Valentin Magnon
email: valentin.magnon@uca.fr

Jordan Mille

Rudy Purkart

Marie Izaute

Pierre Chausse

Frédéric Dutheil

Guillaume T Vallet
email: guillaume.vallet@uca.fr

When does vagal activity benefit to the discrimination of highly overlapping memory traces?

Keywords: autonomic nervous system, heart rate variability, memory discrimination, memory retrieval, pattern separation

Distinguishing among similar events is indeed crucial to memory. In a dangerous context, this ability may be less essential, whereas in a secure context it may provide an adaptative advantage notably in social situations. Vagal activity as a marker of the individual's adaptation in social engagement contexts might predict the ability to discriminate highly similar memories in an unthreatening context. The present study aims to test the relation between vagal activity and memory discrimination by manipulating the visual distinctiveness of stimuli (high vs. low) in an ecological memory discrimination task with humanoid characters. It was expected that vagal activity support adaptive self-regulatory processes which may be needed only when the memory discrimination is challenging (low distinctiveness between true memory and lure). In a study phase, multiple realistic fictional humanoid characters performed the same action on different objects. The characters could be very similar or very distinct one to each other. Then, during a test phase, the participants had to discriminate whether a displayed character on the screen was exactly matched the one performing the given action on a specific object in the study phase (target). Vagal activity was assessed in undergraduate students (n = 40) before the study phase. Higher vagal activity predicted better memory discrimination performance specifically when the distinctiveness between the lure and target was low. These data extend previous work on vagal activity and memory suggesting that heart-brain interactions represent an adaptive psychophysiological mechanism underlying memory discrimination specifically when it is challenging in an unthreatening context.

Introduction

Vagal activity is a marker of the individual's adaptation in social engagement contexts [START_REF] Porges | The polyvagal perspective[END_REF]. Vagal activity is measured by recording the fluctuation in time intervals between succeeding heart beats over time, namely heart rate variability [START_REF] Shaffer | A healthy heart is not a metronome: An integrative review of the heart's anatomy and heart rate variability[END_REF]. According to the polyvagal theory [START_REF] Porges | The polyvagal perspective[END_REF], self-regulatory processes enabling social engagement are underpinned by outflow and functional organization of vagus nerve while sympathetic activity primes cognitive processes associated with "fight or flight" responses. Vagally-mediated heart rate variability (vmHRV) is logically associated with performance on goal directed tasks requiring adaptive regulation and inhibitory control (see [START_REF] Forte | Heart rate variability and cognitive function: A systematic review[END_REF] for a review), whereas tasks inducing mental stress (e.g. certain working memory tasks) are associated with decreased vmHRV [START_REF] Park | When tonic cardiac vagal tone predicts changes in phasic vagal tone: The role of fear and perceptual load[END_REF][START_REF] Thayer | Beyond heart rate variability: Vagal regulation of allostatic systems[END_REF].

Memory also plays a crucial role both for survival [START_REF] Scofield | A meta-analysis of the survival-processing advantage in memory[END_REF] and social interactions [START_REF] Tzakis | Social memory and the role of the hippocampal CA2 region[END_REF]. Therefore, only in the latter case should memory rely on vmHRV since a stress-free social context should promote vagal activity. To date, only one study has reported that vmHRV predicts memory performance [START_REF] Feeling | Resting state heart rate variability and false memories[END_REF]. The relationship was found in the sole case of memory discrimination between true (target) and semantically related lures, but not with memory performance per se. Distinguishing among similar events is indeed crucial to memory [START_REF] Stevenson | Pattern separation and source memory engage distinct hippocampal and neocortical regions during retrieval[END_REF].

Otherwise, new information would overwrite similar memories previously learned (catastrophic interference, [START_REF] Yassa | Pattern separation in the hippocampus[END_REF]. This ability is probably not essential in stressful situations since distinguishing a tiger from a leopard does not matter as far the flight response is activated. On the contrary, this ability might be crucial in stress-free situations, as for instance in a social context in which recognizing the face of a friend within a crowd would bring direct social benefits. Accordingly, rapid visual detection of threats rely on automatic processing of their coarse visual features [START_REF] Lojowska | Unconscious processing of coarse visual information during anticipatory threat[END_REF] while vagal is associated with facial recognition (Park & Thayer, 2014). Regarding memory discrimination, distinguishing highly similar memory traces should also only be relevant in an unthreatening context. Therefore, vagal activity should bring an adaptative advantage to memory discrimination in other types of stress-free situations and unthreatening contexts, as for instance when distinguishing memory traces of people performing certain actions. This advantage should logically (according to this evolutionist perspective) only be observed in a situation where discrimination is challengingthat is, when a strong overlap exists between the memory traces. The present study aims to test this specific hypothesis in an ecological visual memory task.

Method Participants

Forty-seven undergraduate psychology students were enrolled in a larger study focusing on the impact of a breathing exercise on decision-making. They all completed an exhaustive screening (interview and questionnaires) to ensure that they had no condition which may interfere with cardiac and cognitive functioning (see Supplementary materials and Magnon, Dutheil, & Vallet, 2021). Sociodemographic data (age, sex, etc.) and medical histories that may affect cognition and cardiovascular functioning, such as neurological (head trauma, epilepsy...), physiological (hypo-or hyperthyroidism, type 2 diabetes...) or neurodevelopmental factors (autism, dyslexia...). Lifestyle factors, such as binge drinking alcohol (more than five consecutive alcoholic drinks in the last week) and high body mass index (BMI ≥ 30), likely to influence HRV also served as exclusion criteria. Depression and anxiety were assessed by the Hospital and Anxiety Depression Scale (HADS, see [START_REF] Zigmond | The hospital anxiety and depression scale[END_REF]. Two participants were excluded due to a high depression/anxiety score and 1 due to active medication. Four more participants were excluded due to flattened heart rate recordings (wristband too loose for comfort reasons). The final sample thus consisted of 40 participants (MAge= 19.76,SDAge= 1.50,5 men). The research was approved by the Île-de-France Ethics Committee (2021-A01735-36).

Material HRV assessment

The EMPATICA E4 Wristband (Empatica E4, Italy) was used to assessed HRV. This medicalgrade wearable device was validated in comparison to electrocardiography [START_REF] Schuurmans | Validity of the Empatica E4 wristband to measure heart rate variability (HRV) parameters: A comparison to electrocardiography (ECG)[END_REF]. Heart rate was recorded during the whole session and the inter-beat intervals were extracted (as recommended, [START_REF] Li | Spectral analysis of heart rate variability: Time window matters[END_REF] during the test phase of the memory discrimination task (more specifically the last 5 minutes of the test phase) whether the task was administrated before or after the breathing exercise (see procedure). A visual check ensured the quality of the recordings as the control of artefacts or occasional ectopic beats. A very low threshold filter was applied when judged necessary. Kubios (v.3.3.1) was used to compute the High Frequency (HF, range from 0.15 to 0.4 Hz) power. This is most reliable vmHRV index (as others might be affected by sympathetic activity, see [START_REF] Berntson | Filter properties of root mean square successive difference (RMSSD) for heart rate[END_REF], which is the percentage of HF among the total power minus the very low frequency (normal units) [START_REF] Laborde | Heart rate variability and cardiac vagal tone in psychophysiological research -Recommendations for experiment planning, data analysis, and data reporting[END_REF][START_REF] Tarvainen | Kubios HRV-heart rate variability analysis software[END_REF].

Memory discrimination task

Fourteen blocks (2 practice blocks, 12 experimental blocks) of 4 characters each were created (for a total of 48 trials) using The Sims 4 TM video game. The distinctiveness of the characters was configured according to 11 dimensions: gender (man vs. woman), build (heavy vs. light), skin color (blue, red…), age (child vs. adult), hair color (blond, brown…), head covering (beanie, beret…), glasses (with vs. without), sleeve length (short vs. long), pant length (short vs. long), shoes (sneakers vs. flip-flops), and faces. In the "LOW DISTINCTIVENESS" blocks, the characters in each block were visually lowly distinct from each other (several dimensions were identical, see Figure 1). More precisely, within each block in the LOW DISTINCTIVENESS condition, 5 dimensions (gender, build, skin color, age, hair color) were identical for the 4 characters of a given block (the 6 remaining were pseudorandomized across the 4 characters within the block). In the "HIGH DISTINCTIVENESS" condition, the characters in each block were visually highly distinct from each other (11 dimensions were pseudorandomized across the 4 characters within the block). Twelve types of action were selected (e.g., sitting on, playing, etc.) to be performed on the 4 chosen objects per block (e.g., a chair, a stool…). Within each block, the four characters performed the same action, but on a different object. A given action (e.g., sitting on) was provided in the LOW DISTINCTIVENESS condition for half of the participants, and in the HIGH DISTINCTIVENESS condition for the other half of the participants. The selected actions are realistic and socially relevant (a humanoid character sitting on a chair). Each action was presented in a video format lasting approximately 8 seconds (see Figure 2 for details on the task's administration). After a 400ms fixation cross, a voice stated the current action and an object on which the action was performed while a video showing a character preforming the action was played. The participants have to memorize the appearance of a character, the type of action and the object on which the action was performed on. The videos in the LOW and HIGH DISTINCTIVENESS blocks were counterbalanced across participants (using two lists, see Supplementary materials for more details on this task). After the study phase, a filler task (mental arithmetic) occurred before the test phase, one of the audio tracks from the study phase was replayed with a neutral grey background screen for 3,000ms. A photograph of one of a character of the corresponding action's block was then displayed with a black background for 1,000ms or until the participant's responded. Finally, an Inter-Trial Interval (ITI) of 2,000ms was used before the next trial started. The participants had to decide whether the character shown corresponded exactly to the character who performed the action on the specific object (learned in the study phase) by using the keyboard to answer "MATCH" or "MISMATCH" as accurately and quickly as possible. In the "MISMATCH" condition, the displayed character was from the same action's block and had performed the same action, but not on the targeted object stated by the voice. In the "MATCH" condition, the character was the same as the character who had performed the action on the object mentioned in the audio description. The characters were presented only once either in the "MISMATCH" or in the "MATCH" conditions. Accuracy and response times for each trial were collected. This task is considered as robust giving similar results in several studies of our lab [START_REF] Purkart | Playing "guess who?": When an episodic specificity induction increases trace distinctiveness and reduces memory errors during event reconstruction[END_REF]. As discrimination performance was the variable of interest, the main outcome of this task was accuracy.

