Marie Puren

Fanny Lebreton

Aurélien Pellet

Pierre Vernus

From parliamentary history to digital and computational history : a NLP-friendly TEI model for historical and contemporary parliamentary proceedings

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

Introduction

Parliamentary debates have been a valuable source for research in the humanities and social sciences [START_REF] Chester | Questions in Parliament[END_REF][START_REF] Franklin | Parliamentary questions[END_REF][START_REF] Mela | Proceedings of the Digital Parliamentary Data in Action (DiPaDA 2022) Workshop[END_REF]. Sociology [START_REF] Cheng | Islamophobia, Muslimophobia or racism? Parliamentary discourses on Islam and Muslims in debates on the minaret ban in Switzerland[END_REF], political science [START_REF] Van Dijk | Political identities in parliamentary debates[END_REF] or linguistics (de [START_REF] Galembert | Faire parler le parlement: Méthodes et enjeux de l'analyse des débats parlementaires pour les sciences sociales[END_REF][START_REF] Hirst | Argumentation, Ideology, and Issue Framing in Parliamentary Discourse. ArgNLP. Frontiers and Connections between Argumentation Theory and Natural Language Processing[END_REF][START_REF] Rheault | Measuring Emotion in Parliamentary Debates with Automated Textual Analysis[END_REF] have used them as research material. These are also analyzed by political historians -in a national [START_REF] Ouellet | Les débats parlementaires au service de l'histoire politique[END_REF][START_REF] Bouchet | French Parliamentary Discourse[END_REF] or transnational (Ihalainen et al., 2006;[START_REF] Ihalainen | Parliament and parliamentarism: A comparative history of a european concept[END_REF] perspective -but also of social life, economics and religion [START_REF] Marnot | Les ingénieurs au Parlement sous la IIIe République[END_REF][START_REF] Lemercier | Un catholique libéral dans le débat parlementaire sur le travail des enfants dans l'industrie (1840). Parlement[s[END_REF], or even law [START_REF] Fournier | La majorité politique: Étude des débats parlementaires sur la fixation d'un seuil[END_REF]. From a mere disciplinary point of view, a clear dichotomy emerges for the exploitation of parliamentary debates with historians working on older periods, and actors from other disciplines on contemporary proceedings. However, especially when it comes to studying long-term phenomena1 , the continuity between past and present data is of crucial importance.

Working on a transnational corpus from 1803 to 2005, Helen Baker and her co-authors [START_REF] Baker | Ireland in British parliamentary debates 1803-2005. Plotting changes in discourse in a large volume of time-series corpus data[END_REF] have for instance shown how bridging political history and political science could be a fertile research avenue.

At the same time, contemporary parliamentary data is increasingly available on the web, as are historical parliamentary data (Bonin, 2019). Pasi Ihalainen even speaks of the "parliamentary turn" that conceptual and intellectual history has taken as a result of the massive digitisation of debates and the rise of comparative studies based on these documents [START_REF] Ihalainen | Parliaments as Meeting Places for Political Concepts[END_REF]. Parliamentary proceedings constitute indeed a "remarkable corpus" for political and intellectual history, and a perfect material for computational methods (Bonin, 2019). Recent interest in debates held in transnational institutions and the beginning of computational diplomacy [START_REF] Cafiero | Datafying diplomacy: how to enable the computational analysis and support of international negotiations[END_REF] make us anticipate that more data will soon be available. Yet, this new abundance of parliamentary data has not generated in history and political science as much work as hoped for [START_REF] Blaxill | The War of Words : The Language of British Elections, 1880-1914[END_REF]. And while it is becoming increasingly easy to collect transcripts of old parliamentary debates, to our knowledge there is no corpus of historical parliamentary debates encoded in XML-TEI.

In this paper, we present processes developed to allow for that kind of trans-period or transnational research, thinking of the necessary continuity between periods, and allowing for computational methods annotations. We propose an XML-TEI model specifically adapted to the encoding of these documents, be they proceedings from regional and national parliaments or international institutions. To illustrate our model, we work here on the parliamentary debates held in the French Chamber of Deputies 2 , transcribed and published between 1889 and 1893 in the Journal officiel de la République française. Débats parlementaires 3 .

In the first part, we explain the reasons that led us to develop such a model. In the second part, we outline the choices we made to build this model and explain why we had to adapt existing schema. Then, we present the method we adopted to automatically encode that kind of large corpus. Finally, we formulate a proposal to annotate parliamentary debates in TEI with natural language processing analyses, in particular topic modeling.

Interoperable historical data for a transperiod and transnational history

National parliamentary data are increasingly available online, with the growing digitisation of parliamentary, legislative and judiciary publications. In France, for example, the debates of the 2 The Chamber of Deputies was the lower house of the French Parliament during the Third Republic (1870-1940).

3

The chosen corpus includes 10418 images in JPG format. Images have been downloaded via the Gallica Images API: https://api.bnf.fr/fr/api-iiif-de-recuperation-des-images-de-gallica.

National Assembly and the Senate4 are only available online. The need to develop a format for the exchange of "machine-readable" parliamentary data quickly became apparent. One example is the OASIS standard Akoma Ntoso, which developed an XML format for encoding legislative and judiciary documents, including parliamentary proceedings (Ervajec et al., 2022b). After noticing the lack of standardization in XML-TEI for encoding parliamentary debates, the CLARIN research infrastructure initiated ParlaClarin in 2019. This project developed a data structuring schema limiting the TEI encoding options to options specifically applicable to parliamentary debates. ParlaMint was formed as a result of the ParlaClarin initiative and has so far developed 27 corpora of contemporary parliamentary debates in XML-TEI, covering the COVID pandemic, with 16 main languages (Ervajec et al., 2022b).

Significant efforts have thus been made to publish comparable and multilingual Parliamentary Proceedings Corpora (PPCs) in XML-TEI (Ervajec et al., 2021;Ervajec et al., 2022c) by these two initiatives [START_REF] Fišer | CLARIN resources for parliamentary discourse research[END_REF]Erjavec et al., 2023).

The development of an encoding model in TEI for historical proceedings also addresses the need to produce parliamentary data in a machine-readable and interoperable format. The standards for transcribing parliamentary debates have changed little since they were first introduced in the nineteenth century France [START_REF] Gardey | Scriptes de la démocratie: Les sténographes et rédacteurs des débats (1848-2005)[END_REF]. The transcripts of these debates are still published today in a dedicated edition of the Journal officiel 6 . From the outset, we assumed that the schema proposed by CLARIN should be easily adaptable for historical proceedings of parliamentary debates. It also seemed appropriate to produce data in a format that becomes standard not only for parliamentary proceedings, but more generally for the management and exchange of digital data in the human sciences [START_REF] Burnard | What is the Text Encoding Initiative?[END_REF] and for natural language processing [START_REF] Piotrowski | Natural language processing for historical texts[END_REF].

TEI also offers the possibility to select sub-corpuses more easily. Historians often need to work on smaller corpora of data. For example, they may want to work on a particular speaker and select only his or her speeches; they may want to define a specific time period; or they may be interested in a particular topic and collect only debates on it. It seemed to us that well-chosen annotations should facilitate the creation of such sub-corpuses: for example, we will select the <persName> whose value corresponds to the name of the searched deputy.

TEI has also proved to be a particularly interesting technical choice for the project. One of our objectives is to publish a TEI-compliant corpus of historical documents stored in an eXist-db 7 database and published via TEI Publisher 8 . To publish this data via TEI Publisher, it is indeed necessary to encode it in XML-TEI. Since we respect the standard set by ParlaClarin and ParlaMint, it would therefore be possible to integrate the most recent debates into the platform and thus realise our ambition to offer a corpus of debates covering, in time, the whole of the nineteenth and twentieth centuries.

Between modification and adaptation: creating an ODD for historic parliamentary proceedings

From a "proof of concept" perspective, we ourselves worked on a sub-corpus, namely the debates of the 1889-1893 parliamentary cycle 9 . Available on Gallica, the digital library of the Bibliothèque nationale de France, the chosen corpus includes 10418 images in JPG format.

Images have been downloaded via the Gallica Images API 10 .

We designed an encoding model in XML-TEI, formalizing it in a Relax NG schema and documenting it in a One Document Does it all (ODD) 11 [START_REF] Rahtz | Reviewing the TEI ODD system[END_REF]. While we drew on the ODD produced by ParlaClarin 12 and the work carried out by ParlaMint 13 (Erjavec et al., 2022a;[START_REF] Erjavec | ParlaMint: Comparable parliamentary corpora[END_REF] to create it, an important analysis of this ODD was however 7 http://exist-db.org/exist/apps/homepage/index.html 8 https://teipublisher.com/index.html 9 This parliamentary cycle or "Vème législature / 5th legislature" began on 12 November 1889 and ended on 14 October 1893. 10 https://api.bnf.fr/fr/api-iiif-de-recuperation-des-images-de-gallica 11 In order to generate the Relax NG schema and the ODD, we used TEI Roma: https://roma.tei-c.org/. This tool proposed by the TEI Consortium is a web interface for manually creating the specifications of an encoding model. 12 https://clarin-eric.github.io/parla-clarin/ 13 https://clarin-eric.github.io/ParlaMint/ necessary, both on the textual particularities of our source and on the recommendations of the Text Encoding Initiative. If the proceedings of contemporary debates are very close to those of the 19th century, the annotation rules must nonetheless be adapted to the source being treated 14 .

Here we propose to focus on the aspects that we felt were most important in the creation of the ODD and the related schema. One of the challenges was to annotate the names of the speakers, and more broadly the named entities in the proceedings. To recognize these named entities, we use the OCR tool developed in the OCR tool developed within the SoDuCo project 15 which also offers a very efficient NER module on our documents. Currently in private alpha version, this tool has been used, for example, to prepare the data used in Abadie, 2022.

It allows the recognition of named entities using the French language model CamemBERT [START_REF] Martin | CamemBERT: A Tasty French Language Model[END_REF] fine-tuned on data from trade directories. This functionality is an essential asset for us because it allows, at the cost of a low-cost training 16 , to recognise the names of the deputies in the corpus, and thus to consider annotating them and linking them to repositories such the database of French deputies since 1789 17 .

