Keywords: Displacement measurement, Magnetic anisotropy, magnetic flux density, magnetic forces, magnetostriction, soft magnetic materials, transformer cores, vibrations Sinusoidal secondary RMS voltage : Anisotropic angle of applied field

The magnetic material quality of transformer cores influences their emitted acoustic noise. The latter is generated by vibrations resulting from the effects of the Maxwell's forces, located in the corners, and the magnetostriction inside the magnetic steel. In this paper, two annular structures allow differentiating these two phenomena and analyzing the vibration origins. Then, two different materials are compared, leading to conclusions about the influence of the material anisotropy on vibrations. At last, the measured vibration components are fully justified with the local saturation effects.

I. INTRODUCTION

he acoustic noise emitted by transformers results from many contributions. Indeed, the magnetic core composed of electrical steel sheets is source of high concern for the steel producers and the transformer manufacturers [START_REF] Bengtsson | Industrial significance of losses and noise[END_REF]- [START_REF] Majer | The analysis of the vibrations of the cores of model transformers[END_REF]. The noise emission results from vibrations, which have two different origins: the Maxwell's forces due to effects between laminations [START_REF] Weiser | Mechanisms of noise generation of model transformer cores[END_REF] and the magnetostriction due to the specific behaviour of the electrical steel [START_REF] Moses | Measurement of magnetostriction and vibration with regard to transformer noise[END_REF]- [START_REF] Kubiak | Vibration analysis of small power transformer[END_REF]. Thus, it is a challenge to separate the influence of these two factors and to weight their respective contribution to the emitted noise when Grain Oriented (GO) steel is used to build the magnetic circuit [START_REF] Loizos | Core Vibration Analysis in Si-Fe Distributed Gap Wound Cores[END_REF]- [START_REF] Vandevelde | Magnetic forces and magnetostriction in electrical machines and transformer cores[END_REF].

The first part of the paper presents the phenomenon of magnetostriction and the forces arising between the sheets. Then, the second part describes the test bench composed of stacked GO electrical steel rings [START_REF] Penin | Estimation of relative magnetostriction and Maxwell's forces in stacked Grain Oriented steel structures[END_REF]- [START_REF] Brudny | Magnetic core and use of magnetic core for electrical machines[END_REF]. The results are presented and discussed in the third part. The main contribution concerns the effect distinction of the Maxwell's forces and the magnetostriction. Hints are given for understanding the noise generation in transformers. Two qualities of electrical steel, i.e. a conventional GO electrical steel (denoted CGO) and a high permeability GO one (denoted HGO), are also compared to quantify the influences of the anisotropy on the vibrations. In order to explain every vibration components, the authors analyse the impact of the supply voltage harmonics [START_REF] She | Wind Energy System with Integrated Functions of Active Power Transfer, Reactive Power compensation, and Voltage conversion[END_REF]- [START_REF] Iida | Study on magnetic noise caused by harmonics in output voltages of PWM inverter[END_REF] and the effects of local saturations in the magnetic circuit [START_REF] Kefalas | Harmonic Impact on Distribution Transformer No-Load Loss[END_REF].

II. MAGNETOSTRICTION AND MAXWELL'S FORCES

The Maxwell's forces are generated by the presence of magnetic flux across the air gap between the magnetic steel laminations. It is particularly the case in the transformer corners because in this area, the magnetic field has to change direction and, due to the high anisotropy of GO electrical steel, it has to cross the air gaps existing between the latter, creating magnetic discontinuities. Indeed, the field can remain Expression [START_REF] Bengtsson | Industrial significance of losses and noise[END_REF] shows that p components contain a prevailing term of fundamental frequency twice the value of the excitation one. This study will be focused on vibrations induced by all p components, the corresponding induced external displacements being tied to the mechanical response of the structure.

The magnetostriction appears due to the magnetization of the electrical steel sheets. It takes place when the domains have to rotate to align themselves along the direction of the magnetic field. This is particularly the case in the transformer joints where the magnetic field concentrates at the inner part of the corners and makes its magnitude high enough to locally move the domains [START_REF] Arai | Magnetostriction of grain oriented Si-Fe and its domain model[END_REF].

Magnetostriction is generally expressed as a relative deformation l l which depends, for a single crystal, on the magnetostriction coefficients 100 and 111 , respectively 23.7x10 -6 and -4.1x10 -6 for the case of the GO structure [START_REF] Shilling | Magnetic properties and domain structure in grain-oriented 3% Si-Fe[END_REF].

Let us denote the anisotropic angle of applied field with respect to the Rolling Direction (RD), which will be marked with a bold arrow in the figures, and the angle between the measurement direction and the RD. For from 0° to 55°, the magnetostriction is negative, and so the sheet is contracting. On the other hand, for from 55° to 90 , the magnetostriction is positive and so the sheet is stretching. The deformation amplitude varies with [START_REF] Hasenzagl | Magnetostriction of 3% SiFe for 2-D magnetization patterns[END_REF]: it is very low when 0 and it increases with . These behaviours are due to a low field magnetostriction. This specificity can be easily visualized thanks to the so-called "butterfly loop" graphs [START_REF] Yamaguchi | Magnetostriction measurement on the multidimensional magnetization performance of SiFe steel[END_REF]. Fig. 2 shows the link between the magnitude of the relative deformations of a sheet due to the magnetostriction depending on and θ. The displacements in the RD have amplitudes higher than those in the Transverse Direction (TD) and no deformation around 55°. Since magnetostriction is at constant volume, the deformations in the outer plane direction are consequences of the in-plane ones. That means that the lamination presents a deformation according to its thickness [START_REF] Wakabayashi | Measurement of Three-Dimensional Magnetostriction on Grain-Oriented Electrical Steel Sheet[END_REF]. Like Maxwell's forces, prevailing vibrations due to magnetostriction present a fundamental frequency twice the excitation [START_REF] Kitagawa | Structure deformation analysis caused by magnetostriction of magnetic steel sheet using combination of Electromagnetic and Structure analysis[END_REF].

III. TEST BENCH

A. Ring Structure

The experimental device presented in this paper deals with stacks of rings built with GO electrical steel subjected to a circumferential magnetisation. Each ring of the stack is shifted from the previous one by a constant angle (Fig. 3). This structure is able to give an image of the magnetic behavior of a transformer corner joint when the Maxwell's forces in plane are assumed to be null.

Axial (along z) and radial (along y) vibrations are analysed in this study considering 0 and 90 .

Case of axial vibrations:

For 0 , there is no reason for the field to move from one layer to another since all rings are stacked with their RD parallel. In that way no Maxwell's force takes place between the sheets and the only possible deformations is related to magnetostriction.

For 90 between successive layers, the magnetic flux is established according to the RD of lamination and passes through the interlaminar air gap to reach the RD of the adjoining laminations [START_REF] Parent | Analysis of the Magnetic Flux Distribution in a New Shifted Non Segmented Grain Oriented AC Motor Magnetic Circuit[END_REF]. Hence, deformations will result from a mix of Maxwell's forces and magnetostriction. Moreover, Maxwell's force amplitudes will be maximum.