Procedure

Each participant was tested individually in a session that last 1 hour. Firstly, the Empatica wristband was installed on the left wrist of the participant and turned on. During a 10-minute rest period, health and sociodemographic data were collected. Then, participants completed either a decision-making task or the memory discrimination task (Supplementary materials) lasting approximately 10 minutes. Afterward, the participants underwent a 5-minute breathing exercise followed by the completion of the other computerized task counterbalanced with the first one described above (either the memory discrimination task or the decision-making task).

After the HADS and other questionnaires (Supplementary materials), the session ended by a debriefing. This design was chosen to maximize the variability of HF to serve as assessed during the memory discrimination test. In this way, the participants who completed the memory task after the deep and slow breathing should have higher HF than some participants who performed it before. These choices allow to greatly diminish the sample size requirements (see Magnon, Dutheil, et al., 2021 for a similar procedure in a previous study).

Statistical analyses

All the statistical analyses were performed with the R statistical software package (v4.0.2, RStudio Team, 2020). The significance threshold was set at p < .05. An analysis of variance with Time (memory discrimination task completed before vs. after the breathing exercise) confirmed that participants who completed the memory task after the breathing exercise had higher HF power level. To test whether vmHRV is positively associated with memory discrimination specifically in the LOW DISTINCTIVENESS condition, an interaction was tested between HF power as a continuous predictor and DISTINCTIVENESS (LOW vs. HIGH) as a categorical predictor and the memory discrimination index A' (or area under the curve) as dependent variable. Then, to explore further the specificity of the effect, a linear regression with HF power as the predictor and the A' index as dependent variable in both LOW and HIGH DISTINCTIVENESS was performed. Assumptions for linear regression were checked with the R package "car" [START_REF] Fox | car: Companion to Applied Regression (Version 3.0-12[END_REF] and an outlier detection was performed as well revealing no outliers or influential data (the only value detected was not influential, p = .08). To ensure the validity of our findings, we conducted robust (to outliers) interaction analyses and linear regressions using the R package "MASS" [START_REF] Ripley | MASS: Support Functions and Datasets for Venables and Ripley's MASS (Version 7.3-55[END_REF]. Because traditional linear regressions are highly sensitive to outliers and influential data, robust regressions are specifically designed to overcome these limitations while maintaining the sample size [START_REF] Yu | Robust linear regression: A review and comparison[END_REF]. The A' index1 is based on the signal detection theory [START_REF] Macmillan | Detection theory: A user's guide[END_REF]:

A' = 1-(𝑓𝑎𝑙𝑠𝑒𝑎𝑙𝑎𝑟𝑚𝑠 ℎ𝑖𝑡𝑠 ⁄ )+(𝑜𝑚𝑖𝑠𝑠𝑖𝑜𝑛𝑠 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑟𝑒𝑗𝑒𝑐𝑡𝑖𝑜𝑛𝑠 ⁄ ) 4
The lower is the A', the poorer is the discrimination performance. A' is robust and widely used discrimination index [START_REF] Pastore | Signal detection theory analyses of semantic priming in word recognition[END_REF] especially for memory discrimination (see [START_REF] Reagh | Spatial discrimination deficits as a function of mnemonic interference in aged adults with and without memory impairment[END_REF], recognized for its validity [START_REF] Stanislaw | Calculation of signal detection theory measures[END_REF].

Results

As expected, the interaction term between HF power and DISTINCTIVNESS (LOW vs. HIGH, see Table 1 for descriptive statistics) accounted for a significant proportion of the variance in the memory discrimination score A' (F(1, 38) = 4.17, p = .04, R² = .21). Furthermore, a linear regression analysis revealed that HF level explained a significant amount of the variance in A' index in the LOW DISTINCTIVENESS condition (F(1, 38) = 5.81, p = .02, R² = .11). HF level positively predicted discrimination performance A' (β = .005, p = .02, see Figure 3). In contrast, a linear regression analysis showed that HF level did not explain a significant amount of the variance in A' index in the HIGH DISTINCTIVENESS condition (F(1, 38) = -.0005, p > .10, R² = .02, see Figure 4).

Table 1

Mean (SD) of the discrimination performance index A' in the HIGH and LOW DISTINCTIVENESS conditions.

Distinctiveness level HIGH LOW

A' 0.74 (0.15) 0.57 (0.23)

Discussion

The present study aimed to explore whether vmHRV provides a benefit to memory discrimination only when the stimuli are highly overlapping as predicted by the polyvagal theory [START_REF] Porges | The polyvagal perspective[END_REF]. The vagus nerve has indeed been considered as a maker of one's ability to self-regulate and to adjust to new situations and environments in social context [START_REF] Porges | The polyvagal perspective[END_REF][START_REF] Thayer | Heart rate variability, prefrontal neural function, and cognitive performance: The neurovisceral integration perspective on selfregulation, adaptation, and health[END_REF]. As highly overlapping lures are more challenging to discriminate, this process might require self-regulatory adaptative processes associated with vmHRV which would not be needed for lowly overlapping lures. Such discrimination appears crucial in social contexts compared to stressful/survival situations in which fast gist impressions should direct the individual's reactions [START_REF] Scofield | A meta-analysis of the survival-processing advantage in memory[END_REF].

As expected, higher vmHRV predicted better memory discrimination performance but only when the distinctiveness between lures and targets was low. Those results indicate that vmHRV is not always required to discriminate lures from hits as previously suggested [START_REF] Feeling | Resting state heart rate variability and false memories[END_REF]. This result could be applied to cognitive aging in which memory performance is particularly related to discrimination [START_REF] Mille | Low-resolution neurocognitive aging and cognition: An embodied perspective[END_REF]. Maintaining an elevated vmHRV on the long-term might promote the discrimination of highly overlapping memory traces across the lifespan (Magnon et al., In revision). Accordingly, intervention that stimulates vagal activity could prove beneficial to prevent memory aging. Supporting this hypothesis, it was previously reported that transcutaneous vagus nerve stimulation enhances associative memory performance in older individuals [START_REF] Jacobs | Transcutaneous vagus nerve stimulation boosts associative memory in older individuals[END_REF], which highlights the relevance of studying heart-brain interactions to promote healthy aging. Should high vagal activity be found to prevent memory impairment with age, a simple 5-minute breathing exercise that seems sufficient to increase vagal activity in the elderly (Magnon, Dutheil, et al., 2021) could be a promising, cheap and simple clinical lead to promote successful cognitive aging.

While many factors were controlled to ensure internal validity, several lifestyle factors that might influence HRV or cognition were not assessed, such as the practice of physical activity and sedentariness [START_REF] Magnon | Sedentary behavior at work and cognitive functioning: A systematic review[END_REF][START_REF] Magnon | Sedentary lifestyle matters as past sedentariness, not current sedentariness, predicts cognitive inhibition performance among college students: An exploratory study[END_REF]. Furthermore, the sample for this exploratory study is relatively small. However, the design chosen proves to be efficient in increasing variability in vmHRV by using a validated breathing exercise reducing sample size requirements (Magnon, Dutheil, et al., 2021). Finally, despite the validated reliability and accuracy of the medical-grade wristband [START_REF] Schuurmans | Validity of the Empatica E4 wristband to measure heart rate variability (HRV) parameters: A comparison to electrocardiography (ECG)[END_REF], the wristband remains relatively less accurate than an electrocardiogram.

To conclude, the present study replicates and extends the results of the sole study on the association between HRV and memory discrimination. The distinctiveness manipulation between lures and targets indicates that vagal activity is only associated with highly overlapping memory traces corresponding to challenging social situations. Future studies are required to further investigate this relationship with its neural dynamic during memory discrimination. If heart-brain interactions are proven to be involved in memory discrimination, then vagal stimulation, through breathing exercises for instance, might constitute a simple and cheap cognitive intervention.

Fig. 1 .

 1 Fig. 1. Photographs of characters in a block for each distinctiveness condition.

Fig. 2 .

 2 Fig. 2. Procedure and details of the memory discrimination task.

Fig. 3 .

 3 Fig. 3. The relationship between vmHRV (HF level) and discrimination performance (A') in

Fig. 4 .

 4 Fig. 4. The relationship between vmHRV (HF level) and discrimination performance (A') in

With hits = answering "match" when the character shown was the one performing the given action asked; correct rejections = answering "mismatched" when the character shown is different; false alarms = answering "match" when the character shown in the test phase does not exactly match the character performing the action in the study phase; omissions = answering "mismatch" when the character shown during the test phase is exactly the same.

Acknowledgments

Jordan Mille, Valentin Magnon and Guillaume T. Vallet are supported by a grant from the Auvergne-Rhône-Alpes region for the project Vieillissement, Maladie Chronique et Stimulation Cognitive (ViMaCC). The ViMaCC project is co-financed by the European Union within the framework of the Fonds Européen de Développement Régional (FEDER). Marie Izaute is supported by the FACTOLAB framework, sponsored by Michelin Tyres Manufacturer and by the French government research programs "Investissements d'Avenir" through the IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25) and the IMobS3 Laboratory of Excellence (ANR-10-LABX-16-01).