Fig. 2 View of the interface

Another challenge was to annotate the structure of the source. The proceeding always starts with a summary, directly followed by the speeches. It concludes with the next agenda and is completed by additional parts: annexes (results of votes, names of voters, their positions, etc.), reports, erratum, rectifications. The body of the text (the debates themselves) is divided 14 For example, we have removed the <recordingStmt> element which is unnecessary for the debates of the Third Republic. 15 https://soduco.github.io/, https://anr.fr/Projet-ANR-18-CE38-0013 16 CamemBERT was found to require little training data to perform very well on directories [START_REF] Abadie | A Benchmark of Named Entity Recognition Approaches in Historical Documents Application to 19th Century French Directories[END_REF]. 17 https://www2.assemblee-nationale.fr/sycomore/recherche into parts -corresponding to the different topics discussed -introduced with a title. If we want to use TEI annotation to facilitate the selection of sub-corpuses, it is necessary to annotate this structure. For example, many users will want to retrieve only the text of the debates, without the summary or annexes.

ParlaClarin and ParlaMint did not process appendices similar to the ones we were facing. Should they be embedded within the text in the <body> element, or distinguished from the text by including them in the <back> element? We could not determine whether they represented what was said in the sessions, and therefore whether they were an integral part of the spoken discourse, or whether they were included afterwards, as a complement to the proceedings. After consulting several references on the conception of minutes [START_REF] Martin | Variétés. Nouveau manuel de sténographie ou Art de suivre la parole en écrivant[END_REF][START_REF] Coniez | L'Invention du compte rendu intégral des débats en France (1789-1848)[END_REF][START_REF] Gardey | Scriptes de la démocratie: Les sténographes et rédacteurs des débats (1848-2005)[END_REF], we included each of the complementary parts in the <back> element.

Fig. 3 <back> element used for the appendices -and especially the votes

The proposal we make for encoding these appendices could be reused to encode the appendices of contemporary proceedings. Indeed, Fig. 4 shows that the appendices recording the results of votes in 2023 are particularly similar to the appendices dating from the 1880s. Although the proceedings have a homogeneous structure, their content is heterogeneous, and requires the consideration of many different elements18 . The role of annotation is to highlight the structural and semantic elements of the debates (e.g. the named entities) ; but it was very complicated to make all these scientific objectives coexist while respecting the TEI Guidelines. For example, the Transcriptions of Speech module declares the TEI element <incident> to encode descriptions of incidental sounds (applause, exclamations, etc.). But the TEI does not allow <lb>, <pb> or <cb> elements to be put in <incident>, which does not allow line breaks, or page or column changes to be reported. The Transcriptions of speech module was indeed created to directly transcribe spoken speech recorded using contemporary tools (such as a dictaphone), and not to process older transcriptions of spoken discourse. We decided not to keep the <lb> and <cb> elements, but to only annotate the page changes, in order to retain the page number. It is indeed essential for historical analysis to be able to refer to page numbers -for example, to quote the verbatim of a speech. With the help of the TEI list 19 , we have chosen to encode page changes occuring in an <incident> as follows: 19 We especially thank Lou Burnard for his advice.

Developing an automated annotation strategy

As the corpus represents a very large set of texts to be processed, we envisaged automating TEI tagging from the start. We created Python scripts that would transform the texts OCR'd in JSON format into textual data annotated in XML-TEI format, according to the predefined encoding model 20 .

To apply all the tags to the OCR'd data, we needed to find features within the texts. For example, if we wanted to encode speakers' speeches using the <u> element, we needed to be able to find a feature at the beginning and end of the speech in order to locate the opening and closing tags of the <u> element. However, we realized that the proceedings had too few recurring text features and were therefore complex to encode automatically. This led to the creation of an annotation guide 21 , which was the main instrument to help with data conversion.

Its aim was to guide a manual pre-annotation within the OCR tool, by providing a list of tags and defining their context of use so that they could be employed according to a common standard 22 . 20 Scripts available on Github: https://github.com/FannyLbr/Memoire-AGODA-TNAH2022/tree/f18284d45b83248c4d7938b2bf7984b2b006a832/C%20-%20Encodage%20automatique/C2%20-%20Scripts%20Python 21 Annotation guide available online: https://github.com/FannyLbr/Memoire-AGODA-TNAH2022/blob/f18284d45b83248c4d7938b2bf7984b2b006a832/C%20-%20Encodage%20automatique/C1%20-%20Guides/guide_annotations_agoda.pdf 22 The annotation guide we have set up indicates the name of the tags, their context of use, and the modelling of the XML-TEI result of the conversion of the tagged area.

Annotate the proceedings with the results of the topic modeling

The parliamentary debates of the Third Republic constitute a massive corpus 23 . Faced with such a large quantity of documents, it is necessary to develop new ways of reading these sources [START_REF] Clavert | Vers de nouveaux modes de lecture des sources[END_REF] based on "distant reading" as advocated by Franco Moretti [START_REF] Moretti | Distant reading[END_REF].

Distance is here a "condition of knowledge" [START_REF] Moretti | Conjectures on World Literature[END_REF] because it makes it easier to identify anomalies or patterns that a close reading would not have detected. Parliamentary debates lend themselves particularly well to "computer-assisted analyses", in the sense that the computer facilitates changes of scale [START_REF] Ihalainen | European History as a Nationalist and Post-Nationalist Project[END_REF] with the passage from distant to close reading and vice versa.

To analyze parliamentary proceedings, the use of computers is becoming an increasingly indispensable prerequisite (Bonin, 2019;[START_REF] Ihalainen | European History as a Nationalist and Post-Nationalist Project[END_REF][START_REF] Blaxill | Parliamentary Corpora and Research in Political Science and Political History[END_REF], even though its use in the humanities has decreased since the 1970s, particularly in history 23 There were 14 parliamentary cycles or legislatures between 1881 and 1940; the debates of a parliamentary cycle are about 10-12,000 pages long (208532 images). [START_REF] Lemercier | Quantitative methods in the humanities: An introduction[END_REF][START_REF] Kemman | Trading zones of digital history[END_REF][START_REF] Salmi | What is digital history?[END_REF]. If, like the authors of Exploring Big Historical Data, we believe that "Big Data analysis skills are on the verge of no longer being a 'nice to have' for historians but nearly a necessity" (Graham et al., 2014), we also believe that it is essential to provide turnkey tools to facilitate the reading of these large corpora by the general public, as well as researchers whose needs are limited to consulting the documents.

One avenue is to develop interfaces that make it easy to query these corpora [START_REF] Ihalainen | Building and Testing a Comparative Interface on Northwest European Historical Parliamentary Debates: Relative Term Frequency Analysis of British Representative Democracy[END_REF]. This type of platform is essential if historians are to become aware of the research potential offered by this digitized source. This is what we propose to develop in the framework of the AGODA project [START_REF] Puren | AGODA: Analyse sémantique et Graphes relationnels pour l'Ouverture et l'étude des Débats à l'Assemblée nationale[END_REF]. It seems essential to us to propose such a platform in the French context, where digital history suffers from a disaffection for quantitative methods [START_REF] Karila-Cohen | Nouvelles cuisines de l'histoire quantitative[END_REF][START_REF] Lemercier | Quantitative methods in the humanities: An introduction[END_REF], and from a lack of training in this field [START_REF] Ruiz | Former « au numérique » en sciences humaines et sociales? Propositions d'un historien[END_REF]. This platform will offer two levels of reading: on the one hand by creating a corpus that is searchable and readable by humans (close reading); on the other hand, by producing data that can be easily exploited with computational techniques (distant reading). A pitfall of "distant reading" methods is the imposition of inappropriate categories [START_REF] Karila-Cohen | Nouvelles cuisines de l'histoire quantitative[END_REF]. Rather than imposing manually defined categories, we chose to adopt an inductive approach based on the results of topic modeling. This method allows topics to emerge from the texts themselves, without human intervention [START_REF] Lemercier | Quantitative methods in the humanities: An introduction[END_REF][START_REF] Graham | Exploring big historical data: The historian's macroscope[END_REF].

Laurent Klein and his co-authors propose to use the topics generated with the topic modeling to propose a "new way of reading" newspaper collections [START_REF] Klein | Exploratory Thematic Analysis for Digitized Archival Collections[END_REF]. Exploring parliamentary debates by their subjects has already been widely deployed [START_REF] Abercrombie | Sentiment and position-taking analysis of parliamentary debates: A systematic literature review[END_REF]. In a first study [START_REF] Bourgeois | Using Topic Generation Model to explore the French Parliamentary Debates during the early Third Republic (1881-1899[END_REF], we found that Latent Dirichtet Allocation (LDA) [START_REF] Blei | Latent dirichlet allocation[END_REF] performs well on the corpus. It therefore seems relevant to us to create an interface that would allow users to browse the corpus according to the topics that cross it.

We propose to use the topics generated by LDA to enrich the data by annotating the debates with them. Using the <standOff> element, we can store these annotations in XML files.

The topic name is chosen by hand; to attach this semantic annotation to the corresponding word list, we chose to use the <span> element which allows us to attach an analytical note to parts of the text. Each word in the text is annotated by a <w> element accompanied by a unique identifier consisting of the document identifier (formed by the prefix "ps" for parliamentary sitting, followed by the date of the sitting) and a number corresponding to the place of the word in the text. A "ref" attribute then associates the topic with the corresponding words. We have also chosen to group the semantic annotations in the <standOff> element to facilitate their management. These <span> tags are also grouped in a <spanGrp> element associated with a "type" attribute:

Conclusion

In conclusion, this study has demonstrated the value of a novel XML-TEI model for the encoding of historical and contemporary parliamentary proceedings. The model, with its ability to handle diverse temporal and regional contexts, serves as an efficient tool for computational research in the humanities, particularly in historical and political studies. It promotes the continuity of analysis across different periods and political systems, thereby addressing the dichotomy that currently exists in the study of parliamentary debates.

Through the use of our model, as demonstrated in the analysis of debates from the French Chamber of Deputies (1889-1893), we have shown that the massive digitized parliamentary data now available can be made more accessible and analytically tractable. Furthermore, by incorporating natural language processing techniques, especially topic modeling, the model allows for a more comprehensive exploration of the themes and patterns present in the data.

We believe that our XML-TEI model will inspire further advancements in the field, encouraging more scholars to engage with digital and computational approaches in their research. By connecting past and present, and enabling more effective transnational and transperiod research, this model opens up a new research avenue that will enrich our understanding of parliamentary history in its broader context.

