Paul Sethares

William A Klein

Andrew G And Brown

Anh Hoang

Philip Klausmeyer

Herwig Wendt Roux

C Richard ( Johnson

Paul Messier

William A Sethares

Andrew G Klein

Christopher Brown

Patrice Abry

Laurens Van Der Maaten

Eric Postma

James Coddington

Ann Lee

Daffner

Hanako Murata

Henry Wilhelm

Sally Wood

Mark Messier

PURSUING AUTOMATED CLASSIFICATION OF HISTORIC PHOTOGRAPHIC PAPERS FROM RAKING LIGHT PHOTOMICROGRAPHS

ABSTRACT

Surface texture is a critical feature in the manufacture, marketing and use of photographic paper. Raking light reveals texture through a stark rendering of highlights and shadows. Though raking light photomicrographs effectively document surface features of photographic paper, the sheer number and diversity of textures used for historic papers prohibits efficient visual classification. This work provides evidence that automatic, computer-based classification of texture documented with raking light photomicrographs is feasible by demonstrating an encouraging degree of success sorting a set of 120 photomicrographs made from samples of historic silver gelatin paper.

Using this dataset, four university teams applied different image processing strategies for automatic feature extraction and degree of similarity quantification. All four approaches were successful in detecting strong affinities among similarity groupings built into the dataset as well as identifying outliers. The creation and deployment of the algorithms was carried out by the teams without prior knowledge of the distributions of similarities and outliers. These results indicate that automatic classification of historic photographic paper based on texture photomicrographs is feasible and should be aggressively pursued. To encourage the development of other classification schemes, the 120 sample "training" dataset used in this work is available to other academic researchers at www.PaperTextureID.org.

TEXTURE IN PHOTOGRAPHIC PAPER

Texture is a defining attribute of photographic paper. Starting in the early 20th century, manufacturers manipulated texture to differentiate their products and to satisfy the aesthetic and functional requirements of photographers. Especially prior to WWII, when black and white silver gelatin paper was the dominant photographic medium [START_REF] Messier | Les Emulsion Industrielles[END_REF], dozens of manufacturers worldwide produced a wide array of surfaces. From this period a book of specimen prints by the Belgian company Gevaert lists twenty five different surfaces made up of combinations of texture, reflectance, color and paper thickness (Gevaert Company of America c.1935). Around the same time, a sample book from the Defender Company of Rochester New York lists twenty seven surfaces (Defender Photo Supply Company c.1935), Mimosa twenty six (Mimosa AG c.1935) and Kodak twenty two (Eastman Kodak c.1935). Each listed surface was proprietary to the different manufacturers and each was used across their multiple brands of paper with changes, additions, and deletions occurring over a span of many years.

A vital factor in the evaluation of paper surface, texture impacts the visibility of fine detail and thus provides insight into the intent of the photographer and the envisioned purpose of a particular print. For example, prints made for reproduction or documentary functions tend to be better suited to smooth-surface papers that render details with sharpness and clarity whereas more impressionistic or expressive subjects, especially those depicting large unmodulated masses of shadows or highlights, are best suited for papers with rough, broadly open textures (Eastman Kodak Company c.1935). A result of a careful and deliberate manufacturing process, texture applied to silver gelatin paper is designed to be distinct and distinguishable through processing and post-processing procedures. Likewise manufacturer-applied texture endures despite localized defects such as abrasions and deterioration caused by poor handling, storage environment and enclosures. Given these attributes, an encyclopedic collection of surface textures could reveal vital clues about a photographic print of unknown origin. Likewise a method for classifying textures could provide a means to link prints to specific photographers or to other prints of known provenance.

Previous work [START_REF] Parker | Automating Art Print Authentication Using Metric Learning[END_REF]Messier, Messier and Parker 2010) not only established the practicality of the image data collection procedure described in the next section, but also suggested that more sophisticated approaches to automated classification would yield sufficient levels of success such that the ultimate aim of developing a highly reliable texture matching methodology might be achieved. This paper heads in that direction.

TEXTURE IMAGE PREPARATION

Sample papers dating from 1908 to 1977 were selected from a large reference collection of photographic paper. Each sample was identified by manufacturer, brand, date, and manufacturer-assigned surface designation. The reference collection and the methods used to identify the samples have been described elsewhere (Messier et al. 2004;[START_REF] Connors-Rowe | Optical Brighteners in Black-and-White Photographic Paper: Appearance and Degradation[END_REF]).

The texture images were acquired with a microscope system assembled using an Infinity 2-3 imager manufactured by the Lumenera Corporation fitted with an Edmund Optics VZM 200i lens, as shown in Figure 1. The imager incorporates an Interline Sony ICX262 3.3 megapixel color progressive scan CCD sensor producing images that incorporate 1536 x 2080, 3.45µm, square pixels. The imaged area on each sample measured 1.00 x 1.35 cm. Raking light photomicrographs were made using a fixed point illumination source using a 3 inch LED line light manufactured by Advanced Illumination placed at a 25 • raking angle to the surface of the photographic paper. Each raking light photomicrograph generated a 16-bit TIFF. Typical samples are shown in Figure 2. The image capture technique is non-contact / non-destructive and therefore easily adapted for use on photographic prints of high intrinsic value.

COLLABORATIVE COMPETITION

As part of a materials-based characterization project of modernist photographs at The Museum of Modern Art (MoMA), raking light photomicrographs were made from each print from the Thomas Walther Collection to document surface texture. This work stimulated interest in developing an automated scheme to cluster like prints based on surface texture. An appeal was made to university teams with signal processing experience to initiate a collaborative competition to develop methods for sorting texture images.

Four university teams joined this project:

• University of Wisconsin: Sethares

• Worcester Polytechnic Institute: Klein, Brown, Do, and Klausmeyer

• Ecole Normale Superieure de Lyon: Abry, Jaffard, Wendt, Roux, and Pustelnik • Tilburg University: van Noord, van der Maaten, and Postma Each team adopted a different approach to the development of the two standard parts of an automatic classifier: (1) feature vector extraction and (2) degree of similarity quantification. These strategies stem from a broad variety of basic approaches to texture image classification [START_REF] Haralick | Statistical and structural approaches to texture[END_REF] and are described in the following section.

Prototype algorithms were constructed by the four teams using a training set of 50 samples with some known texture matches. This preliminary work established that the orientation of the primary paper fiber direction relative to the raking light had no significant impact on results (this finding does not exclude a priori that photo paper textures may possess other forms of anisotropy). This work also resulted in the design of a 120 sample dataset of raking light photomicrographs of photographic papers with known metadata including manufacturer, brand, date, gloss, and texture classification, and offering varying degrees of self-similarity (the Appendix lists all samples used in this study). The dataset delivered to the teams for testing was largely composed of nine groups of ten paper samples each. Within these groups, there were three similarity subsets: (1) images made from the same sheet of paper, (2) images made from sheets taken from the same manufacturer package of paper and, (3) images from papers made to the same manufacturer specifications over a period of time. The remaining thirty samples were picked without concern for texture similarity but instead were selected to span the range of textures associated with historic silver gelatin paper.

Conventional wisdom suggests that any raking light photomicrograph taken from different spots on a single sheet of paper would appear nearly identical. Likewise, texture images from different sheets of paper taken from the same manufacturer package also should show strong similarity. Furthermore, shots from papers manufactured to the same specifications but made at different times should show strong similarity, but to a somewhat lesser degree. For the thirty remaining samples, selected to demonstrate diversity, some would appear similar to the group of ninety textures and some would appear to be unique. The challenge posed to the teams was to discover these similarity groupings and isolate unique textures by producing a system of texture affinities that described the entire set.

TECHNICAL APPROACHES

The approaches taken by the four teams can be divided into two categories [START_REF] Haralick | Statistical and structural approaches to texture[END_REF][START_REF] Gonzalez | Digital Image Processing[END_REF] based on the approach to feature definition: (1) non-semantic / Wisconsin and Tilburg and (2) multiscale / Lyon and WPI. The fundamental difference is that non-semantic features are derived directly from the image data where multiscale features are based on a structural model presupposed as relevant to the encountered data.

EIGENTEXTURES (WISCONSIN)

In the eigentexture approach, a collection of small patches are chosen from each photographic image. These patches are gathered into a large matrix and then simplified to retain only the most relevant eigendirections using a singular value decomposition (SVD) [START_REF] Moon | Mathematical Methods and Algorithms for Signal Processing[END_REF]. The preparation stage consists of two steps:

1. For each imaged paper j, randomly pick N p × p pixel patches X j,i ∈ R p×p for i = 1, 2, ..., N (with N = 2000 and p = 25 in this case). Lexigraphically reorder the X j,i into column vectors a j,i ∈ R p 2 .

2. Create matrices A j = [a j,1 a j,2 ... a j,N ] consisting of the N column vectors and calculate the SVDs A j = U j ΣV j for all j. Extract the m columns of U j corresponding to the m largest singular values and call this U j (with m selected as 15 in this case).

The U j are the representatives of the classes and may be thought of as vectors pointing in the most-relevant directions. During the classification stage, a number of similarly-sized patches are drawn from the tested photographic paper.

Each of these patches is compared to the representatives of the classes via a least squares (LS) procedure.

3. Select Q (with Q = 2000 used here) p × p pixel patches Q i from the tested paper and reorder into vectors q i ∈ R p 2 . Calculate the distance from the ith patch to the jth class:

d(i, j) = ||q i -U j (U T j q i )|| 2 .
Every patch is closest to one of the classes, and the number of patches closest to the jth class is recorded.

4. For each patch i, f i = argmin j d(i, j) locates the smallest of the d(i, j), indicating that class j is the best fit for patch i. Tally the set of all such f i , i = 1, 2, ..., Q.

The commonest entry among the f i is the most likely class for this image. The second most common entry is the next most likely class for this image, etc.

RANDOM-FEATURE TEXTRON METHOD (TILBURG)

This method combines random features and textons, i.e., the random-feature texton method. This method was developed by Liu and Fieguth [START_REF] Liu | Texture classification from random features[END_REF] and is an adaptation of the texton approach (Varma and Zisserman 2009) using random features. Textons are prototypical exemplar image patches capturing the "essence" of the texture of an image. Random-features (RF) are random projections of image patches with N × N pixels to vectors with D elements (N = 9, D = 20, D < N × N ). More specifically, a random feature (RF) is defined as a D × N 2 matrix, the elements of which are sampled from the standard multivariate normal distribution N (0, 1).