Data Availability Statement

The data used in the present article are available at https://mfr.osf.io/render?url=https://osf.io/pxrcn/?direct%26mode=render%26action=downlo

Author contributions

VM and JM conceived the study. VM designed the protocol and wrote the first version of the manuscript. GTV supervised the conception of the study. JM and RP created the visual discrimination memory task. VM, JM, RP and GTV analyzed the data and interpreted the results. FD and PC gave their insight and expertise for the HRV measure and analyses. All authors participated in the critical revision of the article. All authors read and approved the final version.


Norbert Hoffmann
email: norbert.hoffmann@univ-reims.fr

Heterocyclic Compounds in Enantioselective Photochemical Reactions

Introduction 1.2 Asymmetric catalysis with chiral templates 1.3 Asymmetric photo-enzyme catalysis 1.4 Asymmetric photochemical reactions in crystals 1.5 Crystalline inclusion complexes 1.6 Inclusion in zeolits 1.7 Memory of chirality 1.8 Conclusions and perspective 1.9 References

Introduction

The present chapter deals with different key topics. Heterocyclic compounds play a central role in many domains of chemistry such as the search of new biologically active compounds in pharmaceutical and agricultural chemistry [START_REF] Dinges | Bioactive Heterocyclic Compound Classes -Pharmaceuticals[END_REF]. Also, many new materials such as semiconducting compounds contain heterocyclic moieties [START_REF] Ostroverkhova | Handbook of Organic Materials for Electronic and Photonic Devices[END_REF]. In these domains, a large structural diversity and molecular complexity is highly needed. Here, traditional methods of organic synthesis find their limits. Photochemical reactions extend such limits. As electronic excitation completely change the chemical reactivity of compounds or whole compound families [START_REF] Klán | Photochemistry of Organic Compounds[END_REF], products become accessible which cannot be synthesized by more conventional methods which is of high interest for application in the field of bioactive compounds [START_REF] Lefebvre | Photochemical Rearrangements in Heterocyclic Chemistry[END_REF][START_REF] Hoffmann | Photochemical reactions as key steps in organic synthesis[END_REF]. Furthermore, the outcome of known reactions, especially catalytic reactions can be improved when they are carried out under photochemical conditions. Based on the enormous quantity of recent and past results in the field of photochemical reactions, it makes sense to subdivide chemical reactions into two classes: reactions which occurs in the electronic ground state and reactions in which electronic excitation is involved. From the economic and ecological point of view, photochemical reactions are particularly interesting, since many of them can be carried out without an additional chemical reagent. The photon is considered as a traceless reagent [START_REF] Turro | Photochemical Reactions as a Tool in Organic Synthesis[END_REF][START_REF] Hoffmann | Photochemical reactions of aromatic compounds and the concept of the photon as a traceless reagent[END_REF]. For these reasons, these reactions are now highly appreciated in chemical and pharmaceutical industry [START_REF] Li | Visible-Light Photocatalysis as an Enabling Technology for Drug Discovery: A Paradigm Shift for Chemical Reactivity[END_REF][START_REF] Bonfield | Photons as a 21st century reagent[END_REF][START_REF] Michelin | Les réactions photochimiques à l'échelle industrielle[END_REF].

Stereoselectivity also plays a central role in organic synthesis. Biological activity as well as material properties strongly depend on the stereochemistry of chemical compounds. Sooner or later, almost all synthesis methods will face this problem. In the past, photochemical reactions have been considered as being inherently stereo-unselective. It was thought that the high energy uptake by light absorption induces uncontrolled relaxation processes which leads to unselective reactions with large amounts of degradation either of the substrates or the photoproducts [START_REF] Oelgemöller | Studies in organic and physical photochemistryan interdisciplinary approach[END_REF]. In this regard, it must however be pointed out that stereoselective and stereospecific photochemical reactions have been known from the very beginning of this research area [START_REF] Roth | The Beginnings of Organic Photochemistry[END_REF][START_REF] Rau | Asymmetric Photochemistry and Photochirogenesis[END_REF]. The controlled dissipation of the high electronic excitation energy in photochemical reactions is the reason for the high stereoselectivity in such reactions [START_REF] Oelgemöller | Studies in organic and physical photochemistryan interdisciplinary approach[END_REF]. In particular, photochemical reactions can be conducted enantioselectively in chiral supramolecular structures [START_REF] Ramamurthy | Supramolecular Photochemistry as a Potential Synthetic Tool: Photocycloaddition[END_REF][START_REF] Yang | Supramolecular photochirogenesis[END_REF]. Enantiopure compounds are obtained in different ways: they can be prepared directly from other chiral precursors such as natural products ("chiral pool") or by optical resolution using different types of chromatography or crystallization techniques. Asymmetric syntheses using chiral auxiliaries, which are removed after the stereoselective reaction, also provide enantiopure compounds. Asymmetric catalysis as well as enzymatic catalysis directly yield enantioenriched compounds. A chiral enantiopure environment in a supramolecular structure or in a crystal may be the inductor chirality in asymmetric reactions. In the present chapter, methods will be discussed leading directly to enantiopure heterocyclic compounds via photochemical reactions.

Asymmetric catalysis with chiral templates

Photochemical substrates may be complexed with chiral structures which induce chirality [START_REF] Brimioulle | Enantioselective Catalysis of Photochemical Reactions[END_REF]. A typical example is described in Scheme 1.1 [START_REF] Bauer | Catalytic enantioselective reactions driven by photoinduced electron transfer[END_REF]. The quinolone derivative 1 carrying a pyrrolidine moiety, undergoes an intramolectular cyclization leading to the spirocyclic indolizidine compound 6. The substrate is complexed with the enantiopure Kemp acid derivative (2) via hydrogen bonds between two lactam moieties. In this arrangement, the pyrrolidine approaches the reaction center mainly by one diastereotopic half space. In this complex, the shielding group acts also as an aromatic ketone sensitizer (sens). After photochemical excitation of the latter, electron transfer from the tertiary amine moiety to the ketone leads first to a radical ion pair 3 and after proton transfer to intermediate 4 [START_REF] Bertrand | Diastereoselective tandem addition-cyclization reactions of unsaturated tertiary amines initiated by photochemical electron transfer (PET)[END_REF]. The nuclephilic α-aminoalkyl radical attacks with 70% of stereoselectivity the electrophilic double bond of the quinolone moiety. Thus, an electrophilic oxoallyl radical is generated affording the diradical intermediate 5. The final product 6 results form a hydrogen transfer from the ketyl radical to the oxoallyl radical. It must be pointed out that in the present case this step is favored because it is an intramolecular process. In these radical steps, polar effects play an important role [START_REF] Fischer | Factors controlling the addition of carbon-centered radicals to alkenesan experimental and theoretical perspective[END_REF][START_REF] Roberts | Site-Selective C-H Funtionalization by Decatungstate Anion Photocatalysis: Syntergistic Control by Polar and Steric Effects[END_REF][START_REF] Hoffmann | Electron and hydrogen transfer in organic photochemical reactions[END_REF]. In the corresponding intermolecular stereoselective reactions, these effects contribute essentially to the efficiency of these processes [START_REF] Bertrand | Diastereoselective tandem addition-cyclization reactions of unsaturated tertiary amines initiated by photochemical electron transfer (PET)[END_REF]. The intermolecular addition of tertiary amines to indolone derivatives with an exocyclic electron deficient olefinic double bond has been carried out with similar Kemp acid derivatives [START_REF] Lenhart | Enantioselective Visible-Light-Induced Radical-Addition reactions to 3-Alkylidene Indolin-2-ones[END_REF]. In this case however ruthenium or iridium complexes have been used as external photoredox catalysts which were excited by visible light absorption. Using a similar chiral sensitizer, an intramolecular [2+2] photocycloaddition has been carried out with high enantioselectivity (Scheme 1.2) [START_REF] Alonso | A Chiral Thioxanthone as an Organocatalyst for Enantioselective [2+2] Photocycloaddition Reactions Induced by Visible Light[END_REF]. The quinolone derivative 7 is transformed, under visible light irradiation, into a complex polycyclic compound 8 containing a pyrrolidine moiety. It must be pointed out that the same [2+2] photocycloaddition is also induced by UV-irradiation via direct light absorption but no chiral induction takes place. It is therefore necessaire to choose a sensitizer which absorbs in the visible domain of the light spectrum to ensure enantioselectivity. The thioxanthone derivative 9 absorbs in the visible light region and transfer its triplet energy to the complexed substrate [START_REF] Michelin | Les réactions photochimiques à l'échelle industrielle[END_REF]. Again, this complexation occurs via hydrogen bonds between the two lactams of the substrate and the Kemp acid moiety of the sensitizer. In this structure the olefinic double bond in the side chain approaches the reactive center of the quinolone almost only by one diastereotopic half-space. Similar asymmetric reactions have been performed with 3alkylquinolones carrying a 4-O alkene side chain. In this case, tetrahydrofurane moieties are formed [START_REF] Li | Visible-Light-Mediated Enantioselective Photoreactions of 3-Alkylquinolones with 4-O-Tethered Alkenes and Allenes[END_REF].

Scheme 1.2 Construction of a pyrrolidine moiety using an enantioselective [2+2] photocycloaddition.

The substrate can also be complexed to a metal or a strong coordinating atom. In such a case, chirality is induced by a chiral ligand sphere [START_REF] Zhang | Steering Asymmetric Lewis Acid Catalysis Exclusively with Octahedral Metal-Centered Chirality[END_REF]. In this context, chiral Lewis acid 11 was used to catalyze the asymmetric intramolecular [2+2] photocycloaddition of the dihydropyridinone derivative 12 (Scheme 1.3) [START_REF] Brimioulle | Enantioselective Lewis Acid Catalysis of Intramolecular Enone [2+2] Photocycloaddition Reactions[END_REF]. In this reaction, a δ-valeroolactam moiety (13) is formed. By complexation with a Lewis acid, the absorption maximum of compound 12 is shifted from 290 nm to 350 nm. Using fluorescent lamps with an emission λmax = 366 nm, complex 14 was excited almost exclusively since the non-complexed substrate 12 does not absorb light in this spectral range. Thus, the formation of racemic product as background reaction is suppressed. In the complex 14, the approach of the olefin to the reaction center again occurs by one diastereotopic half-space.