While our work represents a significant step forward, we acknowledge that there are further avenues to explore and challenges to tackle. Future research could refine the model's ability to handle linguistic and semantic complexities inherent in the parliamentary data, or explore ways to incorporate additional NLP techniques to enrich the analysis.

Fig. 1 A

 1 Fig. 1 A page from a parliamentary debate of 26 November 1889 5

Fig 4

 4 Fig 4 On the left, a page with the results of the vote of 26 November 1886; on the right, a page with the results of the vote of 6 April 2023

  Fig. 5 Extract from a speech with an incident at the parliamentary sitting of 26 November

  1889

Fig. 6

 6 Fig. 6 Change of page occurring in <incident>

Fig. 7

 7 Fig. 7 Extract from the annotation guide for the <u> element

Fig. 9

 9 Fig. 9 Example of the use of the <standOff> element

We have found that discussions can start several years before a law is actually passed(Bourgeois et al.,

2022).

The National Assembly is the lower house of the French parliament, the upper house being the Senate.

https://gallica.bnf.fr/ark:/12148/bpt6k63959929/f569.item#

The most recent transcripts of the parliamentary debates can be found in the digital edition of the Journal officiel: https://www.legifrance.gouv.fr/liste/debatsParlementaires.

The proceedings cover all the constituent elements of a sitting, including speeches, votes and their outcome, as well as the opening and closing of the sitting, interruptions and even the atmosphere.

Funding

This work supported by the DataLab of the Bibliothèque nationale de France.


Huu-Tu Dang
email: huu-tu.dang@ut-capitole.fr

Benoit Gaudou
email: benoit.gaudou@ut-capitole.fr

Nicolas Verstaevel
email: nicolas.verstaevel@ut-capitole.fr

A multi-level density-based crowd simulation architecture

Keywords: Agent-based architecture, Multi-level behaviour, Pedestrian modeling, Multi-scale simulation

Large-scale crowd phenomena are complex to model as the behaviour of pedestrians needs to be described at both strategic, tactical, and operational levels and is impacted by the density of the crowd. Microscopic models manage to mimic the dynamics at low densities, whereas mesoscopic models achieve better performances in dense situations. This paper proposes and evaluates a novel agent-based architecture to enable agents to dynamically change their operational model based on local density. The ability to combine microscopic and mesoscopic models for multi-scale simulation is studied through a use case of pedestrians at the Festival of Lights, Lyon, France. Simulation results are compared to different models in terms of density map, pedestrian outflow, and computation time. The results demonstrate that our agent-based architecture can effectively simulate pedestrians in diverse-density situations, providing flexibility for incorporating various models, and enhancing runtime performance while achieving comparable pedestrian outflow results to alternative models.

Introduction

Pedestrian simulation is an effective tool to model and study the behaviour of pedestrian in various contexts, from optimisation of pedestrian flows to safety issues [START_REF] Duives | State-of-the-art crowd motion simulation models[END_REF]. Pedestrian behaviour can be modelled into three-layer level architecture [START_REF] Hoogendoorn | Pedestrian route-choice and activity scheduling theory and models[END_REF]:

-Strategic level: pedestrians determine a list of activities (or targets) and when they want to perform these activities. -Tactical level: pedestrians choose a path to the predefined destinations based on information about the environment. -Operational level: pedestrians adjust their local movements such as collision avoidance to adapt to the surrounding area.

In order to build a pedestrian simulation, a modeller has to effectively instantiate one model for each of the three layers, that is to say at least one strategic model (also referred to as goal selection model) [START_REF] Haghani | Human exit choice in crowded built environments: Investigating underlying behavioural differences between normal egress and emergency evacuations[END_REF][START_REF] Kielar | Modeling pedestrians' interest in locations: A concept to improve simulations of pedestrian destination choice[END_REF], one tactical level model (also known as path planning model) [START_REF] Van Toll | Real-time density-based crowd simulation[END_REF][START_REF] Jiang | Dynamic navigation field in the social force model for pedestrian evacuation[END_REF], and one operational level model [START_REF] Helbing | Simulating dynamical features of escape panic[END_REF][START_REF] Berg | Reciprocal n-Body Collision Avoidance[END_REF]. In this paper, we focus on selecting the proper algorithm for the operational level.

Those models are used to compute the velocities and speed of each pedestrian based on local information. Many types of operational models exist in the literature, each model being built to capture and mimic a specific phenomenon [START_REF] Papadimitriou | A critical assessment of pedestrian behaviour models[END_REF]. Therefore, their performance differs if they are put into context for which they have not been designed. Microscopic models are usually suited to simulate pedestrians in lowand medium-density situations, where the density level is less than 2 ped/m 2 [START_REF] Duives | State-of-the-art crowd motion simulation models[END_REF]. These models encounter specific problems when applied to high-density situations, such as abnormal vibrations with the Social Force Model [START_REF] Pelechano | Controlling Individual Agents in High-Density Crowd Simulation[END_REF], congestion in dense bidirectional flow with the Velocity Obstacle model [START_REF] Zhao | A role-dependent data-driven approach for highdensity crowd behaviour modeling[END_REF], unrealistic results in terms of collision metrics with the data-driven models [START_REF] Korbmacher | Differences in pedestrian trajectory predictions for high-and low-density situations[END_REF]. On the other hand, pedestrian dynamics in high-density situations are similar to fluid flow [START_REF] Helbing | A Fluid Dynamic Model for the Movement of Pedestrians[END_REF], and fluid-like equation models [START_REF] Helbing | A Fluid Dynamic Model for the Movement of Pedestrians[END_REF][START_REF] Treuille | Continuum crowds[END_REF] are particularly appropriate for these scenarios due to their assumption of the continuous of pedestrian flow.

Given that we observe varying performance of pedestrian models in different crowd densities, this contribution proposes and evaluates an agent-based architecture to enable agents to dynamically change their operational model based on local density to adapt to their dynamic environments. Local density estimation is achieved through predefined regions in the environment, which enables contextual and structured adaptation to various environments. Additionally, the use of predefined areas can facilitate the implementation of the model by providing a more intuitive and manageable approach to estimating density.

The paper is organized as follows: In Section 2, a summary of related works is provided. Section 3 describes our proposed agent-based architecture for the multi-level behaviour of pedestrians. Next, Section 4 illustrates the experimental design and presents our simulation results. Finally, the paper ends with a conclusion and discussion in Section 5.

Related Work

Helbing et al. introduced the Social Force Model (SFM) [START_REF] Helbing | Simulating dynamical features of escape panic[END_REF], which describes pedestrian motion as driven by social forces. These forces result from both internal factors, such as an individual's attraction towards a personal goal, and external factors, like repulsion from neighboring pedestrians and walls. The SFM is formulated based on Newton's second law:

m i dv i dt = m i v 0 i (t)e 0 i (t) -v i (t) τ i + j̸ =i f ij + W f iW (1)
where m i , v i , v 0 i , and e 0 i represent the mass, current velocity, preferred speed, and desired direction of pedestrian i, respectively. The first term of Equation 1 accounts for the acceleration from current velocity to the preferred velocity within the reaction time τ i , while f ij and f iW describe interaction forces with other pedestrians and walls, respectively. The SFM has been demonstrated to realistically reproduce self-organization phenomena in low-density situations such as lane formation and arc-shaped clogging at bottlenecks [START_REF] Helbing | Simulating dynamical features of escape panic[END_REF].

On the other hand, Treuille et al. [START_REF] Treuille | Continuum crowds[END_REF] proposed the Continuum Crowds (CC) model which treats pedestrian flow similarly as a continuum without taking into account individual differences. Global navigation is managed by dynamic potential field using the eikonal equation (Equation 2):

||∇ϕ(x)|| = C (2) v = -f (x, θ) ∇ϕ(x) ||∇ϕ(x)|| (3)
where ϕ, C are potential function and unit cost in the direction ∇ϕ, respectively. The cost value to the goals is based on distance, time and discomfort factor. Pedestrian velocity is determined by the direction opposite the gradient of function in Equation 2, with the magnitude dependent on the local area's density field and velocity field. The CC model is suitable for simulating large crowds in extreme-density situations with computational efficiency.

The aforementioned models are only suitable for simulating pedestrians in either low-density or high-density scenarios [START_REF] Pelechano | Controlling Individual Agents in High-Density Crowd Simulation[END_REF][START_REF] Treuille | Continuum crowds[END_REF]. To simulate pedestrians in both low-density and high-density situations simultaneously, hybrid modeling is often employed, typically combining macroscopic and microscopic models. Autoswitch models [START_REF] Xiong | A case study of multi-resolution modeling for crowd simulation[END_REF][START_REF] Xiong | A hybrid model for simulating crowd evacuation[END_REF] move from a macroscopic model to a microscopic model and vice versa on a trigger condition, while region-based model [START_REF] Xiong | Hybrid modelling of crowd simulation[END_REF] specifies certain areas for each model. A mechanism for the aggregation and disaggregation of crowds is required for these models to have a consistent transition between the macroscopic and microscopic levels. However, these models normally ignore a higher level of pedestrian behaviour modeling and are therefore only applied in a simple environment with a single infrastructure. Recently, Curtis et al. [START_REF] Curtis | A modular framework for simulating crowd movement[END_REF] proposed the Menge framework, a crowd simulator that aggregates different modeling algorithms for each level of pedestrian behaviour. Despite its advancements, this framework only allows for one modeling technique to be implemented at the operational level in the whole simulation.

Currently, there is a lack of a general framework for modeling multi-level pedestrian behaviour that allows agents to dynamically change their operational model. To tackle this challenge, our work focuses on developing an agent-based architecture that models multi-level pedestrian behaviour and is compatible with a variety of modeling algorithms at each level. Moreover, this architecture enables pedestrian agents to switch their operational model in response to changes in local density. A detailed description is provided in the following section.

General overview

In dynamic environments where the density state of each area changes over time, the environment is divided into several zones either based on expert knowledge or dynamic segmentation according to crowd density. These zones are classified as high-density zones (as shown by the red dashed area in Fig. 1) and low-density regions (the region outside of the dashed area in Fig. 1). Each zone is associated with specific triggering rules and transition functions.