Case of Radial Vibrations:

The radial vibrations are caused, whatever the shifting, only by the magnetostrictive effect.

B. Test Bench Presentation

The experimental device has been designed to maximize the reliability of the measurements (Fig. 4.). The two stack configurations are mounted inside the same frame structure. The stacks can be rotated around the stacking direction axis.

Two steel qualities have been tested (Fig. 5): the first one is GO "PowerCore C" electrical steel of 0,35mm nominal thickness (174 sheets for each stack) called "CGO" and the second one is GO "PowerCore H" electrical steel of 0,30mm nominal thickness (203 sheets for each stack) called "HGO" [START_REF]Our products-grain oriented steel PowerCore[END_REF]. The HGO permeability in the RD is higher and it is more anisotropic [START_REF] Taguchi | New grain-oriented silicon steel with high permeability "ORIENTCORE HI-B[END_REF]. The sheets are of internal radius 8mm and of outside radius 50mm. The sheet insulation is around 5μm. The sheets are clamped between two non magnetic bakelite A and B plates and maintained with threaded rods. The tightening torque applied on the nuts is controlled with a torque wrench and is the same for each stack and for every test.

Holes have been drilled through the bakelite plates for sensors access in order to record the axial deformations of the stacks (Fig. 4). The measurements points A 1 to A 4 and B 1 to B 4 are presented in Fig. 6(a) (holes face to face are numbered in the same way).

The spatial reference for the radial measurements is the RD of the first sheet of a stack (against the plate A), which is aligned along the x'x axis of the holes noted 1 and 3 (Fig. 6(b)).

The radial measured point is located with the angle with respect to this reference.

For the shifted structure, 0 correspond to RD for every two sheets. The excitation is provided by a primary 28 turn winding creating a circumferential magnetisation. A 10 turn secondary winding allows measuring the average flux density in the core. Those windings are distributed on one half of the structure only (Fig. 4) in order to apply the measuring devices. The primary winding is supplied with a 50Hz sinusoidal voltage provided by an auto-transformer (Total Harmonic Distortion < 3%). Hence, the devices are submitted to a unidirectional field.

Finally, the magnetic flux density in the core is computed from the secondary voltage through the following formula:

2 2 2 V B nS (2)
The measurements have been done for magnetic flux density values of 0.7T, 1.1T and 1.5T. A Bruël and Kjaer PULSE data acquisition system is used in order to analyze signals measured with mono-axial accelerometers.

IV. EXPERIMENTAL RESULTS

A. Measurements of Axial Vibrations

The accelerations of the deformations are presented for the CGO in Fig. 7 for the 100 Hz frequency component only for points A 1 to A 4 of the non-shifted structure with the magnetic flux density value of 1.1T. Theoretically, for both configurations, according to symmetries, points A 1 and A 3 should present the same vibrations. The same remark can also be done for points A 2 and A 4 . But, differences between these points are measured; this comes from the uncertainty of the accelerometer placement. To reduce this, an averaging measured values on points 1 and 3 as well as on points 2 and 4 is made for plates A and B. Fig. 8 shows this averaging for both configurations ( 0 and 90 ) and both qualities of steel denoting, for example, A 1 -A 3 the averaging of measurements made on A 1 and A 3 .

It appears that the non-shifted structure ( 0 ) presents higher vibration levels that the shifted one. It can be explained as follow: The Maxwell's forces arising in the case of the shifted configuration induce a contraction of the interlaminar air gaps which limits the mechanical displacements [START_REF] Demian | AC magnetic circuits using non segmented shifted grain oriented electrical steel sheets: impact on induction machine magnetic noise[END_REF].

The flux density distribution is different in the shifted stacks The authors show in [START_REF] Parent | Analysis of the Magnetic Flux Distribution in a New Shifted Non Segmented Grain Oriented AC Motor Magnetic Circuit[END_REF] that the flux crosses the air gaps to flow mainly in the RD. Hence is low, that lead to lower magnetostriction effects. Both phenomena (magnetostriction and Maxwell's forces) can compensate each other at a given frequency, explaining the observed vibration differences. Concerning the difference between the two qualities, the axial and radial vibrations are lower for the HGO shifted stack than for the CGO shifted stack. It can be explained by the fact that the effects of the magnetostriction and the Maxwell's force effects are better compensated and by the fact that the HGO magnetostriction characteristics lead to lower deformations.

It is interesting to note that, for HGO and CGO, the nonshifted structure for 0.7T have the more important vibrations in positions A 1 -A 3 and B 1 -B 3 . Indeed, these points correspond to zones where the flux is in the TD of the sheets. Hence, the magnetostrictive effect is the highest. For 90 , these vibrations are the same for all positions.

B. Radial Vibration Measurements

For those measurements, the accelerometers are placed on the edges of the stacks for 0 and 90 (Fig. 6(b)). The results are presented in Fig. 9 for the 100Hz frequency component and for 0 and 90 . Whatever the value of the angle is, the radial vibrations are always due to magnetostriction. For the non-shifted setup, the vibrations occurring along the RD are bigger than the ones across the TD. This is in accordance with the "butter-fly" loops previously mentioned. Moreover, vibrations in the shifted setup are fairly identical for both 0 and 90 .

Concerning the difference between the two qualities, the HGO leads to higher vibrations than the CGO for the nonshifted structure. That can be explained by the strong anisotropy of the material, which generates vibrations of high amplitudes when the magnetic flux is established in the TD. On the contrary, the HGO shifted structure presents lower vibrations than the CGO. The Fig. 9 highlights noteworthy acceleration amplitude differences with the shifting. Indeed, the CGO shifted setup for 0.7T shows that amplitudes are divided by 27 and 8.3 respectively for 0 and 90 .

Nevertheless, the ratios decrease when the magnetic flux density value increases: 6.6 and 2.2 for 1.1T and 3 and 0.95 for 1.5T. For the HGO quality, theses ratios are respectively 81 and 27 for 0.7T, 20 and 8.7 for 1.1T and finally 7.2 and 2 for 1.5T. It means that the more anisotropic the sheets are, the higher the shifting effect is. The vibratory behaviour of the whole structure has also been investigated. An "Operating Deflection Shape" has been performed on one fourth of both shifted and non shifted CGO stacks for 1.1T and 1.5T. For 0.7T, the vibrations were too weak to be efficiently analyzed.

The results are presented in Fig. 10(a They put in evidence the reaction differences between the two configurations. The non-shifted stack presents counterphased vibrations between the RD and TD. This is the usual vibration reaction of GO steels. On the other hand, the shifted stack presents in-phase vibrations between the RD and TD. Moreover, the vibrations for 45 are counter-phased.

The phase reference is set for 0 and the measured values corresponding to A 1 , B 1 , and 90 , are presented in Fig. 11. As previously mentioned, it can be noted that the axial and radial vibrations are in phase on both setups.

