Simon Felder

Shiqi Wu

Dr Jules Brom

Dr Laurent Micouin

Erica Benedetti
email: erica.benedetti@parisdescartes.fr

Enantiopure Planar Chiral [2.2]Paracyclophanes: Synthesis and Applications in Asymmetric Organocatalysis

Keywords: [2.2]Paracyclophanes, planar chirality, enantioselectivity, stereoselective synthesis, asymmetric organocatalysis

This short review focuses on enantiopure planar chiral [2.2]paracyclophanes (pCps), a fascinating class of molecules which possess an unusual three-dimensional core and intriguing physicochemical properties. In the first part of the review, different synthetic strategies for preparing optically active pCps are described. Although classical resolution methods based on the synthesis and separation of diastereoisomeric products still dominate the field, recent advances involving the kinetic resolution of racemic compounds and the desymmetrization of meso derivatives open up new possibilities to access enantiopure key intermediates on synthetically useful scales.

Due to their advantageous properties including high configurational and chemical stability, [2.2]paracyclophanes are increasingly employed in various research fields, ranging from stereoselective synthesis to material sciences. The applications of [2.2]paracyclophanes in asymmetric organocatalysis are described in the second part of the review. While historically enantiopure pCps have been mainly employed by organic chemists as chiral ligands in transition-metal catalysis, these compounds can also be used as efficient catalysts in metal-free reactions, and may inspire the development of new transformations in the near future.

Introduction

Since its discovery in 1949 by Brown and Farthing, 1 [2.2]paracyclophane (pCp) has been of great interest to the scientific community due to its interesting structural and chemical properties. Constituted of two benzene rings stacked on top of each other, unable to rotate on their axis, and linked together by two ethylene bridges at their para positions (Figure 1), [2.2]paracyclophane is the smallest compound of the cyclophane family. 2 X-Ray crystallographic analysis by Trueblood et al. in 1972 revealed that the two co-facial phenyl ring of pCp assume an unusual boat-shaped conformation (Figure 1). [START_REF] Hope | The Crystal and[END_REF] Further crystallographic and computational studies since then have extended our knowledge of the structural features, and molecular dynamics of unsubstituted [2.2]paracyclophanes. [START_REF] Walden | Distinctive Normal Harmonic Vibrations of[END_REF][START_REF] Henseler | Theoretical Study on the Molecular Distortions in [2.2Paracyclophane and Cyclobutane[END_REF][START_REF] Grimme | Accurate Computation of Structures and Strain Energies of Cyclophanes with Modern DFT Methods[END_REF] At low temperature, this molecule is nowadays believed to adopt two distinct distorted geometries with a D2 symmetry. The rapid interconversion between the D2 conformers globally affords an achiral D2h strained structure at room temperature. The conformational tension of pCp impacts the conjugation of the π-systems, and consequently the reactivity of its aromatic units. Thus, although classical benzene chemistry (e.g. Friedel-Craft acylations, halogenations, or metal-catalysed cross couplings) is usually employed to decorate [2.2]paracyclophane, an array of transformations, rarely observed for more common aromatic compounds, is also allowed to take place (i. e. Diels-Alder cycloadditions, hydrogenations or ionic additions). [START_REF] Kleinschroth | The Chemical Behavior of Multibridged [2n] Cyclophanes[END_REF][START_REF] Aly | Novel Synthesis of Phenanthrenoparacyclophanes and Phenanthrenophanes and a Study of Their NMR Properties[END_REF][START_REF] Minuti | 5-Nitro[2.2Paracyclophanepyran-6-One-Building Block for the Synthesis of [2.2]Paracyclophanes Containing Condensed Benzofuran Subunits[END_REF][START_REF] Aly | Cycloadditions to Alkenyl[2.2]Paracyclophanes[END_REF] One of the most intriguing features of pCp is the ability of substituents on one deck to modify the reactivity of the system by activating or deactivating other positions on both rings. Examples of this peculiar behaviour include the regioselective pseudo-gem bromination of ester, nitro, and amide derivatives, [START_REF] Reich | Macro Rings. XXXVII. Multiple Electrophilic Substitution Reactions of [2.2]Paracyclophanes and Interconversions of Polysubstituted Derivatives[END_REF][START_REF] Hassan | Bräse S Regioselective Functionalization of [2.2]Paracyclophanes: Recent Synthetic Progress and Perspectives[END_REF] as well as the Vilsmeier-Haack para-formylation of N-substituted amino compounds. [START_REF] Felder | Para-Functionalization of N-Substituted 4-Amino[2.2]Paracyclophanes by Regioselective Formylation[END_REF] The two aromatic units of [2.2]paracyclophane possess a total of eight possible modification sites. As soon as a substituent is introduced onto the pCp framework, planar chiral compounds are obtained (Figure 2). [START_REF] Mcnaught | IUPAC Compendium of Chemical Terminology[END_REF] In the presence of more than one functional group, complex chiral molecules can easily be formed. These derivatives are usually described by taking into account their substitution patterns as exemplified in Figure 2. In addition to ortho-, meta-and para-substituted substrates, showing two different functional groups on the same aromatic ring of the pCp core, pseudo-gem, pseudo-ortho, pseudo-meta or pseudo-para compounds can be obtained when the substituents are located onto the two pCp decks. The first experimental evidence of the inherent chirality of functionalized pCps was brought by Cram and Allinger in 1955 through an elegant diastereoselective crystallization of the brucine salts of 4-carboxy [2.2]paracyclophane. [START_REF] Cram | Macro Rings. XII. Stereochemical Consequences of Steric Compression in the Smallest Paracyclophane1[END_REF] In 1966, Cahn, Ingold and Prelog set up the rules to attribute the absolute configuration of enantiopure pCps. [START_REF] Cahn | Specification of Molecular Chirality[END_REF]

FIGURE 2. Planar chiral molecules derived from [2.2]paracyclophane

According to their proposition, one must place the most substituted ring in the furthest position (backplane) and start numbering atoms from the first out-of-plane carbon of the closest bridge (pilot atom, P) to the highest priority substituent. The Rp and Sp descriptors (p for identifying planar chirality) are nowadays commonly employed in cyclophane chemistry (Figure 3). Beginning in the late 1990s, organic chemists have taken advantage of the uncommon structure and reactivity of pCp to develop original planar chiral substrates that can be useful as ligands or catalysts in asymmetric synthesis. [START_REF] Gibson | 2.2]Paracyclophane derivatives in asymmetric catalysis Org[END_REF][START_REF] Hassan | Planar chiral [2.2]paracyclophanes: from synthetic curiosity to applications in asymmetric synthesis and materials[END_REF][START_REF] Rowlands | Planar Chiral Phosphines Derived from [2.2]Paracyclophane[END_REF][START_REF] Schneider | 2.2]Paracyclophane-Derived Planar-Chiral Hydrogen-Bond Receptors[END_REF][START_REF] Paradies | Paracyclophane Derivatives: Synthesis and Application in Catalysis[END_REF][22] Planar chiral [2.2]paracyclophanes have also been extensively employed as organic building blocks in material sciences. [START_REF] Morisaki | Planar Chiral [2.2]Paracyclophanes: Optical Resolution and Transformation to Optically Active π-Stacked Molecules[END_REF][START_REF] Weiland | Beyond Simple Substitution Patterns -Symmetrically Tetrasubstituted [2.2]Paracyclophanes as 3D Functional Materials[END_REF] However, obtaining enantiopure synthons to chemically design such compounds has been, and remains, quite challenging. In this up-to-date review, we will describe the different possible strategies for obtaining optically pure [2.2]paracyclophanes. We will then present the applications of these compounds in organic chemistry, with a particular focus on their use in asymmetric organocatalysis.

Synthesis of enantiopure pCps

More than seventy years have passed since the first evidence of the chiral properties of pCps. [START_REF] Cram | Macro Rings. XII. Stereochemical Consequences of Steric Compression in the Smallest Paracyclophane1[END_REF] From that moment on, as already pointed out, different substrates bearing one or more substituents have been isolated in their enantiopure forms, and applied in various scientific fields. In many cases, the synthesis of complex planar chiral [2.2]paracyclophanes relies on the preparation and derivatization of optically active key intermediates. To obtain such compounds, different strategies have been developed over the years. In the following part of this review, we present an overview of these techniques, which include i) chromatography purification on chiral stationary phases, ii) classical chemical resolutions involving the formation of diastereomeric products, iii) chemical or enzymatic kinetic resolutions, and iv) the desymmetrization of meso compounds.

CHROMATOGRAPHY PURIFICATION ON CHIRAL STATIONARY PHASES

High performance liquid chromatography (HPLC) on chiral stationary phases is nowadays the most widely used method for determining the purity and enantiomeric excess of optically active compounds. HPLC on semi-preparative or preparative scales can also be employed as an effective way to separate the two enantiomers of racemic mixtures. Although costly, this approach has been extensively applied in paracyclophane chemistry to access a variety of enantiopure products, such as pCp-based acids, [START_REF] Lämmerhofer | Direct High-Performance Liquid Chromatographic Enantiomer Separation of Anti-Inflammatory Planar Chiral [2.2]Paracyclophane-4-Acetic Acid[END_REF] anilines, 26 imines or hydrazones, [START_REF] Masterson | Catalytic enantioselective cyclopropanation of olefins using N-salicylidene-4-amino[2.2]paracyclophane as an asymmetric ligand[END_REF] and sulfur-containing compounds [START_REF] Deschamps | Functionalization of [2.2]Paracyclophanes via a Reductive Sulfanylation Reaction[END_REF] among many others. [START_REF] Furo | Absolute Configuration of Chiral [2.2]Paracyclophanes with Intramolecular Charge-Transfer Interaction. Failure of the Exciton Chirality Method and Use of the Sector Rule Applied to the Cotton Effect of the CT Transition[END_REF][START_REF] Meyer-Eppler | Chiral Resolution, and Absolute Configuration of Dissymmetric 4,12-Difunctionalized [2.2Paracyclophanes[END_REF][START_REF] Anhäuser | Chiral Self-Sorting Behaviour of [2.2]Paracyclophane-Based Bis(Pyridine) Ligands[END_REF][32][START_REF] Oliveras-Gonzalez | Tetrathiafulvalene-[2.2Paracyclophanes: Synthesis, Crystal Structures, and Chiroptical Properties[END_REF] Moreover, even if it is not always easy to implement, preparative chiral HPLC today remains a method of choice for the late-stage separation of complex pCp molecules. [START_REF] Hasegawa | Synthesis and chiroptical properties of stereogenic cyclic dimers based on 2,2'-biselenophene and [2.2]paracyclophane[END_REF] Despite earning less recognition as a resolution method in comparison with HPLC, gas chromatography (GC) is a valuable technique frequently employed for separating and analysing lowmolecular-weight organic compounds. By using modified silica capillary columns and cyclodextrins as chiral selectors, GC has been successfully exploited in 1994 to resolve the two enantiomers of a whole range of mono-substituted pCp derivatives. [START_REF] König | Resolution of Chiral [2.2]Paracyclophanes by Enantioselective Gas Chromatography[END_REF] The temperature has been kept between 140 °C and 150 °C to avoid homolytic cleavage of the molecules. However, while very efficient on an analytical scale, to the best of our knowledge, GC separations of planar chiral [2.2]paracyclophanes have hardly been transferred to preparatory scale up to now. Semi-preparative supercritical fluid chromatography (SFC) can be considered as a valuable alternative to HPLC and GC for the resolution of chiral compounds due to its higher separation speed, and the use of mobile phases that mostly evaporate during fraction collection. In spite of its benefits and high potential, SFC has rarely been used to isolate enantiopure [2.2]paracyclophanes [START_REF] Zhang | Dimesitylboryl)phenyl-Substituted [2.2]Paracyclophanes Featuring Intense and Sign-Invertible Circularly Polarized Luminescence[END_REF] possibly because of the expensive equipment required to perform chiral purifications on synthetically useful scales.