The triggering rules evaluate pedestrian and environmental characteristics within the respective zone to select suitable models for pedestrian simulation. Due to the varying performance of pedestrian models across different crowd densities, these rules identify appropriate models to accurately capture the dynamics within each area. When new individuals enter a zone, the transition functions are activated to adapt their information, facilitating a smooth transition from their previous zone to the current one.

When entering a new zone, pedestrians adjust their tactical and operational models to align with the requirements of the corresponding zone. To ensure a generic and extensible approach for various models, an agent-based architecture is proposed in Section 3.2 to model the multi-level behaviour of pedestrians. This agent-based architecture enables pedestrians to dynamically select the appropriate modeling algorithm at each level of behaviour.

Agent-based architecture for multi-level behaviour

To develop an agent-based architecture for multi-level pedestrian behaviour, an abstraction is first introduced to describe the problem of pedestrian modeling.

The problem is divided into two subproblems: tactical and operational subproblem 3 . Suppose that S = {E, A}, where E and A are the environment and the set of agents in simulation respectively. The tactical subproblem can be formalized by the function T, T : S × t × R 2 → R 2 , which maps the simulation state, time, and the agents' destination to the local target which is in R 2 . The operational subproblem can be formalized as follows:

O : S × t × R 2 × [0, 1] → R 2 .
The function O maps the local simulation state S, time, local target, and check if agents in high-density zones to a feasible velocity, which is then used to update agents' locations in the next simulation step. In general, the problem of pedestrian modeling can be stated by the following mathematical formulation:

v i (t) = O i (T i (t)) (4)
p i (t + 1) = p i (t) + v i (t)∆t (5)
where v i (t) and p i (t) are the velocity and the location of agent i in the xycoordinate system at time t respectively and ∆t is the duration of a simulation step. Each modeling algorithm for each level needs to inherit the corresponding abstract class. The Tactical Level and Operational Level classes return the local target and feasible velocity for pedestrians, respectively. What should be noted here is that in the simulation environment, excluding Pedestrian class, only one single instance of each class is created (singleton pattern). Therefore, our architecture does not increase complexity but improves the flexibility of simulation. Furthermore, it is possible to change the modeling algorithm at each level of behaviour by simply updating the variables t and o in Pedestrian class.

Pedestrian activity diagram

Fig. 3 illustrates the activity diagram of pedestrian agents at each simulation step. Initially, the agents perceive their surrounding environment to identify neighboring agents. Subsequently, if the agents have either reached their current local target or do not have a local target, they execute define local target behaviour which invokes their corresponding inheritance model from the Tactical level class to assign a new local target. Afterward, pedestrian agents perform the move behaviour using the returned velocity from their respective inheritance model in the Operational level.

After moving to new locations, pedestrians check whether they have arrived at their destination. If yes, their movement is completed. Otherwise, they determine if they have transitioned into a new zone. In such cases, pedestrian agents need to update their states by sending a request to the Zone entity to obtain information about the tactical and operational models employed in this new zone. This query returns the results o new and t new , which represent the operational and tactical in use within the zone, respectively. Pedestrians then update their attributes by setting their operational variable o to o new and their tactical variable t to t new .

This continuous process of perceiving the environment, defining local targets, moving, and updating states based on zone transitions enables pedestrian agents to navigate through the simulation space effectively. This dynamic approach allows for the modeling of complex scenarios and interactions between agents, accounting for varying tactical and operational models across different zones.

Experiments and Results

This section presents a set of simulations to demonstrate the capabilities of our proposed architecture. More precisely, the ability to dynamically change the operational model is studied through simulations of crowd exit during the Festival of Lights [START_REF]Festival of Lights[END_REF] at the Place des Terreaux, Lyon, France. Simulation results are compared between our model and different other models in terms of density map, pedestrian outflow, and performance.

Experiment design

The simulation area is the Place des Terreaux, a large plaza located in Lyon, France (highlighted by the red dashed rectangle in Fig. 4a) with an area of 7000 m 2 . During the Festival of Lights [START_REF]Festival of Lights[END_REF], pedestrians gather at the plaza to watch the show, which is projected onto the walls of the surrounding buildings (see Fig. 4b). After the show ends, pedestrians head towards the two preconfigured exits (as illustrated by the bottom-left blue arrows in Fig. 4a) to explore new attractions. To accommodate these movements, the environment is discretized into three distinct zones (illustrated by the three dashed rectangles in Fig. 4a).

During the pedestrian exit process, the plaza experiences high density, while the two exits exhibit lower density. Our proposed architecture can effectively handle such multi-density situations. The experiment configuration is as follows:

-The simulation is conducted with a time step ∆t = 0.1 s, with 6000 agents. Fig. 4: Information of the Festival of Lights [START_REF]Festival of Lights[END_REF].

-The CC model [START_REF] Treuille | Continuum crowds[END_REF] for a single target cell is used to simulate pedestrians in the plaza (high-density zones) due to its effectiveness in dense scenarios. This approach leads to further discretization into cells, each storing information about the environment and the pedestrians, such as average velocity and local density. -The SFM [START_REF] Helbing | Simulating dynamical features of escape panic[END_REF] is applied for those who have exited the plaza as it can realistically simulate pedestrians in low-density situations. -In the CC model, the minimum and maximum speeds are set to 0.05 m/s and 1.4 m/s respectively. -The parameters for the SFM are chosen to be consistent with those in [START_REF] Helbing | Social force model for pedestrian dynamics[END_REF],

where the distribution of the preferred speed is Gaussian with the mean of 1.34 m/s and the standard deviation of 0.26 m/s, with a maximum preferred speed of 1.74 m/s.

The proposed model is compared to three others: SFM-only (which applies the SFM in all three zones), 3-CC-1 model (consisting of three separate CC models, each with one target cell for simulating a single zone), and 1-CC-2 model (using one CC model for simulating the entire environment, with two target cells for two exits in two blue dashed areas in Fig. 4a). These comparisons are based on simulations of crowd exits with 6000 agents. For each simulation, three indicators are computed, and an average is determined from a total of 15 simulations for each model:

-Density map (in ped/m 2 ) of pedestrian density distribution across the simulation area. -Outflow (in ped/simulation step) of pedestrians exiting the plaza to the two exits. -Computation time (in s) required to calculate one simulation step.

All simulations are conducted using the GAMA platform [START_REF] Taillandier | composing and experimenting complex spatial models with the GAMA platform[END_REF] on a M1 Mac-Book Pro with 32 GB of memory.

Results

Fig. 5a displays the simulation of 6000 agents at Place des Terreaux, with pedestrians in high-density zones represented in red and pedestrians in low-density regions shown in blue. In comparison, Fig. 5b presents the histogram of pedestrian speeds for both low-density regions and high-density zones. As expected, the speed distribution in the high-density zones differs from that in the low-density regions. The mean speed of pedestrians in the high-density zones is 0.39 m/s, which is substantially lower than the mean speed of 1.37 m/s in the low-density regions. The observed differences in speed distributions between the two areas suggest that the characteristics of pedestrians in low-and high-density situations can be mimicked by incorporating suitable models at the operational level in our architecture. Next, comparisons of different models are evaluated in terms of the density map and pedestrian outflow. Fig. 6 presents the density maps of various models over different time steps (t = 60 s, 150 s, 240 s). In the SFM-only model, the highest density area, with a maximum density of 3.5 ped/m 2 , is observed in both the plaza and the two exits. Similarly, the 3-CC-1 model and 1-CC-2 model can simulate extremely high densities of 6 -8 ped/m 2 , but these extremely highdensity areas also appear in both the plaza and the two exits. In contrast, our model exhibits a clear difference in density levels between the plaza and the two exits. Moreover, Fig. 7a, which shows the outflow of pedestrians exiting the plaza for each model over the simulation time, shows that our model's pedestrian outflow results are similar to those of the other models, with a maximum outflow of approximately 1.4 pedestrians per simulation step and a total of approximately 6800 simulation steps. These results indicate that our model can effectively simulate pedestrians in environments with a mix of low-and high-density situations while having similar results of the pedestrian outflow with the other models. Furthermore, a key advantage of this architecture is its generic nature and flexibility, as it can accommodate any combination of zones and models, enabling the modeling of various scenarios and crowd dynamics. Finally, various numbers of pedestrian agents (3K to 15K) are used to evaluate the simulation time of different models. The results, shown in Fig. 7b, indicate that the SFM-only and 3-CC-1 models have the longest simulation times, which increase significantly as the number of pedestrians grows, while the 1-CC-2 model has a similar simulation time to them at first but drops as the number of agents increases. In contrast, our model demonstrates superior performance in terms of simulation time, with a large difference from the other models observed as the number of pedestrians grows. This finding indicates that using mesoscopic models for low-density regions is unnecessary, and they can be replaced with microscopic models. These results suggest that having different modeling algorithms at each level not only improves the variety of behaviour observed in various scenarios but also enhances overall performance compared to using only microscopic models or multiple mesoscopic models.

Conclusion and Discussion

Large-scale crowd phenomena are complex to model as the behaviour of pedestrians needs to be described at both strategic, tactical, and operational levels and it is impacted by the density of the crowd. Particularly in the operational level modeling, each model can only simulate a single type of pedestrian dynamic, while pedestrian dynamics vary significantly depending on the density of the environment. This paper proposes an agent-based architecture for simulating the multi-level behaviour of pedestrians, where agents are able to dynamically change their operational model based on local density. Our architecture is evaluated through a use case of pedestrians at the Festival of Lights, Lyon, France with simulation results compared to different models in terms of density map, pedestrian outflow, and computation time. The results demonstrate that our agent-based architecture can effectively simulate pedestrians in diverse-density situations, providing flexibility for incorporating various models, and enhancing runtime performance while achieving comparable pedestrian outflow results to alternative models. Our architecture is not only generic in the domain of pedestrian modeling but also extensible to other domains such as traffic simulation and social simulation.

Our future work aims to incorporate a greater variety of modeling algorithms for each level of behaviour, particularly at the operational level. Then, the selection of an optimal algorithm at each level of behaviour and the criteria for switching these algorithms need to be defined based on the density of each region. Another promising direction involves applying density-based clustering algorithms to pedestrian coordinates in order to dynamically estimate the local density of environments.