Scheme 1.3 Enantioselective Lewis acid catalysis of an intramolecular [2+2] photocycloaddition reaction.

TADDOLs (α,α,α',α'-tetraaryl-2,2-disubstituted 1,3-dioxolane-4,5-dimethanol, e.g. 23) are capable of complexing numerous substrates via hydrogen bonds [START_REF] Seebachn | TADDOLs, Their Derivatives, and TADDOL Analogues: Versatile Chiral Auxiliaries[END_REF]. When a photochemical substrate is complexed with such compounds, chirality can be induced. Under these conditions, when the flavone derivative 15 was irradiated with the diphenyl butadiene 16, a [2+3] cycloaddition took place leading to compounds 17 and 18 (Scheme 1.4) [START_REF] Wang | Total Syntheses of Isomeric Aglain Natural Products Foveoglin A and Perviridine B: Excited-State Intramolecular Proton-Transfer Photocycloaddition[END_REF]. After reduction with NaBH4, compounds 19 and 20 were isolated in good yields. Furthermore, the major diastereoisomer 19 was obtained in high enantioselectivity and recrystallization led to almost enantiopure samples. The high enantioselectivity of the photochemical reaction was explained by the structure of complex 21 which strongly favors the attack of the olefin by only one diastereotopic half-space. Interestingly, when the sterically much less encumbered chiral alcohol 22 was added instead of the TADDOL compound 23, an efficient chiral induction was still observed. The high enantioselectivity observed with hydrogen bond complexes can also be explained by the fact that excited state intramolecular proton transfer (ESIPT) plays a key role in the reaction mechanism [START_REF] Roche | Biomimetic Photocycloaddition of 3-Hydroxyflavones: Synthesis and Evaluation of Rocaglate Derivatives as Inhibitors of Eukariotic Translation[END_REF]. In fact, it was shown that the cycloaddition occurred at the triplet state of 15 and that most probably single electron transfer is involved. Compound 19 was transformed into (+)-foveoglin A which is the enantiomer of a natural product. This compound family plays an important role in medicinal chemistry as they possess anticancer and antiviral activities.

Scheme 1.4 Asymmetric synthesis of (-)forveoglin A using ESIPT promoted [2+3] cycloaddition with a flavone derivative.

The cyclization reaction of two alkyne and one nitrile function is a convenient method for the preparation of pyridine compounds [START_REF] Heller | The fascinating construction of pyridine ring systems by transition metal-catalysed [2+2+2] cycloaddition reactions[END_REF]. It can be carried out under particular mild conditions when simple and inexpensive cobalt catalysts such as 24 are used (Scheme 1.5) [START_REF] Schulz | Photoassisted cocyclization of acetylene and nitriles catalyzed by cobalt complexes at ambient temperature and normal pressure[END_REF]. Under irradiation with visible light, the formation of the cobaltacyclopentadiene intermediate 25 is accelerated and the addition of the nitrile leading to the intermediate 26 or 27 becomes the rate determining step. The consumption of the nitrile substrate becomes first order reaction step. Under these particularly mild conditions, a variety of pyridine derivatives have been synthesized possessing fragile substituents (Figure 1.1) [START_REF] Heller | Facile and Racemization-Free Conversion of Chiral Nitriles into Pyridine Derivatives[END_REF]. Asymmetric catalysis was also successfully performed. The cyclopentadienyl ligand in the cobalt catalyst was replaced by chiral analogues, best results being obtained with catalyst 28 (Scheme 1.6) [START_REF] Gutnov | Asymmetric Synthesis of Axially Chiral 1-Aryl-5,6,7,8tetrahydroquinolines by Cobalt-Catalysed [2+2+2] Cycloaddition Reaction of 1-Aryl-1,7octadiynes and Nitriles[END_REF]. With this reaction axial chirality can be efficiently induced as shown by the transformation of compound 29 into naphthyl pyridine 30. Similar reactions have been carried out starting from 2-alkoxy-1-naphthonitriles 31 using different chiral cobalt complexes as catalysts. Either diines were used leading to tetracyclic compounds 32 or two equivalents of a monoalkyne were employed yielding the corresponding tricyclic products 33. In this part of the study, the chiral catalysts ent-28, 34, 35, 36 and 37 have also been tested. The study of this type of chirality, atropisomerism, has recently gained particular attention in photochemistry [START_REF] Kumarasamy | Tale of Twisted Molecules. Atropslective Photoreactions: Taming Light Induced Asymmetric Transformations through Non-biaryl Atropisomers[END_REF].

Asymmetric photo-enzyme catalysis

Enzyme catalysis is a general method to produce enantiomerically enriched or pure compounds. Many such transformations need multi enzyme systems which complicates the application to organic synthesis. In some cases, however, the replacement of one or more enzyme activities by a chemical transformation facilitates the transformations. As they tolerate a large variety of reaction conditions, photochemical reactions were efficiently applied in this context [START_REF] Ni | Photocatalysis: The Power of Combining Photocatalysis and Enzymes[END_REF].

The Bayer-Villiger monoxygenase (BVMO) asymmetrically catalyzes the transformation of ketones into esters or lactones in the case of cyclic ketones (Scheme 1.7) [START_REF] Hollmann | Biocatalytic synthesis of lactones and lactams[END_REF]. This enzyme contains a flavin adenine dinucleotide (E-FADred) unit which reduces molecular oxygen into hydrogen peroxide capable of oxidizing the ketone substrate. The oxidized Flavin species (E-FADox) is reduced by NADPH. The resulting NADP + is reduced via a glucose dehydrogenase catalyzed reaction. This second enzyme activity can be replaced by adding flavin (FAD) to the reaction mixture. The reduced form of the non-bound FADred is capable of reducing the enzyme bound E-FADox. At the excited state FADox (generated by irradiation with visible light) is easily reduced via electron transfer from a sacrificial electron donor such as EDTA. Scheme 1.7 Replacement of the glucose dehydrogenase by a simple photoredox catalytic system based on non-bound FAD.

Under these conditions the asymmetric Baeyer-Villiger reaction was carried out with the racemic 2-phenylcyclohexanon 38 (Scheme 1.8) [START_REF] Hollmann | A Light-Driven Stereoselective Biocatalytic Oxidation[END_REF]. Only the R-enantiomer was oxidized and the phenylcaprolactone 39 was obtained in high enantioselectivity. The racemic cyclobutanone derivative 40, has been transformed under the same conditions. Interestingly both enantiomers were oxidized but different outcomes were observed. The α-S-enantiomer yields the bicyclic butyrolactone 41 whereas the α-R-enantiomer is transformed into regioisomer 42. Both compounds are formed in high enantioselectivity. Compound 41 is the expected regioisomer of a Baeyer-Villiger reaction. Depending on the stereochemistry of the starting ketone, the enzyme activity is therefore capable of directing the regioselectivity of the reaction [START_REF] Krow | The Baeyer-Villiger Oxidation of Ketones and Aldehydes[END_REF]. It should be pointed out that under such reaction conditions, hydrogen peroxide is generated in low stationary concentration which reduces the enzyme degradation [START_REF] Perez | Visible light-driven and chloroperoxidasecatalyzed oxygenation reactions[END_REF]. Scheme 1.8 Enzyme catalyzed enantioselective Baeyer-Villiger reaction applied to the synthesis of lactones.

Combined photo and enzyme catalysis was also carried out with alcohol dehydrogenases (ADH) [START_REF] Gargiulo | A Photoenzymatic System, for Alcohol Oxidation[END_REF]. These enzymes use NAD(P)H/NAD(P) + as cofactor [START_REF] Kroutil | Biocatalytic Oxidation of Primary and Secondary Alcohols[END_REF]. In order to optimize the enzyme activity, this cofactor system must be regenerated [START_REF] Chenault | Regeneration of Nicotinamide Cofactors for use in Organic Synthesis[END_REF]. One of the hydroxyl functionalities of the achiral diol 43 is dehydrogenated to an aldehyde by horse liver alcohol dehydrogenase (HLADH) with molecular oxygen (Scheme 1.9) [START_REF] Rauch | Photobiocatalytic alcohol oxidation using LED light sources[END_REF]. Cyclization leads to the lactol 44, a reversible step. In a second dehydrogenation also catalyzed by HLADH, the lactol is oxidized to the 4-methyl-δvalerolactone 45, obtained with complete enantioselectivity. In these septs, hydrogen is transferred to NAD + and NADH is formed. NAD + is regenerated by hydrogen transfer to oxygen leading to hydrogen peroxide. This step is photocatalyzed by Flavin mononucleotide (FMN). The irradiation was carried out with blue LED (λmax = 465 nm). Under similar reaction conditions and using a flavin-dependent "ene"-reductase, a variety of lactams have been obtained in high enantioslectivity [START_REF] Biegasiewicz | Photoexcitation of flavoenzymes enables a stereoselective radical cyclization[END_REF].

The combination of enzymatic reactions with photochemical, in particular photoredox processes offers numerous perspectives for organic synthesis [START_REF] Zhang | Quarternary Charge-Transfer Complex Enables Photoenzymatic Intermolecular Hydroalkylation of Olefins[END_REF]. This combination is a particularly efficient approach in connection with sustainable or green chemistry. As these processes are the basis of photosynthesis in green plants, they have been suggested of potential relevance for a sustainable chemical industry by G. Ciamician more than 100 years ago [START_REF] Ciamician | Sur les actions de la lumière[END_REF]. It was the beginning of green chemistry [START_REF] Albini | Green chemistry and photochemistry were born at the same time[END_REF] although it has been almost forgotten over the decades. Scheme 1.9 Enantiospecific oxidation of the achiral diol 43 to 4-methyl-δ-valerolactone 45 using a photobiocatalytic process.