The application of the random-feature texton method on the 120 sample dataset is conducted as follows. A set of X sub-images of M × M pixels is selected for each gray-value texture image in the 120 sample dataset (M = 512). The sub-images are defined to be the central regions of M × M pixels of which the intensity distributions are normalized to zero mean and unit variance. A sample of 45,000 randomly selected N × N (N << M ) patches (represented as vectors of length N 2 ) of the normalised sub-images are contrast-normalised and subsequently multiplied with RFs, yielding RF vectors of length D.

Subsequently, a texton dictionary is created by applying k-means clustering to all RF vectors of the X sub-images of each texture image of the 120 sample dataset. Each image of the dataset is transformed into a texture histogram by comparing all of its patches (represented as RF vectors) to the entries in the texton dictionary. Finally, the histograms are classified using a k-nearest neighbour algorithm using the χ 2 similarity measure.

ANISOTROPIC WAVELET MULTISCALE ANALYSIS (LYON)

This method relies on the use of the Hyperbolic Wavelet Transform (HWT) (De Vore et al. 1998;Roux et al. 2013), which is a variation of the 2D-Discrete Wavelet Transform (2D-DWT) (Mallat 2008). The HWT explicitly takes into account the possible anisotropic nature of image textures. Indeed, instead of relying on a single dilation factor a used along both directions of the image (as is the case for the 2D-DWT), HWT relies on the use of two independent factors a 1 = 2 j 1 and a 2 = 2 j 2 along directions x 1 and x 2 respectively. The Hyperbolic Wavelet coefficients of imaged paper i, denoted as T i ((a 1 , a 2 ), (k 1 , k 2 )) are theoretically defined as:

T i ((a 1 , a 2 ), (k 1 , k 2 )) = i(x 1 , x 2 ), 1 √ a 1 a 2 ψ( x 1 -k 1 a 1 , x 2 -k 2 a 2 ) .
From these HWT coefficients, structure functions, consisting of space aver-ages at given scales a 1 , a 2 , are defined as :

S i ((a 1 , a 2 ), q) = 1 n a k |T i ((a 1 , a 2 ), (k 1 , k 2 ))| q ,
where n a stands for the number of T i ((a 1 , a 2 ), (k 1 , k 2 )) actually computed and not degraded by image border effects.

To measure proximity between two images i and j, a cepstral distance between their structure functions S i ((a 1 , a 2 ), q) and S j ((a 1 , a 2 ), q) is computed. It consists of a classical L p norm computed on log-transformed normalized structure functions: d(i, j) = a | Si (a, q) -Sj (a, q)| p 1 p , with Si (a, q) = ln S i (a,q) a S i (a ,q) .

PSEUDO-AREA-SCALE ANALYSIS (WPI)

Area-scale analysis is a technique which has been applied to various problems in surface metrology (Brown et al. 1993). Much as the measured length of a coastline depends on the scale of observation and therefore the resolvability of small features, the measured area of a surface is also a function of the scale of observation. The area-scale approach uses fractal analysis to decompose a surface into a patchwork of triangles of a given size. As the size of the triangles is increased, smaller surface features become less resolvable and the 'relative area' of the surface decreases. The topological similarity of two surfaces is computed by comparing relative areas at various scales. The technique has traditionally been employed on topographic data sets containing height information over a surface. Though lacking a direct measure, area-scale analysis can be applied to the photomicrographs using light intensity as a proxy for height.

The proposed approach proceeds in three steps: (1) preprocessing, ( 2) feature extraction, and (3) classification. The preprocessing step extracts a square N ×N region from the center of the image (where N was chosen to be 1024), and normalizes the intensity of the resulting extracted image. The N × N grid of equally spaced points (representing pixel locations) is decomposed into a patchwork of 2( N -1 s ) 2 isosceles right triangles where s is a scale parameter representing the length of two legs of each triangle. The pixel values at each of the triangle vertices are then taken as the 'pseudo-height' of each of the vertices. The area of each triangle in 3-D space is then computed and the areas of all triangular regions are summed, resulting in the total relative area A s at the chosen scale s. To conduct feature extraction, the relative area for an image is computed over a range of scale values; in this study, 8 scale values were used ranging from 1 pixel to 34 pixels, which correspond to lengths of 6.51 µm to 0.221 mm, respectively. Finally, to classify and compare the similarity of two images i and j, a χ 2 distance measure d(i, j) is computed via

d(i, j) = s∈S A (i) s -A (j) s 2 A (i) s + A (j) s where A (i)
s is the relative area of image i at scale s and S is the set of chosen scale values. Small values of d(i, j) indicate high similarity between images i and j, while large values indicate low similarity.

RESULTS AS AFFINITY MAPS

From the metadata and each teams' automatic classifiers the degree of similarity (affinity) was tabulated for each possible pairing of images in the 120 sample dataset. These scores were then converted to a grey-scale with the darkest intensities indicating the greatest affinity and the lightest the least affinity.

To visualize these values a table containing 120 rows and 120 columns was created, one row and column for each sample in the data set. Each of the resulting 14,400 cells in the table was shaded according to the similarity of compared samples with black describing an exact match, white a total mismatch and gray-scale values in between describing a range of better or worse similarities. For example, the top diagram in Figure 3, shows predicted similarities within the sample group suggested by the metadata including manufacturer, texture, brand, and date. As expected, the nine dark blocks starting in the upper left and continuing down along the diagonal show a high degree of affinity (dark gray and black) as these blocks depict the nine groups of similar textures. Lesser degrees of similarity are scattered throughout the figure with the 30 samples selected to show diversity (poorer levels of similarity) falling in the lower right quadrant and along the right side and bottom edge.

Gray-scale affinity maps produced to display the results from each of the four teams are also shown in Figure 3. The principal similarity among the five affinity maps in Figure 3 are the nine dark squares along the upper left to lower right diagonal. Given the construction of the dataset, these blocks should be dark due to the high affinity of the samples in these groups. The light stripes in the right and bottom quarters of the affinity maps, due to some relatively matchless textures among samples 91-120, are also shared by all five affinity maps. While small local differences among the five maps indicate that work remains to find an ideal automated scheme, striking fundamental similarities between the metadata-based affinity map and the four produced by automated schemes validate raking light photomicrographs as having sufficient texture information to support the automated classification of historic photographic papers.

OBSERVATIONS

As shown in Figure 3 there is a relatively high level of agreement between the affinity pairings prepared by the classification algorithms and those derived from metadata and subject-matter expertise. As discussed in the previous section, the principal correspondence among the five affinity maps is the nine dark squares along the diagonal running from upper left to lower right. Given the construction of the dataset, the samples in these blocks are very similar and these texture affinities were recognized both by a subjective metadata sort and by the four automated solutions. In addition, both 'human' and automated solutions are sensitive to the increased levels of diversity within samples 81-90 (ninth dark block on the diagonal) that track a manufacturer's surface over a fairly extended period of twenty two years. Besides the nine similarity groupings added to the dataset by design, both the human observer and the automated solutions discovered another strong affinity between subsets 11-20 and 31-40 (shown along the cross diagonal axis adjoining the third dark square on the diagonal). As shown in the Appendix, these samples have the same manufacturer, brand, surface designation and date but are taken from different paper packages. These findings are reinforced by Figure 4 which shows a normalization of the distances between each texture pairing within the tested groups. The shape of the curves are remarkably consistent with the automated solutions and the human observer detecting very similar degrees of affinity across the groups. The chart confirms there is no measurable difference between texture images made from the same sheet of paper as compared to images made from different sheets from the same manufacturer package. Further, depending on the technique, textures within the same manufacturing standard produced over time show fair to good levels of similarity. These results, though not a surprise given high levels of manufacturing regularity, are important for the possible development of future systems that rely on indices of known 'exemplar' textures to identify unknowns.

CONCLUSIONS AND NEXT STEPS

This project opens a path toward a machine vision system that provides meaningful results for the study of photographic prints. To have meaning, an automated classification system cannot produce results simply based on an internal, self-referential 'sameness/difference' parameter but instead must render results that are relevant to trained practitioners such as conservators and curators. For example, the photomicrographs made from ten spots on the same sheet of paper, though totally different images, need to be recognized as the 'same'. Likewise the two other similarity groups made from different sheets from the same manufacturer package and from papers manufactured to the same standard must be recognized as related. A useful system needs to reliably cluster these groups together though be discriminating enough to set these groups apart from others that might have similar textures but, for example, are made by different manufacturers. Using different techniques, each of the four teams met this standard. The fundamental outcome of this experiment is the intuitive 'human / expert observer' conception of a classification system based on sameness / difference can be replicated through imaging and signal processing techniques.

The techniques described in this work could engender new modes of scholarship based on the discovery of materials-based affinities. Work at MoMA is underway to determine how these techniques might meaningfully be applied to prints in its Thomas Walther collection. Moving forward, reference libraries of surface textures, containing papers grouped by photographer or paper manufacturer can be assembled and used as a basis of comparison. This work is already underway through the assembly of large paper reference collections categorized by manufacturer, brand, surface finish and date as well as for individual artists including Man Ray (1890-1976) and Lewis Hine (1874-1940). With standardized imaging techniques and a networked infrastructure, conservators could query such texture libraries to detect similar papers held by other collections, potentially characterizing and identifying works in their collection as well as revealing relationships within an artist's body of work and between artists. These methodologies are being applied to other media, including ongoing work with ink-jet papers and the platinum papers of F. Holland Day (1864Day ( -1933)).

A website, www.PaperTextureID.org, has been created to distribute the dataset of silver gelatin textures used as the basis for this study. In addition, an ink-jet paper dataset composed using similar specifications is also posted at this site. The availability of these image sets should encourage and assist other teams to develop their own automated classification and sorting schemes.

Paul Messier is an independent conservator of photographs working in Boston Massachusetts, USA. Founded in 1994, his studio provides conservation services for private and institutional clients throughout the world. The heart of this practice is unique knowledge and ongoing research into photographic papers. The Messier Reference Collection of Photographic Papers plays a vital role in this work. Messier is the corresponding author. Address: 103 Brooks Street, Boston, MA 02135. Email: pm@paulmessier.com

Figure 1 :

 1 Figure 1: Raking-light photomicrograph acquisition

Figure 3 :

 3 Figure 3: Top: Affinities (dark: strong, light: weak) from metadata and expert domain knowledge. Bottom Quartet: Wisconsin (upper left), Tilburg (upper right), Lyon (lower left), WPI (lower right).