Fig. 1 :

 1 Fig. 1: General overview.

Fig. 2 :

 2 Fig. 2: Agent-based architecture for multi-level pedestrian behaviour.

Fig. 3 :

 3 Fig. 3: Pedestrian's activity diagram at each simulation step.

  (a) Main circulation of the exiting crowd. (b) Pedestrians watching the show at the Place des Terreaux 4 .

  (a) Simulation of 6000 agents at the Place des Terreaux. (b) Comparison of speed histogram of two regions.

Fig. 5 :

 5 Fig. 5: Simulation results.

Fig. 6 :

 6 Fig. 6: Density maps among different models with 6000 simulated agents.

  (a) The outflow over the simulation time. (b) Model performance.

Fig. 7 :

 7 Fig. 7: Comparison results of different models.

The integration of strategic subproblem is not in the scope of this paper.

https://www.fetedeslumieres.lyon.fr/fr/oeuvre/grand-mix-au-musee-des-beaux-arts-de-lyon

Acknowledgements The authors acknowledge the Franco-German research project MADRAS funded in France by the Agence Nationale de la Recherche (ANR, French National Research Agency), grant number ANR-20-CE92-0033, and in Germany by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant number 446168800.


Marie-Laure Courboules
email: laure.courboules@departement13.fr

Daniela Peloso
email: daniela.peloso@ipsofacto.coop

Vincent Dumas
email: vdumas@mmsh.univ-aix.fr

Preliminary results of the Monitoring Preservation Program (2015-2017) of the Roman barge Arles Rhône 3 (Arles, France). An innovative approach for the monitoring of ancient wooden shipwrecks in Museum exhibitions

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

In a first step, the geometrical and 3D photogrammetric survey records and measures the gradual deformations in the barge's wood. The derived technical data is combined with the observations registered during the annual condition report or after occasional incidents.

In a second protocol step, all available data are organized in a relational and explorable database management system. The objective is to link and interrelate the 3D monitoring results with the archaeological data, the restoration treatments and the preventive conservation data within the museum exhibition context. Given the purpose of our study that is to know the behaviour of the wreck materials as regards the environmental conditions in the museum and thus prevent the risk of acidification and the deformation to ensure long-term preservation, we gathered some preliminary data. The barge Arles Rhône 3 reached equilibrium conditions after one year within his exposition environment. We observed some movements along X and Y axis and also between each section, independently to the general movement that affects the barge on his total length. The movements in Z axis have generally negative values (this trend is not easy to be interpreted).

The

The critical areas are now identifiedm and they are the object of a microphotogrammetrical survey. In the same time, during the annual reports, the wood and metal pieces are monitored and the database is improved (technical glossary, layer/attribute table link) to develop a fully adapted monitoring tool for the museums hosting in their collections such fragile artefacts as ancient wooden ships.

  Arles-Rhône 3 (AR3) barge raised from the Rhône river in 2013 and exhibited in 2014 in the new hall of the Musée Départemental Arles Antique (MdAa), has been the subject of an intense maintenance and monitoring program developed by the conservation team of the Museum, in collaboration with Arc'Nucléart and A'Corros specialists (French private companies specialized in conservation of waterlogged and metal artifacts). The study of the conservation conditions and of the constraints presented in the Roman maritime-fluvial barge AR3 demonstrated the necessity to develop adapted monitoring tools. This project development was initiated in 2015 and it is conducted based on the work of the Museum conservators and on the technical expertise of Ipso Facto (private underwater archeological company), as well as the technical laboratory of the Centre Camille Jullian (Aix-Marseille University, CNRS). Preliminary results of the Monitoring Preservation Program (2015-2017) of the Roman barge Arles Rhône 3 (Arles, France). An innovative approach for the monitoring of ancient wooden shipwrecks in Museum exhibitions Marie-Laure Courboulès*, Daniela Peloso**, Vincent Dumas*** The protocol can be summarized in two steps :

  Photogrammetry survey (zone)


domain: math info sdv

text:
Élisabeth Remy
email: elisabeth.remy@univ-amu.fr

Paul Ruet
email: ruet@irif.fr

From multivalued to Boolean functions: preservation of soft nested canalization

Keywords: nested canalizing functions, Boolean functions, multivalued functions, regulatory network modelling

Nested canalization (NC) is a property of Boolean functions which has been recently extended to multivalued functions. We study the effect of the Van Ham mapping (from multivalued to Boolean functions) on this property. We introduce the class of softly nested canalizing (SNC) multivalued functions, and prove that the Van Ham mapping sends SNC multivalued functions to NC Boolean functions. Since NC multivalued functions are SNC, this preservation property holds for NC multivalued functions as well. We also study the relevance of SNC functions in the context of gene regulatory network modelling.

Introduction

Dynamical properties of biological systems, such as stability and robustness, are often associated with the canalyzing property. This notion is at the basis of the work of Waddington in embryology, which describes, in the 1940s, epigenetic landscapes representing embryogenesis by canalizing configurations [START_REF] Waddington | Canalization of development and the inheritance of acquired characters[END_REF]. In the 1990's, S. Kauffman introduced the class of canalizing Boolean functions [START_REF] Kauffman | The origins of order: Self organization and selection in evolution[END_REF][START_REF] Kauffman | Random Boolean network models and the yeast transcriptional network[END_REF] to formalize the canalizing behaviour observed in gene regulatory networks. In short, canalizing Boolean functions are functions f : (Z/2Z) n → Z/2Z (or R) such that at least one input variable, say x i (1 ⩽ i ⩽ n), has a value a = 0 or 1 which determines the value of f (x). Nested canalizing (NC) functions provide a "recursive" version of canalizing functions: an NC function f is canalizing and, moreover, its restriction f ↾ x i ̸ =a is itself NC. NC functions are particularly interesting because of their "low complexity": in particular, their average sensitivity (a measure of complexity related to spectral concentration, learning properties, decision tree complexity [START_REF] Donnell | Analysis of Boolean functions[END_REF] and stability properties [START_REF] Kauffman | The origins of order: Self organization and selection in evolution[END_REF][START_REF] Kauffman | Genetic networks with canalyzing Boolean rules are always stable[END_REF]) is bounded above by a constant [START_REF] Li | Boolean nested canalizing functions: A comprehensive analysis[END_REF][START_REF] Klotz | Bounds on the average sensitivity of nested canalizing functions[END_REF].

A natural question is the extension of this canalizing property to multivalued functions, involving more than two expression levels, which are often needed in the modeling of biological systems to circumvent the too restrictive Boolean representation [START_REF] Thomas | Regulatory networks seen as asynchronous automata: a logical description[END_REF]. The present paper investigates the property of nested canalization for multivalued functions f : (Z/kZ) n → Z/kZ or R for some k ⩾ 2. A definition of nested canalizing (NC) multivalued function has been proposed in [START_REF] Murrugarra | Regulatory patterns in molecular interaction networks[END_REF][START_REF] Murrugarra | The number of multistate nested canalyzing functions[END_REF]. In [START_REF] Remy | Average sensitivity of nested canalizing multivalued functions[END_REF], we introduced a more general class of multivalued functions, which we called weakly nested canalizing (WNC), for which we proved that the average sensitivity is bounded above by a constant.

Multivalued modelling allows to distinguish different thresholds of action of a gene on its targets, and is of course more complex to analyse than the Boolean case. Boolean mappings have been proposed, associating one Boolean variable to each threshold. This gives access to a large number of theoretical works and numerical tools specific to the analysis of Boolean models. Thus we are interested in the relationship between the notions of canalization for multivalued and Boolean functions. Clearly an NC Boolean function is NC multivalued, but what can we say about the other direction? More precisely, a mapping β from multivalued functions to Boolean functions have been defined by Van Ham in [START_REF] Van Ham | How to deal with variables with more than two levels. In Kinetic logic: a Boolean approach to the analysis of complex regulatory systems[END_REF], and it have been proved in [START_REF] Didier | Mapping multivalued onto Boolean dynamics[END_REF][START_REF] Tonello | On the conversion of multivalued gene regulatory networks to Boolean dynamics[END_REF] to be in some sense unique (it is the only injective such mapping preserving neighbours in the state space, and for which the regulatory graphs are unchanged).

We prove here that β maps NC multivalued functions to NC Boolean functions. To this end, we introduce in Section 2 the notion of softly nested canalizing (SNC) multivalued functions, and prove that NC ⇒ SNC ⇒ WNC (Proposition 2 and Section 2.3). Then we actually prove in Theorem 3 that SNC functions are mapped by β to NC Boolean functions. On the other hand, we show that Theorem 3 does not extend to WNC functions: we shall see in Section 3.5 an example of a WNC function which is mapped by β to a non NC Boolean function.

NC functions appear as appropriate rules in Boolean models of gene regulatory networks. Indeed, Boolean networks have been proposed as models for gene regulatory networks [START_REF] Thomas | Boolean formalization of genetic control circuits[END_REF], with different nodes corresponding to different genes. The activity of a gene is regulated by the activity of other genes to which it is connected through a Boolean function. These networks are built from biological data and knowledge from the literature, and their associated Boolean functions are far from random. It turns out that NC functions are predominant in the large databases of Boolean gene networks [START_REF] Subbaroyan | Minimum complexity drives regulatory logic in Boolean models of living systems[END_REF]. In Section 4 we extract the multivalued genes from this database, and we study the canalizing properties of their multivalued functions and their corresponding Boolean functions.

Nested canalizing multivalued functions

The concept of nested canalization, originally considered for Boolean functions [START_REF] Kauffman | The origins of order: Self organization and selection in evolution[END_REF], has been extended to multivalued functions in [START_REF] Murrugarra | Regulatory patterns in molecular interaction networks[END_REF][START_REF] Murrugarra | The number of multistate nested canalyzing functions[END_REF][START_REF] Kadelka | Multistate nested canalizing functions and their networks[END_REF]. We first recall here this definition, before introducing soft nested canalization in Section 2.1.

Let k, n be positive integers, k ⩾ 2. Z/kZ is the ring of integers modulo k. A function f : (Z/kZ) n → Z/kZ is said to be canalizing with respect to coordinate i and (a, b) ∈ Z/kZ × Z/kZ if there exists a function g :

(Z/kZ) n → Z/kZ different from the constant b such that f (x) = b if x i = a g(x) if x i ̸ = a.
We shall simply say that f is canalizing if it is canalizing with respect to some i, a, b.

