro by r 1 since we conclude a valid sales contract, I delivered but defendant did not pay' makes explicit four argumentative elements to support his assertion 'Defendant owes me 500 euro'. These argumentative elements are: a law article r 1 ; the fact that both parties concluded a valid contract; the fact that plaintiff delivered his service and at last, the fact that defendant did not pay 500 euros. By means of this dialogue act, we indicate that the dialogue may continue using loci L 31 , . . . , L 34 respectively associated with these four elements.

I1 I2 I3 L3 1 . . . . L3 2 . . . . L3 3 . . . . L3 4 L2 L1 L0 . . . . (L3 1 ,)L3 2 ,L3 3 ,L3 4 L2 L1 L0
Plaintiff Defendant Intervention I 4 is more complex, it is not given into account by a unique dialogue act but by a sequence of several ones. Indeed, in intervention I 4 , defendant concedes two plaintiff's elements: 'plaintiff delivered' and 'I did not pay'. These two concessions are successively represented, each of them being a positive dialogue act followed by a negative one with an empty ramification, at first above L 34 then above L 33 . At last, intervention I 4 ends by negating the factual proposition 'The parties concluded a valid sales contract', this is represented by a positive dialogue act focused on L 32 and with a singleton as ramification. The argumentative dialogue until this fourth intervention is represented Fig. 4.

I4 L3 1 . . . . L4 3 L3 2 L4 2 ∅ L3 3 L4 1 ∅ L3 4 L2 L1 L0 . . . . L4 3 L3 2 L4 2 L3 2 ∅ L3 2 ,L3 3 L4 1 L3 2 ,L3 3 ∅ (L3 1 ,)L3 2 ,L3 3 ,L3 4 L2 L1 L0 Plaintiff Defendant

Other Speech Acts

Providing an information or asking a wh-question may also be represented. In such cases, dialogue acts must be combined with operations on a knowledge base.

• When a locutor gives an information i, she not only produces a dialogue act (+, L i , {0}, v), by means of which she expresses the value v of the information, but also she relocates this information from her knowledge base to the locus of the dialogue.

• When a locutor asks a wh-question, she not only formulates her question but she is also ready to receive and register an answer. Even if the speech act starts with a first positive dialogue act (+, ξ, {0}, qwh), where qwh is the expression of the question, it is not reduced to the dialogical expression. The speech act continues with other internal calculi. The previous dialogue act is followed by all the possibilities to receive an answer and then to register it by means of a fax.

We develop in the next section a way to compute inferences5 by means of internal calculi performed by locutors.

Inferences in Dialogues

In dialogues, interactions occur not only at the surface level, i.e. the distribution of turn of speech, but also between the contents of interventions or with knowledge. We focus in this section on inference modelling. We use such a term to refer to calculi that make explicit elements that are exchanged during a dialogue.

For that purpose, we have to give an account for the state of knowledge (dialogue occurring for exchanging information) or the state of commitments (argumentation dialogue) for each locutor and which both constitute the context of the dialogue. We give an account for these contextual elements by means of two context (of dialogue) states: one for each locutor. These context states are increased during the dialogue by elements arising from the dialogue. Then a dialogue between two protagonists S and A is represented similarly to, say, [START_REF] Gazdar | Speech act assignment[END_REF][START_REF] Ginzburg | Integrating conversational move types in the grammar of conversation[END_REF] or [START_REF] Prescott | Process and policy: Resource-bounded nondemonstrative reasoning[END_REF], that is by a sequence of dialogue states. The specificity here is the definition of the dialogue state: it is a triple of data, the context state of S, the context state of A, and the net of two designs associated with the current dialogue.

Information Exchanges

By means of the notion of context state, we may complete the account of speech acts performed during a dialogue with exchange of information. The dialogue acts that we pointed in section 2.2.3 are only the publicly useful parts of more complex elements contained in (private) context states.