CLASSICAL CHEMICAL RESOLUTIONS

Already employed by Pasteur more than 150 years ago, [START_REF] Gal | Pasteur and the art of chirality[END_REF] classical resolutions are the oldest way of separating two enantiomers. These processes are based on the isolation of diastereomeric salts formed by an acid-base reaction between a racemate and an optically pure partner. Successive recrystallizations typically yield enantiopure products thanks to the different physical properties of the salts. Similarly, chemical resolutions involve reversible covalent modifications of racemic mixtures with chiral reagents. The resulting diastereomeric products can then be separated by chromatography. In both cases, an additional chemical step is required to recover the initially sought compounds in their enantiopure forms: hydrolysis of the salt in the first case, and removal of the chiral auxiliary in the second case. Despite being costly and time-consuming, classical and chemical resolutions can still be considered valuable methods for obtaining enantiopure chemicals, and are, again today, the most frequently employed techniques to access optically active pCps. In the next sections, the different approaches used to isolate planar chiral pCps will be subdivided according to the class of compounds that undergo the resolution procedure.

Carboxylic acids

As mentioned previously, the first classical resolution of 4carboxy[2.2]paracyclophane was described as early as 1955, when the racemic compound was resolved by successive recrystallizations of its brucine salt. [START_REF] Cram | Macro Rings. XII. Stereochemical Consequences of Steric Compression in the Smallest Paracyclophane1[END_REF] In 1998, an improved synthesis of enantiopure 4carboxy [2.2]paracyclophane has been reported (Scheme 1). Following this approach, a lithium/halogen exchange, and consequent CO2 quench can be employed to convert commercially available 4-bromo[2.2]paracyclophane (±)-1 into its carboxylate analogue. The racemic acid (±)-2 is then resolved through a reaction with α-(p-nitrophenyl)ethylamine (NPEA) and separation of the obtained diastereoisomeric salts by filtration ((Sp,S)-3 and (Rp,S)-3, Scheme 1). After hydrolysis of the salts under acidic conditions, the optically pure products (Sp)-2 and (Rp)-2 have been isolated in 45% and 48% yields respectively. In 2004, a dibromo substituted carboxylic acid derived from pCp ((±)-4) has been resolved using ()-cinchonidine to selectively isolate the Sp enantiomer in 52% yield and 97% ee after recrystallization from ethanol and hydrolysis of the salt (Scheme 2). 39 SCHEME 2. Resolution of a dibromo substituted carboxylic acid derived from pCp A chemical resolution of 4,12-dicarboxy[2.2]paracyclophane has been described in 2005. [START_REF] Jiang | Resolution of (±)-[2.2]Paracyclophane-4,12-Dicarboxylic Acid[END_REF] In this case, the racemic compound (±)-5 is first treated with thionyl chloride, and then reacted with a chiral primary alcohol derived from (1S)-()-camphanic acid ((S)-6, Scheme 3). After the separation of the diastereoisomeric esters (Sp,S)-7 and (Rp,S)-7 by column chromatography and subsequent hydrolysis, acids (Sp)-5 and (Rp)-5 have been isolated with 99% and 97% ees respectively (Scheme 3).

Aromatic halides

Halogenated aromatic compounds are very precious reactants in organic synthesis as they offer an entry way for simple chemical modifications, notably through reactions with alkyl lithium reagents and electrophiles. This type of process has been exploited in paracyclophane chemistry to resolve racemic compounds incorporating one or more bromine atoms by introducing chiral tolyl sulfoxide moieties onto the pCp scaffold. For example, in 1991, 4-bromo[2.2]paracyclophane (±)-1 has been submitted to a reaction with n-BuLi and ()-menthyl ()-(S)-p-toluenesulfinate. [START_REF] Reich | Asymmetric Induction in the Oxidation of [2.2]Paracyclophane-Substituted Selenides[END_REF]

SCHEME 4. Resolution of 4-bromo[2.2]paracyclophane

Separation of the diastereoisomeric products by silica gel column chromatography, and treatment of the pure compounds with n-BuLi, CO2 and diazomethane led to the formation of enantioenriched methyl esters (Scheme 4). A similar strategy has been used in 2010 [START_REF] Parmar | Towards a Flexible Strategy for the Synthesis of Enantiomerically Pure [2.2]Paracyclophane Derivatives: The Chemistry of 4-Tolylsulfinyl[2.2]Paracyclophane[END_REF] and 2012 [START_REF] Morisaki | Practical Optical Resolution of Planar Chiral Pseudo-Ortho -Disubstituted [2.2[END_REF] to isolate a variety of planar chiral compounds, including aldehydes, alcohols, carboxylic acids, anilines and sulfoxides.

Anilines

The two enantiomers of 4-amino[2.2]paracyclophane (±)-10 have been separated for the first time in 1997 using (S)-(+)camphorsulfonic acid as the chiral resolving agent. Fractional crystallization of the obtained diastereoisomeric salts leads to enantiopure compound (Rp)-10 after treatment with sodium hydroxide (0.1 M aqueous solution, Scheme 5). Starting from product (Rp)-10 a series of transformations can be easily carried out to access a variety of planar chiral compounds. 44

SCHEME 5. Resolution of 4-amino[2.2]paracyclophane

Aldehydes and ketones

Because of their chemical versatility, carbonyl derivatives are commonly employed as intermediates in organic synthesis. Aldehydes and ketones derived from [2.2]paracyclophane are no exception to that rule. Indeed, over the years, these compounds have repeatedly proved to be valuable precursors of complex pCp molecules both in their racemic and enantiopure forms. Chemical resolutions of formylated or acetylated pCps bearing different substituents have mainly been carried out by using (R)or (S)-α-phenylethylamine (PEA) as chiral reagents. [START_REF] Antonov | Synthesis, Resolution and Absolute Configuration Determination of (S)-and (R)-4-Formyl-5-Hydroxy[END_REF][START_REF] Rozenberg | Regioselective Fries Rearrangement and Friedel-Crafts Acylation as Efficient Routes to Novel Enantiomerically Enriched Ortho -Acylhydroxy[2.2]Paracyclophanes[END_REF][START_REF] Sergeeva | Resolution Approaches to Enantiomers of 4-Formyl[2.2]Paracyclophane[END_REF][START_REF] Vorontsova | Novel Ligands Based on Bromosubstituted Hydroxycarbonyl [2.2]Paracyclophane Derivatives: Synthesis and Application in Asymmetric Catalysis[END_REF][START_REF] Friedmann | Improved Synthesis of Enantiopure 4-Hydroxy[2.2Paracyclophane[END_REF][START_REF] Cakici | Modular Synthesis of Planar-Chiral Para -Substituted Paracyclophanes by Double Suzuki Coupling[END_REF][START_REF] Antonov | Iminophenol Ligands Derived from Chiral Regioisomeric Hydroxy[2.2]Paracyclophane-Carbaldehydes: The Influence of the Substitution Pattern on Asymmetric Induction[END_REF][START_REF]Formyl-3-hydroxy[END_REF] For example, 4-formyl-5-hydroxy[2.2]paracyclophane (±)-12 readily reacts with (R)-PEA to form two diastereosomeric imines that can be separated by recrystallization in MeOH. After acid hydrolysis, the enantiopure product (Sp)-12 has been isolated in 33% yield (Scheme 6). Partially resolved aldehyde (Rp)-12, on its side, needs to be submitted to a second resolution cycle, this time in the presence of (S)-PEA, to access the enantiopure isomer (Rp)-12 in 31% overall yield (Scheme 6). [START_REF] Antonov | Iminophenol Ligands Derived from Chiral Regioisomeric Hydroxy[2.2]Paracyclophane-Carbaldehydes: The Influence of the Substitution Pattern on Asymmetric Induction[END_REF]

SCHEME 6. Resolution of 4-formyl-5-hydroxy-[2.2]paracyclophane

In addition to the reactions with PEA, 4acetyl[2.2]paracyclophane (±)-14 can be resolved using (S)-()-SAMP (Ender's chiral auxiliary). [START_REF] Minuti | An Improved Synthesis of (S)-(+)-and (R)-(-)-4-Ethenyl[2.2]Paracyclophane[END_REF] After the separation and hydrolysis of the diastereomeric hydrazones (Sp,S)-15 and (Rp,S)-15, the two enantiomers of ketone 14 could be isolated in 22-25% yields and >99% ee.

SCHEME 7. Resolution of 4-acetyl-12-hydroxy[2.2]paracyclophane

Phenols

Enantiopure mono-and di-hydroxy-substituted pCps can be obtained through the separation of diastereoisomeric ester adducts formed upon treatment with activated camphanoyl derivatives. [START_REF] Vorontsova | Planar Chiral Hydroxy Derivatives of [2.2]Paracyclophane as Auxiliaries for Asymmetric Allylboration[END_REF][START_REF] Jiang | A Simple and Efficient Resolution of (±)-4,12-Dihydroxy[2.2]Paracyclophane[END_REF][START_REF] Morisaki | New Types of Planar Chiral [2.2]Paracyclophanes and Construction of One-Handed Double Helices[END_REF][57][START_REF] Morisaki | Planar Chiral Tetrasubstituted [2.2]Paracyclophane: Optical Resolution and Functionalization[END_REF][START_REF] Sawada | Synthesis of Enantiopure Planar Chiral Bis-(Para)-Pseudo-Meta -Type [2.2]Paracyclophanes[END_REF][START_REF] Rozenberg | Resolution and Novel Reactions of 4-Hydroxy[2.2]Paracyclophane[END_REF] This quite general approach can be exemplified by the chemical resolution of 4-hydroxy[2.2]paracyclophane (±)-16. 57 This compound has been submitted to a reaction with (1S)-()-camphanoyl chloride ((S)-17, Scheme 8) to form a couple of diastereomers which can be easily separated by silica gel column chromatography (Scheme 8). After saponification with KOH in MeOH, products (Sp)-16 and (Rp)-16 have both been obtained in up to 38% yield and >99% ee (Scheme 8).

SCHEME 8. Resolution of 4-hydroxy[2.2]paracyclophane

Thiols

In 2016, an interesting classical resolution of [2.2]paracyclophane-4-thiol (±)-18 has been described. [START_REF] Vincent | Enantiomerically Pure [2.2]Paracyclophane-4-Thiol: A Planar Chiral Sulfur-Based Building Block Readily Available by Resolution with an Amino Acid Chiral Auxiliary[END_REF] Based on the screening of a wide range of chiral auxiliaries, activated Lisoleucine ((2S,3S-19, Scheme 9) appears to be the reagent of choice to access enantioenriched thiols (Sp)-18 and (Rp)-18 through the formation of diastereoisomeric salts. Indeed, after the synthesis, separation, and hydrolysis of compounds (Sp, 2S,3S)-20 and (Rp,2S,3S)-20, products (Sp)-18 and (Rp)-18 could be obtained in approximately 40% yield, with 93% and >99% ees respectively (Scheme 9). This procedure appears to be quite appealing as the resolution can be run on a large scale without any loss in efficiency.