Asymmetric photochemical reactions in crystals

Asymmetric synthesis is often based on selection of conformations. An efficient method to do this is to carry out reactions at the crystalline state. Molecular symmetry and crystallography are strongly linked [START_REF] Jacques | Enantiomers, Racemates, and Resolutions[END_REF]. Particularly impressive photochemical reactions have been reported with achiral substrates which crystallize in Sohncke space groups [START_REF] Nespolo | Crystallographic shelves: space-group hierarchy explained[END_REF][START_REF] Levkin | Homo-and Heterochirality in Crystals[END_REF]. When achiral compounds crystallize in these space groups, most frequently, one enantiopure conformer is present in such homochiral crystals [START_REF] Sakamoto | Spontaneous chiral crystallization of achiral materials and absolute asymmetric photochemical transformation using the chiral crystalline environment[END_REF][START_REF] Kuroda | Circular Dichroim in the Solid State[END_REF].

When α-ketoamides 46 are irradicated at the solid state, the hydroxy-β-lactams 47 are obtained in high entatioselectivity (Scheme 1.10) [START_REF] Toda | Formation of a chiral β-lactam by photocyclisation of an achiral oxo amide in its chiral crystalline state[END_REF]. After light absorption, a hydrogen atom is transferred from an isopropyl group to the α-keto function leading to the diradical 48. Radical cyclization yields the final product 47. All these reaction steps occur under conformational control exerted by the crystal environment. In most cases, enantiomeric excesses (ee) were higher than 90%. In the case of Ar = Ph, the substrate crystallized in the chiral space group P212121 [START_REF] Sekine | X-ray Structural Studies of Chiral β-Lactam Formation from an Achiral Oxo Amide Using the Chiral-Crystal Environment[END_REF]. Irradiation of the homochiral crystals yields product 47 (Ar = Ph) in 93% ee which crystallized in the same space group P212121. Both enantiomers have been selectively prepared choosing the corresponding homochiral crystals of the substrate. Under similar conditions the achiral α,β-unsaturated thioamide 49 was transformed into the thio-β-lactam 50 [START_REF] Sakamoto | Crystal-to-Crystal Solid-State Photochemistry: Absolute Asymmetric β-Thiolactam Synthesis from an achiral α,β-Unsaturated Thioamide[END_REF]. After photochemical excitation, an hydrogen was transferred from one of the benzyl positions into the β-position of the cyclohexene moiety [START_REF] Sakamoto | Spontaneous chiral crystallization of achiral materials and absolute asymmetric photochemical transformation using the chiral crystalline environment[END_REF]. Radical combination yields the final product 50. The starting product crystallized in the P21 space group and when homochiral crystals were irradiated one enantiomer of 50 was obtained in high enantiomeric excess. It is noteworthy that, when compound 49 was irradiated in solution, the same product 50 was isolated but as a racemic mixture along with side products [START_REF] Sakamoto | Photochemical Reaction of N,N-Dialkyl-α,β-unsaturated Thioamides[END_REF]. Various other examples of this approach to chiral compounds have been reported [START_REF] Sakamoto | Solid-State Photochemistry: Absolute Asymmetric β-Thiolactam Synthesis from Achiral N,N-Dibenzyl-α,β-unsaturated Thioamides[END_REF].

This method for the production of only one enantiomer without external chiral induction is part of absolute asymmetric synthesis. Using particular crystallization methods, e.g. seeding crystallization with the desired homochiral crystal, the achiral starting compound can be selectively transformed into only one of the homochiral crystals [START_REF] Kellogg | Practical Stereochemistry[END_REF]. It should further be pointed out that such solid-state photochemical reactions can also be carried out on larger scale when suspensions are irradiated [START_REF] Veeman | Large-Scale Photochemical Reactions of Nanocrystalline Suspensions: A Promising Green Chemistry Method[END_REF]. Scheme 1.10 Synthesis of β-lactams using absolute asymmetric synthesis with homochiral crystals of the achiral substrates. Each enantiomer has been selectively produced from the corresponding homochiral crystal of the substrate.

A certain control of the crystal symmetry can be affected by attaching a homochiral element to the substrate. In this case, the number of possible space groups is reduced to 65 (Soencke groups). A defined chiral environment is thus created around the photochemical substrate [START_REF] Jacques | Enantiomers, Racemates, and Resolutions[END_REF][START_REF] Nespolo | Crystallographic shelves: space-group hierarchy explained[END_REF][START_REF] Levkin | Homo-and Heterochirality in Crystals[END_REF][START_REF] Sakamoto | Spontaneous chiral crystallization of achiral materials and absolute asymmetric photochemical transformation using the chiral crystalline environment[END_REF]. In order to obtain enantiomerically pure or enriched photochemical products, the chiral element should not be covalently bonded to the photochemical substrate [START_REF] Scheffer | Asymmetric Induction in Organic Photochemistry via the Solid-State Ionic Chiral Auxiliary Approach[END_REF]. In this context, ammonium salts of chiral amines and α-ketoamide 52 carrying a carboxylate function have been prepared and the crystalline phase was irradiated (Scheme 1.11) [START_REF] Natarajan | Control of Enantioselectivity in the Photochemical Conversion of α-Oxoamides into β-Lactam Derivatives[END_REF]. In most cases, the resulting hydroxyl β-lactams 53 have been obtained in high yield and enantioselectivity. The oxooxazolin derivatives 54 were formed in minor amounts. Both enantiomers of compounds of carboxylates 53 or 54 have been obtained as major stereoisomers depending the configuration of the chiral ammonium cation. The absolute configuration of the carboxylates has not been determined in this study.

Scheme 1.11 Synthesis of β-lactams by irradiation of crystalline chiral ammonium salts.

A similar norbornene 55 derivative has been transformed under the same conditions (Scheme 1.12) [START_REF] Xia | Asymmetric Synthesis of Dihydrofurans via a Formal Retro-Claisen Photorearrangement[END_REF]. Upon irradiation, the carbonyl is added to the alkene function leading to the triplet 1,4diradical intermediate 56 which is a typical intermediate of the Paternò-Büchi reaction [START_REF] Frénau | The Paternò-Büchi reaction-Mechanisms and application to organic synthesis[END_REF]. Such intermediates may undergo C-C bond formation leading to oxetanes. Bond cleavage of the newly formed C-O bond (b) can also take place regenerating the starting compound. This reaction step plays a key role in the stereoselective Paternò-Büchi reaction in solution [START_REF] Oelgemöller | Studies in organic and physical photochemistryan interdisciplinary approach[END_REF][START_REF] Buschmann | Chiral induction in photochemical reactions. 10. The principle of isoinversion: A model of stereoselection developed from the diastereoselectivity of the Paternò-Büchi reaction[END_REF]. However, in the present case, a C-C bond (a) of the norbornene moiety is cleaved yielding the final product 57, a bicyclic dihydrofuran derivative. In most cases, high stereoselectivity was observed. The reaction is a photo-Claisen rearrangement [START_REF] Galindo | The photochemical rearrangement of aromatic ethers A review of the Photo-Claisen reaction[END_REF] with intersystem crossing taking place. Scheme 1.12 Asymmetric photo-Claisen rearrangement by irradiation of crystalline chiral ammonium salts.

Crystalline inclusion complexes

As pointed out above, TADDOLs (α,α,α',α'-tetraaryl-2,2-disubstituted 1,3-dioxolane-4,5dimethanol) are auxiliaries which efficiently induces chirality without being covalently bond to the reacting molecule [START_REF] Seebachn | TADDOLs, Their Derivatives, and TADDOL Analogues: Versatile Chiral Auxiliaries[END_REF]. Using co-crystallization of these compounds with substrates of photochemical reactions is an interesting strategy for asymmetric synthesis [START_REF] Toda | Reaction Control of Guest Compounds in Host-Gest Inclusion Complexes[END_REF][START_REF] Koshima | Chiral Solid-State Photochemistry Including Supramolecular Approaches[END_REF]. Crystals of TADDOLs are host structures with cavities which can be filled by guest molecules. When cocrystallized with at TADDOL derivative in a 1:1 ratio, the furoic acid amide 58 undergoes enantioselectively photocycloaddition yielding the quinolinone compound 59 (Scheme 1.13) [START_REF] Toda | Enantioselective photocyclization of N-alkylfuran-2-carboxanilides to trans-dihydrofuran derivatives in inclusion crystals with optically active host compounds derived from tartaric acid[END_REF]. Due to its polar mesomeric structure 60, the amide constitutes a 6π system with benzene and furan moieties. Therefore, it undergoes conrotatory photocylization leading to 61. The final product 59 is formed via a tautomerization step. The same reaction was carried out with the acrylanilide derivative 62 (Scheme 1.13). Co-crystals of a 1:1 ratio with the same TADDOL have been irradiated. Again, the quinolinone compound [START_REF] Frénau | The Paternò-Büchi reaction-Mechanisms and application to organic synthesis[END_REF] was obtained in high enantiomeric excess. Guest molecules can interact principally in two different ways with the TADDOL host matrix. They can approach the host molecules with their polar face which may lead to the formation of hydrogen bonds. This was observed for the furoic acid amide 58 (Figure 1.2a). Guest molecules may also interact with the aryl substituents of the TADDOLs. In such cases, van der Waals, π-π-stacking or edge-to-face interactions [START_REF] Jennings | Attractive Intramolecular Edge-to-Face Interactions in Flexible Organic Molecules[END_REF] can be observed. This may be the case in the reaction of compound 62 (Figure 1.2b). Numerous photochemical reactions have been performed using cyclodextrins inclusion complexes [START_REF] Yang | Supramolecular photochirogenesis[END_REF][START_REF] Rekharsky | Complexation Thermodynamics of Cyclodextins[END_REF]. In water solution, cyclodextrins form such complexes with a variety of organic molecules. However, in many cases when β-cyclodextrin is used, the inclusion complexes are less soluble and the corresponding suspensions, powders or films are irradiated. In this context, a suspension of the 1:1 complex of β-cyclodextrin and the azepinone 64 has been irradiated with UV light (Scheme 1.14) [START_REF] Mansour | β-Cyclodextin-Mediated Enantioselective Photochemical Electrocyclization of 1,3-Dihydro-2H-azepin-2-one[END_REF]. The photochemical product 65 was hydrogenated and the corresponding bicyclic γbutyrolactam 66 was isolated in good yields and moderate enantioselectivity. Similar results have been obtained when films of the inclusion complex were irradiated. When the photochemical reaction of 64 in the presence of β-cyclodextrin was carried out in solution, followed by hydrogenation, 66 was isolated as a racemic mixture. Obviously, under these conditions, no inclusion of 66 takes place. Scheme 1.14 Photocyclization of the dihydroazepinone 64 as part of an inclusion complex with βcyclodextrin.