Figure 4 :

 4 Figure 4: Normalized image pair distances for the dataset of texture images (Diagonal blocks 1-3 same sheet, 4-6 same package, 7-9 same manufacturing standard and, 10-12 diverse samples)

ACKNOWLEDGEMENTS

The authors wish to thank: Jill Sterret and Theresa Andrews of the San Francisco Museum of Modern Art for providing a meeting venue during the summer of 2012, Andrew Messier, Lincoln Laboratories, Massachusetts Institute of Technology, for help designing the imaging system and Ian Holland, Lumenera Technical Assistance Center, for assistance with imaging specifications.

Appendix: PAPER SAMPLES USED IN THE SILVER GELATIN DATASET

For the following tables ID is the sequential numbering system suggested by the teams following image processing. Date refers to the paper expiration dates applied to manufacturer packages or estimates made based on packaging, M ID is the Messier Reference Collection catalog number. Other descriptors, such as brand and paper characteristics are taken directly from the manufacturer packaging.


Benoit Gaudou
email: benoit.gaudou@alumni.enseeiht.fr

Andreas Herzig
email: herzig@irit.fr

Dominique Longin
email: longin@irit.fr

Group belief and grounding in conversation

, and augments it by a modal operator expressing groundedness in a group. We show that groundedness for the single-agent group {i} corresponds to belief of i. Thus a particular individual belief operator is superfluous. We neither develop here temporal aspects nor dynamics between action and mental attitudes.

Introduction

When we study dialogue between artificial agents, two traditions have emerged: a subjective one with for example [START_REF] Cohen | Rational interaction as the basis for communication[END_REF][START_REF] Rao | Modeling rational agents within a BDI-architecture[END_REF][START_REF] Rao | An abstract architecture for rational agents[END_REF][START_REF] Sadek | A study in the logic of intention[END_REF]] and an objective one with [START_REF] Conte | Cognitive and Social Action[END_REF][START_REF] Walton | Commitment in Dialogue: Basic Concepts of Interpersonal Reasoning[END_REF][START_REF] Munindar | A Social Semantics for Agent Communication Languages[END_REF][START_REF] Fornara | Operational Specification of a Commitment-Based Agent Communication Language[END_REF][START_REF] Verdicchio | A Logical Model of Social Commit-ment for Agent Communication[END_REF]4].

The former approach, often named mentalist approach, considers that dialogue is function of the agents' mental attitudes, usually formalized with BDI (Belief Desire Intention) modal logics. The speech acts that an agent can perform depend on his mental attitudes and are seen as modifying cognitive state of agents. Its most important application is the definition of FIPA-ACL (Agent Communication Language) semantics [START_REF]FIPA Communicative Act Library Specification[END_REF], the most used Agent Communicative Language. This approach is also based on the recognition of mental attitudes (in particular intention). Linguistically, statement significance does not depend only on the statement but also on the speaker's intentions. That is what Grice names non-natural significance [START_REF] Grice | Meaning[END_REF]. The example of indirect speech acts is the most obvious. But nothing proves that the hearer well recognizes the correct mental state and the speaker has no warranty to have been correctly understood (even if the communication is perfect).

The mentalist approach has a lot of predictive power (dialogue can be viewed as a driven by intentions and that can be planed [6,1]). But in exchange, it needs very strong hypotheses on agents' behavior: e.g. sincerity, cooperation... This approach has often been criticized (see e.g. [START_REF] Munindar | A Social Semantics for Agent Communication Languages[END_REF][START_REF] Fornara | Operational Specification of a Commitment-Based Agent Communication Language[END_REF]) because hypotheses are too strong in open and heterogeneous multi-agents systems (MAS for short). Moreover no verification on the dialogue can be performed because it would need to access to private mental attitudes of agents, and such hypothesis is not realistic in open MAS.

To get round this problem objective approaches, also called conventional or structural approaches, take into account only what is public in the dialogue. The semantics of speech acts is described in terms of commitments, i.e. propositions that are registered in the commitment store, a kind of public black board for every agent taking part in the dialogue. This approach is much more descriptive in particular because commitments are not related to private mental attitudes. But it does not need hypotheses on agents anymore: agents can be insincere, uncooperative and based on approaches other than the BDI paradigm, in particular they may have no intentions at all (such as automata used in most nowadays "dialogue systems").

The first aim of this paper is to bridge the gap between these two approaches by extending a BDI-like logical framework with an operator formalizing what is public in the dialogue.

We will use a small example to illustrate some properties of the notion we will study. Consider three rational agents in a company. The agent 0 thinks privately for some reasons that his boss (agent 2) is smart. But this idea is not widespread in his department: agent 0 meets agent 1, a very charismatic agent who often claims publicly that his boss (agent 2) is dumb. They discuss about their boss and agent 0 asserts that he is really a moron (for some social reasons) and of course agent 1 confirms. At this moment the boss comes and enters the conversation. Soon he oriented the discussion on himself and agent 0 congratulates him by asserting he is smart. And agent 2 and agent 1 (given the boss' attendance) express their agreement.

It is interesting to see that agent 0 expressed fully different points of view depending on his hearers. As we are interested in what is public in a dialogue, we have to make precise which group of agents constitutes the public because a speaker's behavior depends on who can hear what he says as illustrated above. Thus the second aim of this paper is to formalize such a dialogue with different groups of hearers. Firstly we will philosophically ground this notion and define its major features (Section 2). Then we will formalize it by introducing it into a logic of belief, choice and action (Section 3). Afterwards we will show some applications of our new notion. We can apply it to formalize commitments and dialogue games à la Walton & Krabbe (Section 4). Finally we will show its link with group belief (Section 5).

The present paper is part of Benoit Gaudou's PhD thesis ``Formalizing social attitudes in modal logic'' (PhD thesis of the University of Toulouse, July 2008).

Notion of Grounding

We name our central notion grounding 1

Philosophical foundations

in reference to works of Traum [START_REF] Traum | Computational theory of grounding in natural language conversation[END_REF]. He defined it as "the process of adding to the common ground between conversational participants". We use it in a more general sense: for us grounding refers to what is public in the dialogue. We give a more precise definition below.

Our notion stems from speech act theory, where Searle's expression of an Intentional state [START_REF] Searle | Intentionality: An essay in the philosophy of mind[END_REF] concerns a psychological state related to the state of the world. Even if an utterance was insincere an Intentional state has been expressed, and that state corresponds to a particular belief of the speaker in some way.

Vanderveken [START_REF] Vanderveken | Principles of language use, volume 1 of Meaning and Speech Acts[END_REF][START_REF] Vanderveken | Formal semantics of success and satisfaction, volume 2 of Meaning and Speech Acts[END_REF] has captured the subtle difference between expressing an Intentional state and really being in such a state by distinguishing success conditions from non-defective performance conditions, thus refining Searle's felicity conditions [START_REF] Searle | Speech acts: An essay in the philosophy of language[END_REF][START_REF] Searle | Expression and Meaning. Studies on the Theory of Speech Acts[END_REF][START_REF] Searle | Foundation of illocutionary logic[END_REF]. According to Vanderveken, when we assert p we express that we believe p (success condition), while the speaker's belief that p is a condition of non-defective performance.

The notion of groundedness is also behind Moore's paradox, according to which one cannot successfully assert "p is true and I do not believe p". The paradox follows from the fact that: on the one hand, the assertion entails expression of the sincerity condition about p (the speaker believes p); on the other hand, the assertion expresses the speaker believes he ignores that p. If we accept introspection then this expresses that the speaker does not believe p, and the assertion is contradictory (if we accept that beliefs are consistent).

Definition and features

We view grounded information as information that is publicly expressed or accepted as being true by all the agents participating in a conversation.

A piece of information might be grounded even when some agents privately disagree, as long as they do not publicly manifest their disagreement.

If we consider the above example, by asserting that his boss is a moron in front of agent 1, agent 0 has expressed that he believes that his boss is a moron (although 0 does not think so) and thus this proposition is grounded for agents 1 and 2. After acceptance by the second agent, it becomes grounded for the group of agents consisting of 0 and 1 that their boss is a moron. The contrary is grounded afterwards for the group of agents {0,1,2}.

Groundedness is an objective notion: it refers to what can be observed, and only to that. It is different from other objective notions such as that of social commitment of [START_REF] Munindar | Agent Communication Languages: Rethinking the Principles[END_REF][START_REF] Munindar | A Social Semantics for Agent Communication Languages[END_REF][START_REF] Fornara | Operational Specification of a Commitment-Based Agent Communication Language[END_REF][START_REF] Verdicchio | A Logical Model of Social Commit-ment for Agent Communication[END_REF]. In our approach we do not try to determine whether j must do such or such action or not: we just establish the facts, without any hypothesis on the agents' beliefs, goals, intentions... or commitments.

As we will show in Section 4, grounding and commitment paradigms are equivalent in the case of an assertion. To see the difference for requests consider the speech act where agent i asks agent j to pass him the salt. Thereafter it is established (if we assume that the speech act is well and completely understood) that i has the intention that j passes him the salt and nothing is grounded for j. In contrary, in a commitment-based approach this typically leads to a conditional commitment (or precommitment) of j to pass the salt, which becomes an unconditional commitment upon a positive reaction, whereas the requester is not committed to the fact that he performed a request.

We make also the hypothesis that grounding is a rational notion, i.e. a proposition and its contrary cannot be grounded in the same group. But this can be the case for two different groups, see the example.

In a previous paper [START_REF] Gaudou | A Logical Framework for Grounding-based Dialogue Analysis[END_REF], we presented a modal logic of belief and choice augmented by the modal operator G to express the notion of grounding. But this operator was a bit too restricted: G φ expresses that φ is publicly grounded, where "publicly" means for all agents. Thus in a given group of agents, we cannot distinguish a private dialogue between two agents from a public debate. In the former a piece of information could be grounded between only two agents and stay secret for the other agents of the group.

In the next section we will present the logical framework of the grounding operator: it is the one of [START_REF] Gaudou | Grounding and the expression of belief[END_REF] augmented with the mutual belief operator.

Semantics

Let AGT = {i, j...} be a finite set of agents. A group of agents (or a group for short) is a nonempty subset of AGT. We use I, J, K... to denote groups. When I' ⊆ I we say that I' is a subgroup of I. Let ATM = {p, q...} be the set of atomic formulas.

Complex formulas are denoted by φ, ψ… A model includes a set of possible worlds W and a mapping V: W → (ATM → {0, 1}) associating a valuation V w to every w ∈ W. Models moreover contain accessibility relations that will be detailed in the sequel.