A segment is a subset of Z/kZ of the form {0, . . . , i} or {i, . . . , k-1}. Let σ ∈ S n be a permutation, A 1 , . . . , A n be segments, and c 1 , . . . , c n+1 ∈ Z/kZ be such that c n ̸ = c n+1 . Then f is said to be nested canalizing (NC) with respect to σ, A 1 , . . . , A n , c 1 , . . . , c n+1 if

f (x) =                  c 1 if x σ(1) ∈ A 1 c 2 if x σ(1) / ∈ A 1 , x σ(2) ∈ A 2 . . . . . . c n if x σ(1) / ∈ A 1 , . . . , x σ(n-1) / ∈ A n-1 , x σ(n) ∈ A n c n+1 if x σ(1) / ∈ A 1 , . . . , x σ(n-1) / ∈ A n-1 , x σ(n) / ∈ A n .
We shall simply say that f is NC if it is NC with respect to some σ, A 1 , . . . , A n , c 1 , . . . , c n+1 .

Note that the segment A i plays the role of a threshold for the variable x σ(i) . For an NC multivalued function f as defined in [START_REF] Murrugarra | Regulatory patterns in molecular interaction networks[END_REF][START_REF] Murrugarra | The number of multistate nested canalyzing functions[END_REF][START_REF] Kadelka | Multistate nested canalizing functions and their networks[END_REF], each variable x σ(i) is used exactly once in the process of canalization (and the value of f if determined by the threshold corresponding to A i ). It is possible to relax the definition by allowing canalization along successive thresholds for a same variable. This is essentially what we shall do in defining softly nested canalizing functions below. In doing so, we shall also remove the condition c n ̸ = c n+1 .

Softly nested canalizing multivalued functions

Let n be a positive integer. For each i ∈ {1, . . . , n}, let k i > 0, Ω i be a set of k i integers, Ω = i Ω i , and f : Ω → R. Note that we do not require k i ⩾ 2 for all i. If k j = 1 for some j, f could be viewed as a function with one less variable, i.e. as a function on i̸ =j Ω i , but we still consider it as a function defined on i Ω i .

We shall say that f is softly canalizing with respect to coordinate i and

(a, b) ∈ Ω i × R if f (x) = b whenever x i = a,
and simply that it is softly canalizing if it is softly canalizing with respect to some i, a, b.

Note that this definition differs slightly from the usual definition by the absence of condition on the values of f for x i ̸ = a: we do not require the existence of some x such that x i ̸ = a and f (x) ̸ = b. In particular, constant functions are softly canalizing, though not canalizing.

If f is canalizing with respect to i, a, b and k i ⩾ 2, we shall consider

f ↾ x i ̸ =a : Ω ∩ {x | x i ̸ = a} → R,
the restriction of f to the set of x ∈ Ω such that x i ̸ = a. The class of softly nested canalizing on Ω = i Ω i is then defined by induction on the cardinality

|Ω| = i k i of Ω. If |Ω| = 1, i.e. k i = 1 for all i, any f : Ω → R is softly nested canalizing (SNC) on Ω. If |Ω| > 1, f : Ω → R is SNC on Ω if there exist i, a, b such that • k i ⩾ 2,
• a is either the min or the max of Ω i ,

• f is softly canalizing with respect to i, a, b,

• f ↾ x i ̸ =a is SNC on Ω ∩ {x | x i ̸ = a}, a strict subset of Ω.
Intuitively, a function f : Ω → R is SNC if its domain Ω can be "peeled" by successively removing coordinate hyperplanes (defined by equations of the form x i = a with a minimal or maximal) whose points are mapped by f to the same value, whence the following characterization:

Proposition 1. Letting K = i k i -n, f is SNC if and only if there exist a function v : {1, . . . , K} → {1, . . . , n}, numbers a i ∈ Ω v(i) and b i ∈ R for each i ∈ {1, . . . , K} such that: f (x) =            b 1 if x v(1) = a 1 b 2 if x v(1) ̸ = a 1 , x v(2) = a 2 . . . . . . b K if x v(1) ̸ = a 1 , . . . , x v(K-1) ̸ = a K-1 , x v(K) = a K and for each i ∈ {1, . . . , K}, Ω ′ v(i) = Ω v(i) \ {a j | j < i, v(i) = v(j)
} is not a singleton and a i is either the min or the max of Ω ′ v(i) .

In decomposing an NC function f : (Z/kZ) n → Z/kZ, each coordinate i ∈ {1, . . . , n} is considered exactly once (in some order prescribed by a permutation σ) and the value of f is fixed for x σ(i) in some segment A i . This can be realized by successively fixing the value of f for each α ∈ A i , and therefore, the class of SNC functions contains the class of NC functions, as stated in the following Proposition:

Proposition 2. If f : (Z/kZ) n → Z/kZ is NC, then it is SNC.
Proof. Assume f is NC with respect to σ, A 1 , . . . , A n , c 1 , . . . , c n+1 . For each i ∈ {1, . . . , n}, let

A i = {α 1 i , . . . , α |A i | i } (Z/kZ) \ A i = {α 1+|A i | i , . . . , α k-1 i }.
More precisely, the K = n(k -1) numbers α j i ∈ Z/kZ are defined according to the following ordering which depends on the nature of the segment A i :

• either 0 ∈ A i , and then we choose the numbers α j i so that 0 =

α 1 i < • • • < α |A i | i and k -1 = α 1+|A i | i > • • • > α k-1 i ; • or k -1 ∈ A i , and we choose the α j i so that k -1 = α 1 i > • • • > α |A i | i and 0 = α 1+|A i | i < • • • < α k-1 i .
If both 0 and k -1 ∈ A i , i.e. if A i = Z/kZ, we choose any of the two orders. Now, for each i ∈ {1, . . . , n} and j ∈ {1, . . . , k -1}, let

β j i = c i if j ⩽ |A i | c n+1 otherwise.
To comply with the characterization of SNC functions (Proposition 1), we relabel the numbers α j i , β j i by identifying the list

α 1 1 , . . . , α |A 1 | 1 , . . . , α 1 n , . . . , α |A i | n , α 1+|A 1 | 1 , . . . , α k 1 , . . . , α 1+|An| n , . . . , α k-1 n
as the list a 1 , . . . , a K , and by identifying similarly the list

β 1 1 , . . . , β |A 1 | 1 , . . . , β 1 n , . . . , β |An| n , β 1+|A 1 | 1 , . . . , β k 1 , . . . , β 1+|An| n , . . . , β k-1 n
as the list b 1 , . . . , b K . Call φ this relabelling, which maps r ∈ {1, . . . , K} to the pair φ(r) = (i, j) such that a r = α j i and b r = β j i . For instance, φ(1) = (1, 1) and φ(K) = (n, k -1). Then finally, a function v : {1, . . . , K} → {1, . . . , n} is defined by v(r) = σ(i) if φ(r) = (i, j). Then f clearly enjoys the characterization of SNC functions, with the choice of function v and numbers a r , b r .

Some examples

• We have already noticed that constant functions from (Z/kZ) n to Z/kZ are SNC but not NC.

• An easy induction on k shows that the functions min and max : (Z/kZ) 2 → Z/kZ are SNC. For instance, min = min k : {0, . . . , k -1} 2 → {0, . . . , k -1} is softly canalizing with respect to 1, 0, 0, then min k ↾ x 1 ̸ =0 is softly canalizing with respect to 2, 0, 0, and min k ↾ x 1 ̸ =0,x 2 ̸ =0 is identical to the function min k-1 : {1, . . . , k -1} 2 → {1, . . . , k -1}, which is SNC.

However, they are not NC [START_REF] Kadelka | Multistate nested canalizing functions and their networks[END_REF]. Intuitively, there is more freedom in the construction of SNC functions, which can be built by successively "peeling" coordinate hyperplanes defined on some coordinate i (i.e. by some equation x i = a), then a coordinate hyperplane defined on some other coordinate j, and later a coordinate hyperplane defined on i again.

• For the same reason, the identity from (Z/kZ) n to (Z/kZ) n is SNC but not NC. As we shall see in Section 4, this applies to the functions governing the regulation of genes such as Raf, Dsor, Drk, Stat92E.

• In Section 4, we shall see other examples of SNC and non SNC functions occurring in multivalued gene regulatory networks found in the literature.

WNC multivalued functions and average sensitivity

It is possible to consider a slightly more general notion of multivalued canalization, by removing the condition on canalizing values in the definition of SNC functions. This is what we do in [START_REF] Remy | Average sensitivity of nested canalizing multivalued functions[END_REF]: weakly nested canalizing (WNC) functions f : (Z/kZ) n → Z/kZ are defined like SNC functions, but the values a used to define f (x) for x i = a need not be extreme values (initially 0 or k -1), they can be intermediate values: 0 < a < k -1.

By Proposition 2, NC ⇒ SNC ⇒ WNC, and we prove in [START_REF] Remy | Average sensitivity of nested canalizing multivalued functions[END_REF] that WNC (hence NC and SNC) multivalued functions have "low complexity" in the sense that their average sensitivity (see [START_REF] Donnell | Analysis of Boolean functions[END_REF]Chapter 8]) is bounded above by a constant (independent of n), while the average sensitivity of an arbitrary multivalued function is of order O(n).

It is worth noticing that the above implications are strict. For instance, the function f : Z/3Z × Z/3Z → Z/3Z defined by the following table:

f (x, y) = x\y 0 1 2 0 2 0 0 1 1 1 1 2 2 0 2
is WNC but not SNC: the first canalization has to take the intermediate value x = 1 (and determines f (x, y) = 1), then the second canalization takes y = 0 (determining f (x, y) = 2) or y = 1 (determining f (x, y) = 0), and finally x = 0 or x = 2.

Preservation of nested canalization under Boolean mapping

Mapping multivalued functions to Boolean functions

Let n, Ω i = Z/k i Z and Ω = n i=1 Ω i be as above. The injective mapping β : Ω → {0, 1} k , with k = i k i -n, proposed by Van Ham [START_REF] Van Ham | How to deal with variables with more than two levels. In Kinetic logic: a Boolean approach to the analysis of complex regulatory systems[END_REF] is defined as follows. For i ∈ {1, . . . , n} and 1 ⩽ a ⩽ k i -1, let β i,a : Ω → {0, 1} be defined by

β i,a (x) = 1 if x i ⩾ a 0 otherwise, and
β i (x) = (β i,1 (x), . . . , β i,k i -1 (x)) = 1 x i 0 k i -1-x i ,
where 1 ℓ = (1, . . . , 1) is the ℓ-tuple of 1's (and similarly for 0 ℓ ), and tuples of 0's and 1's are represented by words (without commas and parentheses).