SCHEME 9. Classical resolution of 4-thiol[2.2]paracyclophane

Phosphine oxides

Starting from commercially available 4,12dibromo[2.2]paracyclophane (±)-21 a double lithium-halogen exchange, followed by transmetalation with magnesium, and subsequent reaction with diphenylphosphinic chloride leads to the formation of phosphine oxide (±)-22. [START_REF] Pye | A New Planar Chiral Bisphosphine Ligand for Asymmetric Catalysis: Highly Enantioselective Hydrogenations under Mild Conditions[END_REF] This compound can be resolved with dibenzoyl-D-tartaric acid as described in Scheme 10.

SCHEME 10. Synthesis and classical resolution of pCp-based phosphine oxides

The resulting enantiopure products have been used as key intermediates in one of the first syntheses of PhanePhos, a wellknown pCp-based planar chiral ligand (vide infra). Various substituents can be introduced onto the phosphorous atoms simply by using different diarylphosphinic chloride reagents (Ar2POCl), thus opening the pathway for the preparation of a wide range of optically active products. Unsymmetrical phosphorous derivatives [START_REF] Cheemala | New Paracyclophane Phosphine for Highly Enantioselective Ruthenium-Catalyzed Hydrogenation of Prochiral Ketones[END_REF] and mono-substituted substrates [START_REF] Zhang | Synthesis of planar chiral [2.2]paracyclophane monophosphine ligands and their application in the umpolung allylation of aldehydes[END_REF] have been prepared as well following this synthetic strategy.

CHEMICAL OR ENZYMATIC KINETIC RESOLUTIONS

Among many classes of asymmetric reactions, kinetic resolutions (KR) are certainly attractive tools for organic chemists. In kinetic resolutions, two enantiomers undergo a transformation in the presence of a chiral catalyst or an enzyme and react with different reaction rates. Theoretically, at 50% of conversion, one enantiomer is isolated as the reaction product; the other is recovered as starting material. KR may constitute a valuable alternative to both stoichiometric resolutions and chromatographic separations that are generally costly in terms of reagents, may be time-consuming or require pricy high-pressure equipment to deliver enantiopure material on a large scale. After an early proof-of-concept described in 1968, [START_REF] Falk | The Absolute Configuration of[2.2]Paracyclophanecarboxylic Acid[END_REF] kinetic resolutions have been successfully implemented in paracyclophane chemistry to access enantiopure compounds showing different reactive functions. In the next section of this review, the diverse methods developed so far will be described by ordering them according to the nature of the pCp derivatives submitted to the KR processes.

Aromatic halides

Enantioenriched 4,12-dibromo[2.2]paracyclophane (Rp)-21 has been obtained in 1997 through a kinetic resolution process based on a palladium-catalysed Buchwald-Hartwig coupling (Scheme 11). 66 Interestingly, PhanePhos (Sp)-23 has been identified as the best chiral ligand to perform this transformation. Indeed, after 16 h of reaction at 50 °C, compound (Rp)-21 could be isolated in 42% yield and 93% ee together with small amounts of mono-and di-substituted pCp-based amines. More recently, a diastereoselective Buchwald-Hartwig coupling between 4-bromo[2.2]paracyclophane (±)-1 and (S)-PEA has been reported to proceed smoothly in the presence of Pd2(dba)3, and chiral ligands. Using 0.5 equivalents of amine and (Sp,R)-Walphos I as chiral ligand, a kinetic resolution of compound (±)-1 has been carried out to isolate the aminated product (Rp,S)-24 in 46% yield and 86% de (Scheme 12). Unreacted enantioenriched derivative (Sp)-1 could also be recovered after the reaction.

Aldehydes and ketones

In addition to the stoichiometric resolution methods described above, aldehydes and ketones derived from [2.2]paracyclophane can be obtained in their enantiopure form using different kinetic resolution approaches. Optically active 4-formyl2.2]paracyclophane can easily be prepared on synthetically useful scales through asymmetric transfer hydrogenation. [START_REF] Delcourt | Efficient and Scalable Kinetic Resolution of Racemic 4-Formyl[2.2Paracyclophane via Asymmetric Transfer Hydrogenation[END_REF] Indeed, when the racemic compound (±)-25 is treated with RuCl(p-cymene)[(R,R)-Ts-DPEN] (Noyori's catalyst) and t-BuOK in a 1:1 i-PrOH-MeCN mixture at 10 °C, the unreacted aldehyde (Rp)-25 and reduced alcohol (Sp)-26 can be resolved to 99% ee and 68% ee respectively at 56% conversion (Scheme 13). Pseudo-gem and pseudo-para bromosubstituted pCp-based aldehydes can be obtained as well in high yields and enantioselectivities following this efficient strategy. 69 SCHEME 13. Kinetic resolution of 4-formyl[2.2]paracyclophane 4-Formyl2.2]paracyclophane (±)-25 has also been kinetically resolved using a yeast strain (DSM 11285) to reduce the carbonyl group. Nonetheless, in this case, a high volume of culture medium (~2 L) is required to convert small quantities of starting material (80 mg), and only few milligrams of enantiopure products can be obtained. 70 The kinetic resolution of 4-acetyl2.2]paracyclophane (±)-27 can also be performed through catalytic asymmetric reductions. For example, methyl ketone (Sp)-27 and benzylic alcohol (Rp,R)-28 have been isolated in good yields, with >99% and 61% ees respectively, using borane as the reducing agent together with commercially available Corey-Bakshi-Shibata oxazaborolidine catalyst ((S)-CBS, Scheme 14). [START_REF] Dorizon | A Practical Kinetic Resolution of 4-Acetyl[2.2]Paracyclophane[END_REF]

SCHEME 14. Kinetic resolution of 4-acetyl[2.2]paracyclophane

Very recently, 4-acetyl2.2]paracyclophane (±)-27 has been efficiently resolved on a synthetically useful scale by asymmetric hydrogenation in the presence of a ruthenium-based catalyst ((S)-RUCY ® -XylBINAP). High hydrogen pressures are, however, required for the reaction to work. [START_REF] Zippel | Multigram-scale Kinetic Resolution of 4 -Acetyl[2.2Paracyclophane via Ru-Catalyzed Enantioselective Hydrogenation: Accessing [2.2Paracyclophanes with Planar and Central Chirality[END_REF] α,β-Unsaturated ketones can be submitted to kinetic resolution as well. [START_REF] Akagawa | Kinetic Resolution of a Planar-Chiral [2.2]Paracyclophane Derivative by Helical-Peptide-Catalyzed Michael Addition of Nitromethane[END_REF] Starting from racemic pCp (±)-29, Michael addition of nitromethane promoted by a chiral helical peptide affords nitromethylated product (Rp,S)-30 in 40% yield and 90% ee after 6 d at 30 °C. Unreacted compound (Sp)-29 is recovered as well after the reaction in 53% yield and 63% ee (Scheme 15).

Imines and N-sulfonylimines

Very recently, palladium-catalysed Suzuki couplings have been successfully exploited to perform the kinetic resolution of imines [START_REF] Zhao | Synthesis of Paracyclophanes with Planar and Central Chirality: Kinetic Resolution of [2.2]Paracyclophane Aldimines via Palladium-Catalyzed Addition of Arylboronic Acids[END_REF] and cyclic N-sulfonylimines. [START_REF] Zhao | Kinetic Resolution of [2.2]Paracyclophane-Derived Cyclic N-Sulfonylimines via Palladium-Catalyzed Addition of Arylboronic Acids[END_REF] Following these procedures, the racemic pCp-based derivatives (±)-31 and (±)-33 are submitted to a reaction with various aryl-boronic acids, Pd(TFA)2, and a planar chiral ferrocene-derived phosphinooxazoline ligand. At approximatively 50% conversion, one of the enantiomers is recovered as unreacted starting material in up to 99% ee, while the other enantiomer is isolated as an arylated product in high dr and er (up to 20:1, and 99:1 respectively).

SCHEME 16. Examples of kinetic resolution of pCp-based imines and cyclic

N-sulfonylimines

Amines

The synthesis of palladacycles has elegantly been applied in paracyclophane chemistry to realize the kinetic resolution of N,N-dimethylaminomethyl [2.2]paracyclophane. [START_REF] Dendele | Synthesis of a [2.2]Paracyclophane Based Planar Chiral Palladacycle by a Highly Selective Kinetic Resolution/C-H Activation Reaction[END_REF][START_REF] Dunina | A similar approach has been followed to perform the classical resolution of pCp-based palladacycles[END_REF]

SCHEME 17. Kinetic resolution of N,N-dimethylaminomethyl[2.2]paracyclophane

When reacted with (R)-N-acetylphenyalanine and Pd(II) salts, the racemic substrate (±)-35 affords the metalated product (Sp)-36 in 34% yield and 90% ee, together with unreacted compound (Sp)-35, which can be obtained in 41% yield and 98% ee (Scheme 17). While amine (Sp)-35 can serve as a precursor of more complex pCp derivatives, the enantioenriched pCp-based palladacycle may be useful as catalyst to promote asymmetric transformations.

Esters

Enzymes have frequently been exploited by organic chemists as biocatalysts to resolve synthetically useful compounds since they usually induce high levels of enantioselection and are environmentally friendly. Together with oxidoreductases, lipases are possibly the most frequently used enzymes in asymmetric catalysis. Among many other substrates, these biomolecules can catalyse the kinetic resolution of 4-acetoxy[2.2] paracyclophane (±)-37. [78][START_REF] Pamperin | Chemoenzymatic Synthesis of Optically Pure Planar Chiral (S)-(-)-5-Formyl-4-hydroxy[2.2]Paracyclophane[END_REF][80] In the presence of commercially available Candida rugosa lipase (CRL) optically active 4hydroxy[2.2]paracyclophane (Rp)-16 has indeed been obtained in 47% yield and 96% ee through ester hydrolysis. Unreacted derivative (Sp)-37 has been recovered as well in high yield and selectivity (Scheme 18). [START_REF] Cipiciani | Influence of PH and Temperature on the Enantioselectivity of Propan-2-Ol-Treated Candida Rugosa Lipase in the Kinetic Resolution of (±)-4-Acetoxy-[2,2]-Paracyclophane[END_REF] The same enzyme can be employed to access enantiopure 4,12-dihydroxy[2.2]paracyclophane through similar biotransformations. 82

SCHEME 18. Enzymatic kinetic resolution of 4-hydroxy[2.2]paracyclophane

DESYMMETRIZATIONS

As a logical consequence of separating racemic mixtures, one of the main limitations of all resolution methods, unrelated to how efficient they are, is their maximum theoretical yield of 50%. The desymmetrization of meso compounds, on the other hand, can theoretically afford enantiopure products in up to 100% yield. This technique, which relies on the differentiation of the prochiral sites of achiral molecules, has widely been exploited in stereoselective synthesis, but rarely applied to [2.2]paracyclophanes.