Grounding.

To each possible world w and each non-empty I ⊆ AGT, we associate the set of possible worlds that are consistent with all propositions grounded in world w for the group I. This set is characterized by the mapping: G: 2 AGT → (W → 2 W ) associating an accessibility relation to each non-empty subgroup of AGT. G I (w) contains those worlds where all grounded propositions hold.

G I φ reads "it is publicly grounded for group I that φ is true" (or for short: "φ is grounded for I"). When I is a singleton, G {i} φ reads "φ is grounded for (agent) i" and G {i} is identified with the standard belief operator Bel i à la Hintikka [START_REF] Hintikka | Knowledge and Belief: An Introduction to the Logic of the Two Notions[END_REF]. We write G i φ for G {i} φ.

The truth condition for G I stipulates that φ is grounded in w, noted w ⊩ G i φ, if and only if φ holds in every world that is consistent with the set of grounded propositions:

w ⊩ G i φ iff w ⊩ φ for every w ∈ G I (w).
We assume that:

[SC1] G I is serial. Thus, groundedness is rational: at least one world that is consistent with the set of grounded propositions exists.

Furthermore we postulate the following constraints on accessibility relations, for groups I and I' such that I' ⊆ I:

[SC2] if u G I′ v and v G I w then u G I w; [SC3] if u G I′ v and u G I w then v G I w; [SC4] if u G I v and v G I w 1 then there is w 2 such that u G I w 2 and • V (w 1 ) = V (w 2 ), • G K (w 1 ) = G K (w 2 ) for all K such that K ∩ I = ∅, • C K (w 1 ) = C K (w 2 )
for all k such that k ∉ I, where C is the accessibility relation for choice to be defined below;

[SC5] G I ⊆ ⋃ i∈I G I ⃘ G i .
Constraint [SC2] stipulates that agents of a subset I' of the set I are aware of what is grounded in the group I: whenever w is a world for which it is grounded for I' that all I-grounded propositions hold in w, then all I-grounded propositions indeed hold in w. This is a kind of attention property: each subgroup taking part in a conversation is aware of what is grounded in the group.

Similarly [SC3] expresses that subgroups are aware of what is ungrounded in the group, too.

[SC2] and [SC3] together make that if

u G I' v then G I (u) = G I (v), i.e. if u G I' v
then what is grounded for I at u is the same as what is grounded for I at v. From [SC2] and [SC3] it also follows that G I is transitive and Euclidian.

[SC4] says that if an information "about something outside group I" (see the definition in the following subsection) is grounded for I then it is grounded for I this information is grounded for every subgroup of I.

[SC5] says that if it is grounded for a set I that a proposition is established for every agent then it is grounded for I, too.

Mutual belief.

From individual belief (i.e. grounding for singleton groups), we define the notion of mutual belief of a group of agents. Semantically we have the mapping MB: 2 AGT → (W → 2 W ) associating an accessibility relation MB I to each I ⊆ AGT. MB I (w) denotes the set of possible worlds compatible with mutual beliefs of the group I. For each group I, MB I is defined as the transitive closure of the set of accessibility relations associated to the I's members beliefs (i.e. G i for each i ∈ I):

[SC6] MB I = (⋃ i∈I G i ) +
MBel I φ reads "it is mutual belief for the group I that φ is true". It means that every member of the group believes individually that φ and that it is mutual belief for the group that φ is true2

Figure 1. Grounding and choices

Choice. Among all the worlds in G i (w) that are possible for agent i, there are some that i prefers. Semantically, these worlds are identified by yet another mapping C : AGT → (W → 2 W ) associating an accessibility relation C i to each i ∈ AGT . C i (w) denote the set of worlds the agent i prefers.

Ch i φ reads "agent i chooses that φ". Choice can be viewed as a preference operator and we sometimes also say that "i prefers that φ". Note that we only consider individual choices, group choices being beyond the scope of the present article.

The truth condition for Ch i stipulates that w ⊩ Ch i φ if φ holds in all chosen worlds: w ⊩ Ch i φ iff w ⊩ φ for every w ∈ C i (w).

We assume that:

[SC7] C i is serial, transitive and Euclidian. 3 (See [START_REF] Herzig | C&L intention revisited[END_REF] for more details about the logic of choice, and the definition of intention from choice.) Choice and grounding. As said above, an agent only chooses worlds he considers possible (see Figure 1):

[SC8] C i (w) ⊆ G i (w).
Hence what is grounded for an agent must be chosen by him, and choice is a mental attitude that is logically weaker than groundedness. 3 The choice operator used is not fully classical. In particular it refers to what Cohen and Levesque named "goal" with stronger properties (Cohen and Levesque only assumed seriality). A more detailed comparison with other choice operators is developed in [START_REF] Herzig | C&L intention revisited[END_REF].

We moreover require that worlds chosen by i are also chosen from i's "grounded worlds", and vice versa.

[SC9] if w G i w' then C i (w) = C i (w').
This constraint means that agent i is aware of his choices.

Action.

Let ACT = {α, β...} be the set of actions. Sometimes we write (i:α) to denote that i is the author of (i.e. performs) the action α. 4The model contains a mapping R: ACT → (W → 2 W ) associating an accessibility relation R α to every α ∈ ACT. R α (w) is the set of worlds accessible from w through the execution of α. Just as Cohen and Levesque we suppose here that there is at most one possible execution of α. Hence R α can also be viewed as a partial function on W.

The formula After α φ reads: "φ holds after every execution of α". As there is at most one possible execution of α, the dual operator Happens α φ ≝ ¬After α ¬φ reads: "α is happening and φ is true just afterwards".

Hence After α ⊥ expresses that α does not happen and Happens α ⊤ that α happens. We often write Happens(α) for Happens α ⊤.

The truth condition is:

w ⊩ After α φ iff w' ⊩ φ for every w' ∈ R α (w)
The formula Before α φ reads: "φ holds before every execution of α". The dual Done α φ ≝ ¬ Before α ¬φ expresses that the action α has been performed before which φ held. Hence Done α ⊤ reads: "α has just happened". We often write Done(α)

for Done α ⊤.

The accessibility relation for Before α is the converse of the above relation R α . The truth condition is:

w ⊩ Before α φ iff w' ⊩ φ for every w' ∈ R α -1 (w).
As said above, we do not detail here the relationship between action and mental attitudes (belief and choice) and refer the reader to [START_REF] Herzig | C&L intention revisited[END_REF]. We only consider here the link between action and grounding.

Action and grounding.

As it is often the case in the structural approaches to dialogue, we consider in this paper that actions are public for attending agents, in the sense that they are completely and soundly perceived by them. For example, when agent i performs an assertive speech act only towards agent j then j will perceive the assertion. If no other agent perceives this action then the attentive group is limited to K = {i, j}, and the action is public for exactly this group. But when agent i performs a speech act towards agent j in front of an assistance L then the attentive group is extended to K = {i, j} ∪ L.

Let α be an action performed by agent i in front of attentive group K (of which i is a member). The hypothesis of public actions (for group K) corresponds to the constraint:

[SC10] R α -1 (w) = ∅ if and only if (G K ⃘ R α -1
)(w) = ∅

Axiomatics

Grounding. The logic of the grounding operator is a normal modal logic of type KD:

(D GI ) G I φ → ¬ G I ¬φ
(D GI ) expresses that grounded information in a group are consistent: it cannot be the case that both φ and ¬φ are simultaneously grounded.

In accordance with the preceding semantic conditions the following logical axioms respectively correspond to the semantic constraints [SC2] and [SC3]. Thus, for each I' ⊆ I:

(SR+) G I φ → G I' G I φ (SR-) ¬ G I φ → G I' ¬ G I φ
The axioms of strong rationality (SR+) and (SR-) express that if a proposition φ is grounded (resp. ungrounded) for group I then it is grounded for each subgroup that φ is grounded (resp. ungrounded) for I. This is due to the public character of the grounding operator. 5The next axiom must be restricted to particular formulas, viz. objective formulas for a group, that we define as follows.

Definition. The set of formulas that are objective for a group I is defined inductively to be the smallest set such that:

• every atomic formula p is objective for I;

• G K φ is objective for I if K ∩ I = ∅, for every formula φ;
• Ch j φ is objective for I if j ∉ I, for every formula φ;

• if φ and φ' are objective for I then ¬φ, φ ∧ φ' are objective for I.

With respect to the semantic constraint [SC4], our third axiom, a weak rationality axiom, stipulates that if I is a subgroup of I and φ is objective for I then:

(WR) G I φ → G I G I' φ
(WR) expresses that if φ is objective for group I and grounded for I then it is necessarily grounded for I that for each subgroup I the formula is grounded.

Note that this does not imply that for every subgroup φ is actually grounded, i.e. (WR) does not entail G I φ → G I' φ. In particular, the fact that φ is grounded for group I does not imply that the members of I believe that φ.

It is very important to note that (WR) concerns only formulas φ that are objective for I. Indeed, if we applied (WR) to some mental states of an agent of the group, we would restrict the agents' autonomy.

For example, when an agent i asserts to another agent j that φ in presence of group I, he publicly expresses that he believes φ [START_REF] Searle | Speech acts: An essay in the philosophy of language[END_REF][START_REF] Vanderveken | Principles of language use, volume 1 of Meaning and Speech Acts[END_REF] and thus he socially commits himself on the fact that he believes φ, as we will develop in Section 4. Thus his belief that φ is immediately and without discussion grounded for the group. Now if agent i asserts that G j φ in presence of group I, then the formula G I G i G j φ holds afterwards, and if (WR) applied unrestrictedly then j could not express later that he ignores whether φ, or believes ¬φ. If he made this last speech act, the formulas G I G j ¬φ and, thanks to (WR), G I G i G j ¬φ would hold, which is negative) introspection axioms commonly accepted for mental attitudes (like belief, choice...): each agent i member of the group I if aware of what is grounded (resp. ungrounded) for the group I:

G I φ → G i G I φ and ¬ G I φ → G i ¬ G I φ. inconsistent with the above formula G I G i G j φ [16].
This restriction highlights that a formula φ can be grounded in two different manners: either φ is objective for group I and it must be discussed by all the agents of I, or φ is not and it is grounded directly by being expressed. We will discuss this distinction in the sequel (Section 5).