The maps can be combined to define β : Ω → {0, 1} k by

β(x) = (β 1,1 (x), . . . , β 1,k 1 -1 (x), . . . , β n,1 (x), . . . , β n,kn-1 (x)) = 1 x 1 0 k 1 -1-x 1 • • • 1 xn 0 kn-1-xn .
The image β(Ω) of β is a strict subset of {0, 1} k (unless k i = 2 for all i), and a point x ∈ {0, 1} k is said admissible when x ∈ β(Ω). Given F : Ω → Ω, by injectivity of β, the equality

F β • β = β • F, defines F β : β(Ω) → β(Ω), the Booleanization of F . Clearly, if F is Boolean (i.e. k i = 2 for all i), then β(Ω) = Ω = (Z/2Z) n and F β = F .
Let us mention that other injective mappings from Ω to {0, 1} k may be defined but β is the only one preserving neighbours and regulatory graphs [START_REF] Didier | Mapping multivalued onto Boolean dynamics[END_REF][START_REF] Tonello | On the conversion of multivalued gene regulatory networks to Boolean dynamics[END_REF]:

• two neighbouring states (i.e. states x, y ∈ Ω whose Hamming distance equals 1) are mapped by β to neighbouring states in β(Ω),

• the global regulatory graphs G(F ) and G(F β ) underlying the dynamics F and F β are isomorphic. We refer to, e.g., [START_REF] Remy | On differentiation and homeostatic behaviours of Boolean dynamical systems[END_REF][START_REF] Remy | Graphic requirements for multistability and attractive cycles in a Boolean dynamical framework[END_REF][START_REF] Ruet | Local cycles and dynamical properties of Boolean networks[END_REF] for the definition of G(F ).

The purpose of the remainder of this section is to prove that β maps SNC multivalued functions to NC (Boolean) functions. Before doing this in Theorem 3 we need to recall the definition of nested canalization for Boolean functions.

Nested canalizing Boolean functions

Let k be a positive integer, σ ∈ S k be a permutation, and a 1 , . . . , a k , b 1 , . . . , b k ∈ {0, 1}. We recall that a Boolean function f : {0, 1} k → {0, 1} is said to be nested canalizing (NC) with respect to σ, a

1 , . . . , a k , b 1 , . . . , b k if f (x) =            b 1 if x σ(1) = a 1 b 2 if x σ(1) ̸ = a 1 , x σ(2) = a 2 . . . . . . b k if x σ(1) ̸ = a 1 , . . . , x σ(k-1) ̸ = a k-1 , x σ(k) = a k . ( * )
Note that we slightly modify the usual definition by not requiring f

(x) = 1 -b k when x σ(1) ̸ = a 1 , . . . , x σ(k-1) ̸ = a k-1 , x σ(k) ̸ = a k .
We shall simply say that f is NC if it is NC with respect to some σ, a 1 , . . . , a k , b 1 , . . . , b k , and that a function

F = (F 1 , . . . , F k ) : {0, 1} k → {0, 1} k is NC if F j : {0, 1} k → {0, 1} is NC for all j.

Nested canalizing partial Boolean functions

In Section 3.4, we shall consider functions f : X → {0, 1} defined on a subset X ⊆ {0, 1} k . We extend the notion of nested canalization to such partial functions, by simply saying that f : X → {0, 1} is NC with respect to the above data if condition ( * ) holds for all x ∈ X.

Relation between β and nested canalization

Theorem 3. If F = (F 1 , . . . , F n ) : Ω → Ω is SNC, then F β : β(Ω) → β(Ω) is NC.
Proof. As in the definition of β, we shall use double indices for Boolean functions, and let

F β = (F β 1,1 , . . . , F β 1,k 1 -1 , . . . , F β n,1 , . . . , F β n,kn-1 ) = (F β 1 , . . . , . . . , F β n ),
where F β j,a = β j,a • F : β(Ω) → {0, 1} and

F β j = β j • F = (F β j,1 , . . . , F β j,k j -1 ) : β(Ω) → {0, 1} k j -1 .
Therefore F β j (β(x)) = 1 F j (x) 0 k j -1-F j (x) and F β j,a (β(x)) = 1 ⇔ F j (x) ⩾ a. Let us fix j ∈ {1, . . . , n} and 1 ⩽ a ⩽ k j -1 and prove that F β j,a is NC. By assumption, F j : Ω → Ω j is SNC, hence by Proposition 1, there exist a function v : {1, . . . , K} → {1, . . . , n}, with K = i k i -n, and numbers a i ∈ Ω v(i) and b i ∈ Ω j for each i ∈ {1, . . . , K} such that, for all x ∈ Ω:

F j (x) =            b 1 if x ∈ Ω (1) and x v(1) = a 1 b 2 if x ∈ Ω (2) and x v(2) = a 2 . . . . . . b K if x ∈ Ω (K) and x v(K) = a K ,
where for all i ∈ {1, . . . , K}, Ω (i) ⊆ Ω is the set of x such that

x v(1) ̸ = a 1 , . . . , x v(i-1) ̸ = a i-1 .
Note that Ω = Ω (1) ⊃ Ω (2) ⊃ • • • ⊃ Ω (K) form a decreasing sequence of subsets of Ω. Moreover, by Proposition 1, the numbers a i are assumed to satisfy the following constraint:

either a i = min Ω (i) v(i) or a i = max Ω (i) v(i) .
Let ε i = 0, a ′ i = a i + 1 in the first case (min case) and ε i = 1, a ′ i = a i in the second case (max case). Letting y = β(x), we observe that in the min case, when x ∈ Ω (i) we have

x v(i) = a i ⇔ x v(i) ⩽ a i ⇔ y v(i),a i +1 = β v(i),a i +1 (x) = 0,
and that in the max case, we have

x v(i) = a i ⇔ x v(i) ⩾ a i ⇔ y v(i),a i = β v(i),a i (x) = 1.
These two equivalences are summerized in the following property:

x v(i) = a i ⇔ y v(i),a ′ i = ε i whenever x ∈ Ω (i) . (P )
Let us prove by induction on i that

x ∈ Ω (i)        y v(1),a ′ 1 ̸ = ε 1 . . . y v(i-1),a ′ i-1 ̸ = ε i-1 . (Q i ) Q 1 is trivial because Ω (1)
= Ω, and by property P , we have

x ∈ Ω (i+1) ⇔ x ∈ Ω (i) and x v(i) ̸ = a i ⇔ x ∈ Ω (i) and y v(i),a ′ i ̸ = ε i , hence Q i entails Q i+1
, and we have shown that Q i holds for any i. Combining properties P and Q i , we then obtain:

x ∈ Ω (i) x v(i) = a i ⇔            y v(1),a ′ 1 ̸ = ε 1 . . . y v(i-1),a ′ i-1 ̸ = ε i-1 y v(i),a ′ i = ε i .
On the other hand, F β j,a (y) = 1 ⇔ F j (x) ⩾ a. Hence, letting χ b⩾a = 1 if b ⩾ a and 0 otherwise, we have F β j,a (y) = χ F j (x)⩾a . In particular

F j (x) = b i ⇒ F β j,a (y) = χ b i ⩾a .
Therefore, the condition that F j is SNC gives, for any y ∈ β(Ω):

F β j,a (y) =            χ b 1 ⩾a if y v(1),a ′ 1 = ε 1 χ b 2 ⩾a if y v(1),a ′ 1 ̸ = ε 1 , y v(2),a ′ 2 = ε 2 . . . . . . χ b K ⩾a if y v(1),a ′ 1 ̸ = ε 1 , . . . , y v(K-1),a ′ K-1 ̸ = ε K-1 , y v(K),a ′ K = ε K .
This means that F β j,a is NC in the sense of Section 3.3. Since this holds for all j, a, F β is NC.

Counterexample for WNC functions

Let f : Ω = Z/3Z × Z/3Z → Z/3Z be the function defined in Section 2.3. It is WNC but not SNC, and where pairs in {0, 1} 2 are written without commas and parentheses. This table is simply obtained from the one defining f by the substitutions 0 → 00, 1 → 10, 2 → 11. Now, the two Boolean functions

β(Ω) ⊆ {0, 1} 2 × {0, 1} 2 . The function f β : β(Ω) → {0,
f β 1 , f β 2 : β(Ω) → {0, 1} such that f β = (f β 1 , f β 2 )
are therefore:

f β 1 (x, y) = 00 10 11 00 1 0 0 10 1 1 1 11 1 0 1 f β 2 (
x, y) = 00 10 11 00 1 0 0 10 0 0 0 11 1 0 1 and it is easy to see that f β 1 is NC but that f β 2 is not NC. Therefore Theorem 3 does not extend to WNC functions.

Examples in the regulatory network literature

Boolean NC functions appear predominant in databases of Boolean genetic networks [START_REF] Subbaroyan | Minimum complexity drives regulatory logic in Boolean models of living systems[END_REF]. If this canalyzing property reflects physical characteristics of biological systems, we wonder whether this property is also present in multivalued functions that model these same systems. Thus, in order to have an insight into the characteristics of multivalued functions specified in the context of biological systems, we have selected the multivalued genes present in the logical models explored in [START_REF] Subbaroyan | Minimum complexity drives regulatory logic in Boolean models of living systems[END_REF] (45 genes endowed with a ternary variable) and studied the canalizing properties (NC, SNC, WNC) of their logical functions. We also applied the Boolean mapping β and studied the canalizing property of the resulting Boolean functions. The results of the analysis are collected in Table 1, reflecting four different qualitative situations: Remark that in this set of functions, all WNC functions are actually SNC.