C. Radial Deformations

In the previous measurements, only the accelerations have been presented. In this part, the deformations of the CGO stacks are considered. In order to determine their nature, i.e. contraction or extension, the acceleration and the secondary voltage are used.

The waveforms of the flux density and deformations have been evaluated from formula 3 and 4 respectively. changes, it explains that, for 90 , the nature of the displacement has changed: an elongation for the shifted setup and a contraction for the non-shifted device (Fig. 13). The amplitude of the displacements is lowered with the shifting. This is the case of the step lap transformer joint, where the GO laminations RD are crossed. The HGO stack presents the same phenomena but with a different level of vibration.

D. Local Analysis of the Flux Density

In previous measurements, the authors have presented the vibration variations of the 100 Hz component. However, the spectra in Fig. 14 show that other components occur for the CGO non-shifted stack, although the supply is nearly sinusoidal. In order to understand the origin of these components, two experiments have been done to estimate the contribution of local saturations of the sheets and voltage harmonic components.

First, small coils, shown in Fig. 15, have been placed to analyse the flux density distribution in the sheets at β=0°, a zone where the magnetic flux is in the TD giving the higher magnetostrictive vibrations. The displacements are measured at 0 . Second, to evaluate the supply influence, a voltage component has been superimposed to the 50Hz fundamental. The chosen frequency of this harmonic is 170Hz in order to distinguish clearly the impact of the additional voltage from eventual magnetic saturations components of frequency multiple of 50Hz [START_REF] Lecointe | Distinction of toothing and saturation effects on magnetic noise of induction motors[END_REF]. This frequency, close to the third excitation harmonic, can come from the grid or the power electronic [START_REF] Chaturvedi | Carrier Based Neutral Point Potential Regulator with Reduced Switching Losses for Three-Level Diode Clamped Inverter[END_REF]. The amplitudes of the sinusoidal 50Hz and 170Hz lead to average magnetic flux densities of 0.4T and 0.2T respectively, as shown in Fig. 16. The local measurement of the flux density show, for = 0° in Fig. 17(a) and for =90° in Fig. 18(a), a higher spectral content: components of major amplitude appear at 150Hz (due to the saturation) and also the frequency created with the intermodulation products (IM3) of the excitation caused by the non-linearity of the material: 70Hz (170-2x50), 270Hz (2x50+170), 290Hz (2x170-50) and 390Hz (2x170+50). As the displacements due to the magnetostriction are proportional to the square of the flux density, the square of magnetic flux density spectrum has been determined, as shown in Fig. 17(b) and in Fig. 18(b).

There are great similarities between the acceleration (Fig. 14) and the square of magnetic flux density calculated from the local flux density (Fig. 17 The second concerns components of different frequencies: 20Hz (70Hz -50Hz), 200Hz (50Hz + 150Hz), 240Hz (170Hz + 70Hz) and 320Hz (170Hz + 150Hz).

As a consequence, the local sensors allow measuring local saturations responsible of vibrations components as well as those due to the voltage harmonics. It can be noted in Fig. 17

V. CONCLUSION

The measurement results presented in this paper provide interesting information about magnetostriction and Maxwell's forces effects. Although, in this study, the in plane Maxwell's forces at the corner joints of the step lap have not been modelled, it seems to give some hints towards explanation for vibration mechanisms inside transformer core, particularly at the joints. The important result, which can be extracted from this analysis, is that both Maxwell's force and magnetostriction effects are, concerning the vibrations, of same order of magnitude. That leads to questions about the contribution of noise generated by the transformers. The legs generate magnetostrictive noise whereas the Maxwell's forces and the magnetostriction which occur in the corners may be partially compensated. Then, small coils allow measuring local saturations. It has been shown that they lead to vibrations of non negligible amplitude. Moreover, the vibration spectral may be greatly enhanced if the supply is not sinusoidal.

Grain

  Oriented Steel Rings for an Experimental Comparison of Relative Magnetostriction and Maxwell's Forces Effects R. Penin, J-Ph. Lecointe, G. Parent, J-F.Brudny, T. Belgrand T in the same plane or pass from one layer to another depending on permeability situation [13]. Moreover, two potential sources for Maxwell forces exist: in plane (Fig. 1(b) way 1) due to the distributed air gap of the step lap arrangement (Fig. 1(a)) and out of the plane (Fig. 1(b) way 2) due to the air gap brought by the insulation layer intrinsic to the steel [14]-[15]. This paper is focused on the second case. The p pressure corresponding to the Maxwell's forces interacting between two successive laminations can be expressed as follows:

Fig. 3 .Fig. 2 .

 32 Fig. 3. Stack of shifted rings

Fig. 1 .

 1 Fig. 1. Joint region in conventional single step lap assembling

Fig. 6 .Fig. 5 .Fig. 4 .

 654 Fig. 6. Axial and Radial measuring points

  ) ( 0 ) and Fig. 10(b) ( 90 ) Fig. 11(a) and Fig. 11(b) show the 3D acceleration

Fig 7 :Fig. 8 . 1 BFig. 10 .

 78110 Fig 7: Measurements of axial vibrations for plate A ( 1.1T B)

Fig. 13 .Fig. 14 .

 1314 Fig. 13. Radial displacements along

  (a) for 0 and in Fig.14(b) for = 90°

Fig. 16 .

 16 Fig. 16. Flux density for =0° and =90°

  (b)). The frequencies of the displacement components can be clearly explained by considering two flux density harmonic components of frequency 1 f and 2f . When multiplied, two components of explains that: -The components visible in the average flux density spectrum generate vibrations at 100Hz (2x50Hz), 120Hz (170Hz-50Hz), 220Hz (170Hz+50Hz) and 340Hz (2x170Hz

Fig. 17 .Fig. 18 .

 1718 Fig. 17. Spectrum of the local flux density (a) and its square (b) for =0°

  (a) and 18(b) that all the vibrations measured in the TD are, as for the 100Hz component, greatly reduced compared to the RD.

supervised by the French national technological research cluster on electrical machine efficiency increase. This program, including ThyssenKrupp Electrical Steel UGO, is sponsored by the region Nord Pas-de-Calais (France), the French ministry and the European founds (FEDER)

For this evaluation, the integration constant is 0 0 ) t ( l at 0 0 ) t ( B T.