Inclusion in zeolits

Zeolits are inorganic crystalline porous materials which absorb small or medium sized molecules depending on the cavity size [START_REF]Catalysis and Zeolites[END_REF]. They are frequently used as catalysts. Concerning photochemical reactions, they have been used as a host structure for organic molecules [START_REF] Ramamurthy | Achiral Zeolites as Reaction Media for Chiral Photochemistry[END_REF][START_REF] Scaiano | Intrazeolite Photochemistry: Toward Supramolecular Control of Molecular Photochemistry[END_REF]. The photocyclization of pyridones 67 to the bicylic β-lactams 68 has been performed in super cages of MY zeolites, where M are different alkali metal ions (Scheme 1.15) [START_REF] Sivasubrmanian | Value of zeolites in asymmetric induction during photocyclization of pyridines, cyclohexadiones and naphthalenones[END_REF]. In order to create a chiral environment, inclusion into the super cages of MY zeolite of (-)-norephedrine or (-)-ephedrine together with the substrate 67 was carried. In order to assure a maximum of interaction with the chiral inductor, the aminoalcohols were used in a 10-fold excess. In cases of larger substituents R, the bicylic β-lactams 68 has been obtained with enantiomeric excesses up to 53%. The confinement of the non-covalently bond chiral inductor with the pyridone substrates reduces the number of conformers. The proximity of the chiral inductor directs this selection in an enantioselective way.

It must be pointed out that with the same reactions when carried out in solution, almost no chiral induction was observed. The reaction was also performed with pyridone substrates in which chiral amine derivatives were covalently bond. In such cases, significant diastereoselectivity was observed when compounds were absorbed by Y zeolites.

Memory of chirality

In many stereoselective reactions, the chiral information is transferred from a chiral center, for example a chiral carbon atom, to the reaction center [START_REF]For such cases models have been developed to quantify chiral induction: Ruch E, Ugi I. The Stereochemical Analogy Model -A Mathematical Theory of Dynamic Stereochemistry[END_REF]. The resulting products are formed with diastereoselectivity. In some cases, however, such a chiral center is destroyed and the chiral information is conserved (memory of chirality) in more or less stable conformers or other noncovalent interactions. In this way, such reactions become stereo-or enantioselective. Such phenomena are part of chiral memory effects [START_REF] Alezra | Recent Progress in Memory of Chirality (MOC): An Advanced Chiral Pool[END_REF][START_REF] Kramer | The Same and Not the Same: Chirality, Topicity, and Memory of Chirality[END_REF]. In this context, the proline derivative 69 has been irradiated in an acetone/water mixture (Scheme 1.16) [START_REF] Griesbeck | Spin-Orbit Coupling in Biradicals. Ab Initio MCSCF Calculations on Trimethylene and the Methyl-Methyl Radical Pair[END_REF]. Under these conditions, sensitization occurs via triplet energy transfer from photochemical excited acetone to 69.

Intramolecular electron transfer occurs from the carboxylate functionality to the phthalimide moiety [START_REF] Rekharsky | Complexation Thermodynamics of Cyclodextins[END_REF]. Immediate decarboxylation takes place and an α-aminoalkyl radical (71) is generated. In this reaction step, the chiral information at the former proline moiety is lost. Radical cyclization leads to the final product 72 with high enantioselectivity. It must be pointed out that the chiral information was conserved in relatively rigid conformations and transferred with inversion of configuration. This effect is explained by the fact that the spin multiplicity changes in this step.

The triplet diradical 73 is transformed into the final product in its singlet state 74. Radical combination and intersystem crossing concomitantly occur when the radical carrying p orbitals in the diradical are orthogonally oriented as depicted in structure 73 [80]. Such an arrangement increases spin-orbit coupling. As shown for the present and other examples, such an interaction has a significant impact on the stereoselectivity of these reactions [START_REF] Griesbeck | Selectivity Control in Electron Spin Inversion Processes: Regio-and Stereochemistry of Paternò-Büchi Photocycloadditions as a Powerful Tool for Mapping Intersystem Crossing Processes[END_REF]. In a similar electrochemical reaction, which takes place at the singlet ground state, almost the same enantioselectivity was observed but with retention of configuration at the pyrrolidine moiety [START_REF] Kramer | The Same and Not the Same: Chirality, Topicity, and Memory of Chirality[END_REF][START_REF] Wanyoike | O. Aliphatic Nitrogen-Containing Compounds, Amines, Amino Alcohols, and Amino Acids[END_REF]. Memory of chirality effects have been observed in a variety photochemical reactions involving radical intermediates [START_REF] Ramljak | Memory of chirality in the phthalimide photocyclization[END_REF].

1. [START_REF] Brimioulle | Enantioselective Catalysis of Photochemical Reactions[END_REF] Observation of a memory of chirality effect in a decarboxylative photocyclization.

Conclusion and perspectives

Enantioselective synthesis of heterocyclic compounds plays an important role in many domains such as the search of new biological active compounds medicine or agriculture, or for the preparation of new materials. Photochemical reactions significantly contribute to this research field. Currently, template supported or organometallic catalysis are intensively investigated with considerable impact in photoredox catalysis and photosensitization [START_REF] Michelin | Photosensitization and Photocatalysis -Perspectives in Organic Synthesis[END_REF][START_REF] Stephenson | Visible Light Photocatalysis in Organic Chemistry[END_REF]. Enzyme catalysis represent an efficient method for the preparation of enantiopure compounds, among them many heterocyclic compounds. Photochemical reactions can simplify the catalytic systems. In many cases, the use of coenzymes has been replaced by simple photochemical processes. Solid state photochemistry is an efficient method to control the equilibrium of conformers in an enantioselective way. Consequently, many of such reactions are carried out with high or complete enantioselectivity. This research domain provides interesting perspectives for material science. Photochemistry of heterocyclic compounds also contributes to basic understanding of chemical reactivity. Memory of chirality is observed in some of these reactions. For example, the influence of spin-orbit coupling, the conformational rigidity on enantioselectivity has been discussed in this context. It should also be pointed out that supramolecular structures [START_REF] Ramamurthy | Supramolecular Photochemistry as a Potential Synthetic Tool: Photocycloaddition[END_REF] or supramolecular catalysis [START_REF] Bhattacharyya | Supramolecular Engineering and Self-Assembly Strategies in Photoredox Catalysis[END_REF] play a key role in most of these reactions. Among the reaction discussed in this chapter, asymmetric photoredox catalytic reactions and photochemically modified enzymatic reactions are certainly the most attractive methods for application to organic synthesis. In the context of sustainable chemistry, photochemical reaction with crystalline substrate are particularly interesting since no solvent is needed [START_REF] Toda | Reaction Control of Guest Compounds in Host-Gest Inclusion Complexes[END_REF][START_REF] Mishra | Mechanical property of molecular solids[END_REF].

Scheme 1 . 1

 11 Scheme 1.1 Enantioselective synthesis of a spirocyclic indolizidine compound induced by a photochemical electron transfer.

Scheme 1 . 5

 15 Scheme 1.5 Visible light supported cobalt catalyzed [2+2+2] cycloaddition applied to the synthesis of pyridines.

Figure 1 . 1 Scheme 1 . 6

 1116 Figure 1.1 Pyridines that have been synthesized under mild conditions using photochemically promoted cobalt catalyzed [2+2+2] cycloaddition.

Scheme 1 . 13

 113 Scheme 1.[START_REF] Rau | Asymmetric Photochemistry and Photochirogenesis[END_REF] Photocyclization in TADDOL co-crystals yielding quinolones in high enantioselectivity.

Figure 1 . 2 X

 12 Figure 1.2 X-ray sturcutes of compound 58 (a) and 62 (b) in 1:1 cocrystals with a TADDOL (compare Scheme 1.13).

Scheme 1 . 15

 115 Scheme 1.15 Photochemical transformation of pyridones 67 to bicylic β-lactams 68 co-absorbed with chiral aminoalcohols in Y zeolites.


Nyoman Juniarta
email: nyoman.juniarta@loria.fr

Olivier Bonami
email: olivier.bonami@linguist.univ-paris-diderot.fr

Nabil Hathout
email: nabil.hathout@univ-tlse2.fr

Fiammetta Namer
email: fiammetta.namer@univ-lorraine.fr

Yannick Toussaint
email: yannick.toussaint@loria.fr

Organizing and Improving a Database of French Word Formation Using Formal Concept Analysis

Keywords: derivation, formal concept analysis, morphology, subgraph matching

We apply Formal Concept Analysis (FCA) to organize and to improve the quality of Démonette2, a French derivational database, through a detection of both missing and spurious derivations in the database. We represent each derivational family as a graph. Given that the subgraph relation exists among derivational families, FCA can group families and represent them in a partially ordered set (poset). This poset is also useful for improving the database. A family is regarded as a possible anomaly (meaning that it may have missing and/or spurious derivations) if its derivational graph is almost, but not completely identical to a large number of other families.