And finally, corresponding to the semantic constraint [SC5], we have the last axiom of common grounding:

(CG) (⋀ i∈I G I G i φ) → G I φ
It expresses that if a proposition is established for every agent in I, then it is grounded for the group I. Together, (WR) and (CG) stipulate that for formulas φ that are objective for I we have:

(1) (⋀ i∈I G I G i φ) ↔ G I φ
From axioms (SR+) and (SR-), we can prove that we have the modal axioms ( 4) and ( 5) for G I operators as theorems of our logic:

(4GI) G I φ → G I G I φ (5GI) ¬ G I φ → G I ¬ G I φ
Thus operator G I is in a normal modal logic of type KD45. Hence for individual groundedness we obtain the standard logic of belief KD45.

We can moreover show that if I' ⊆ I then:

(2) G I φ ↔ G I' G I φ (3) ¬ G I φ ↔ G I' ¬ G I φ
These theorems express that subgroups of a group are aware of what is grounded (resp. ungrounded) in the group. The formula (⋀ I'⊆I G I G I' φ) → G I φ is provable from our axiom (CG). Moreover we can prove that:

(4) G I φ ↔ G I G I' G I φ (5) ¬ G I φ ↔ G I G I' ¬ G I φ
These theorems say that if φ is (not) grounded for a group, it is grounded for this group that it is grounded for every subgroup of this group that φ is (not) grounded for the group.

Even if I' is a subgroup of I we do not necessarily have G I φ → G I' φ. Such a principle would be too strong because it would restrict the autonomy of subgroups I' of I: a proposition can be grounded for a group I while there is a dissident subgroup I' of I, i.e. a group where the contrary is grounded:

G I φ ∧ ¬ G I' φ is consistent in our logic even if I ∩ I' ≠ ∅.

Mutual belief.

Axiomatically mutual belief is defined by the Fixpoint Axiom, which specifies that a mutual belief about φ holds if and only if every agent believes that φ and that the mutual belief holds (see [START_REF] Fagin | Reasonning about Knowledge[END_REF] for more details):

(FP MBelI ) MBel I φ ↔ ⋀i∈I G i (φ ∧ MBel I φ)
and the Least Fixpoint axiom, that will be used in the sequel:

(LFP MBelI ) ⋀i∈I G i φ ∧ MBel I (φ → ⋀i∈I G i φ) → MBel I φ
From this axiomatics, we can deduce that MBel I is a normal modal operator of type KD4:

MBel I φ → ¬MBel I ¬φ MBel I φ → MBel I MBel I φ

Mutual belief and grounding. By definition, it follows from the Fixpoint

Axiom that when it is mutual belief for a group I that φ holds then necessarily every member of I believes individually that φ:

(6) MBel I φ → ⋀i∈I G i φ
Now we will show the link between mutual belief of a group I and the grounding for the whole group I: Theorem 3.1 We have the equivalence:

(7) G I φ ↔ MBel I G I φ
That means that a formula φ is grounded in a group I if and only if there is mutual belief in the group that φ is grounded. This property is mainly due to the public nature of the grounding operator.

Proof.

1. ⊢ MBel I G I φ → G i G I φ, by theorem (6) 2. ⊢ MBel I G I φ → G I φ, from 1. by theorem (2) 3. ⊢ G I φ → G i G I φ, by theorem (2), for every i ∈ I 4. ⊢ G I φ → ⋀ i ∈I G i G I φ, from 3. because it holds for every i ∈ I 5. ⊢ MBel I (G I φ → ⋀ i ∈I G i G I φ) from 4.

by the Rule of Necessitation for MBel

I 6. ⊢ MBel I (G I φ → ⋀ i ∈I G i G I φ) → (⋀ i ∈I G i G I φ → MBel I G I φ), from Axiom (LFP MBelI ) 7. ⊢ ⋀ i ∈I G i G I φ → MBel I G I φ,
(D Chi ) Ch i φ → ¬ Ch i ¬ φ (4 Chi ) Ch i φ → Ch i Ch i φ (5 Chi ) ¬ Ch i φ → Ch i ¬ Ch i φ
Choice and Grounding. Due to the semantic constraint [SC8] we have the following axiom:

(8) G i φ → Ch i φ
which means that every formula grounded for agent i must necessarily be chosen by this agent.

Our semantics also validates the principles:

(9) Ch i φ ↔ G i Ch i φ (10) ¬ Ch i φ ↔ G i ¬ Ch i φ that correspond with constraint [SC9]
. This expresses that agents are aware of their choices.

Action.

With respect to the semantic constraints, the action operators After α and its converse Before α are defined in a K t logic, i.e. a normal modal logic with following conversion axioms:

(I Afterα,Doneα ) φ → After α Done α φ (I Beforeα,Happensα ) φ → Before α Happens α φ
These axioms characterize the fact that the relation R α -1 is the converse of R α .

Action and grounding.

As we have said above we only consider public actions and α be an action performed by an agent i in front of attentive group K (of which i is member). Thus we have following axioms of public actions corresponding to the semantic constraint [START_REF] Durkheim | The rules of Sociological Method[END_REF], for each group K observing an action α:

(PA K,α ) G K Done α ⊤↔ Done α ⊤ (NA K,α ) G K ¬Done α ⊤↔ ¬ Done α ⊤
To sum it up, an action has been (resp. has not been) performed by a member of group I if and only if it is grounded for the group that it has been (resp. has not been) performed.

Action laws

Action laws come in two kinds: executability laws describe the preconditions of the action, and effect laws describe the effects. The preconditions of an action are the conditions that must be fulfilled in order that the action be executable. The effects (or postconditions) are properties that hold after the action because of it. For example, to toss a coin, we need a coin (precondition) and after the toss action the coin is heads or tails (postcondition).

The set of all action laws is noted LAWS, and some examples are collected in Table 2. The general form of an executability law is (Int Chi,i:α ) Ch i Happens(i:α) ∧ precond(i:α) ↔ Happens(i:α) This expresses a principle of intentional action: an action happens exactly when its preconditions hold and its author chooses it to happen [START_REF] Lorini | Introducing the attempt in a modal logic of intentional action[END_REF]. The general form of an effect law is φ → After α postcond(α). In order to simplify our exposition we suppose that effect laws are unconditional and therefore the general form of an effect law is here:

After α postcond(α)
A way of capturing the conventional aspect of interaction is to suppose that these laws are common to all the agents. Formally they are thus global axioms to which the necessitation rule applies [START_REF] Chris | Proof Methods for Modal and Intuitionistic Logics[END_REF].

Example

To highlight our proposal for the semantics of grounding we will formalize the introduction example. Let consider the example where there are three agents AGT = {0, 1, 2}:

1. Agent 0 (privately) believes that 2 is smart, formally written G 0 smart 2 .

2. Now suppose that in private conversation agent 0 tells 1 that 2 is not smart.

The illocutionary effect is G {0,1} G 0 ¬ smart 2 .

3. If 1 publicly adopts ¬ smart 2 (e.g. by confirming publicly that ¬ smart 2 ) we moreover obtain G {0,1} ¬ smart 2 .

4. Then agent 2 joins in the conversation, and later on 0 informs 1 and 2 that 2 is smart: the illocutionary effect is G {0,1,2} G 0 smart 2 .

5. Then if both 1 and 2 publicly adopt smart 2 we moreover obtain

G {0,1,2} smart 2 .
This illustrates that even for nested groups J 0 = {0} ⊂ J 1 = {0, 1} ⊂ J 2 = {0, 1, 2} we might have states of public groundedness for the different groups which are about propositions that are mutually inconsistent, viz. here:

G J 0 smart 2 G J 1 ¬ smart 2 G J 2 smart 2

Application to Walton & Krabbe's dialogue games

We now apply our formalism to a particular kind of dialogue, viz. persuasion dialogues defined by Walton&Krabbe (W&K for short). [START_REF] Walton | Commitment in Dialogue: Basic Concepts of Interpersonal Reasoning[END_REF] presents a dialogue type hierarchy based on the notion of conflict. Among them, a detailed study is made for persuasion dialogues, with quite precise descriptions of game rules and speech act semantics in term of commitments. These works mainly follow from Hamblin's works [START_REF] Charles | Fallacies[END_REF]. This section adapts the case study of [START_REF] Gaudou | A Logical Framework for Grounding-based Dialogue Analysis[END_REF] to the present logical framework.

A persuasion dialogue takes place when there is a conflict between two agents' belief. The goal of the dialogue is to resolve this situation: an agent can persuade the other party to concede his own thesis (in this case he wins the dialogue game) or concede the point of view of the other party (and thus lose the game).

W&K define two types of persuasion dialogue: the Permissive Persuasion Dialogue (PPD for short) and the Rigorous Persuasion Dialogue (RPD). RPD is asymmetric (participants have different roles viz. proponent and opponent), and is analytic (the initial proposition is decomposed during the dialogue), while PPD is symmetric and non analytic (allows to introduce new arguments).

We show, by characterizing PPD 0 , how our formally well-grounded operator can be used to define speech acts semantics and game rules instead of the informal commitments à la W&K.

In order to simplify our exposition we suppose with W&K that there are only two agents (but the account can easily be generalized to n agents).

Strong and Weak commitments

W&K distinguish two kinds of commitment: those which must be defended by a proof or a justification when challenged, called assertions, and those which in contrarily does not need, called concessions. We formalize this distinction with the notions of strong commitment (SC i,K ) and weak commitment (WC i,K ). They are linked by the fact that a strong commitment to a proposition implies a weak commitment to it [48, p. 133]. We use the logical framework presented above to formalize these two notions, and apply it to PPD 0 . In relation with this logical framework, we define:6

(Def SC i,K ) SC i,K φ = G K Bel i φ (Def WC i,K ) WC i,K φ = G K ¬Bel i ¬φ
logical" than W&K's a(φ): W&K allow both a(φ) and a(¬ φ) to be the case simultaneously, while for us

G K G i φ ∧ G K G i ¬ φ is inconsistent.
In the case of weak commitment, we agree with W&K's works: in our framework, WC i,K φ ∧ WC i,K ¬ φ is consistent.

In terms of the preceding abbreviations we can prove: [START_REF] Engel | Believing, holding true, and accepting[END_REF] shows the rationality of the agents: they cannot commit both on φ and ¬φ. [START_REF] Fagin | Reasonning about Knowledge[END_REF] and ( 13) account for the public character of commitment. With these three theorems, we can show that SC i,K is an operator of a normal modal logic of type KD45, too. 7(14

(11) SC i,K φ →¬ SC i,K ¬φ (12) SC i,K φ ↔ SC i,K SC i,K φ (13) ¬ SC i,K φ ↔ SC i,K ¬ SC i,K φ (
) G K φ ↔ SC i,K G K φ (15) ¬ G K φ ↔ SC i,K ¬ G K φ (16) SC i,K φ ↔ SC j,K SC i,K φ (17) ¬ SC i,K φ ↔ SC j,K ¬ SC i,K φ
These theorems express the public character of the commitment. ( 14) and [START_REF] Fornara | Operational Specification of a Commitment-Based Agent Communication Language[END_REF] entail that it is grounded that the agents are committed to the grounded (resp. ungrounded) propositions. ( 16) and ( 17) mean that each agent is committed to the other agents' commitments, and non-commitments.