The magnetostriction effect value with the magnetic flux density is presented in Fig. 13 for 0 and 90 and the CGO quality. The non-shifted setup presents a contraction along the RD and an elongation across the TD. This is the case for all the considered magnetic flux density values. For 0 , the maximal displacement values of the shifted configuration are 26 (0.7T), 7 (1.1T), and 3 (1.5T) times lower than with the non shifted setup. As for = 55°, the magnetostriction effect


Francis Pardal

Vincent Lapinte

Jean-Jacques Robin

Kinetics of cotelomerization of 3-(trimethoxysilyl)propylmethacrylate and perfluorodecylacrylate

Keywords: copolymerization, fluorinated polyacrylate, kinetic, reactivity ratio, telomerization

come

Introduction

Reactive end-group oligomers have attracted considerable interest during 20 years owing to the possibility of further reaction to give block and graft copolymers. Telomerization is a polymerization technique providing the synthesis of such macromolecular architectures. The term telomerization defined the reactive process where a molecule YZ, named telogen, reacts onto a polymerizable compound M to form telomers of general formula Y(M) n Z. The difference between telomerization and polymerization is, in the first case, the low number of monomer units in the final compound (n<100) which involves an importance of the end-group of the polymer chains. Telomerization has been performed by all polymerization mechanisms: anionic, group transfer, cationic, free-radical, polyaddition and polycondensation. Radical telomerization is the most useful technique to synthesize reactive α-functional polymers. The end-group functionality comes from the transfer agent nature (Scheme 1). Transfer agents have to present an easily radically cleavable bond such C-I, C-Br, C-Cl, S-H (mercaptans) and P-H (phosphonates). Mercaptans have been extensively studied: thioglycolic acid, [1][2][3] 2mercaptoethylamine, [4] 3-mercaptopropionic acid, [4] and 2-mercaptoethanol. [5][6][7][8] The efficiency of 2-mercaptoethanol in transfer reactions has been outlined and explored in several previous investigations. [6][7] Its reactivity has been well studied in the case of the cotelomerization of methyl methacrylate with styrene in order to obtain α-hydroxyl oligomers with random type copolymer structures for a further utilization as macromonomers. [7] Fluorinated polymers are unique materials since they combine chemical inertness (to acids, bases and solvents), low water absorptivity, excellent weartherability, good resistance to oxidation, aging, and very interesting surface properties with low surface energy. Among the fluorinated monomers, acrylates with long perfluoroalkyl (R f ) side chains, such as perfluorodecylacrylate (PFDA) monomer, offer interesting properties. [9][10][11] Fluorinated acrylate based polymers are most often synthesized by emulsion polymerization [11][12] and to our knowledge, no telomerization reaction involving fluorinated acrylate monomer has been attempted yet. In opposition to acrylates with long perfluoroalkyl (R f ) side chains, some ethylene units substituted by fluorine atoms were telomerized. For instance, Boutevin et al.

studied radical telomerization of chlorotrifluoroethylene and cotelomerization with butyl vinyl ether and mercaptans (2-hydroxy ethyl mercaptan, C 6 F 13 C 2 H 4 SH or perfluorochloroalkyl iodides). [13] Vinylidene fluoride was also telomerized with perfluoroalkyl iodides by thermal and redox processes. [14] The main problem in the solution polymerization of this type of monomer is their low solubility in usual hydrocarbonated solvents. The cotelomerization with hydrophilic hydrogenated methacrylate comonomers can solve this lack.

Thus, this paper is focused on the synthesis on additives for applications in organic/inorganic hybride materials. Oligomers bearing two functionalities were prepared: a fluorinated part for surface properties and a silane one for grafting onto silica particles. The oligomers were choosen since they allow low viscosities and facilitate their processing. So, this paper descibes the cotelomerization of heptadecafluorodecyl acrylate with a reactive acrylate offering grafting onto inorganic matrix. [15][16][17][18][19] A well-known candidate for the organic/inorganic hybrid applications is 3-(trimethoxysilyl)propyl methacrylate (TMSPMA).

None investigation on copolymerization of TMSPMA with PFDA has already been reported in the literature. Although, few references exist on homopolymerization of TMSPMA [20][21][22] and copolymerization [22][23] with others acrylate and methacrylate comonomers. Koh et al. [23] have synthesized amphiphilic block copolymers by atom-transfer radical polymerization (ATRP) with TMSPMA, methyl methacrylate and poly(ethylene oxide) methyl ether methacrylate in order to form organic/inorganic hybrid nanocapsules used in the encapsulation of substances into confined spaces. Ritz et al. reported the copolymerization by ATRP of TMSPMA with methyl methacrylate and generated gradient copolymers for applications as compatibilizers for polymer blends or pressure sensitive adhesives. [24] D'Agosto et al. also synthesized amphiphilic block copolymers by RAFT technique which can self-assemble into a variety of nano-objects and can be used as colloid stabilizers, nanoreactors, or templates for making inorganic solids. [20] The aim of the work was firstly, the determination of the transfer constant (C T ) of TMSPMA in the presence of 2-mercaptoethanol by telomerization and the check of non-attendance of the side reactions between trimethylsilyl groups and the functional telogen. Secondly, the free-radical cotelomerization of TMSPMA with PFDA in the presence of 2-mercaptoethanol was studied in order to prepare α-hydroxy terminated statistical oligomer-type backbone which may be further used to obtain graft copolymers. Finally, the monomer reactivity ratios of TMSPMA (monomer M1) and PFDA (monomer M2) were determined according to

Analytical techniques

1 H NMR spectra were recorded using a Bruker AC 200 with CDCl 3 as solvent. Chemical shifts were referenced to the peak of residual CHCl 3 at 7.26 ppm. 13 C NMR spectra were recorded using a Bruker AC 300 with CDCl 3 as solvent. Chemical shifts were referenced to CDCl 3 at 77 ppm. Fourier Transform Infrared (FTIR) spectra were recorded with a Perkin

Elmer Spectrum 100 spectrometer equipped with an attenuated total reflectance (ATR) crystal.

Synthesis of the TMSPMA telomer: P(TMSPMA)

Telomerizations were performed with TMSPMA monomer in acetonitrile with different molar ratios R 0 (R 0 =n telogen /n monomer ) from 0.05 to 0.3. In a typical telomerization of TMSPMA using 2-mercaptoethanol, 12.41 g (0.05 mol) of TMSPMA, 0.39 g (0.005 mol) of 2mercaptoethanol were introduced in a 50 mL two necked flask equipped with a condenser and a septum and the solution was diluted to 50 mL with anhydrous acetonitrile. The solution was bubbled with nitrogen for 30 min before heating at 80 °C. Finally, 0.041 g (2.5.10 -4 mol) of AIBN (C 0 =n initiator /n monomer =0.5%) in 1 mL of anhydrous acetonitrile were added through the septum with a syringe. After 10 h of reaction, the polymer was washed three times with anhydrous n-hexane to remove the unreacted monomer and telogen, and was dried under vacuum.

Synthesis of P(TMSPMA-stat-PFDA) cotelomer

Cotelomerizations of TMSPMA with PFDA comonomer were performed in acetonitrile with various initial molar contents of PFDA (5%, 10%, 15%). In a typical cotelomerization, 2.235 g (0.009 mol) of TMSPMA, 0.518 g (0.001 mol) of PFDA, 0.078 g (0.001 mol) of 2mercaptoethanol were introduced in a 10 mL two necked flask equipped with a condenser and a septum and the solution was diluted to 10 mL with anhydrous acetonitrile. The solution was bubbled with nitrogen for 30 min before heating at 80°C. Finally, 0.0082 g (5.10 -5 mol) of AIBN (C 0 =n initiator /n monomer =0.5%) in 0.2 mL of anhydrous acetonitrile were added through the septum with a syringe. At the conclusion of the reaction, the copolymer was washed three times with anhydrous n-hexane to remove the unreacted monomer and telogen, and was dried under vacuum.