Aldehyde derivatives

The desymmetrization of achiral meso compounds derived from pCp has been described recently. Noyori asymmetric transfer hydrogenations under the conditions previously employed to perform the kinetic resolution of racemic 4formyl[2.2]paracyclophane can afford enantioenriched products in high yields and selectivities starting from centrosymmetric dialdehydes. In fact, when using t-BuOK as a base and RuCl(pcymene)[(R,R)-Ts-DPEN] as the catalyst in a 1:1 i-PrOH/DMF mixture at 0 °C, 4,16-diformyl[2.2]paracyclo-phane is reduced smoothly to afford the product (Rp)-39 in 74% yield and >99% ee (Scheme 19). Interestingly, only small amounts of diol 40 have been isolated after the reaction. [START_REF] Delcourt | Highly Enantioselective Desymmetrization of Centrosymmetric Pseudo-Para-Diformyl[2.2Paracyclophane via Asymmetric Transfer Hydrogenation[END_REF] Bromine atoms can be introduced para to the aldehyde groups to obtain optically active tetra-substituted compounds in a rapid and efficient fashion following this convenient strategy. [START_REF] Delcourt | Highly Enantioselective Asymmetric Transfer Hydrogenation: A Practical and Scalable Method To Efficiently Access Planar Chiral [2.2Paracyclophanes[END_REF]

SCHEME 19. Desymmetrization of 4,16-diformyl[2.2]paracyclophane

Applications

As mentioned above, the chirality of [2.2]paracyclophanes was reported by Cram and Allinger as early as 1955. [START_REF] Cram | Macro Rings. XII. Stereochemical Consequences of Steric Compression in the Smallest Paracyclophane1[END_REF] At first, nearly all stereochemichal studies involving planar chiral pCps focused on the synthesis of mono-substituted derivatives and the determination of their absolute configuration. Because of their advantageous properties, such as a high configurational stability (200 °C), good tolerance to the action of acids and bases, chemical stability towards light or oxidants, and ease of handling, from the early 1990s onward, chiral paracyclophanes have gradually emerged as powerful ligands or catalysts in asymmetric synthesis, and several poly-substituted compounds have been prepared. Nowadays, enantiopure paracyclophanes are increasingly employed as precursors for the development of materials with original chiroptical behaviours. The synthesis of planar chiral pCp ligands and their use in enantioselective metal-catalysed reactions have been extensively reviewed. [START_REF] Gibson | 2.2]Paracyclophane derivatives in asymmetric catalysis Org[END_REF][START_REF] Hassan | Planar chiral [2.2]paracyclophanes: from synthetic curiosity to applications in asymmetric synthesis and materials[END_REF][START_REF] Rowlands | Planar Chiral Phosphines Derived from [2.2]Paracyclophane[END_REF][START_REF] Schneider | 2.2]Paracyclophane-Derived Planar-Chiral Hydrogen-Bond Receptors[END_REF][START_REF] Paradies | Paracyclophane Derivatives: Synthesis and Application in Catalysis[END_REF][22] Reviews on the preparation and properties of optically active materials derived from [2.2]paracyclophane have also appeared in recent years. [START_REF] Morisaki | Planar Chiral [2.2]Paracyclophanes: Optical Resolution and Transformation to Optically Active π-Stacked Molecules[END_REF][START_REF] Weiland | Beyond Simple Substitution Patterns -Symmetrically Tetrasubstituted [2.2]Paracyclophanes as 3D Functional Materials[END_REF] In the next sections, we will therefore mainly focus our attention on the recent applications of pCps in the field of asymmetric organocatalysis. [START_REF]The applications of enantiopure planar chiral pCps in asymmetric organocatalysis have been only partially reviewed[END_REF]

PLANAR-CHIRAL PCP-BASED ORGANOCATALYSTS

A variety of planar-chiral organocatalysts derived from [2.2]paracyclophane have been synthesized over the last three decades. These compounds have been successfully employed to promote different types of asymmetric transformations with various degrees of selectivity. In the following part of this review, the pCp-based organocatalysts will be subdivided into several classes according to the nature of their reactive functions. The synthesis of these compounds will be described below, together with representative examples of their use in various metal-free reactions.

Phosphines

PhanePhos is without doubt the most well-known C2symmetrical diphosphine derived from [2.2]paracyclophane. First synthesized in an industrial laboratory in 1997, [START_REF] Pye | A New Planar Chiral Bisphosphine Ligand for Asymmetric Catalysis: Highly Enantioselective Hydrogenations under Mild Conditions[END_REF] this compound is nowadays commercially available through major chemical suppliers. PhanePhos can be prepared starting from enantiopure phosphine oxide (Rp)-22 through a reduction with trichlorosilane (Scheme 20). In addition to their role as ligands in metal catalysis, [START_REF] Rowlands | Planar Chiral Phosphines Derived from [2.2]Paracyclophane[END_REF] enantiopure PhanePhos and its derivatives have been employed as stoichiometric reductants to promote the kinetic resolution of hydroperoxides. In particular, the sterically hindered compound (Rp)-43 provides high enantioselectivities in this transformation (up to 99% ee, Scheme 21). 85 SCHEME 20. Synthetic pathway for the preparation of PhanePhos and its derivatives SCHEME 21. Kinetic resolution of hydroperoxides pCp-Based monophosphines obtained starting from enantiopure mono-substituted phosphine oxides proved to be effective organocatalysts in asymmetric allylic substitution reactions. The direct functionalization of Morita-Baylis-Hillman products have indeed been achieved with good degrees of enantioselectivity (up to 71% ee) in the presence of pCps derivatives carrying aliphatic substituents on their phosphorous atom (e.g. (Rp)-46, Scheme 22). 86

SCHEME 22. Enantioselective allylic substitution reactions

In addition to PhanePhos and its derivatives, planar chiral pCpbased bifunctional phosphine-phenol compounds have been prepared and used as catalysts in metal-free reactions (Scheme 23). 87-90

SCHEME 23. Synthesis of planar chiral pCp-based bifunctional phosphinephenol catalysts

To access these compounds, 4-bromo-12hydroxy[2.2]paracyclophane (Sp)-47 is first isolated through a mono metalation of enantiomerically pure 4,12-dibromo[2.2]paracyclophane (Sp)-21, followed by a reaction with trimethylborate, and oxidation of the resulting product with hydrogen peroxide and sodium hydroxide. The obtained key intermediate (Sp)-47 is then engaged into a Suzuki-Miyaura coupling in the presence of phosphinylboronic esters. Subsequent reduction of the phosphine oxide group finally leads to the expected products (Sp)-49 in good overall yields (Scheme 23). Different bifunctional phosphine-phenol catalysts can also be prepared by submitting 4-bromo-12hydroxy[2.2]paracyclophane (Sp)-47 to triflation, followed by stepwise successive Pd-catalysed cross-coupling reactions. [START_REF] Kitagaki | Synthesis of planar chiral pseudo-ortho-substituted aryl[2.2]paracyclophanes by stepwise successive palladiumcatalyzed coupling reactions[END_REF] Bifunctional phosphine-phenol pCp derivatives such as compound (Sp)-49 have been employed to catalyse asymmetric aza-Morita-Baylis-Hillman reactions of aldimines with vinyl ketones (Scheme 24). [START_REF] Kitagaki | Planar chiral [2.2]paracyclophane-based phosphine-Brønsted acid catalysts bearing exceptionally high reactivity for aza-Morita-Baylis-Hillman reaction[END_REF][START_REF] Takenaga | Planar chiral [2.2]paracyclophane-based phosphine-phenol catalysts: application to the aza-Morita-Baylis-Hillman reaction of Nsulfonated imines with various vinyl ketones[END_REF] The same catalysts can also be used to promote enantioselective [3+2] annulations of unsubstituted allenic esters and N-tosylimines (Scheme 24). 87 In both transformations, bifunctional pCps provide better yields and enantioselectivities in comparison with PhanePhos or other planar chiral ligands.

SCHEME 24. Asymmetric transformations promoted by bifunctional phosphine-phenol pCp catalysts

Phosphoric acids

Only few examples of planar chiral phosphoric acids derived from [2.2]paracyclophane have been reported in the literature to date.

Starting from enantiopure 4-bromo-12hydroxy[2.2]paracyclophane (Sp)-47, a first pCp-based phosphoric acid has been prepared in 2012 through a Suzuki coupling, and consequent reaction with phosphoryl chloride (Scheme 25). [START_REF] Enders | Synthesis and Application of the First Planar Chiral Strong Brønsted Acid Organocatalysts[END_REF]

SCHEME 25. First synthesis of a pCp-based phosphoric acid

More recently, a different family of phosphoric acid derivatives showing a pCp core has been obtained in four steps using enantiopure 4,12-dibromo-[2,2]-paracyclophane (Rp)-21 as a key precursor. [START_REF] Xie | Design of Planar Chiral Phosphoric Acids with a [2.2]Paracyclophanyl Backbone as Organocatalysts for the Highly Enantioselective Aza-Friedel-Crafts Reaction[END_REF] The synthetic procedure employed to access these compounds starts with a double Suzuki coupling which occurs in the presence of arylboron reagents featuring a Me-protected hydroxyl function at their meta position, and different substituents in para position. Demethylation with BBr3 affords [2.2]paracyclophane-based bisphenol intermediates (Rp)-55, which can be converted to the corresponding phosphoric acids through a cyclization with bis(diisopropylamino)(2cyanoethoxy)phosphine, and subsequent oxidation of the resulting phosphites with tert-butyl hydroperoxide. Removal of the cyanoethyl substituent of phosphates (Rp)-56 under basic conditions finally gives access to the desired products (Rp)-57 (Scheme 26).

SCHEME 26. Access to a different family of pCp-based phosphoric acid derivatives

The pCp-based planar chiral phosphoric acids have been employed to promote the synthesis of enantiopure 3indolylmethanamine derivatives through asymmetric aza-Friedel-Crafts reactions (Scheme 27). [START_REF] Enders | Synthesis and Application of the First Planar Chiral Strong Brønsted Acid Organocatalysts[END_REF][START_REF] Xie | Design of Planar Chiral Phosphoric Acids with a [2.2]Paracyclophanyl Backbone as Organocatalysts for the Highly Enantioselective Aza-Friedel-Crafts Reaction[END_REF] SCHEME 27. Asymmetric aza-Friedel-Crafts reaction High selectivities (up to 99% ee) have been achieved in the presence of catalysts (Rp)-57 (Scheme 27). These compounds show superior catalytic activities compared to other planar chiral Brønsted acids, including derivative (Sp)-54.

Thioureas

A few planar-chiral hydrogen-bond catalysts have been described in recent years. The great majority of these compounds feature either one or two thiourea groups onto the aromatic rings of [2.2]paracyclophane. Different mono-thiourea derivatives have been synthesized starting from enantiopure precursors. Unsubstituted catalysts such as (Sp)-62 are, for example, easily prepared through the reaction of enantiopure 4-amino[2.2]paracyclophane (Sp)-10 with activated isothiocyanates (Scheme 28). 93 SCHEME 28. Synthesis of unsubstituted thiourea catalysts 4-Bromo-12-hydroxy[2.2]paracyclophane (Sp)-47 can be employed as starting material to access pseudo-ortho hydroxysubstituted catalysts (Scheme 29). [START_REF] Schneider | Planar-Chiral Thioureas as Hydrogen-Bond Catalysts[END_REF] In this case, the hydroxy group of the molecule first needs to be protected as MEM ether. A halogen-lithium exchange can then occur, and the resulting metalated pCp derivative can undergo a reaction with TsN3 to form an azido-substituted product. Reduction of this compound to the corresponding amine, deprotection of the hydroxy group, and subsequent reaction with 3,5-bis(trifluoromethyl)phenyl isothiocyanate finally afford the expected thiourea (Sp)-65 in 24% overall yield (Scheme 29).