Introduction

Démonette2 [START_REF] Hathout | Giving lexical resources a second life: Démonette, a multi-sourced morpho-semantic network for French[END_REF][START_REF] Namer | Demonette2 -Une base de données dérivationnelle du franc ¸ais à grande échelle : premiers résultats[END_REF][START_REF] Namer | ParaDis and Démonette -from theory to resources for derivational paradigms[END_REF]) is a derivational database that systematically describes the derivational properties of a fragment of the French lexicon. In the database, an entry corresponds to a pair of lexemes from the same derivational family. A derivational family is a set of lexemes connected by derivational relations. Fig. 1 shows examples of five derivational families. Each family is shown with its pair(s) of lexemes and the morphological pattern relating each pair. The family of cramer, for example, has only one pair describing the direct derivation from the verb cramer V to the noun cramage N using the morphological pattern X-Xage. The set of derivations in the family of cramer is similar to that of haubaner, allowing them to be grouped in the same paradigm. This similarity also poses some questions. Is it possible that a derivational relation (cramer V crame N ) is missing in the family of cramer? Or conversely, is the derivation haubaner V -hauban N erroneous? It is also possible that these derivations are correct as such, the two families belong then to two different morphological paradigms. This paper deals with the following questions:

1. How to systematically represent the relation among families.

2. How to detect families having anomalies, i.e. families having either missing or incorrect derivations.

The presence of anomalies in Démonette2 is not surprising, given that it is built from heterogeneous resources. This merge generated some mistakes that should be corrected.

The improvement of derivational databases of numerous languages has been studied: Bulgarian [START_REF] Dimitrova | Coping with derivation in the Bulgarian WordNet[END_REF], Croatian [START_REF] Filko | Expansion of the derivational database for Croatian[END_REF][START_REF] Filko | Redesign of the Croatian derivational lexicon[END_REF], Czech ( Ševčíková et al., 2018), Hungarian [START_REF] Trón | Morphdb. hu: Hungarian lexical database and morphological grammar[END_REF], and Polish [START_REF] Dziob | Enriching plWordNet with morphology[END_REF]. A method based on graph theory has been proposed to improve a German derivational database [START_REF] Papay | Evaluating and improving a derivational lexicon with graphtheoretical methods[END_REF]. This method focuses on the fingerprint of each family, which is the graph describing the family's derivational relations. A fingerprint contains labelled edges with "anonymized" vertices. A family with incorrect derivations is assumed to have a fingerprint that is similar but not identical to the fingerprint of a large number of families.

In this paper, we go further by including part of speech information (verb, noun, adjective, etc.) in each vertex in a fingerprint. Moreover, we also take into account that a fingerprint may be a subgraph of another larger fingerprint, given that a set of derivations may be a subset of another.

Our hypothesis is as follows. Among the families that contain the smaller fingerprint, if a large percentage of them also have d, then the families that do not have d are likely to be incomplete. Conversely, if only a small percentage of them also have d, then this node d is likely to be spurious. We can consequently use the subgraph relation among fingerprints to define a partial order among derivational families. The systematic representation of families using this partial order can be performed using Formal Concept Analysis (FCA). The partially ordered set (poset) generated by FCA allows us to study the grouping of the families and the overlap among families' set of derivations.

Grouping families according to their paradigm or fin-

Formal Concept Analysis

FCA is a mathematical framework based on lattice theory and used for classification, data analysis, and knowledge discovery [START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF]

m 1 m 2 m 3 m 4 m 5 m 6 m 7 g 1 × g 2 × × × × × × g 3 × × × × × g 4 × × × × g 5 × × g 6 × × × g 7 × × × × × Table 1: A formal context corresponds to a maximal rectangle of crossed cells. A concept (A 1 , B 1 ) is subconcept of (A 2 , B 2 ), writ- ten as (A 1 , B 1 ) ≤ (A 2 , B 2 ), iff A 1 ⊆ A2. In this case, (A 2 , B 2 ) is superconcept of (A 1 , B 1 ).
With the subsumption relation ≤, the set of all concepts is partially ordered, and can be represented as a concept lattice.

Concept Lattice

The concept lattice of

downward path from C a to C b , then C a is superconcept of C b , and consequently C b is subconcept of C a .
The lattice in Fig. 2 left is illustrated in a full form. All objects and attributes of the extent and intent, respectively, are listed. A simplified but equivalent form consists in listing only objects and attributes the first time they occur in the lattice. Inheritance from top to bottom for attributes, and from bottom to top for objects, allows us to calculate the complete extent and intent for each concept. This simplified version is illustrated in Fig. 2 middle. Thus, an object and an attribute are introduced in the lowest and the highest concept, respectively, where they are present. In Fig. 2 middle, m 6 is only shown in C 8 , the first time it occurs from top to bottom. The extent of C 8 is g 3 and its inherited object: g 2 , while its intent is m 6 and its inherited attributes: m 1 , m 2 , m 3 , and m 5 .

AOC-Poset

A concept lattice may have 2 min(|G|,|M |) concepts. The large number of concepts that can be found in a formal context could render the lattice too complex to be explored. The use of an Attribute-Object-Concept (AOC) poset [START_REF] Dolques | AOC-posets: a scalable alternative to concept lattices for relational concept analysis[END_REF] is one approach to reduce this complexity.

Instead of retrieving all concepts, the AOC-poset is restricted to object-concepts (which introduce at least one object) and attribute-concepts (which introduce at least one attribute), keeping the ≤ relation among selected concepts. In Fig. 2 1.

We define the number of levels in an AOC-poset as the longest top-down path. The AOC-poset in Fig. 2

right contains 5 levels (C 1 → C 2 → C 5 → C 8 → C 11 ).
We also define a concept C a as a child of C b if C a ≤ C b and there is no concept between them. The concept C 9 in Fig. 2 right is a lower neighbor of C 4 and C 5 . Those two concepts are called parents of C 9 .

Implication and Association Rules

Concept lattice and AOC-posets are both useful for extracting implication and association rules, since they provide hierarchical information among objects and attributes. For example, in Fig. 2 middle,m 1 and m 5 are introduced in C 5 and C 2 , respectively, with C 5 ≤ C 2 . This means that any object having m 1 also has m 5 , written as the implication m 1 → m 5 . Conversely, we can also find the association rule m 5 → m 1 with confidence 3/4 = 0.75, since there are 4 objects in C 5 and 3 objects in C 2 . This means that among 4 objects having m 5 , 3 of them have also m 1 . The rules present in the lattices can be extracted from the corresponding AOC-poset. Consequently, the AOC-poset, which is simpler, is preferred. Furthermore, an association rule can also highlight anomalies among objects and attributes, by looking at the confidence value. In the previously mentioned rule m 5 → m 1 , we see that among 4 objects having m 5 , only one of them does not have m 1 . This may indicate that this one object, g 5 , is an anomaly. This anomaly detection is the basis of our detection of incorrect derivations in Démonette2.

Methodology

In this section we explain how to use FCA to represent derivational families in Démonette2 and to find possible inaccuracies in it. Démonette2 is a database whose entries are pairs of derivationally related lexemes. Each entry has 38 columns describing the two lexemes and their relation. In this paper, we are focusing only on: the two lexemes and their part of speech (noun, verb, etc.), the orientation of the relation (direct, indirect, undecidable), and the morphological patterns relating the lexemes (X-Xion, X-Xment, etc.)

Démonette2 as Graphs

In a graph, we assign the lexemes as the labels of vertices, while the morphological patterns serve as the label of edges. The orientation of the derivation determines the edge type. Direct derivations are represented by solid directed edges, while indirect and undecidable derivations are represented by dashed and dotted undirected edges respectively. The graphs of families of the cramer and haubaner are depicted in Fig. 3. A fingerprint [START_REF] Papay | Evaluating and improving a derivational lexicon with graphtheoretical methods[END_REF] is a graph that illustrates the structure of a family, obtained by keeping the label of edges without labeling any vertice. Here we extend this notion by keeping part of speech information in the vertices. The fingerprints of cramer and

haubaner_V haubanage_N X-Xage hauban_N X-X cramer_V cramage_N X-Xage
Figure 3: The graphs of cramer (top) and haubaner (bottom). A solid directed edge and a dashed edge represent a direct derivation and an undecidable derivation respectively. haubaner are depicted in Fig. 4. A fingerprint can correspond to several families since a fingerprint does not contain the lexemes. This is the case of the family of cramer and roder, who share the fingerprint shown in Fig. 4 top. The graph g 1 is a subgraph of g 2 (written g 1 ⊂ g 2 ) if g 1 is included in g 2 . In Fig. 4, the top fingerprint is a subgraph of the bottom fingerprint, since the latter contains all vertices and edges of the former.

V N X-Xage N X-X V N X-Xage

Creation of AOC-Poset

To build an AOC-poset, first we have to construct a formal context. As explained in Section 2, a formal context is a table of objects and their attributes. In this section, we present how to build formal context using fingerprints. In the formal context, each object corresponds to a family, and each attribute corresponds to a fingerprint f ∈ F . The set of fingerprints F is partially ordered by subgraph inclusion. A family g possesses its own fingerprint f g and any fingerprint f i such that f i ∈ F and f i ⊂ f g . The formal context of families in Fig. 1 is shown in Table 2. The family of roder and cramer share the same fingerprint f 1 . The family of haubaner has not only its own fingerprint f 2 , but also f 1 as f 1 ⊂ f 2 . The family of ajouter has f 4 , and also f 1 , f 2 , and f 3 since they are subgraphs of f 4 . The corresponding AOC-poset in simplified form is depicted in Fig. 5, together with the fingerprint of each concept. Any resulting concept is an attribute concept, introducing exactly one fingerprint and one or more families

Family

Fingerprints

f 1 f 2 f 3 f 4 roder × cramer × haubaner × × jaunir × × ajouter × × × × Table 2:
The formal context of the 5 families in Fig. 1 and the 4 fingerprints shown in Fig. 5.

sharing that fingerprint. Consequently, from the resulting AOC-poset, we can see how a family's fingerprint extends to another fingerprint, and how two fingerprints combine to form another fingerprint. From Fig. 5, we see that f 1 grows by adding an indirect X-X derivation to become f 2 or by adding two derivations to become f 3 . We also see that f 2 and f 3 combine to form f 4 .