The following theorems show links between strong and weak commitments. [START_REF] Gilbert | Modelling collective belief[END_REF] says that strong commitment implies weak commitment. [START_REF] Gilbert | On Social Facts[END_REF] expresses that if agent i is weakly committed to φ then i is not strongly committed to ¬φ.

(18) SC i,K φ → WC i,K φ (19) WC i,K φ → ¬ SC i,K ¬φ
(20

) WC i,K φ ↔ SC j,K WC i,K φ (21) ¬ WC i,K φ ↔ SC j,K ¬ WC i,K φ (20)
expresses that weak commitment is public. ( 21) is similar for absence of weak commitment.

Speech acts and grounding

In our framework, speech acts are just particular actions. In a general way [START_REF] Gaudou | Grounding and the expression of belief[END_REF], they are 5-tuples of the form 〈i, J, K, FORCE, φ〉 where i ∈ AGT is the author of the speech act (i.e. the speaker), K ∈ AGT the group of agents attentive to the conversation, J ⊆ K \ {i} the set of its addressees, FORCE its illocutionary force, and φ a formula denoting its propositional content. As said above we consider dialogues between only two agents, thus the hearers group J is restricted to the singleton {j} and the attentive agents group K is limited only to {i, j}. In the sequel we will use a simplified notation for speech acts: 〈s, h, FORCE, φ〉, where s is the speaker and h the hearer and K will denote the set {s, h}.

Figure 2. (Additional) possible moves after each act

The dialogues that we want to formalize (W&K-like dialogues) are controlled by some conventions: the rules of the game, which describe the allowed sequences of speech acts. The allowed sequences of acts are those of W&K's PPD 0 (cf. [48, p. 150-151]). They are depicted in Figure 2 and will be discussed below. For example, after a speech act 〈s, h, Assert, p〉, the hearer can only challenge p or concede it. We formalize them in our logic by expressing that an act grounds that the hearer's choices are limited only to some acts. Speech acts have two different effects: one is on the commitment store in terms of weak and strong commitments (cf. Table 1) and the other one is the set of acts the hearer can perform in response (cf. Table 2).

We suppose that initially nothing is grounded, i.e. the belief base is {¬ G I φ : for every I ∈ 2 AGT such that Cardinality( I ) > 1 and where φ is a formula such that ⊬ φ}. 8The Assert act on p can only be used by the two parties in some preliminary moves of the dialogue to state the theses of each participant. The effect of the act is that it is grounded that its content p holds for the speaker: he has expressed a kind of strong commitment (an assertion for W&K) on p in the sense that he must defend his commitment by an argument if it is challenged.

Table 1. Preconditions and effects of speech acts (with commitments)

Precond(α) Α Effects(α) ¬SC s,K p 〈s,h, Assert, p〉 SC s,K p SC s,K p 〈s,h, SRetract, p〉 ¬ SC s,K p WC s,K p 〈s,h, WRetract, p〉 ¬ WC s,K p SC s,K p ∧ ¬ WC h,K p 〈s,h, Argue, (q 1 ,…,q n SOp)〉 ⋀ 1 ≤ i ≤ n SC s,K q i ∧ SC s,K (⋀ 1 < i < n q i → p) ¬ WC s,K p 〈s,h, Concede, p〉 WC s,K p ¬ WC s,K p 〈s,h, RefuseConcede, p〉 ¬ WC s,K p SC s,K q ∧ ¬ WC h,K q ∧ ¬ WC h,K p 〈s,h, RequestConcede, p〉 ∅ ¬ WC s,K p∧ SC h,K p ∧ ¬G K Done〈 s,h,Challenge,p〉 ⊤ 〈s,h, Challenge, p〉 ∅ ¬ WC h,K p 〈s,h, Serious, p〉 ∅ WC h,K p ∧ WC h,K q ∧ (p ↔ ¬ q) 〈s,h, Resolve, p〉 ∅
To Concede p means to admit that p could hold, where p is a strong commitment of the other party (e.g. p has been asserted). The effect of this act is that it is grounded that the speaker has taken a kind of commitment on p. But the nature of this commitment seems weaker than the former one: this one has not to be defended when it is attacked. W&K call it concession and it corresponds to our notion of Weak Commitment.

The Challenge act on p forces the other participant to either put forward an argument for p, or to retract the assertion p. For a given propositional content this act can only be performed once.

To defend a challenged assertion p, an argument, expressed by Argue, must have p as conclusion and a set of propositions q 1 ...q n as premises. We write it as follows:

(Def SO ) q 1 ...q n SO p ≝ q 1 ∧ ... ∧ q n ∧ (q 1 ∧ ... ∧ q n → p)

The effect of this act is that the speaker is strongly committed on all premises q 1 , ..., q n and on the implicit implication q 1 ∧ ... ∧ q n → p. It follows that the agent who has been challenged must explicitly take position in the next move (challenge or concede) on each premise and on the implicit implication. To challenge one premise means that the argument cannot be applied, while to challenge the implicit implication means that the argument is incorrect. If he does not challenge a proposition, he (implicitly) concedes it. But as soon as he has conceded all the premises and the implication, he must also concede the conclusion. To avoid some digressions, W&K suppose that an unchallenged assertion cannot be defended by an argument. Moreover, we took over their form of the support of arguments, viz. φ → ψ, although we are aware that more complex forms of reasoning occur in real world argumentation. 〈s, h, Argue, (q 1 ,...,

q n SO p) 〉 ⋀ 1 ≤ i ≤ n G K (Ch h Happens (〈h, s, Challenge, q i 〉) ∨ Ch h Happens (〈h, s, Concede, q i 〉)) ∧ G K (
Ch h Happens (〈h, s, Challenge, q 1 ∧...∧ q n → p〉) ∨ Ch h Happens (〈h, s, Concede, q 1 ∧...∧ q n → p〉)) At any time, the speaker may request more concessions (with a RequestConcede act) from the hearer, to use them as premises for arguments. The hearer can then accept or refuse to concede. W&K use the same speech act type to retract a concession and to refuse to concede something (the act nc(p)). But it seems to us that it is not the same kind of act, and we decided to create two different acts: 〈s, h, WRetract, p〉 to retract his own weak commitment on p, and 〈s, h, RefuseConcede, p〉 to decide not to concede p. A strong commitment can be retracted with a 〈s, h, SRetract, p〉. This act removes the strong commitment from the commitment store, but not the weak commitment, whereas the 〈s, h, WRetract, p〉 act removes the weak commitment and, if it exists, the strong commitment, too.

In our logic, WC i,K φ ∧ WC i,K ¬φ is satisfiable, but not SC i,K φ ∧ SC i,K ¬φ. Thus we are more restrictive than W&K: in the following, a contradiction in an agents' commitment store is only due to contradictory Weak Commitments. 9When an agent chooses to challenge a proposition p or to refuse to concede it, his opponent can query him to reassess his position. Finally the speech act Serious(p) imposes that the agent must concede p or refuse to concede it.

When a party detects an contradiction in the other party's commitment store, it can ask him to resolve it (with the act Resolve(p,q) where "p and q are explicit contradictories" [48, p. 151].). The other party must retract one of the inconsistent propositions. W&K do not make any inference in the commitment store, so Resolve only applies to explicit inconsistency (that is: Resolve(p,¬p)). We will write Resolve(p) instead of Resolve (p,q) where q is ¬p. Resolve(p) and Resolve(¬p) are thus equivalent. To perform the speech act Resolve(p), we can show that it is necessary and sufficient that the propositions p and ¬p are weak commitments of the opponent. In our formalism, the act Resolve holds only to weak commitments. Moreover the two contradictory weak commitments cannot be derived from two inconsistent strong commitments (which W&K allow), because such are consistent in our logic.

Note that W&K define another commitment store that contains what they call dark-side commitments. Whereas assertions and concessions (light-side commitments) are public, no agent is necessary aware of these commitments. They characterize deep features of agents. If p is a dark-side commitment, it must be revealed after a Serious(p) and the agent must concede p and cannot retract it. 10We do not consider such commitments here because, we focus on what is observable and objective in the dialogue: so if an agent chooses to concede p, we do not know if it was a dark-side commitment or not, consequently the agent may, even if it had a dark-side commitment on p and contrary to W&K's theory, retract it in a subsequent dialogue move dialogue.

The action preconditions are not mutually exclusive. This gives the agents some freedom of choice. We do not describe here the subjective cognitive processes that lead an agent to a particular choice.

Figure 3. Example of dialogue (from [48, p. 153])

As we have said, in order to stay consistent with our logical framework, we have to add an effect to the W&K speech act of concession: when i concedes a proposition p, every strong commitment of i on ¬p is retracted. Agent i is then weakly committed on both p and ¬p. We thus weaken the paraconsistent aspects of W&K, viz. that an agent can have assertions or concessions that are jointly inconsistent, in order to keep in line with standard properties of the modal operator G K . Now we can establish formally that our logic captures W&K's PPD 0 -dialogues. For example we have:

Table 3. Commitment stores in the example dialogue

Grounded propositions

SCi WCi SCj WCj

∅ p1 p2 WCi,K p3 SCi,K p3, SCi,K p3 → p1 p1, p3, p3 → p1 p2 WCj,K p3 → p1, SCj,K p4, SCj,K p5, SCj,K p4 ∧ p5 → p2 p2, p4, p5, p4 ∧ p5 → p2 p3 → p1 WCi,K p5, SCi,K ¬ p4, SCi,K p5, SCi,K ¬ p4 ∧ p5 → p3, WCi,K p4 ∧ p5 → p2 p1, p3 → p1, p3, ¬ p4, p5, ¬ p4 ∧ p5 → p3 p5, p4 ∧ p5 → p2 ¬ SCj,K p3 → p1, WCj,K p3, WCj,K ¬ p4 ∧ p5 → p3, SCj,K p3, SCj,K p3 → p4, WCj,K ¬ p4 p2, p4, p5, p3, p3 → p4, p4 ∧ p5 → p2 ¬ p4, ¬ p4 ∧ p5 → p3 SCi,K ¬ p4, SCi,K ¬ p4 → (p3 → p1) p3, p3 → p1, p1, ¬ p4, p5, ¬ p4 ∧ p5 → p3 ¬ p4 → (p3 → p1) p5, p4 ∧ p5 → p2 ¬ SCj,K p4, ¬ WCj,K p4 ¬ WCj,K p3 → p4, ¬ SCj,K p3, ¬ SCj,K p5, ¬ SCj,K p3 → p4, ¬ WCj,K p4 ∧ p5 → p2, ¬ SCj,K p4 ∧ p5 → p2 WCj,K p3 → p1, WCj,K p1, WCj,K ¬ p4 → (p3 → p1) SCj,K p6, SCj,K p6 → p2 p2, p6, p6 → p2 ¬ p4, ¬ p4 → (p3 → p1), p3, p5, p3 → p1, p1, ¬ p4 ∧ p5 → p3
Thus after an assertion of p the only possible reactions of the hearer are to either challenge or concede p, under the condition that he has not doubted that ¬p, and that he has not challenged p in the preceding move.