SCHEME 29. Synthesis of pseudo-ortho di-substituted thiourea catalysts

Pseudo-gem hydroxy-substituted catalysts can be prepared through the derivatization of enantiopure 4carboxy[2.2]paracyclophane (Sp)-2. [START_REF] Schneider | Planar-Chiral Thioureas as Hydrogen-Bond Catalysts[END_REF] Following the synthetic pathway reported in Scheme 30, this precursor is first converted to the corresponding methyl ester (Sp)-66 through the formation of an acyl chloride, and consequent reaction with methanol. A regioselective electrophilic aromatic substitution is then performed to functionalize the pCp framework with a bromine atom at the pseudo-gem position. Saponification of the resulting bromo ester affords a carboxylic acid ((Rp)-67) that can be submitted to Curtius rearrangement. Boc-protected amine derivative (Rp)-68 is obtained by trapping the thermally generated isocyanate with tert-butyl alcohol. Metalation of this compound, followed by addition of trimethylborate, oxidative work-up, and removal of the protecting group with TFA leads to the formation of a substituted amino-alcohol ((Rp)-69), which can react smoothly with isothiocyanates to afford various thiourea derivatives, including product (Rp)-70 (Scheme 30). SCHEME 30. Synthesis of pseudo-gem di-substituted thiourea catalysts Chiral amino thioureas have been synthesized starting from 4bromo-12-amino[2.2]paracyclophane (Rp)-71. [START_REF] Lu Y | Synthesis of planar chiral [2.2]paracyclophane-based thioureas and their application in asymmetric aldol reactions of ketones with isatins[END_REF] The synthetic leading to these compounds involve the introduction of both a benzhydrylideneamine moiety (S)-N-Boc-prolyl group onto the molecules through coupling reactions.

Synthesis of a planar chiral amino thiourea derivative

After the deprotection of the benzhydrylideneamino group under basic conditions, the thiourea motif is again introduced via a reaction between the free amino group and an activated isothiocyanate. Boc-deprotection with TFA finally generates the expected bifunctional product (Rp,S)-74 in 80% yield (Scheme 31). In 2013, a bis-thiourea pCp organocatalyst showing a C2 symmetry has been synthesized starting from optically pure bromo [2.2]paracyclophanyl triflate Rp-75 (Scheme 32). [START_REF] Kitagaki | Planar chiral [2.2]paracyclophane-based bis(thiourea) catalyst: application to asymmetric Henry reaction[END_REF] This precursor has successfully been submitted to a double Buchwald-Hartwig coupling using N-tert-butyl carbamate as an ammonia equivalent in the presence of a palladium-based catalyst and JohnPhos. Subsequent removal of the Bocprotecting groups with TFA leads to the diamine key intermediate (Rp)-77, which easily reacts with 3,5bis(trifluoromethyl)phenyl isothiocyanate to afford the desired bis-thiourea derivative (Rp)-78 in 74% yield (Scheme 32). SCHEME 32. Synthesis of a bis-thiourea pCp-based organocatalyst Planar-chiral hydrogen-bond catalysts incorporating a single thiourea moiety have been employed to promote enantioselective Friedel-Crafts alkylations [START_REF] Schneider | Planar-Chiral Thioureas as Hydrogen-Bond Catalysts[END_REF] and asymmetric transfer hydrogenations, [START_REF] Schneider | Planar-Chiral Thioureas as Hydrogen-Bond Catalysts[END_REF] as well as asymmetric aldol condensations of ketones with isatins. 94 SCHEME 33. Asymmetric Friedel-Crafts alkylations, organocatalytic transfer hydrogenations, and aldol condensations In most cases, however, moderate yields and/or enantioselectivities have been obtained for these reactions (Scheme 33). The pCp-based pseudo-ortho bis-thiourea derivative (Rp)-78, on the contrary, has proven to be a powerful and effective catalyst for enantioselective Henry reactions. [START_REF] Kitagaki | Planar chiral [2.2]paracyclophane-based bis(thiourea) catalyst: application to asymmetric Henry reaction[END_REF] In the presence of this compound, several arylaldehydes have indeed been converted to the corresponding nitroalcohols in good yields and high enantiomeric excesses (up to 97% ee, Scheme 34).

SCHEME 34. Examples of asymmetric Henry reactions

Diols and silanols

Aside from functionalized thiourea derivatives, different planar chiral H-bonding organocatalysts, which incorporate diol or bissilanol moieties, have been developed taking advantage of the rigid conformation of the [2.2]paracyclophane scaffold. [START_REF] Beemelmanns | Planar-Chiral Bis-silanols and Diols as H-Bonding Asymmetric Organocatalysts[END_REF] These compounds have been prepared starting from enantiopure 4,12dibromo-[2.2]-paracyclophane through a double metal-halogen exchange, followed by a reaction with either monochlorosilanes or methyl carbonate (Scheme 35). Different oxidation conditions have then been employed to convert the sililated intermediates (Sp)-87 to the corresponding bis-silanols (Sp)-89. At the same time, the reduction of ester derivative (Sp)-88 with various Grignard reagents has been carried out to access the desired diol products (Sp)-90 (Scheme 35).

SCHEME 35. Synthesis of planar-chiral bis-silanols and diols

The catalytic activity of bis-silanols and diols has been tested in the hetero-Diels-Alder reaction of Rawal's diene with aliphatic or aromatic aldehydes. [START_REF] Beemelmanns | Planar-Chiral Bis-silanols and Diols as H-Bonding Asymmetric Organocatalysts[END_REF] Both types of catalysts can promote the asymmetric transformation, even if, in most cases, moderate enantioselectivities have been observed (44% ee at most, Scheme 36). 2]paracyclophane and its derivatives have been employed to promote asymmetric Diels-Alder reactions, [START_REF] Braddock | Planar Chiral PHANOLs as Organocatalysts for the Diels-Alder Reaction via Double Hydrogen-Bonding to a Carbonyl Group[END_REF][START_REF] Braddock | Planar Chiral PHANOLs as Double Hydrogen Bonding Donor Organocatalysts: Synthesis and Catalysis[END_REF] as well as the organocatalytic ring-opening of epoxides. [START_REF] Braddock | Planar Chiral PHANOLs as Double Hydrogen Bonding Donor Organocatalysts: Synthesis and Catalysis[END_REF] However, in these cases, racemic products or low enantiomeric excesses (<5% ee) have been obtained.

N-Heterocyclic Carbenes (NHCs)

Planar chiral bicyclic 1,2,4-triazolium salts derived from [2.2]paracyclophane have been prepared and successfully applied in asymmetric organocatalysis. [START_REF] Zhao | Asymmetric β-Boration of ,β-Unsaturated N-Acyloxazolidinones by [2.2]Paracyclophane-Based Bifunctional Catalyst[END_REF][100][101] The synthesis of these compounds starts from enantiopure 4amino(2.2]paracyclophane (Scheme 37). The aniline derivative (Sp)-10 is first converted to 4-iodo[2.2]paracyclophane (Sp)-93 through the formation of a diazonium salt, and subsequent reaction with KI. A palladium-catalysed cross-coupling reaction then allows the preparation of the pCp-based benzophenone hydrazine (Sp)-94 which can react with acetic anhydride and formic acid to generate the formylated product (Sp)-95 in 68% yield (Scheme 37). Acid hydrolysis of this compound affords a formyl-protected hydrazine hydrochloride salt that can undergo a reaction with various imidate derivatives to generate different 1,2,4-triazolium salts (Sp,S)-97 in high overall yields (Scheme 37). SCHEME 37. Synthesis of 1,2,4-triazolium salts derived from [2.2]paracyclophane While products (Sp,S)-97 have frequently been used as planar chiral NHC precursors in asymmetric transition metal catalysis, [START_REF] Gibson | 2.2]Paracyclophane derivatives in asymmetric catalysis Org[END_REF][START_REF] Paradies | Paracyclophane Derivatives: Synthesis and Application in Catalysis[END_REF] only few examples of organocatalyzed transformations promoted by such compounds have been reported to date. These include the enantioselective β-boration of acyclic enones 98 [START_REF] Zhao | Asymmetric β-Boration of ,β-Unsaturated N-Acyloxazolidinones by [2.2]Paracyclophane-Based Bifunctional Catalyst[END_REF]100 as well as the asymmetric 1,2-silylation of aromatic aldehydes 100 (Scheme 38). 101 Both reactions proceed smoothly in the presence of strong bases to afford the expected products in moderate to good yields and up to 77-80% ees.

SCHEME 38. Examples of asymmetric organocatalyzed transformations promoted by pCp-based NHC precursors

Pyridine N-Oxides

Although pyridine N-oxides have more often been employed as ligands for metal-based catalysis 102 or precursors for the development of photoelectronic dyes 103 and metal-organic frameworks, 104 these compounds have also revealed to be effective Lewis base organocatalysts in various transformations. 105 Planar chiral pyridine N-oxides derived from [2.2]paracyclophane have been synthesized through the direct arylation of enantiopure 4,12-dibromo[2.2]paracyclophane (Rp)-21 (Scheme 39). This compound can indeed undergo a reaction with differently substituted pyridine N-oxides, in the presence of tri-tert-butylphosphonium tetrafluoroborate, palladium(II) acetate, and potassium carbonate, to afford mono-and di-substituted products (Rp)-102 and (Rp)-103 (Scheme 39). 106 SCHEME 39. Synthesis of pCp-based pyridine N-oxide organocalaysts pCp-Based pyridine N-oxides (Rp)-103 been employed to promote the Lewis base-mediated allylation of with allyltrichlorosilane (Scheme 40). 106 Even if the organocatalysts a good activity, only modest enantioselectivities have been observed after the reactions (~28-38% ees). Interestingly, electronic factors appear to play a crucial role in these transformations: unsubstituted and electronrich pyridine N-oxides furnish products with opposite absolute configuration, and comparable enantiomeric excess (Scheme 40).

Iodoarenes

Recently, planar chiral iodoarenes derived from [2.2]paracyclophane have been successfully applied in hypervalent iodine catalysis. 108 The preparation of these derivatives, which are more frequently employed as key synthetic intermediates rather than as iodine-based organocatalysts, involves the derivatization of racemic 4hydroxy[2.2]paracyclophane (±)-16 (Scheme 43). After the protection of the free hydroxy group of this molecule as a MOM ether, metalation, followed by quenching with I2 leads to the introduction of an iodine atom ortho to the oxygenated function. Removal of the protecting group under acidic conditions, and optical resolution by HPLC on a chiral stationary phase, afford enantiopure intermediates (Rp)-111 or (Sp)-111 that can undergo alkylation or acylation reactions to generate differently functionalized products. Selected examples of these compounds ((Rp)-112 or (Sp)-112) are reported in Scheme 43. The enantiopure pCp-based iodoarenes can be used to promote the asymmetric fluorination of β-ketoesters (±)-113 with HF and m-CPBA. After a screening of paracyclophanes incorporating diverse functional groups, benzoylated derivatives have been identified as the best catalysts for this transformation (Scheme 44). 108 The nucleophilic fluorination tolerates cyclic β-ketoesters with both electron-donating and electron-withdrawing substituents. The reaction products have been isolated in up to 89% ee while using sterically hindered pCp catalysts. Acyclic compounds do not react, thus representing the limitation of this interesting approach.

SCHEME 44. Examples of nucleophilic fluorination reactions promoted by pCp-based iodoarenes

Conclusion

In the first part of this review, we have described the different approaches that can be followed to enantiopure [2.2]paracyclophanes. Although classical resolution methods relying on the preparation and separation of diastereoisomeric compounds or HPLC separations still dominate this research field, practical examples of kinetic resolutions and desymmetrization reactions have started to emerge in recent years. We expect these transformations, which constitute valuable alternatives to previously described time-consuming and costly processes, to be increasingly employed as key synthetic procedures in paracyclophane chemistry in the near future. Enantiopure [2.2]paracyclophanes have been extensively employed as building blocks for the development of planar chiral compounds with interesting physicochemical properties. In the second part of this review, we have outlined their use as catalysts in several metal-free asymmetric reactions. Considering their broad range of applications, we are firmly convinced that planar-chiral [2.2]paracyclophanes will continue to attract attention and grow in importance in the years to come.

FIGURE 1 .