Telomerization kinetics

All the kinetics were studied following the monomer and telogen concentrations versus reaction time. Each reaction was monitored by sampling and each aliquot was quenched in ice in order to stop the reaction. The thiol conversion of each sample was evaluated using titration of SH groups with a 0.002 mol.L -1 iodine solution prepared from iodine standard solution (0.1049 mol.L -1 ) according to the following equation:

+ I 2 RCH 2 SSCH 2 R + + 2H + 2I - 2RCH 2 SH (1)
The thiol concentration ([RSH]) versus time is given by the following equation:

[ ] ( ) [ ] ( ) t V RSH x 0 V RSH eq 0 eq t = (2)
where V eq (0) and V eq (t) are equivalent to the volume of iodine solution added to the thiol solution at the begining of the reaction and at t min, respectively. [RSH] t , [RSH] 0 are the thiol concentrations at different times of reaction and at t=0, respectively. [I 2 ] is the concentration of the iodine solution used for the titration.

The monomer consumption was calculated by 1 H NMR with CDCl 3 as deuterated solvent.

Each sample was analyzed by comparing the integration ratio of the monomer to that of polymer one.

Results and discussion

Telomerization of TMSPMA: P(TMSPMA)

Free-radical telomerization of TMSPMA: P(TMSPMA) was performed with AIBN as initiator in acetonitrile at 80°C (Scheme 2, way 1). In the kinetic study of radical telomerization of TMSPMA, two functional mercaptans were tested as telogen: mercaptoacetic acid and 2-mercaptoethanol. The influence of the carboxylic acid or hydroxyl end-group of the two telogens on the control of molecular weights was undertaken. Boutevin et al. have shown that mercaptoacetic acid has a high transfer constant for acrylic acid and methyl methacrylate even if the transfer process could be minimized using a mixture of solvents such as water/THF (80/20). [1][2][3] In our case, the utilization of mercaptoacetic acid as telogen agent to polymerize TMSPMA yielded monoadduct and nonpolymeric products. Furthermore, side reactions based on trimethylsilyloxy groups reactivity occured like their hydrolysis and the reaction with carboxylic acid groups. In addition, the cleavage of S-H bond is less important with an alcohol en-group than an acid owing to its smaller electron withdrawing effect. Consequently, transfer reactions are less significant for 2-mercaptoethanol and allow a better control of molecular weights.

CH 2 C C O O CH 2 CH 2 CH 2 Si OCH 3 OCH 3 OCH 3 CH 3 HO CH 2 CH 2 SH AIBN, 80°C CH 3 CN CH 2 C C CH 3 O O CH 2 CH 2 CH 2 Si OCH 3 OCH 3 OCH 3 HO CH 2 CH 2 S n H CH 2 C C O O CH 2 CH 2 CH 2 Si OCH 3 OCH 3 OCH 3 H CH 2 CH C O O CH 2 CH 2 C 8 F 17
In the kinetic study, P(TMSPMA)s were synthesized using various R 0 ratios (R 0 =n telogen /n monomer ). The chain transfer constant C T (C T =k tr /k p ) is the driving parameter for the control of the molecular weights and their distribution. In systems with C T >1, "telomerization" will result in a complex mixture of telomers and polymers whereas a value smaller than 1 involves a better consumption of monomer and desired oligomers are obtained.

Among the various methods for the determination of the C T , [1] Mayo's method [25] requires the plot of the inverse of the 0 n DP versus R 0 as follows:

0 T o o T n R C [M] [T] C DP 1 0 = = (3)
where [T] 0 and [M] 0 are the concentration of telogen and monomer at t = 0, respectively.

Thus, if 1/ 0 n DP is plotted as a function of the R 0 ratio, C T value can be directly deduced.

Experimentally, C T values are determined at very low conversion. The O'Brien method [26] takes into account the conversion rate of the reactants and consists in following the monomer and the telogen conversions during the reaction. C T value was deduced at high conversion yield from the straight line of ln 3). The C T values range from 0.57 for R 0 =0.3 to 0.73 for R 0 =0.05 with a correlation coefficient of the linear relationship (R 2 ) close to 0.99-0.98 (Table 1). The C T value of TMSPMA telomerization with 2-mercaptoethanol (C T =0.65 ± 0.08) means that transfer reactions occur and that a control of the molecular weights can be achieved. Moreover a C T value inferior to 1 means that the relative rate of consumption of the transfer agent is lower than the propagation rate which is in agreement with the shift between the plots in Figure 2. Chain transfer constant depends on the structures of monomer and telogen [4] and it is therefore difficult to compare our results with the C T values found in the literature. However, the values measured (C T =0.65) are in the same order of magnitude as those found by Robin et al. [6] who reported a C T value of 0.72 for 2-(dimethylamino)ethyl methacrylate in acetonitrile and 0.57 in benzene at 70°C. Teodorescu et al. also obtained a C T value of 0.62 for methyl methacrylate in benzene at 60°C. [7] Finally acetonitrile was a solvent able to provide an efficacious transfer avoiding strong hydrogen bonds or dipole-dipole interactions between the solvent and the monomer. Moreover acetonitrile is a good solvent for PFDA co-units for further study of copolymerisation of TMSPMA with PFDA. The theoretical number-average polymerization degree th n DP was calculated as the molar ratio of initial monomer and transfer agent i.e. the inverse of R 0 . The equation ( 7) could be used because the calculations were achieved at the end of the reaction after 10 hours of reaction for conversion rate close to one:

([T] 0 /[T] t ) versus ln([M] 0 /[M] t ): [ ] [ ] t 0 T t o [M] [M] ln C T T ln = (4
0 0 0 th n [T] [M] R 1 DP = = (7)
The theoretical ( th n DP ) and experimental ( exp n DP ) values are listed in Table 1 where it can be seen that higher experimental values in respect to theoretical ones are obtained. These results are in good agreement with the C T values which are lower than 1 meaning that the transfer process is less efficient than the propagation process (Figure 2).

Cotelomerization of TMSPMA with PFDA: P(TMSPMA-stat-PFDA)

To determine the higher content in PFDA fluoromonomer that can be incorporated in the P(TMSPMA-stat-PFDA) copolymer, cotelomerizations were carried out with different initial molar ratios in PFDA fluoromonomer (5%, 10% and 15%) (Scheme 2, way 2). Experiments were performed using 2-mercaptoethanol as transfer agent and AIBN as initiator in acetonitrile at 80°C. A kinetic study of the homotelomerization of PFDA and the calculation of C T value could not be realized owing to the poor solubility of fluorine moieties in organic solvents and their precipitation. Consequently, these copolymers could not be analysed by size-exclusion chromatography because of their insolubility in the solvents usually employed as eluent. Perrier et al. observed the same phenomenon when studying the synthesis of statistical copolymers of methyl methacrylate with PFDA where a 20% maximum content could be reached. [9][10] In our case, the phase separation did not occur until 10% of PFDA in P(TMSPMA-stat-PFDA). Thus, TMSPMA co-units might have a twofold beneficial role, that is, the improvement of solubility in usual solvents and the availability of SiOMe 3 groups for further grafting from process.