Proof. LAWS contains (see Table 2) the formula

After〈 s, h, Assert, p〉 G K ( Ch h Happens(〈h, s, Challenge, p〉) ∨ Ch h Happens(〈h, s, Concede, p〉) )
The precondition for 〈h, s, Challenge, p〉 is ¬WC h,K p ∧ SC s,K p ∧ ¬Done(〈h, s, Challenge, p〉) Now the postcondition of 〈s, h, Assert, p〉 is SC s,K p. Hence we have by the law of intentional action (Int Chi,i:α ):

LAWS ⊧ After〈 s, h, Assert, p〉 ( ¬WC h,K p ∧ ¬ Done(〈h, s, Challenge, p〉) → ( Ch h Happens(〈h, s, Challenge, p〉) → Happens(〈h, s, Challenge, p〉) ) )
Similarly, for concede we have:

LAWS ⊧ After〈 s, h, Assert, p〉 ( ¬WC h,K p → ( Ch h Happens(〈h, s, Concede, p〉) → Happens(〈h, s, Concede, p〉) ) )
Combining these two with the law of intentional action for Assert we obtain our theorem.

 Similar results for the other speech acts can be stated. They formally express and thus make more precise further properties of W&K's dialogue games. For example, the above theorem illustrates something that remained implicit in W&K's PPD 0 dialogues: the hearer of an assertion that p should not be committed that p himself because, if he were the dialogue would no more be a persuasion dialogue and no rule would apply.

Similarly, in a context where h's commitment store contains SC h,K (p ∨ q), SC h,K ¬p, and SC h,K ¬q (and is thus clearly inconsistent), W&K's dialogue rules do not allow s to execute 〈h, s, Resolve, p ∨ q, ¬p ∧ ¬q). This seems nevertheless a natural move in this context. Our formalization allows for it, the formal reason being that our logic of G K is a normal modal logic, and thus validates (SC i,K p ∧ SC i,K q) → SC i,K (p ∧ q).

Formalization of group belief

As expressed above, our grounding operator formalizes what has been "publicly expressed and accepted as being true by all the agents participating in a conversation". In the previous section, we developed the expression aspect of this definition, in particular by describing the link between our operator G I and speech acts. In the sequel we will investigate relations between G I and the notions of group acceptance and group belief, in particular with the notion of proper group belief as defined by Gilbert [START_REF] Gilbert | Modelling collective belief[END_REF] and Tuomela [START_REF] Tuomela | Group beliefs[END_REF] opposed to the so-called we-belief or shared belief. The aim is to show that our grounding operator can also be viewed as a proper group belief operator.

In this paper, we will not consider refinements of the notion of belief (at individual and collective levels): in particular we ignore the distinction between belief, acceptance and holding true. These distinctions were the subject of many investigations, for example [5,[START_REF] Engel | Believing, holding true, and accepting[END_REF][START_REF] Tuomela | Belief versus acceptance[END_REF]. At the collective level, we will not take part either in the debate between the authors who attribute beliefs to groups (like [START_REF] Gilbert | Belief and acceptance as features of groups[END_REF][START_REF] Perron | Rejecting rejectionism[END_REF]) and the "rejectionnists" (e.g. [START_REF] Gilbert | Belief and acceptance as features of groups[END_REF][START_REF] Perron | Rejecting rejectionism[END_REF]) who consider that a group can only accept propositions. (See also [START_REF] Hakli | Group beliefs and the distinction between belief and acceptance[END_REF] for a very interesting discussion about this distinction). We are aware that this is an important philosophical debate. However, from a logical point of view, these are quite subtle distinctions that we do not take account for the time being.

Remark.

In our framework, we have assimilated the peculiar operators G i to private beliefs. To take into account the distinction between belief and acceptance in the individual layer, we could drop this hypothesis and study relations between G i and a belief operator à la Hintikka [START_REF] Hintikka | Knowledge and Belief: An Introduction to the Logic of the Two Notions[END_REF]. But this is out of the scope of this paper.

Thus we consider with Tuomela [START_REF] Tuomela | Group beliefs[END_REF] that we can relate a belief to a whole group. And with Gilbert [START_REF] Gilbert | Modelling collective belief[END_REF] we will use indifferently the words group belief and group acceptance. Our aim is not to formalize all the subtle distinctions of the domain but only to highlight links between our "grounding" operator and "proper group belief". Firstly we will summarize the commonly admitted features of group beliefs and show that our operator verifies them, and secondly we will detail its links with two important group belief approaches.

Common features of proper group beliefs

We base our analysis on the works of Margaret Gilbert [START_REF] Gilbert | Modelling collective belief[END_REF] and Raimo Tuomela [START_REF] Tuomela | Group beliefs[END_REF]. These are the main authors considering group belief as a proper object studies (the term proper group belief is used to name group belief in this sense) and not only as an aggregation of individual beliefs. Indeed, before Gilbert's papers, group belief was rather formalized in a "summative approach" [START_REF] Tuomela | Group beliefs[END_REF] (e.g. Quinton [START_REF] Quinton | Social objects[END_REF]): a group believes that φ (or has the group belief that φ) if every agent or at least the majority of the group members believes individually that φ. A more complex summative approach defines the group belief with the common/mutual belief [START_REF] Lewis | Convention[END_REF]. These approaches are criticized because they relate too deeply individual and collective belief: there a collective belief cannot hold without private beliefs, which discards group belief as the result of a consensus. Thus this constraint is too restrictive and motivated other approaches to emerge. Before studying in details their specificities, we will stress their common features.

Proper group belief is in no case related to individual beliefs. This property is likely the major criticism against the summative approaches. Already Durkheim in [START_REF] Durkheim | The rules of Sociological Method[END_REF] expressed that any proper group belief must be "external to individual consciousness". In fact a group belief can be the effect of a negotiation, a deliberation or a persuasion process and thus a consensus between two or more parts with very different viewpoints. It can even be the result of more or less ethical processes as propaganda or threat. Thus in borderline cases it can be the case that the group has a group belief while no group member individually has the corresponding belief. Tuomela gives the following example [START_REF] Tuomela | Group beliefs[END_REF]: "The Communist Party of Ruritania believes that capitalist countries will soon perish (but none of its members really believes so)." He names these cases "spurious beliefs".

As said above, our grounding operator G I is linked in no way to the private beliefs, and in particular for every agent i, member or not of the group I, neither

G I φ → G i φ nor G i φ → G I φ is a theorem of our logic. 11
There is a kind of commitment on the proper group belief. As soon as the group belief has been established, even if some group members disagree with this belief, they must act in compliance with it, i.e. they are committed in some way to this belief. When they violate it, they are liable for sanctions, ranking from blames of the group [START_REF] Gilbert | Modelling collective belief[END_REF] to the exclusion of the group [START_REF] Tuomela | Group beliefs[END_REF]. This property is an axiom of our logic: it corresponds to Axioms [START_REF] Engel | Believing, holding true, and accepting[END_REF] and (WR).

If it is grounded for a group that φ, then no group member can assert that he believes ¬φ (in the case of φ objective, i.e. when the grounding arises from a discussion between every agent and not only form the assertion of one). 12 The group members share a mutual belief about the proper group beliefs. One of the major criticisms against the "simple summative approach" of group belief is that every group member can believe individually that φ without any collective belief on φ because agents are not aware of what other agents believe. A kind of mutual belief is thus necessary, but not about the content of the group belief (as in the "complex summative approach"), but rather on the group belief itself. Tuomela [START_REF] Tuomela | Group beliefs[END_REF] defends this thesis arguing that group belief is grounded due to a joint and intentional group action. 11 Nevertheless G I φ ∧ G i φ is a consistent formula, i.e. proper group belief does not ban the existence of a mutual belief. 12 Indeed, if it is grounded that φ for the group I, then due to (WR), in particular G I G i φ holds for every member i of the group I and with Axiom (11), ¬ G I G i ¬φ holds too. No member of I can perform an assertive speech act with propositional content ¬ φ [START_REF] Gaudou | Grounding and the expression of belief[END_REF].

This feature is a theorem of our logic: as proven above, the formula G I φ → MBel I G I φ is a theorem. Our logic is stronger because we even have the equivalence.

After this first examination our grounding operator has good properties to represent a kind of group belief. We will examine more deeply its link with Gilbert's and Tuomela's approaches.

Details on Gilbert's and Tuomela's approaches

The Plural Subject Account (Gilbert [START_REF] Gilbert | Modelling collective belief[END_REF][START_REF] Gilbert | On Social Facts[END_REF]). In her book [START_REF] Gilbert | On Social Facts[END_REF], in opposition to "summative" approach, Gilbert gives the following characterization of proper group belief:

1. A group G believes that p if and only if the members of G jointly accept that p.

The members of G jointly accept that p if and only if it is common knowledge in G

that the members of G individually have intentionally and openly expressed their willingness jointly to accept that p with the other members of G.

From this standpoint, group belief à la Gilbert seems equivalent to the common belief in a group on the public and intentional expression by each group member of his joint acceptance.