 1 FIGURE 1. Structure of unsubstituted [2.2]paracyclophane

FIGURE 3 .

 3 FIGURE 3. Description of the absolute configuration of pCps

SCHEME 3 .

 3 SCHEME 3. Resolution of 4,12-dicarboxy[2.2]paracyclophane (±)-5

SCHEME 11 .

 11 SCHEME 11. Kinetic resolution of 4,12-dibromo[2.2]paracyclophane via Buchwald-Hartwig coupling

67

 67

SCHEME 12 .

 12 SCHEME 12. Kinetic resolution of 4-bromo[2.2]paracyclophane via Buchwald-Hartwig coupling

SCHEME 15 .

 15 SCHEME 15. Kinetic resolution of pCp-based α,β-unsaturated methyl ketone

SCHEME 36 .

 36 SCHEME 36. Hetero-Diels-Alder reactions promoted by pCp-based bissilanols and diols Enantiopure 4,12-dihydroxy[2.2]paracyclophane and its derivatives have been employed to promote asymmetric Diels-Alder reactions,[START_REF] Braddock | Planar Chiral PHANOLs as Organocatalysts for the Diels-Alder Reaction via Double Hydrogen-Bonding to a Carbonyl Group[END_REF][START_REF] Braddock | Planar Chiral PHANOLs as Double Hydrogen Bonding Donor Organocatalysts: Synthesis and Catalysis[END_REF] as well as the organocatalytic ring-opening of epoxides.[START_REF] Braddock | Planar Chiral PHANOLs as Double Hydrogen Bonding Donor Organocatalysts: Synthesis and Catalysis[END_REF] However, in these cases, racemic products or low enantiomeric excesses (<5% ee) have been obtained.

SCHEME 40 .SCHEME 41 .SCHEME 42 .

 404142 SCHEME 40. Examples of Lewis base-mediated asymmetric reactions promoted by pCp-based pyridine N-oxides

SCHEME 43 .

 43 SCHEME 43. Synthesis of pCp-based iodoarenes.

38 SCHEME 1.

	Synthesis	and	classical	resolution	of	4-
	carboxy[2.2]paracyclophane

Planar chiral[2.2]paracyclophanes (pCps) constitute a fascinating class of molecules which possesses an unusual threedimensional core and intriguing physicochemical properties. This short review focuses on the synthesis of enantiopure pCps and their application in asymmetric organocatalysis.

Acknowledgements

We gratefully thank the CNRS, Université de Paris (ANR-18-IDEX-0001) and the Ministère de l'Enseignement Supérieur et de la Recherche for financial support.


domain: spi phys

text:
Husson Xavier
email: xavier.husson@onera.fr

Nicolas Trouve

Boust Fabrice

Modeling Windblown Forest Clutter For SAR simulations

In this paper we present a complete radar modeling pipeline for windblown forest clutter. Because the forest is a set of trees, as a first approach, we model an isolated tree. For validation purposes our pipeline is adapted to SAR simulations. For this model the emphasis was put on 3 characteristics : tree animation due to wind, usage of frequencies higher than the S-band and possibility to be representative of the environment. To evaluate the backscattered field an asymptotic EM method was used combining geometrical and physical optics.

Introduction

SAR imaging allows the imaging of the Earth's surface with nearly no disadvantage due to meteorological conditions. It is used in many domains such as measuring melting ice speed, forecasting earthquakes... Nevertheless, it is a method that needs a lot of means (airborne systems, antennas, operators). Another limitation is that it can be difficult to acquire specific areas with these airborne systems. Therefore simulations are used to evaluate the robustness and the precision of processing algorithm or to increase databases size. In order to be able to realize a representative simulation, one should be able to model each components of the scene, either targets (cars, tanks, boats...) or clutters (sea, forest). One of the characteristics of forest clutter is the Doppler shift induced by the wind. This aspect compromises a lot of processing algorithms. This is the focus of this paper.

Since the sixties and the emergence of FOPEN [START_REF] Edward | Foliage penetration radar : detection and characterization of objects under trees[END_REF] (Foliage Penetration) many vegetation models were developed [START_REF] Eom | A scatter model for vegetation up to Ku-band[END_REF] but there seems to be no models adapted to both animation and SAR imaging. For frequencies higher than the S-band (C, X, Ku...), the canopy cannot be considered transparent anymore. In that regard it is needed to model trees as a whole. Different approaches exist in the literature to design trees as presented in the survey from Zhang and Pang [START_REF] Zhang | A survey of modeling and rendering trees[END_REF]. In order to generate typical trees fitted for animation and with a fine level of precision, we decided to use a 3D surfacic mesh representation [START_REF] Idv | Speedtree[END_REF]. To render tree animation and evaluate the backscattered field, a hybrid asymptotic approach is chosen based on the geometrical and physical optics approximations.

First we present the mean used to generate 3D surfacic tree mesh. Then we present our asymptotic approach. A focus is then made on tree animation time dependency. All these aspects allow us to compose our modeling pipeline for simulation purposes. Lastly we present simulation results in the form of SAR images.

3D surfacic mesh modeling

To model our trees and to be able to generate typical trees (in opposition to a specific tree) we decided to use a tree modeling software : SpeedTree [START_REF] Idv | Speedtree[END_REF]. This software allows us to generate a 3D surfacic mesh representation for each modeled tree. Figure 1 shows an example of a tree generated by SpeedTree. SpeedTree major advantages for our model are as follow. Because SpeedTree modeling is based on a hierarchical construction concept, it allows a procedural approach and therefore a large diversity of trees modeling possibilities (broad leaf, conifer, palm tree...). Besides, the use of randomization process allows to easily generate multiple trees of a same type. In addition, SpeedTree includes a wind-based animation model.

High frequency backscattering

Asymptotic method

In our approach we uncovered 2 levels of complexity due to the usage of a 3D mesh-based representation : one on a global scale and one on a local scale. For the global parameter we looked at 2 aspects of the representation : spatial and time. Spatially, the dimensions of the tree, as a whole, are largely greater than the wavelength (3cm for the X-band for example). For the time-based aspect the problem is as follow, the more our mesh is animated the more meshes have to be generated. This can be considered as another meshed object being added. In that regard, one of the constraint of our model is the use of a fast numerical method. In addition, in our 3D surfacic mesh, the smaller elements are either leaves or branches on the rim. Their characteristic dimensions are greater than the wavelength for the X-band or higher frequencies band. Therefore it is possible to use an asymptotic method. For all these reasons, we chose a numerical EM asymptotic method able to render the animation composed of 2 steps : a visibility step based on geometrical optics and a scattering calculation based on physical optics.

Backscattering

From the well-known Maxwell equations, one can reformulate the scattering problem by using the Stratton-Chu equations. Under the physical optics approximations these equations can be further simplified to obtain for the electric field E Equation (1) (similar results can be obtain for the magnetic field H)

- → E s ( - → r ) = e -jkR 0 4πR 0 -→ α E R, -→ E 0 S e -j( k i 0 -k s 0 ). r dr (1)
with -→ E s the scattered electric field, -→ E 0 the incident wave polarization, R the Fresnel reflectivity coefficients matrix, k i 0 and k s 0 the incident and scattered wave vectors, α E a function of 2 variables. This equation can be split into 3 factors. The first factor can be associated to a spherical wave. The second factor α E is material and polarization dependent. The third factor, the double integral, is geometry dependent.

For the second factor, the polarization parameter can be determine using acquisition parameters. The reflectivity ma-trix is linked to the Fresnel's coefficients. These coefficients can be determine using the vegetation permittivity. In our model we used an empirical permittivity model based on Debye-Cole dual-dispersion model [START_REF] Ulaby | Microwave Dielectric Spectrum of Vegetation[END_REF] from the literature [START_REF]Electrical characteristics of the surface of the Earth[END_REF]

ε v = ε dv + v f w ε f w + v bw ε bw (2)
with ε dv the real part of the relative permittivity of bulk vegetation, v bw and v f w are the bound and free water volume fraction, ε bw , and ε f w correspond to bound and free water permittivities [START_REF]Electrical characteristics of the surface of the Earth[END_REF].

To evaluate the double integral, a method was proposed in the case of a closed polygonal surface by Gordon [START_REF] Gordon | Far-Field Approximations to the Kirchhoff-Helmholtz Representations of Scattered Fields[END_REF]. This approach was later reformulated with a more general expression by McInturff and Simon [START_REF] Mcinturff | The Fourier Transform of Linearly Varying Functions with Polygonal Support[END_REF]. In our study, we are working with a surfacic 3D meshed object, thus the use of such a method is suitable.

Time dependency

The two main time constants of SAR acquisitions (or simulations) are the pulsed duration pD and the Pulse Repetition Interval (PRI or its frequency equivalent PRF). In case of SAR acquisition, objects are usually considered static in regards to the pulse duration.

In our model, to render the Doppler Shift induced by the wind, our trees should be animated at the same frequency as the PRF. For trees, experimental campaigns measured natural frequencies for trees movements between 0.1 and 10 Hertz [START_REF] Baker | Measurements of the Natural Frequencies of Trees[END_REF] [10] (depending on the tree characteristics and the measured part position). While comparing tree natural frequencies and the PRF, one cannot consider trees static between pulses but it should also be noted that while it should have an impact on the Doppler shift, the visibility between consecutive pulses should not change drastically.

In that regard, when no analytic formulation exists to calculate the positions of the mesh nodes at the PRF, we decided to interpolate these position in-between the animation frames. As mentioned in section 2, we used the SpeedTree animation model [START_REF] Idv | Speedtree[END_REF]. Because both the positions and the speed of the mesh nodes are known, a second order interpolation was implemented.

Modeling pipeline

We present here the main steps of our modeling pipeline.

Figure 2 provides an illustration. First a 3D surfacic tree mesh is generated with the SpeedTree software [START_REF] Idv | Speedtree[END_REF]. Second the tree mesh is animated to render wind-related movements using the software Blender [START_REF] Community | Blender, Stichting Blender Foundation[END_REF].

Because we decided to use a hybrid electromagnetic asymptotic approach the 2 next steps focus on geometrical and physical optics. Third is indeed the use of ray-tracing to extract the visible contributors of our mesh. The POV-Ray [START_REF]Persistence of Vision Raytracer[END_REF] software was used because we had prior knowledge of this software. Fourth consists in evaluating the scattered field based on the physical optics approximations.

Because we want a model adapted to SAR simulations, two more steps are added. Fifth a SAR airborne system is simulated in order to emulate an acquisition to generate raw SAR data. Sixth the raw data are compressed in azimuth and distance to form a SAR image.

Results

Figure 3 shows a static SAR image obtained with our pipeline. Here the term static means that our tree is not under wind constraint and therefore is motionless. Indeed to simulate a SAR acquisition our fictional carrier is still animated throughout the simulation. In both case the same acquisition parameters were used as in the static case.

As expected in both cases, the trunk base is visibly similar to the static image for both wind strengths. Indeed it can be verified on the animation that the trunk base does not move during the acquisition. Another expected result is that the defocus on the images, induced by along-track and cross-track movements assimilated to the Doppler shift, is correlated to the movements amplitude and frequency and therefore the wind strength.

The stronger the wind the larger the blur is. In order to further study our model, Figure 6 shows a real SAR image including an isolated tree subject to wind. The data were acquired by the ONERA during an acquisition campaign in 2004. An in-depth comparison cannot be made between this image and our simulation because in order to emulate this acquisition all parameters should be known (not only the carrier parameters but also the wind speed, the tree specificities...) which is not the case here. Nevertheless we can observe the azimuthal blur illustrated in our model.