In both case, the conversion of PFDA and TMSPMA monomers versus time were linear throughout the first hour of reaction with a faster consumption of PFDA than TMSPMA (Figure 5). After this reaction time a gradual decrease of PFDA consumption has been observed. For 10% of initial PFDA content, the final composition of P(TMSPMA-stat-PFDA) (at t=5h) is 92% of TMSPMA moieties and 8% of PFDA moieties. It confirms that for high monomers conversion TMSPMA reacts preferentially than PFDA and thus the copolymer is enriched in TMSPMA units. Monomer reactivity ratios are very important parameters for the elucidation of copolymer structure (copolymer composition, monomer sequence distribution) and kinetics (propagation rate coefficients), so, the reactivity ratios of TMSPMA and PFDA were determined. Given the diminution of PFDA consumption, kinetic study was limited to the first hour of reaction (Figure 5). The r 1 and r 2 ratios indicate the reactivity of a given free radical face to monomers 1 and monomer 2. The ratios depend on the structure of the two monomers and of the corresponding radicals. They are independent of the copolymerization rate and of the structure of the initiator. Among the numerous calculation methods of the reactivity ratio in radical copolymerization, Jaacks [27] and Macret [28] methods were selected to evaluate the reactivity ratios of TMSPMA and PFDA (r TMSPMA =r 1 , r PFDA =r 2 ). Jaacks's method is based on the approximation of the quasi-stationnary state where the concentrations of both radicals are quasi-constant. Furthermore, this method requires a great excess in one monomer in respect with the other one to consider only one type of radical. If M 1 is in excess in regard to M 2 , the instantaneous composition of the copolymer is described by the following equation:

2 1 1 2 1 ) ( ) ( M M r M M = δ δ (8)
that can be integrated under the following form:

0 2 2 1 0 1 1 ] [ ] [ ln ] [ ] [ ln M M r M M t t = (9)
Thus, plotting

0 1 1 ] [ ] [ ln M M t versus 0 2 2 ] [ ] [ ln M
M t allows to calculate r 1 as the slope of the straight line. Jaacks's method was only used to determine the reactivity ratio of TMSPMA owing to polymer precipitation which prevents the study of the cotelomerization with an excess of PFDA and the calculation of the reactivity ratio of PFDA. The reactivity ratio r 1 was obtained from a monomer molar ratio TMSPMA/PFDA: 90/10. The evolution of the reaction versus time was monitored using 1 H NMR analysis, measuring the integration of double bonds of each monomer signal (H d cis and H e ), and calculating their corresponding theoretical values according to the reference peak (Si-OCH 3 : H f ) at 3.57 ppm (Figure 6). In the equation (11) The reactivity ratio r 1 was given by the value of the plot of ln([TMSPMA] t /[TMSPMA] 0 ) versus ln([PFDA] t /[PFDA] 0 ) as illustrates in Figure 7. A value of 0.65 for r 1 was found which means that radicals from TMSPMA moieties react preferentially on PFDA monomer. Macret's method was also used to confirm this r 1 value and to determine the r 2 value. [28] This graphical method is issued from Ezrielev et al. works [START_REF] Ezrielev | Vysokomol Soedin[END_REF] with the following equation:

( ) 2 1 r Y X r 1 Y Y X 2 - = - (12)
where ] [ ] [ In contrast with Y value, X value can be easily calculated. Assuming that consumption of monomer is a first order reaction, one can write equations 13 and 14:

t α i i i 0 t e ] [M ] [M - = (13)
] [ ] [ values were obtained using the slope (r TMSPMA ) and the origin of the curve (r PFDA ) of Macret's equation (Figure 8). In addition, r TMSPMA ×r PFDA value close to 1 indicates that statistical copolymers are synthesized. The comparison of the r TMSPMA values ensuing of Macret's and Jaacks' methods reveals a good agreement between these two methods (Table 2). For cotelomerization of TMSPMA with PFDA, Macret's method provides r TMSPMA <1 and r PFDA >1. The first value means that the TMSPMA terminated propagating chain reacts preferentially with PFDA monomer according cross-propagation reaction whereas the r PFDA value means PFDA terminated propagating chain reacts preferentially with PFDA monomer according homopolymerization reaction. Consequently, the copolymer composition should be enriched in PFDA content at the beginning of the reaction. Moreover we have shown that after one hour of reaction, consumption of PFDA decreases (Figure 5). In conclusion at the early stage of the polymerization (first hour), the resulting chains are enriched in PFDA monomer, forming short sequences of PFDA units. At higher monomer conversion, the monomer feed is enriched in less reactive TMSPMA radicals, so that longer sequences of TMSPMA units are formed. All of these results indicate a drift of the copolymer composition with increasing conversion. The determination of the reactivity ratios of TMSPMA or PFDA with other monomer has not been attempted yet. One example of copolymerization of 2-(trimethoxysilyloxy)ethyl methacrylate (TMSEMA) with PFDA by ATRP has been found. [24] The extended Kelen-Tüdos method was used to determine r TMSEMA =1.58 and r MMA =0.59 and illustrated that the copolymers had only a soft gradient composition. Perrier et al. reported the kinetic study of copolymerization of MMA with PFDA and they showed that conversion of each monomer was almost identical throughout the reaction indicating similar reactivity ratios. [9][10] Thus, it is difficult to compare these results according to the different experimental conditions and the different methods to determine the reactivity ratios.

Conclusion

Hydroxyl terminated copolymers were prepared by free-radical telomerization of monomers bearing perfluoro pendant chains and SiOCH 3 reactive groups. Homotelomerization of TMSPMA reveals the efficiency of 2-mercaptoethanol as transfer agent in acetonitrile and the inert nature of SiOCH 3 groups since no by-products were observed. Further, synthesis of copolymers based on TMSPMA and PFDA was successfully achieved. The incorporation of PFDA moieties in P(TMSPMA-stat-PFDA) was limited to 10% due to the insolubility of copolymers with higher contents. Then, the reactivity ratios were determined by Jaacks and Macret's methods. These methods show that r PFDA >1 and r TMSPMA <1: a statistical P(TMSPMA-stat-PFDA) copolymer. Thus, telomerization seems to be a convenient way to synthesize functionalized end-group copolymers by incorporating fluorine groups. Thus, these preliminary results will be applied to the surface treatment of silica particles in order to modify their properties and their processing. These results will be published in a further paper.