Poset Exploration

Currently, the Démonette2 database contains 51,830 unique pairs of lexemes and their description. From these pairs, we obtain 13,897 families and 4,849 different fingerprints. The number of families associated to each fingerprint is varied: 4,181 fingerprints are only associated to one family, while one small fingerprint (having only two nodes) shared with 1,381 families. The formal context contains 13,897 objects and 4,849 attributes. The resulting AOC-poset has 8 levels and 4,849 concepts -one for each fingerprint -while 69 concepts are isolated, i.e. having neither subconcept nor superconcept. There are 209 "top" concepts -having no parent and at least one child -which can be regarded as the beginning of a new paradigm. The maximum number of parents and children of a given concept is 83 and 124 respectively. Among concepts having at least 1 parent, the average number of parents is 5.06; while among the concepts having at least 1 child, the average number of children is 10.08. To detect possible incorrect derivations in a family, we focus on pairs of neighboring concepts with their corresponding association rule. Consider the concepts C 1 and C 2 in Fig. 6, with the fingerprint f 1 ⊂ f 2 . The association rule f 1 → f 2 has confidence 14/15, meaning that among 15 families having f 1 , only one of them does not have the larger f 2 . Therefore, this one family may be missing some derivations.

On the other hand, consider two concepts C 3 and C 4 in Fig. 6, in simplified form, with the fingerprint f 3 ⊂ f 4 . The association rule f 3 → f 4 has confidence 1/16, meaning that 15 families have exactly f 3 , and only one has f 3 and f 4 . Therefore, this one family may have some incorrect additional derivations.

Families with missing derivations. Fig. 7 presents one case of missing derivation found in Démonette2. 2 and to ease understanding we draw in blue the fingerprints involved in the intent of the concepts .

We see concept C 3812 introduces fingerprint f 3606 , and one of its subconcepts C 3205 introduces f 3842 . These two fingerprints are illustrated beside the concepts, and we see that f 3606 is a subgraph of f 3842 , with the latter having two more derivations (direct X-Xage and X-Xeur). Among 708 families having indirect derivation Xage-Xeur, 695 of them also have the direct derivations X-Xeur and X-Xage (e.g. the family survolter with lexemes survolter V , survoltage N , and survolteur N ).

The remaining 13 families could be missing the X-Xage and X-Xeur derivations and should be checked by hand. An actual example of a missing derivation is that of the family orpailleur, which has only two lexemes orpaillage N and orpailleur N , and an indirect derivation between them, without the lexeme orpailler V . We then propose the addition of the missing lexeme and the two missing derivations.

The family radeau, illustrated in Fig. 8, is another relevant example among the 13 families. This family has the indirect Xage-Xeur and direct X-Xage, but is missing the direct derivation X-Xeur from radeler V to radeleur N . Notice that f 3606 is a subgraph of radeau's fingerprint. This shows the usefulness of FCA and its capacity to detect an anomaly in a family's subgraph.

Families with incorrect derivations. To find possibly incorrect derivations in Démonette2, we focus on two neighboring concepts where the subconcept introduces a low number of families compared to its superconcept. This is the case of concepts C 2376 and C 1766 shown in Fig. 9. There are 41 families that have the fingerprint f 3683 , while only one family has f 1972 , which is f 3683 with one additional X-Xion derivation. This one family is détracter, shown in Fig. 10. This family has the extra direct X-Xion derivation from détracter V to détractation N , which is a rarely used synonym of détraction N .

Limitations of association rules. From Fig. 6 right, the example of f 4 introducing only one family may not mean that this one family has incorrect derivations. This is often the case when a family has extra valid derivations for different spelling, like essuyementessuiement, debuscage-debusquage, etc., which should not be regarded as incorrect derivations. Furthermore, many derivational families are indeed incomplete in the language, and should thus be docu-

Validation

The anomalies found by the method explained in Section 4 can then be presented to linguists to check whether are actually missing or incorrect derivations. In order to do this validation, we designed a simple web page to visualize our findings.

Missing derivations. A screenshot of the web page for the validation of missing derivation is shown in Fig. 11. In this example, the family turban may be missing an X-Xment derivation (in blue). We provide the prediction of the supposedly missing lexeme (enturbannement N ) by performing a formal analogy [START_REF] Lavallée | Unsupervised morphological analysis by formal analogy[END_REF][START_REF] Lepage | Solving analogies on words: an algorithm[END_REF][START_REF] Stroppa | Analogical learning and formal proportions: Definitions and methodological issues[END_REF] from the two families, e.g.:

emmitoufler V : emmitouflement N = enturbanner V : ?

Figure 11: A web-based tool of revision of the possibly missing derivation. The reference family is the "normal" family, while the question is the family that is missing some derivations in blue. Each lexeme is accompanied by its category and its frequency from FR-COW corpora.

From this analogy, we generate the predicted lexeme enturbannement by replacing the affix according to the reference. We then check the frequency of the enturbannement in the French web corpus FRCOW (Bildhauer and Schäfer, 2016) to help the linguists decide whether the lexeme exists or not. The linguist/validator can choose one of the three options:

1. keep: the prediction is correct and will be added to the database;

2. delete: the family in question is correct as it is, no change needed in the database;

3. other: for other cases, e.g. when the family in question is indeed missing some derivations, but the given prediction is incorrect.

It should be noted that affix replacement will not always give the correct prediction. This is the case in hydrocyclique Adj : hydrocycliser V = muet Adj : ?, where hydrocyclique and muet do not share any affix. In this case, we put "?" in the prediction, and the linguists can provide the intended lexeme (mutiser) in the comment.

Incorrect derivations. A page for the validation of incorrect derivations is shown in Fig. 12. The "delete" option here removes some elements in the family and updates the database, while the "keep" option does not update the database.

Tracing the source. To trace the origin of a derivation, additional information is added for labeling the edges. Beside the morphological pattern, we provide a code that corresponds to a source file. For example, the label of the edge between palefrenat N to palefrenier N in Fig. 12 is "Mb: Xat-Xier". While "Xat-Xier" is the morphological construction, "Mb" is a code that refers to the source file of the pair.

Update iterations. The result of these validations updates the database, hence updating the AOC-poset.

Given a new AOC-poset, we find a new set of anomalies, and a new validation process is needed. The update of Démonette2 is thus iterative, but we can not know beforehand how many iterations it takes to converge.

Conclusion

We presented in this paper the potential of FCA in detecting both missing and spurious derivations in Démonette2. This detection is important in improving the quality and completeness of the database. Our methodology can easily be adapted to other word formation databases. Our hypothesis, stated in Section 1, may need some refinements, but it is a good starting point in detecting anomalies in a database. However, the creation of formal context we presented in this paper is not the only possible way to build a formal context. Instead of calculating the subgraph relation among fingerprints, we may be interested in calculating all frequent subgraphs and study the grouping of families based on which subgraphs they share, leading to a more complex structure. Moreover, in addition to the morphological construction (X-Xier, X-Xion, etc.), the semantic information for each derivation may give us a better grouping of families.

The iterative update of Démonette2 should also be investigated, since the AOC-poset structure enables more than one update per iteration. It is also important to filter out the anomalies found in each iteration, since an anomaly that is already treated as valid should not be presented again as an anomaly.

Figure 2 :

 2 Figure2: The concept lattice in full form (left), the concept lattice in simplified form (middle), and the AOC-poset in simplified form (right) of the formal context in Table1.

Figure 4 :

 4 Figure 4: The fingerprint of cramer and roder (top), which is a subgraph of the fingerprint of haubaner (bottom).

Figure 5 :

 5 Figure 5: AOC-poset (in black) of Table2and to ease understanding we draw in blue the fingerprints involved in the intent of the concepts .

Figure 6 :

 6 Figure 6: Illustration of two pairs of neighboring concepts. The pair in the left is shown in the full form, while the pair in the right is in the simplified form.

Figure 7 :Figure 8 :

 78 Figure 7: Two neighboring concepts and their fingerprints (f 3606 and f 3842 ). They differ in X-Xage and X-Xeur.

Figure 9 :Figure 10 :

 910 Figure 9: Two neighboring concepts C 2376 and C 1766 . C 2376 introduces the fingerprint f 3683 , shown in upper right, while C 1766 introduces f 1972 , shown in lower right.

Figure 12 :

 12 Figure 12: A web-based tool of revision of the possibly incorrect derivation, similar to the web page shown in Fig. 11.

  , g 3 }, {m 1 , m 2 , m 3 , m 5 , m 6 }) is a conceptfrom Table 1. In the cross-table depiction, a formal concept

	. From
	a formal context, FCA builds all formal concepts, and
	arranges them in a concept lattice.
	A formal context is a binary table of objects and
	their attributes, usually depicted as a cross-table with
	set of objects G as rows and set of attributes M as
	columns. A cross in the table signifies that object
	g ∈ G has attribute m ∈ M . An example of a formal
	context with 7 objects (g 1 , g 2 , • • • g 7 ) and 7 attributes
	(m 1 , m 2 , • • • m 7 ) is given in Table 1.
	A formal concept is a pair (A ⊆ G, B ⊆ M ), where
	A and B are called extent and intent, respectively.
	A formal concept corresponds to the maximal set of
	objects A sharing a set of attributes B, while B is
	the maximal set of attributes shared by A. The pair
	({g 2