• To ask a wh-question. In such a speech act, the locutor not only formulates her question but she is ready to receive and register an answer. The full speech act (see Fig. 5) may be taken into account by a design which contains not only the dialogue act corresponding to the formulation of the question but also the inference tools to compute the reception and the registration of possible answers. The base of such a design contains two positive loci: one for anchoring the dialogue act and one for registering the answer. It is built as follows: The speech act starts with a first positive dialogue act κ 1 = (+, L 0 , {wh}, qwh), where qwh is the expression of the question, something like What is the value of i ?. It is followed by all the anticipation of possible answers, i.e. several negative actions, namely the (-, L wh , {k})'s for each value k of the data 6 . Then the design contains several actions which enable to register on another locus L i the value v of the data expected. Indeed when a sequence of dialogue act

L k0 ∅ L k0=data L k0 ∅ L val=k0 , L k0 L val L k0 L k0 , L i . . . L kn ∅ L kn=data L kn ∅ L val=kn , L kn L val L kn L kn , L i L wh L i L 0 , L i κ 1
(+, L k , {0}, on -data -i)(L val , {0}, )(+, L val=v , ∅, val = v)
is given by the other locutor, the questioner perfoms internally the interaction between this answer-design and his question-design and the result is exactly the design corresponding to the information the value of the data i is equal to v, that is the following design: (+, L i , {0}, thedatais)(-, L wh , )(+, L v , ∅, equal -to -v).

Argumentative Disputes

In case of controversies, we model the commitment states of each locutor; that is the part of context states consisting in the statements which are publicly assumed by each locutor (often as propositions). Indeed, one of the issue of an argumentation dialogue is to evaluate the correctness of such commitments and to eventually update them. The commitment state is composed by commitment elements (C.E.). Here, for simplicity reasons, we only consider them occurring as factual propositions and inferential propositions. Moreover we give a minimal account of a factual proposition 'F ': a C.E. reduced to a unique design based on L F . The minimal account of an inferential proposition 'F implies G' is a C.E. also reduced to a unique design based on L F L G 7 . The management of commitment states, their updating and their use in the dialogue in progress make use of the logical aspect of Ludics. We give below an account of the negation of propositions.

• Negation. We distinguish between a positive factual proposition: something is the case and a negative factual proposition: something is not the case. The C.E. associated with the negative factual proposition is the design which expresses the logical negation of the positive one. More precisely the following designs are respectively associated with the factual propositions 'F is the case' and 'F is not the case', when F is an atomic factual proposition (L F , resp. L notF , is the location of F , resp. not F ):

L F ∅ † L F ∅ L notF
Factual propositions that are logically decomposable may also be represented as C.E. In such cases designs start with actions corresponding to the adequate logical decomposition.

As an example, the following designs are respectively associated with the factual propositions: 'the conjunction of F and G is the case' (located at L F andG ) and 'the conjunction of F and G is not the case' (located at L not(F andG) ):

L F ∅ L notF L G ∅ L notG L F andG † L F ∅ L notF L G L notF ∅ L notF , L notG L F andG L not(F andG) † L G ∅ L notG L F L notG ∅ L notF , L notG L F andG L not(F andG)

Remarks :

7 Notice that we do not suppose that a factual proposition is necessarily a logical formula. In Ludics the notion of logical formulas is replaced by the more general notion of behaviour, i.e. a closed set of designs with same base. Here we follow a still more general setting by putting in a commitment state some sets of designs not necessarily closed. Even if we do not completely exploit this in this paper, it should give the opportunity of a fine-grained account of knowledges. Indeed some knowledges are assumed as fully explicit propositions while others may be only partially assumed: one or several logical propositions may be associated with a given utterance without their full (until atomic ones) decompositions being known.

1. Loci are indexed by their intuitive interpretation. These addresses have not to be confused with the logical notation of propositions.

2. For a strictly commutative interpretation of the conjunction, we need both the two designs (in the right in previous figure) in the C.E. associated with the conjunction of F and G is not the case.