The logical rules corresponding to these genes are collected in the Appendix. To simplify the notation in the rules, the name of the gene is written to represent its activity variable (for instance A : 1 stands for x A = 1), and the value of F A is defined by means of the usual logical connectives (conjonction ∧, disjunction ∨ and negation ¬). For example, the first and third lines of Table A.2 mean:

F Drk (x) =      1 if x Der = 1 2 if x Der = 2 0 otherwise F RI (x) =      1 if x Dsor1 = 1 and x Msk = 1 2 if x Dsor1 = 2 and x Msk = 1 0 otherwise
This means that gene Drk can be activated at level 1 (resp. 2) when its (unique) regulator Der is at level 1 (resp. 2). Gene RI has two regulators, Msk (which is Boolean) and Dsor1, and it can be activated at level 1 (resp.

2) if Msk is present (x Msk = 1) and Dsor1 is at level 1 (resp. 2). Some situations are more complex. For instance, gene MadMed (extracted from the model described in [START_REF] Mbodj | Logical modelling of Drosophila signalling pathways[END_REF]) has three regulators: a multilevel activator Tkv, a Boolean activator Sax and a Boolean inhibitor Dad. The rule for MadMed is the following (cf Table A.5):

F MadMed (x) =     
1 if (x Tkv = 1 or x Sax = 1) and x Dad = 0 and x Tkv < 2 2 if x Tkv = 2 and x Dad = 0 0 otherwise. Thus, in the absence of the inhibitor Dad, the presence of one of the two activators at level 1 (max level for Sax and intermediate for Tkv) allows MadMed to be activated at its level 1. MadMed can be activated at its maximum level as soon as its activator Tkv is at its maximum level 2 and the inhibitor Dad is absent.

The structure of multivalued logical rules varies greatly. A quite current situation is when the level of only one regulator determines the target value of the gene:

A = 1 if B ∧ φ A = 2 if ¬B ∧ φ (S)
with B a regulator of A, and φ a "context" (conditions on the presence or absence of other regulators). This concerns the genes listed in Tables A.2, A.3, A.4. We can see in Table 1 that the rules satisfying this structure (S) generally have strong canalyzing properties: situations (a) or (b). In particular, their Booleanizations are all NC. Then, depending on the context, we distinguish three situations:

• if φ depends on multivalued variables and is expressed only with ∧, then the function behaves according to situation (b); this is the case of, e.g., Stat92E; these functions are listed in Table A 1: Multivalued genes occurring in logical models from database considered in [START_REF] Subbaroyan | Minimum complexity drives regulatory logic in Boolean models of living systems[END_REF]. The two rightmost columns indicate whether the Booleanized functions are nested canalizing, and which rules satisfy structure (S).

The rules that do not satisfy structure (S) are collected in Tables A. [START_REF] Kauffman | Random Boolean network models and the yeast transcriptional network[END_REF] and A.6. In particular, Table A.6 gathers all the rules in situation (d), for which even the Booleanized functions are not NC.

The study of these multivalued functions suggests a certain relevance of canalization, notably SNC, among models arising in the modelling of gene networks. But it should be noted that we have not specified the assumptions for updating the system while they play an important role in the dynamical behavior of the system and its properties. Implicitly, we here considered that the gene directly reached its target value: for instance, Drk can be activated at level 2 when Der is at level 2, and if the current state of Drk is 0, it will update directly to value 2. However, in such context of biological systems, it is not a realistic hypothesis, steps of 1 are more common: in the previous example, if Der is at level 2 and Drk is not active, its activity will first reach level 1. The introduction of the "steps of 1" hypothesis in the definition of the function would obviously change the canalizing properties. A.4: Rules associated to multivalued genes from databases considered in [START_REF] Subbaroyan | Minimum complexity drives regulatory logic in Boolean models of living systems[END_REF], satisfying structure (S) and behaving according to situation (a). VIM ∧ ¬CDH1 Table A.5: Rules associated to multivalued genes from databases considered in [START_REF] Subbaroyan | Minimum complexity drives regulatory logic in Boolean models of living systems[END_REF], not satisfying structure (S) and behaving according to situations (a), (b) or (c).

Gene i x i Rule Ref

  1} 2 is given by the following table:

  (a) functions F which are NC (hence SNC, WNC, and with NC Booleanization F β ); (b) functions which are not NC, but SNC (hence WNC, and with NC Booleanization); (c) functions which are not WNC (hence neither NC nor SNC) but with NC Booleanization; (d) functions which are not WNC and with non NC Booleanizations.

Ras 1 ( 1 ∨ 2 Dpp : 2 ∧: 2 ∧: 1 ∨ 2 cCa 1 ( 2 1 1 ((Tkv : 1 ∨ 2 Tkv : 2 ∧ ¬Dad : 1 E 2 IL4R : 2 ∨ IL2R : 2 VIM 1 (

 1122212121112212221 Sos : 1 ∧ ¬(Sty ∧ Gap1)) ∨ (Gap1 ∧ Sty ∧ Sos : 2) Scw ∨ Gbb) ∧ Punt ∧ ¬(Sog ∨ Tsg) Punt ∧ ¬(Sog ∨ Tsg) mdH : 1) ∨ (mNADH NAD : 2 ∧ mdH : 2) ∨ (mNADH NAD : 1 ∧ mdH : 2) mNADPH 1 (¬mNNT ∧ mIDH2) ∨ (mNNT : 1 ∧ ¬mIDH2) NADP 2 mNNT : 2 ∨ (mNNT : 1 ∧ mIDH2) mNADPH NADP : 2) ∧ [19] (¬mGSH GSSG ∨ mGSH GSSG : 1) 2 (mNADPH NADP : 1 ∨ mNADPH NADP : 2) ∧ mGSH GSSG : 2 mTR 1 (mNADPH NADP : 2 ∧ (¬mTRX ∨ mTRX : 1)) ∨ mNADPH NADP : 1 2 mTRX : 2 ∧ mNADPH NADP : IP3R ∧ ¬ORAI1 ∧ ¬TRPM2 ∧ ¬PMCA) ∨ (¬IP3R ∧ ¬ORAI1 ∧ TRPM2 ∧ ¬PMCA) ∨ (¬IP3R ∧ ORAI1 ∧ TRPM2 ∧ ¬PMCA) ∨ (¬IP3R ∧ ORAI1 ∧ ¬TRPM2 ∧ ¬PMCA) 2 (IP3R ∧ ORAI1 ∧ TRPM2 ∧ ¬PMCA) ∨ (IP3R ∧ ORAI1 ∧ ¬TRPM2 ∧ ¬PMCA) ∨ (IP3R ∧ ¬ORAI1 ∧ TRPM2 ∧ ¬PMCA) cGR 1 (cNADPH NADP : 1 ∨ cNADPH NADP : 2) ∧ (¬cGSH GSSG ∨ cGSH GSSG : 1) 2 (cNADPH NADP : 1 ∨ cNADPH NADP : 2) ∧ cGSH GSSG : 2 cTR 1 (cNADPH NADP : 2 ∧ (¬cTRX ∨ cTRX : 1)) ∨ cNADPH NADP : 1 2 cTRX : 2 ∧ cNADPH NADP : Hop Dome ∧ ¬ET ∧ [(Stam ∧ Hrs ∧ Socs36E) ∨[9] (¬(Stam ∧ Hrs) ∧ ¬Socs36E)] 2 Dome ∧ ¬ET ∧ Stam ∧ Hrs ∧ ¬Socs36E MadMed Sax : 1) ∧ ¬Dad : 1 ∧ ¬Tkv : 2) ∨ (Tkv : 2 ∧ Dad : 1) Spl 1 ¬(Nicd ∧ Mam) 2 Nicd ∧ Mam ∧ ¬H ∧ ¬Gro ∧ ¬CtBP STAT5 1 ¬(IL4R : 2 ∨ IL2R : 2) ∧ [12] (IL4R : 1 ∨ IL2R : 1 ∨ IL15R) SNAIL1 ∨ ZEB1) ∧ ¬(ZEB1 ∧ SNAIL1)

  .2;• if φ is expressed with at least one ∨, then the function behaves according to situation (c) (functions listed in TableA.3);• if φ depends on Boolean variables and is expressed only with ∧, then the function behaves according to situation (a) (functions listed in TableA.4). Note that nodes IL4RA and E2F3 are controlled by a multivalued gene (resp. STAT5 and CHEK), but their level depends only on one threshold (to be above or below level 2).

	Gene	NC SNC WNC Bool. NC Struct.
	E Spl	Yes Yes	Yes	Yes
	mQH2 Q	Yes Yes	Yes	Yes	(S)
	mdH	Yes Yes	Yes	Yes	(S)
	mGR	Yes Yes	Yes	Yes
	mGSH GSSG	Yes Yes	Yes	Yes	(S)
	mTRX	Yes Yes	Yes	Yes	(S)
	cGSH GSSG	Yes Yes	Yes	Yes	(S)
	cGR	Yes Yes	Yes	Yes
	E2F3	Yes Yes	Yes	Yes	(S)
	IL12RB1	Yes Yes	Yes	Yes	(S)
	IL4RA	Yes Yes	Yes	Yes	(S)
	Drk	No	Yes	Yes	Yes	(S)
	Dsor1	No	Yes	Yes	Yes	(S)
	Pnt	No	Yes	Yes	Yes	(S)
	Stat92E	No	Yes	Yes	Yes	(S)
	Raf	No	Yes	Yes	Yes	(S)
	RI	No	Yes	Yes	Yes	(S)
	Sos	No No	Yes No	Yes No	Yes No	(S)
	Tkv Table	No	Yes	Yes	Yes
	mNNT	No	Yes	Yes	Yes
	mCa	No	Yes	Yes	Yes	(S)
	mGPX	No	Yes	Yes	Yes	(S)
	mTR	No	Yes	Yes	Yes
	cGPX	No	Yes	Yes	Yes	(S)
	cTR	No	Yes	Yes	Yes
	cTRX	No	Yes	Yes	Yes	(S)
	STAT5	No	Yes	Yes	Yes

cGSH GSSG : 2 ∧ ¬cROS Table A.2: Rules associated to multivalued genes from databases considered in [START_REF] Subbaroyan | Minimum complexity drives regulatory logic in Boolean models of living systems[END_REF], satisfying structure (S) and behaving according to situation (b). A.6: Rules associated to multivalued genes from databases considered in [START_REF] Subbaroyan | Minimum complexity drives regulatory logic in Boolean models of living systems[END_REF], not satisfying structure (S) and behaving according to situation (d).

Gene