Scheme 1 :

 1 Scheme 1: Mechanism of the radical telomerization using AIBN and 2-mercaptoethanol as

Scheme 2 :

 2 Scheme 2: Telomerization of TMSPMA using AIBN as initiator and 2-mercaptoethanol as

)Figure 1 : 1 H

 11 Figure 1: 1 H NMR spectra (CDCl 3 ) of P(TMSPMA) (R 0 =1) versus time, before precipitation.

Figure 2

 2 Figure 2 reports monomer and telogen concentrations versus time. The monomer

Figure 2 :

 2 Figure 2: Evolution of monomer and telogen concentrations versus time (R 0 =0.1).

Figure 3 :

 3 Figure 3: Ln([ME] 0 /[ME] t ) versus ln([TMSPMA] 0 /[TMSPMA] t ) for the polymerization of

  hexane and dried under vacuum to remove the both unreacted monomer and telogen. The experimental number-average polymerization degree ( exp n DP ) was determined by 1 H NMR analysis (Figure 4) through the integration of the peaks located at 3.91 ppm (H b : CH 2 in α position of the ester function of the telomer) and at 2.68 ppm (H c : CH 2 in β position of hydroxyl function of the telomer) following the equation:

Figure 4 : 1 H

 41 Figure 4: 1 H NMR spectrum of P(TMSPMA) in CDCl 3 after precipitation in n-hexane

Figure 5 :

 5 Figure 5: Ln[M] 0 /[M] t as a function of time for TMSPMA and PFDA monomers (10 mol. %

Figure 6 : 1 H

 61 Figure 6: 1 H NMR spectrum of P(TMSPMA-stat-PFDA) in CDCl 3 before precipitation.

Figure 7 :

 7 Figure 7: ln[TMSPMA] 0 /[TMSPMA] t versus ln[PFDA] 0 /[PFDA] t for the copolymerization of

  concentration ratio in the medium and the monomer concentration ratio inserted in the copolymer respectively as a function of time.

α 1 From

 1 and α 2 representing the conversion rate of M 1 and M 2 respectively. a function of time, α TMSPMA =0.0045 and α PFDA =0.0068 were deduced from the coefficient of the slopes as shown in Figure5. These values allow to calculate Y and thus to report the plot X(Y-1)/Y versus X 2 /Y. Thus, r 1 and r 2

Figure 8 :

 8 Figure 8: X/Y(Y-1) as a function of X 2 /Y for the copolymerization of TMSPMA with PFDA

Table 1 :

 1 C T and

	n DP values according to R 0 for the telomerization of TMSPMA by 2-
	mercaptoethanol.
	R 0	C T	R 2	DP	th n	exp n DP *
	0.05	0.73	0.999	20	26.3
	0.1	0.70	0.983	10	12.8
	0.2	0.60	0.984	5		6.7
	0.3	0.57	0.988	3.3	4.3
	*obtained from 1 H NMR analysis.

After 10h of reaction, the synthesized oligomers were washed three times with anhydrous n-

H d CIS H e CIS H e CIS H d CIS

														the
	integration from H d cis to H f was successively divided by 9 corresponding to the three Si-OMe
	by molecule and by 9 because the initial monomer molar ratio TMSPMA/PFDA is 90/10. The
	conversion rates for TMSPMA and PFDA were expressed by these equations:
			[TMSPMA] [TMSPMA] 0 t	=	H H f ∫ ∫	ppm) ppm) 3.57 5.54 (at (at /9 e	(10)
			[PFDA] [PFDA] 0 t	=	(at (at /81 cis d f H H ∫ ∫	3.57 6.40	ppm) ppm)	(11)
								H
								C			CH 3
						H			C					CH 3
								O	C		O		HO	CH 2	CH 2	S	CH 2	CH	n	CH 2	C	H m
		H	H					CH 2	CH 2 CH 2	Si OCH 3 OCH 3 OCH 3	H f	O CH 2	C C 8 F 17 O CH 2	O CH 2	C CH 2 CH 2	O Si	OCH 3	H f
	H	C	CH											H f	OCH 3 OCH 3
			C
		O	O
		CH 2
			CH 2
			C 8 F 17
			7	6,5			6				5,5	5	4,5	4	3,5	3
													(ppm)

d TRANS H e TRANS H d' H e TRANS H d' H d TRANS

Table 2 :

 2 Comparison of the reactivity ratios of TMSPMA and PFDA.

	Method	r TMSPMA	r PFDA	r TMSPMA ×r PFDA
	Macret	0.66	1.51	0.99
	Jaacks	0.65	---	---

Acknowledgements:

The authors thank the financial support by CREAT -Centre de Recherche de la Division Chargeurs Interlining du groupe Chargeurs.


Emiliano Lorini

Franc ¸ois Schwarzentruber

A logic for reasoning about counterfactual emotions

 Horty, 2001]. STIT logic has been proposed in the domain of formal philosophy in the nineties and, more recently, it has been imported into the field of theoretical computer science where its formal relationships with other logics for multiagent systems such as ATL and Coalition Logic (CL) have been studied. STIT is a very suitable formalism to reason about choices and capabilities of agents and groups of agents. Unfortunately, the version of STIT with agents and groups has been recently proved to be undecidable. In this work we study a decidable fragment of STIT with agents and groups which is sufficiently expressive for our purpose of formalizing counterfactual emotions.

Introduction

A major objective of AI is to develop interactive cognitive systems that are more attractive and closer to the users and that can be considered as believable interlocutors. In this perspective, a challenge for AI is to build artificial agents which are capable: to reason about emotions, to predict and understand human emotions, and to process emotions in reasoning and during their interaction with a human user. With the aim of creating a new generation of emotional interaction systems, the study of affective phenomena has become a "hot" topic in AI where the domain of Affective Computing [Picard, 1997] has emerged in the last few years.

Recently, some researchers have been interested in developing logical frameworks for the formal specification of emotions (see [Meyer, 2006;[START_REF] Steunebrink | [END_REF]El-Nasr et al., 2000] for instance). Their main concern is to exploit logical methods in order to provide a rigorous specification of how emotions should be implemented in an artificial agent. The design of agent-based systems where agents are capable to reason about and to display some kind of emotions can indeed benefit from the accuracy of logical methods.

Although the application of logical methods to the formal specification of emotions has been quite successful, there is still much work to be done in the field of computational and logical modeling of 'counterfactual emotions'. In line with psychological theories of 'counterfactual emotions', we use this term to denote those emotions such as regret which arise during 'counterfactual thinking', that is, when "[...] reality is compared to an imagined view of what might have been." [Kahneman and Miller, 1986, p. 136]. In other terms, counterfactual emotions are based on an agent's alteration of a factual situation and in the agent's imagination of an alternative situation that could have realized if something different was done [START_REF] Roese | [END_REF].

The aim of our work is to advance the state of the art on computational modeling of emotions by providing a logic which supports reasoning about this kind of emotions. Our major concern here is to find a fair trade off between expressivity and complexity of the formalism. We want a logic which is sufficiently expressive to capture the fundamental constituents of counterfactual emotions and, at the same time, with good mathematical properties in terms of decidability and complexity. To this aim, we exploit a well-known logic called STIT [Belnap et al., 2001;Horty, 2001]. STIT logic has been proposed in the domain of formal philosophy in the nineties and, more recently, it has been imported into the field of theoretical computer science where its formal relationships with other logics for multi-agent systems have been studied (see [START_REF] Broersen | [END_REF] for instance). It is a very suitable formalism to make counterfactual reasoning about choices of agents and of groups of agents. Unfortunately, the version of STIT with agents and groups proposed in [Horty, 2001] has been recently proved to be undecidable [START_REF] Herzig | Properties of logics of individual and group agency[END_REF]. In this work we study a decidable fragment of this logic which is sufficiently expressive for our purpose of formalizing counterfactual emotions.