Conclusion

We presented here a first step towards the realization of a windblown forest clutter model, namely that of an isolated tree. This model accounts for different tree aspects : procedural tree generation, wind animation, suitable for radio frequencies higher than the S-band.

A modeling pipeline was produced in order to implement our model. This pipeline is characterize by its modularity. In addition, this model and each of its steps are evolutive (each step can be freely complexified). It allowed us to adapt it to SAR imaging for validating purposes. Indeed a first visual comparison with real SAR images showed the capability of our model to render movement blur due to wind.

The perspectives of our model are as follows. First no multi-bounce were accounted for. It should be ensured to take them into account for our next model. Second we only represented a tree in our approach. Another focus should be on the modeling of entire forest to render a clutter sized component.

Figure 1 .

 1 Figure 1. Example of a broadleaf tree modeled with SpeedTree[4]. The chart under the 3D model represents the hierarchical levels of conception.

Figure 2 .

 2 Figure 2. Modeling pipeline. Green modules correspond to the pipeline core, while blue modules provide the input 3D mesh and the output SAR image.

Figure 3 .

 3 Figure 3. Simulated SAR image of a static tree f 0 = 9.6 GHz, B = 3 GHz, image resolution : 5 cm by 5 cm

Figure 4 .

 4 Figure 4. Simulated SAR image of a dynamic tree for a weak wind

Figure 5 .

 5 Figure 5. Simulated SAR image of a dynamic tree for a strong wind

Acknowledgements

We would like to thanks the TSRE unit from the Departement of Electromagnetism and Radar of the ONERA and more specifically Joseph Martinot-Lagarde and Philippe Martineau for the SAR image of an isolated tree.


Yassir Idmessaoud

Didier Dubois

Jérémie Guiochet

Quantifying confidence of safety cases with belief functions

Keywords: Safety cases, Goal Structuring Notation (GSN), Dempster-Shafer theory (DST), Belief elicitation, Confidence assessment

Structured safety argument based on graphical representations such as GSN (Goal Structuring Notation) are used to justify the certification of critical systems. However, such approaches do not deal with uncertainties that might affect the merits of arguments. In the recent past, some authors proposed to model the confidence in such arguments using Dempster-Shafer theory. It enables us to determine the confidence degree in conclusions for some basic GSN patterns. In this paper, we refine this approach and improve the elicitation method for expert opinions used in previous papers.

Introduction

GSN (Goal Structuring Notation) is a graphical formalism used to represent argument structures (assurance cases, dependability cases, etc). Originally, GSN structures do not include a representation of uncertainty in the arguments. Several independent works, e.g., [START_REF] Cyra | Support for argument structures review and assessment[END_REF][START_REF] Wang | D-S Theory for Argument Confidence Assessment[END_REF] proposed to augment this approach to argumentation with confidence assessment methods. They design numerical confidence propagation models for some GSN patterns. However, in [START_REF] Cyra | Support for argument structures review and assessment[END_REF][START_REF] Wang | Safety Case Confidence Propagation Based on Dempster-Shafer theory[END_REF], the data collection method, that enable these mathematical models to be fed with initial confidence values, allowing the computation of the overall confidence in the system, needs improvement. The previous elicitation methods also present some technical defects. In this paper, after reviewing previous work in Section 2, we introduce an extensive confidence propagation method in Section 3, starting with a brief description of argument types. In Section 4, we present an improved expert opinion elicitation method. Finally, in section 5, we illustrate our approach on a small example.

Related work

The issue of confidence assessment in argument structures has been addressed in multiple works. Safety cases, which represent one type of such structures, aim at proving the safety of a system by producing several pieces of documented evidence. In this paper, we focus on safety cases modeled by the so-called Goal Structuring Notation (GSN) defined in [START_REF] Kelly | Arguing Safety -A Systematic Approach to Safety Case Management[END_REF]. Safety argumentation is a core activity in safety critical systems development. Such an argumentation can be carried out using structured notations. It decomposes the safety requirements of the system into elementary pieces as presented in Figure 1, called goals. Strategies are the components that justify this decomposition. Each goal is supported by one or multiple pieces of evidence called Solutions. The example of GSN in Figure 1, is a classical pattern of an argument that treats the hazards existing in a system, and listed in the context box. However, many other patterns exist. This type of representation does not consider the uncertainty that may pervade each premise or the support relation between solutions and goals. Moreover, it is important to note that GSN models are non formal, and no explicit formal logical relations is expressed between elements.

In [START_REF] Idmessaoud | Belief functions for safety arguments confidence estimation: A comparative study[END_REF], we compared some works that deal with confidence assessment in GSN and give some recommendation to improve these methods. For instance, we showed why it is more adequate to use implication instead of equivalence (used in [START_REF] Wang | D-S Theory for Argument Confidence Assessment[END_REF][START_REF] Wang | Modelling Confidence in Railway Safety Case[END_REF][START_REF] Wang | Safety Case Confidence Propagation Based on Dempster-Shafer theory[END_REF]) to represent argument types. We also discussed why Dempster rule of combination is more suitable for combining evidence, in our case, than other methods used in [START_REF] Cyra | Support for argument structures review and assessment[END_REF] for instance. Building such a confidence model relies

C1

List of identified hazards for {System X}

G1

{System X} is acceptably safe

G3 {Hazard H2} has been addressed

…

Gn

{Hazard Hn} has been addressed {Hazard H1} has been removed.

Sn1

A given prevention risk control is implemented.

Sn2

A given protection risk control is implemented.

Sn3 {Hazard H1} has been addressed

G2 S1

Argumentation by claiming addressed all identified plausible hazards Fig. 1. GSN example adapted from Hazard Avoidance Pattern [START_REF] Kelly | Safety case construction and reuse using patterns[END_REF] on input values, usually provided by experts in qualitative form, and transformed into quantitative values. Such an activity could be called "Expert opinion elicitation". This method is more often used with probabilistic models. For instance, in [START_REF] De Persis | Quantitative system risk assessment from incomplete data with belief networks and pairwise comparison elicitation[END_REF], authors used an expert elicitation procedure in a risk assessment approach in fault trees. However, it can also be used in evidence theory. Ben Yaghlane et al. [START_REF] Yaghlane | Elicitation of expert opinions for constructing belief functions[END_REF], generate belief functions from a preference relation between events provided by experts. In relation to our framework, few authors augmented their confidence assessment method by such a data elicitation procedure in order to provide quantitative values for their models. Only some authors such as [START_REF] Cyra | Support for argument structures review and assessment[END_REF][START_REF] Wang | Safety Case Confidence Propagation Based on Dempster-Shafer theory[END_REF] used an elicitation method that transforms expert opinions given in the form of qualitative values, into quantitative ones. Uncertainty propagation can be addressed by standard existing belief function software based on results in [START_REF] Shenoy | Axioms for probability and belief-function propagation[END_REF] (e.g., the belief function machine implemented in MatLab), but the GSNs we study have a particular tree-like structure that enable an explicit symbolic calculation of the belief function on the conclusion space. The explicit formulas, that can be obtained from approaches such as the one we propose, make the calculation more efficient and we can predict the effect of changing selected inputs, thus better explaining the obtained results, and validating the approach.

Argument types

In this section, we define the argument types used in our confidence assessment method. Remember that an argument type represents the relationship between premises and a conclusion. For instance, in Figure 1,(G 2 ) to (G n ) are the premises of (G 1 ) and they are all necessary to support it. This logical relation could be assimilated to the strategy component in a GSN (e.g., S1). When adopting a logical viewpoint [START_REF] Idmessaoud | Belief functions for safety arguments confidence estimation: A comparative study[END_REF], we then speak of a rule. Unlike the types of arguments proposed in [START_REF] Wang | Modelling Confidence in Railway Safety Case[END_REF], which use the equivalence connective to model such rules, we decided to break down this equivalence into two implications. Each implication brings a single piece of information about the conclusion, given premises. For instance, in the case of a single premise (P ) supporting one conclusion (C), P ⇒ C (that we call direct rule) can only infer the acceptance of the conclusion if the premise is true. On the other hand, the rule ¬P ⇒ ¬C (that we call reverse rule) can only infer the rejection of the conclusion when the premise is false. We believe that, when assessing uncertainty of the relationship between P and C, this separate handling makes the resulting models more accurate and easy to assess. We also decided to associate a simple support mass function to each rule to avoid dependence in the confidence between premises. Below are uncertainty propagation formulas for various argument types; all calculation details are available in [START_REF] Idmessaoud | Calculation of aggregation formulas for GSN argument types using belief functions[END_REF].

D-Arg (Disjunctive):

In this situation, each premise can support alone the whole conclusion. We formally define this argument by : ∧ n i=1 (p i ⇒ C). Rules that infer the rejection of the conclusion (¬C) can be deduced from this argument type by reversing this rule to obtain : (∧ n i=1 ¬p i ) ⇒ ¬C. To get formulas (1) and ( 2), we first assign to each rule a simple support mass function (resp. m i dir for p i ⇒ C and m rev for the reverse rule). We also assign one mass function m i p to each premise p i . This function uses three masses on p i , ¬p i and the tautology ( ) summing to one. Then, we combine, using DS rule of combination, all masses on premises together (m p = m 1 p ⊕ m 2 p ⊕ ... ⊕ m n p ) and masses on rules together

(m r = [m 1 dir ⊕ m 2 dir ⊕ ... ⊕ m n dir ] ⊕ m rev )
. Finally, we combine the resulting masses on the rules and premises (m = m p ⊕ m r ). We obtain degrees of belief and disbelief in C:

Bel C (C) = 1 - n i=1 [1 -Bel i p (p i )Bel i ⇒ (p i ⇒ C)] (1)
Disb C (C) = Bel ⇐ (∧ n i=1 [¬p i ] ⇒ ¬C) n i=1 Disb i p (p i ). (2)
We can notice that (1) expresses a "Multivalued Disjunction". To have maximal belief in the conclusion, it is enough that the degree of belief in one single premise equals 1 (assuming that the mass on the direct rule also equals 1). Formula (2), in contrast, expresses a ' 'Multivalued Conjunction". To have a maximal disbelief in the conclusion, all the disbelief degrees on premises should be equal to 1, supposing that the mass on the reverse rule equals 1 too. We can also notice that, when Bel C (C) is maximal, Disb C (C) is minimal.

C-Arg (Conjunctive): This argument type describes the situation when two premises or more are jointly needed to support a conclusion. Following the same reasoning as in the previous type, we define it formally by two rules : (∧ n i=1 p i ) ⇒ C and its reverse ∧ n i=1 (¬p i ⇒ ¬C). Following the same calculation in the disjunctive type, we get the formulas below :

Bel C (C) = Bel ⇒ ([∧ n i=1 p i ] ⇒ C) n i=1 Bel p (p i ) (3)
Disb C (C) = 1 - n i=1 [1 -Disb i p (p i )Bel i ⇐ (¬p i ⇒ ¬C)] (4)
We can notice that, in contrast with formulas obtained for D-Arg, (3) and (4) respectively express a "Multivalued Conjunction" and "Multivalued Disjunction".