In the sequel, we will show how our grounding operator is close to this definition thank to Axioms (WR) and (CG). In particular in the case of objective formulas φ, we have the equivalence:

(22) G I φ ↔ (⋀ i ∈ I G I G i φ)
Due to the formula (7), we can deduce the equivalence:

(23) G I φ ↔ MBel (⋀ i ∈ I G I G i φ)
This equivalence is very close to Gilbert's characterization of the group belief. In fact, formula G I G i φ expresses that agent i expressed in front of group I that he believes φ (i.e. i asserted φ), or with the abbreviation introduced in the former section, that he involved a strong commitment face of I on φ. Thanks to Axiom (CG) ((⋀ i ∈ I G I G i φ) → G I φ), by asserting φ, agent i expresses also implicitly his acceptance that φ to be grounded in the group. 13 Thus formula (23) can be read: "an objective formula φ is grounded for group I if and only if it is mutual belief in group I that every group member publicly expressed that they believe the objective formula φ". 14 It follows from this informal proof that for objective formulas our grounding operator matches group belief as defined by Gilbert. Thus our grounding operator is an admissible formalization of Gilbert's account of group belief (for objective formulas).

The Positional Account (Tuomela [42]). Tuomela in [42] discusses

Gilbert's group belief definition and shows in particular that in the cases of structured groups (for example with representatives) Gilbert's approach [START_REF] Gilbert | Modelling collective belief[END_REF] needs some updates. Tuomela takes the following example: "the United States believe that the [Soviet] invasion of Afghanistan was an unconscionable act" [START_REF] Gilbert | On Social Facts[END_REF]. Not all Americans have accepted this utterance, but only a small subgroup called the government, that we could name also the leaders. 15 G believes that p in the social circumstances C if and only if in C there are operative members A1, ..., Am of G in respective positions P1, ..., Pm such that: Tuomela gives the following analysis of the proper group belief, being inspired by his group intention formalization based on the distinction between "operative" and "non-operative" members (the ones who form the intention and the others who accept it tacitly) and definition of a "right social and normative circumstances" (a kind of institution composed by norms, roles, social rules and tasks...):

(1) the agents A1, ..., Am, when they are performing their social tasks in their positions P1, ..., Pm and due to exercising the relevant authority system of G, (intensionally) jointly accept that p, and because of this exercise of authority system, they ought to continue to accept and positionnally believe it;

(2) there is a mutual belief among the operative members A1, ..., Am to the effect that (1); 13 In particular, because he is aware of this theorem (thanks to Rule of Necessitation of the grounding operator), he knows that by asserting φ, G I φ could hold if the other agents asserted their acceptance. 14 which is equivalent in our framework to accept that φ could be grounded in the group. 15 Tuomela's purpose is not to give an example of a subgroup imposing its beliefs to the whole group. Every member of the government could believe the opposite of the grounded group belief but decide to accept it as the group belief. On the contrary, the other members of the group could be in accordance with it, and only have to tacitly accept it.

(3) because of (1), the (full-fledged and adequately informed) non operative members of G tend tacitly to accept -or at least ought to accept-p, as members of G;

(4) there is a mutual belief in G to the effect that (3). This characterization seems much more realistic and complex than the Gilbert's one. It is also more general: if we consider that every agent is an operative agent and we ignore the social and normative circumstances, Tuomela's approach is brought down to Gilbert's one. With our simple framework, we obviously cannot formalize the whole complexity of the Tuomela's definition. In particular, we introduce neither notions of roles, institution or norms, nor do we distinguish between operative/nonoperative agents. But following Tuomela, we will simplify a bit his formalism to show an interesting property: we will ignore the social and normative circumstances (assuming that they do not impose any restrictions).

As said above, the formula G I G i φ means: it is grounded for the group I that i believes φ. But another reading that we have previously given is: agent i has expressed in front of I that he believes φ. This formula is in fact the effect (in the sense of the action theory) of an assert speech act. Thus some non-objective formulas can be immediately grounded as soon as an agent performs the corresponding speech act. These formulas neither require discussion not explicit acceptance by every agent of the group. Only a single agent is needed to ground such formula for the whole group. It appears clearly that such formulas do not correspond to group belief à la Gilbert. We will show that it corresponds in fact to a group belief in Tuomela's approach.

In our logic, an agent is operative (in Tuomela's sense) in what concerns his own beliefs: G I G i φ holds whenever i asserts φ. This implies in our logic that he ought to continue to accept publicly that he believes φ. By theorem (7), there is mutual belief that G I G i φ. The other agents (called non-operative in this case) ought to accept that it is grounded that agent i believes φ. Moreover there is mutual belief about this grounded formula. Due to what we have said above about Tuomela's group belief, we have that G I G i φ implies group belief that i believes φ (in Tuomela's sense).

We could be closer to Tuomela's approach by introducing the concept of leaders of a group about a proposition in our framework. The leaders would be a subgroup of the group of agents I, verifying properties such as: G I G leaders(I,φ) φ ↔ G I φ. This means that if it is grounded for the whole group I that it is grounded for leaders that φ, then φ is de facto grounded for the whole group (i.e. if leaders have jointly accepted φ, then other agents have to accept it tacitly and thus φ becomes a proper group belief à la Tuomela). The group of leaders could be for example the government for every decision concerning the whole nation (thus leaders get their power from citizen's votes) or a group of specialists of a domain for every fact concerning their competence domain (they are leaders due to their knowledge and skills).

To summarize we have proven that, despite our simple logical framework without complex social constructions as norms, roles or institutions, our grounding operator is an operator of proper group belief. It is moreover more general than that of Gilbert's approach but still less than that of Tuomela's characterization, because of the specific process of grounding for Bel i φ-like formulas.

Conclusion

Related works

A way to bridge the gap between mentalist and structural approaches and to adapt BDI based agent to social speech act semantics has been recently the object of many works from a logical point of view 16 Firstly Nickles et al. [START_REF] Nickles | A Framework for the Representation of Opinions and Ostensible Intentions[END_REF] developed a logic of ostensible mental attitudes (Ostensible Belief and Ostensible Intention) that represents the public counterpart of mental attitudes and shows links between private mental attitudes (beliefs and intentions) and ostensible mental attitudes. As we discussed in [START_REF] Gaudou | Grounding and the expression of belief[END_REF], our two approaches are from a semantic point of view more or less equivalent. Nickles et al.'s ostensible belief of i w.r.t. j that φ corresponds in our logic to G {i,j} G i φ. Our logic is completely formalized in particular with a semantics, whereas theirs one is not.

, among which Boella et al. [2], Pasquier et al. [START_REF] Pasquier | Modelling the links between social commitments and individual intentions[END_REF] and Nickles et al. [START_REF] Nickles | A Framework for the Representation of Opinions and Ostensible Intentions[END_REF].

Pasquier et al. [START_REF] Pasquier | Modelling the links between social commitments and individual intentions[END_REF], following their works on dialogue games and commitments [4], brings to light links between private intention and commitment. Following the distinction of Bratman [3] between intention-to-do (about an action) and intention-to-be (about a proposition), they consider that commitments in action are the social counterpart of private intention-to-do whereas propositional commitments are the one of the intention-to-be. Their work is not yet logically well-founded and they do not consider any links between private beliefs and a public counterpart. Boella et al. [2] use what they name the "role metaphor" to bridge the gap between mentalist and social approaches of the ACL speech acts. Every agent taking part in a dialogue plays a role and is assumed to act in accordance with his role. They consider that agents must be sincere and cooperative not as individuals but as playing a role. Mental attitudes involved in the dialogue are associated to roles but not to agents. Thus, these attitudes are public and can be checked by every agent. With this concept they formalize both FIPA-ACL speech act semantics and commitments-based semantics.

Summary and perspectives

We have defined the modal operator of grounding to characterize what is public in the dialogue and more particularly what has been openly expressed or accepted by every agent. It can bridge the gap between the public and the private layer of the agent, because it is public, objective and linked to the belief. It is well grounded philosophically and formally.

Moreover we have shown how it can be used to formalize speech acts semantics and game rules, in particular in the case of Walton&Krabbe dialogue games.

In addition the grounding operator is very close of the notion of proper group belief although our logical framework does not contain any concept such as norm, role or institution.

Finally a very important improvement of the framework could be to add roles and institution to permit a much more precise analysis of the proper group belief. Moreover, one could extend the framework to take into account proper group intentions and obtain a unified framework for a group BDI logic.

  from 5. and 6. by Modus Ponens 8. ⊢ G I φ → MBel I G I φ, from 4. and 7. 9. ⊢ G I φ ↔ MBel I G I φ, from 2. and 8.  Choice. With respect to the semantic constraint [SC7], the choice operator is defined in a normal modal logic of type KD45 and we have the axioms (D Chi ), (4 Chi ) and (5 Chi ):

Theorem 4. 1 .

 1 LAWS ⊧ After〈 s, h, Assert, p〉 ( (¬WC h,K p ∧ ¬Done(〈h, s, Challenge, p〉)) → G K (Happens(〈h, s, Challenge, p〉) ∨ Happens(〈h, s, Concede, p〉)) )

Table 2 . Additional effects of speech acts α Constraints on the possible actions following α 〈s

 2

, h, Assert, p〉 G K (Ch h Happens (〈h, s, Challenge, p〉) ∨ Ch h Happens (〈h, s, Concede, p〉)) 〈s, h, SRetract, p〉 ∅ 〈s, h, WRetract, p〉 ∅ 〈s, h, RequestConcede, p〉 G K (Ch h Happens(〈h, s, RefuseConcede, p〉) ∨ Ch h Happens (〈h, s, Concede, p〉))

This notion has nothing to do with Wooldridge's computational grounding notion[START_REF] Wooldridge | Computationally grounded theories of agency[END_REF].

This definition is a recursive one. We can define the mutual belief with an infinite disjunction: every agent believes φ, that other agents believe φ, that other agents believe that every agent believe φ... . (See[START_REF] Fagin | Reasonning about Knowledge[END_REF] for more details about the logic of mutual belief.)

In particular for a speech act α, this notation allows to specify the author of α without mentioning the addressee, the illocutionary force...

In particular (SR+) and (SR-) axioms are a generalization to a group of the (positive and

This is an approximation of W&K's assertion, noted a. Indeed, our G K G i φ is "more

From the definition of G K we can prove that the modal property K is a theorem for SC i,K and that the necessitation rule can be applied to it.

This is an infinite set. In practice one would resort to default reasoning here.

W&K allow the agents to have some contradictory concessions (WC i,K ) and assertions (SC i,K ) in their commitment store (i.e. SC i,K φ and SC i,K ¬φ or WC i,K φ and WC i,K ¬φ can hold simultaneously).

Thus W&K consider agents can be publicly inconsistent but they cannot hide dark-side commitments if the opponent insists with a Serious.