Let us illustrate how such an argumention dialogue modelling works on the example presented in section 2.2.2: In the juridic dialogue, intervention I 3 results from a logical calculus. Let us detail how we interpret this intervention with respect to the commitment state of plaintiff P . Plaintiff gives several informations: there is a valid contract between P and D, P delivered, and P applies the law r 1 , the implicit content of r 1 is 'If there is a valid sales contract between a vendor and an emptor and if the vendor delivers then the emptor has to pay'. The following C.E. are added to the commitment state of P by this third intervention:

• The design associated with the law r 1 is based on L Cont-and-Del L to.pay . This is a rather elaborated design: Then the interaction between this latter design and D not.paid results in a logical contradiction.

The following of the dialogue will consist in giving arguments until the contradiction may be solved.

Conclusion

Ludics is a theory of Logic however its objects are more general than formulas and proofs. This possibility to move around the logical frontier is interesting for studying argumentation as done in [START_REF] Fouqueré | Un cadre formel issu de la théorie de la démonstration pour la théorie de l'argumentation[END_REF]. In particular, we are not obliged neither to associate a logical formula to each dialogical intervention nor to grasp the streams of arguments by means of already logically correct inferences. Since we may separate the dialogical dimension, according to which the streams of arguments take place, and the inferential dimension, we may grasp one of the main issue of controversies: go towards the elucidation of a statement which was not a priori given and take into account correctness/incorrectness of asserted deductive inferences without requiring non monotonic logics. Finally, such a framework enables to model cross-effects between the various levels of dialogue analysis as one kind of objects, i.e. designs, may represent dialogues viewed as interacting processes as well as explicitation processes, or in other words as interaction is at the core of Ludics and gives an account for cut as well as cut-elimination. Our future research will focus on improving the relation between surface vs pragmatical and rhetorical levels, and on extending this model to other levels of dialogue analysis.

Figure 2 :

 2 Figure 2: A divergent dialogue about Kpelle.

Figure 3 :

 3 Figure 3: A juridical example: Interventions I 1 to I 3

Figure 4 :

 4 Figure 4: A juridical example: Intervention I 4

Figure 5 :

 5 Figure 5: Speech act for a wh-question

L•

  Del L to.pay ∅ L notDel , L to.pay L Cont L notDel , L to.pay ∅ L notCont , L notDel , L to.pay L Cont-and-Del L to.pay• The designs respectively associated with the assertions 'there is a valid contract between P and D' and 'P delivered' are the following:D cont = L Cont ∅ D del = L Del∅ There is also in the commitment state of P the following design associated with the fact that defendant did not pay (interpretation of intervention I 1 ):D not.paid = † L to.pay ∅ L notP ayLoci L 31 , . . . , L 34 respectively associated with the argumentative elements in intervention I 3 are delocations of loci on which the four designs D r1 , D cont , D del and D not.paid are anchored8 . A computation using these designs yields a contradiction: indeed, the result of the interaction between designs D cont , D del and D r1 (by using additional logical steps) is the following design: D to.pay = L to.pay ∅

Obviously, since we cannot consider literally simultaneously a proof of A and a proof of ¬A, the notion of design relaxes the one of proof.

A special positive action, which is not justified and which necessarily ends paths.

Note that an initial claim may also be a negation. There is a difference between beginning a dialogue with, e.g., "Paul did not come yesterday" and saying "You are wrong. Paul did not come." as a reply to "Paul came to the party yesterday."

Making facts explicit as well as making deductive inferences.

Infinite number of answers may be taken into account nicely using Terui's c-designs[START_REF] Terui | Computational ludics[END_REF]

We may notice by this way the continuity from internal computations to the interactive level of a dialogue.

 [START_REF] Asher | Logics of Conversation[END_REF]This work was completed with the support of the ANR project LOCI number 0212 02.