H-Arg (Hybrid): This argument describes the case when it is difficult to choose between the conjunctive or disjunctive types. Each premise supports the conclusion to some extent, and the conjunction of the premises does it to a larger extent. We obtain degrees of belief and disbelief in C:

Bel C (C) =Bel ⇒ ([∧ n i=1 p i ] ⇒ C) × n i=1 Bel i p (p i )[1 -Bel i ⇒ (p i ⇒ C)] + {1 - n i=1 [1 -Bel i p (p i )Bel i ⇒ (p i ⇒ C)]}. ( 5
)
Disb C (C) =Bel ⇐ ([∧ n i=1 ¬p i ] ⇒ ¬C) × n i=1 Disb i p (p i )[1 -Bel i ⇐ (¬p i ⇒ ¬C)] + {1 - n i=1 [1 -Disb i p (p i )Bel i ⇐ (¬p i ⇒ ¬C)]}. (6)
We can notice from ( 5) and ( 6) that these formulas subsume those in conjunctive and disjunctive types. On the one hand, if masses on p i ⇒ C are zero, it becomes the formula of the conjunctive type. On the other hand, if the mass on [∧ n i=1 p i ] ⇒ C is zero, we get the disjunctive type formula. This argument provides a general framework that allow as to calculate belief and disbelief values in different situations. D-Arg and C-Arg represent extreme cases where the value of some rules is null.

Note that moving away from these extreme cases may lead to encounter situations of conflict. A contradiction may appear when we have opposite opinion about two premises along corresponding direct and reverse rules. Formally, it always takes the form of a combination of four items of the form: {p i , p i ⇒ C, ¬p j , ¬p j ⇒ ¬C}. The sum (Bel C (C)+Disb C (C)) is then greater than 1. This may indicate something wrong in the GSN or in the way the experts replied questions, or yet on the reported experiments. Equation ( 7) represents the conflict calculation formula :

m(⊥) = n i=1,j =i [Bel i p (p i )Bel i ⇒ (p i ⇒ C) × Disb j p (p j )Bel j ⇐ (¬p j ⇒ ¬C)] (7)
To address this issue, we choose to subtract the mass of the conflict m(⊥) from Bel C (C) and Disb C (C) in ( 5) and ( 6), and get contradiction-free degrees bel

(C) = Bel C (C) -m(⊥) and disb C (C) = Disb C (C) -m(⊥).
We choose not to normalize the results (dividing by 1 -m(⊥)) as proposed in the usual DS rule of combination because this operation will eliminate the conflict and proportionally increase the contradiction-free degrees of beliefs and disbelief bel C (C) and disb C (C) in a misleading way in the case of strong conflict. On the other hand, keeping m(⊥) and subtracting it from Bel C (C) and Disb C (C) will legitimately increase uncertainty (i.e., bel C (C)+disb C (C) is small) and show that the system is not that safe because of the presence of a conflict.

Expert opinion elicitation

In the previous section, we defined three argument types and proposed analytical formulas to calculate the belief and disbelief degrees in conclusions. However, using these models requires the presence of two types of information: belief degrees in premises (e.g., Bel i p ) and the belief degrees for rules (e.g., Bel i ⇒ ). In this section, we are first going to see how we can transform an expert opinion about a premise into belief, uncertainty and disbelief degrees. Then, in the second part, we provide some hints on how we can identify masses on rules.

Elicitation of belief and disbelief on premises

In order to directly obtain belief and disbelief degrees in a premise p from an expert, authors in [START_REF] Cyra | Support for argument structures review and assessment[END_REF] consider asking two pieces of information : one, called decision index Dec(p), describes which side the expert leans towards, acceptance or rejection of p; the other, called confidence Conf (p), reflects the amount of information an expert possesses that can justify his opinion. Namely Dec(p) = 1 (resp., 0) indicates the certainty that p is true (resp false), while Conf (p) = 1 (resp., 0) indicates the expert has full (resp. no) information supporting the choice of Dec(p). This is represented on Fig. 2.

The problem is then to define the belief and disbelief degrees in a proposition p in terms of Dec(p) and Conf (p). In [START_REF] Cyra | Support for argument structures review and assessment[END_REF], it is proposed to let Bel(p) = Dec(p) • Conf (p) and Disb(p) = (1 -Dec(p)) • Conf (p), which implies a natural result:

Conf (p) = Bel(p) + Disb(p). (8)
However, it also implies that Dec(p) =

Bel(p)
Bel(p)+Disb(p) . Note that this formula, presents a discontinuity in case of no information (Bel(p) + Disb(p) = 0). The expression is then completed by assuming Dec(p) = 1 [START_REF] Cyra | Support for argument structures review and assessment[END_REF] or 0 [START_REF] Wang | Safety Case Confidence Propagation Based on Dempster-Shafer theory[END_REF], in this case, which sounds arbitrary.

It is more convincing to use the Pignistic transform [START_REF] Smets | Decision making in the tbm: the necessity of the pignistic transformation[END_REF] that turns a mass function m on a set Ω (the frame of discernment) into a probability, changing the focal sets into uniform distributions. When Ω = {p, ¬p} has two possible states, Dec(p) is the midpoint between belief and plausibility of p, which reads :

Dec(p) = 1 + Bel(p) -Disb(p) 2 (9)
Note that when Bel(p) = Disb(p) = 0, we get Dec(p) = 1/2. Some authors suggest to define Dec(p) by renormalising the pair (P l(p), P l(¬p), where P l(p) = 1 -Bel(p), dividing them by P l(p) + P l(¬p) (plausibility transformation method [START_REF] Cobb | On the plausibility transformation method for translating belief function models to probability models[END_REF]). This method is in agreement with Dempster rule of combination. However, we do not get the midpoint between belief and plausibility, which is intuitively surprizing, and in case of more than 3 elements in the frame, such a transformation may give probability values outside the range [Bel, P l] [START_REF] Dubois | Practical methods for constructing possibility distributions[END_REF].

Using equations ( 8) and ( 9) and the knowledge of Dec(p) and Conf (p), we can calculate belief and disbelief values :

Bel(p) = Conf (p)-1 2 + Dec(p), Disb(p) = Conf (p)+1 2
-Dec(p). Viewing Bel(p) as a lower probability, the pignistic transformation computes the center of gravity of the convex set of probabilities {P : P ≥ Bel}.

However, the pignistic transform also presents one issue for the elicitation procedure. Some values of the pair (Dec, Conf ) provided by the expert may lead to negative values of belief Bel(p) or disbelief degrees Disb(p), which makes no sense. This is because there are constraints relating Conf (p) and Dec(p): (8) and ( 9) imply 1 -Conf (p) ≤ min(2Dec(p), 2(1 -Dec(p)), which is known as Josang triangle [START_REF] Jøsang | Subjective logic[END_REF]. To fix this problem, we can express the range of Dec(p) for a given confidence level as:

1 -Conf (p) 2 ≤ Dec(p) ≤ 1 + Conf (p) 2 (10)
For instance, a strong decision (full acceptance or rejection) should only be made when we have a very high level of confidence, since when Conf (p) = 1, . Choosing scales for (Conf, Dec) and translating such pairs into numerical degrees is not trivial, we thus make the equidistance assumption for simplicity and to be comparable to previous works.

Determination of belief weights for rules

Now, consider the mass functions for rules. Unlike belief and disbelief degrees in premises, belief degrees in rules are more difficult to obtain directly from an expert. Remember that a rule is representing a support relation between a conclusion and its premises. As a first approach, Wang et al. [START_REF] Wang | Modelling Confidence in Railway Safety Case[END_REF] proposed to exploit this relation so as to identify these masses. To this end, they propose certain values of the pair (Dec, Conf ) on premises, as inputs for the model, and ask the expert his opinion about the conclusion using the matrix of Figure 2. Then, for each type of argument, they use a non-linear least square method to estimate the values of parameters (belief in rules).

In the 3 rd and 6 th column, decision levels (resp. acceptable and tolerable) were maintained because the divergence is located in a D-Arg. Due to its disjunctive nature, this argument favors the propagation of the premises that maximally support the conclusion. However, confidence levels were slightly decreased because of a C-Arg, which cumulates the uncertainty present in each premise and propagates it to the conclusion. In the 4 th and 5 th columns the divergence is located in a C-Arg. Unlike D-Arg, this argument favors the propagation of the premises that support the conclusion with the least strength. Thus, we end up with a mildly negative ("opposable") decision level in the 4 th column and a very low level of confidence in the 5 th column.

Conclusion

In this article, we propose a method for confidence assessment in GSN. It covers both the definition of argument types (belief propagation formulas) and data transformation (from elicited qualitative data to belief and disbelief pairs). We also illustrate this approach on a toy example. First results show that it was possible to improve previous work on uncertainty propagation and elicitation issues. We still need to conduct a full experiment for assessing beliefs in rules. We will investigate the expert questionnaire. We also want to propose an approach for automatic rule type identification. In the long range, we also plan to do away with the qualitative to quantitative transformation that contains some arbitrariness, by developing the purely qualitative approach to information fusion outlined in [START_REF] Dubois | A possibilistic counterpart to shafer evidence theory[END_REF], and compare it to the quantitative one.

Fig. 2 .Fig. 3 . 2 ( 2 . 2 (

 23222 Fig. 2. Expert opinion extraction matrixG

A second approach, which is under study, aims to determine these masses and the argument type (C, D or H-Arg) through a series of questions, assuming clear-cut knowledge for premises (Bel = 1 or 0 and Disb = 1 or 0). For example, If decision and confidence on a premise are maximal (acceptable for sure), what is your decision and confidence on the conclusion ? When belief or disbelief in premises are maximal, the mass on the conclusion is the mass of the rule. For instance, if in (3) we let ∀i,

) is obtained from the expert.

Toy example

We apply our approach in this section to a simple GSN example presented in Figure 3, including two types of arguments. It presents a top goal (G) supported by two sub-goals (G 1 ) and (G 2 ). (G 1 ) is supported by two premises (P 1 ) and (P 2 ). On the other hand, (G 2 ) is supported by a single premise (P 3 ). For simplicity, we chose a C-Arg for the argument type used to calculate (G) and a D-Arg in the calculation of (G 1 ).

In order to see how the variation of belief and disbelief degrees in the premises affect the conclusion, we show six different configurations in table 1, where we set for each premise (P 1 , P 2 and P 3 ) a qualitative pair (decision, confidence) (and the corresponding pair (Bel, Disb)), and calculate the conclusion (G) by means of formulas ( 1), ( 2), ( 3) and (4). We also set the values of the masses on the rules to 1. As a result, the values of belief and disbelief in the conclusion will depend only on the masses on the premises.

Table 1. Qualitative (decision, confidence) and quantitative (belief, disbelief) pairs for the example (see Fig. 2 for the meaning of symbols)

P1

(R;C6) (A;C5) (A;C5) (T;C5) (A;C5) (T;C6) (0 ; 1) (0.8 ; 0) (0.8 ; 0) (0.65 ; 0.15) (0.8 ; 0) (0.75 ; 0.25)

P2

(R;C5) (A;C6) (R;C5) (T;C6) (A;C6) (T;C1) (0 ; 0.8) (1 ; 0) (0 ; 0.8) (0.75 ; 0.25) (1 ; 0) (0 ; 0) P3 (R;C6) (A;C6) (A;C6) (O;C6) (A;C2) (T;C6) (0 ; 1) (1 ; 0) (1 ; 0) (0.25 ; 0.75) (0.2 ; 0) (0.75 ; 0.25)

G

(R;C6) (A;C6) (A;C5) (O;C6) (A;C2) (T;C4) (0 ; 1) (1 ; 0) (0.8 ; 0) (0.23 ; 0.76) (0.2 ; 0) (0.56 ; 0)

We can notice, on Table 1, that when we have either three rejectable or three acceptable premises with high levels of confidence (1 st and 2 nd columns), the models maintain the same decision with the same high level of confidence. On the other hand, when we have divergent opinions on the premises, either by opposite decisions (3 rd and 4 th columns) or opposite confidence levels (5 th and 6 th columns), the results will depend on the nature of the argument involved.