The paper is organized as follows. In Section 2 we introduce a fragment of the version of STIT with agents and groups proposed in [Horty, 2001]. Differently from Horty's logic, we prove that this fragment is decidable. Section 3 is devoted to characterize in our STIT fragment counterfactual statements of the form "group J (or agent i) could have prevented χ to be true". These are indeed fundamental constituents of counterfactual emotions. In Section 4 we provide an extension of our STIT fragment with knowledge opera-tors. This is a necessary step in order to capture the subjective dimension of the affective phenomena we intend to analyze. The last part of the paper (Section 5) is devoted to the formalization of two kinds of counterfactual emotions: regret and rejoicing.

2 A decidable fragment of STIT STIT (the logic of "Seeing to it that") is a modal logic framework dealing with what agents and groups of agents do and can do. More precisely, STIT supports reasoning about the effects of actions of agents and groups, and about the capabilities of agents and groups to ensure certain outcomes. In [Belnap et al., 2001] the language of STIT without groups is studied: a complete axiomatization of STIT without groups is provided and the logic is proved to be decidable. The extension of STIT with groups has been proposed in [Horty, 2001]. Unfortunately, in [START_REF] Herzig | Properties of logics of individual and group agency[END_REF]] it has been proved to be undecidable.

We here introduce a decidable fragment of STIT with agents and groups called df STIT which is sufficiently expressive to formalize counterfactual emotions.

Syntax

Let n be a strictly positive integer. Let ATM be a countable set of atomic propositions and let AGT = {1, . . . , n} be a countable set of agents. The language L STIT of the logic STIT with agents and groups proposed by Horty [Horty, 2001] is defined by the following BNF:

ϕ ::= p | ϕ ∧ ϕ | ¬ϕ | [J]ϕ
where p ranges over ATM and J over 2 AGT . J ϕ is an abbreviation of ¬[J]¬ϕ. Operators of type [J] are used to describe the effects of the action that has be chosen by J. If J is a singleton we refer to J as an agent, whereas if J has more than one element we refer to J as a group. We call joint actions the actions chosen by groups. If J has more than one element the construction [J]ϕ means "group J sees to it that ϕ no matter what the other agents in AGT \ J do". If J is a singleton {i} the construction [{i}]ϕ means "agent i sees to it that ϕ no matter what the other agents in AGT \ {i} do". For notational convenience, we write [i] instead of [{i}]. [∅]ϕ can be shorten to "ϕ is necessarily true". The dual expression ∅ ϕ means "ϕ is possibly true". Note that the operators ∅ and [J] can be combined in order to express what agents and groups can do: ∅ [J]ϕ means "J can see to it that ϕ whatever the other agents in AGT \ J do".

The STIT fragment we are interested in here is called df STIT and is defined by the following BNF: -to-it and "can" formulas) where p ranges over ATM and J over 2 AGT \ ∅.

χ ::= ⊥ | p | χ ∧ χ | ¬χ (propositional formulas) ψ ::= [J]χ | ψ ∧ ψ (see-to-it formulas) ϕ ::= χ | ψ | ϕ ∧ ϕ | ¬ϕ | ∅ ψ (see

Models

Here we give two semantics of STIT. It is proved in [START_REF] Herzig | Properties of logics of individual and group agency[END_REF]] that these two semantics are equivalent. The first one corresponds to the original semantics of STIT with agents and groups given in [Horty, 2001].

The other one is based on the product logic S5 n [Gabbay et al., 2003] and is used in the proof of decidability of the satisfiability problem of a df STIT -formula (Theorem 1). Let us give first the original semantics of STIT. Definition 1. A STIT-model is a tuple M = (W, {R J } J⊆AGT , V ) where:

• W is a non-empty set of possible worlds or states;

• For all J ⊆ AGT, R J is an equivalence relation over W such that:

1. R J ⊆ R ∅ ; 2. R J = j∈J R {j} ; 3. for all w ∈ W , for all (w j ) j∈AGT ∈ R ∅ (w) n , j∈AGT R {j} (w j ) = ∅; 4. R AGT = id W . • V is a valuation function, that is, V : W → 2 ATM .
As in the previous Constraint 3, it is convenient to view relations on W as functions from W to 2 W , that is, for every J ∈ 2 AGT , R J (w) = {v ∈ W | wR J v}. R J represents the actual action chosen by J: if wR J v then v is an outcome of the action chosen by J at w. We recall that R ∅ is the relation over all possible outcomes: if w is the current world and wR ∅ v then v is a possible outcome at w. Thus, Constraint 1 on STIT models just means that all outcomes brought about by J are possible outcomes. Constraint 2 just says that the set of outcomes brought about by J at a given world w is equal to the pointwise intersection of the sets of outcomes brought about by the agents in J at w. Constraint 3 expresses a socalled assumption of independence of agents: if w 1 , . . . , w n are possible outcomes at w then the intersection of the set of outcomes that agent 1 brings about at w 1 , and the set of outcomes that agent 2 brings about at w 2 ,..., and the set of outcomes that agent n brings about at w n is not empty. More intuitively, this means that agents can never be deprived of choices due to the choices made by other agents. Constraint 4 expresses an assumption of determinism: the set of outcomes brought about by all agents is a singleton.

Truth conditions for atomic formulas and the boolean operators are entirely standard. For every J ∈ 2 AGT , the truth condition of the modal operator [J] is:

M, w |= [J]ϕ iff M, v |= ϕ for all v ∈ W such that wR J v.
The alternative semantics of STIT is based on the product logic S5 n . It is defined as follows:

Definition 2. A product STIT-model is a tuple M = (W, V ) where: • W = W 1 × • • • × W n where W i are non-empty sets of worlds or states; • V is a valuation function, that is, V : W → 2 ATM .
The truth conditions for the modal operators [J] in product STIT-models are:

M, (w 1 , . . . , w n ) |= [J]ϕ iff M, (v 1 , . . . , v n ) |= ϕ for all (v 1 , . . . , v n ) ∈ W such that v j = w j if j ∈ J. A formula ϕ is STIT-valid (noted |= STIT ϕ) iff ϕ is
true in every world of every STIT-model (or product STIT-model).

Decidability

Our fragment df STIT of STIT with agents and groups has interesting computational properties. In particular: Theorem 1. The problem of satisfiability of df STIT is NPcomplete. This is implied by the following fact: Theorem 2. If a df STIT -formula is satisfiable, then it is in a polynomial sized model. Sketch of proof. The main idea of the proof is a selection-ofpoints argument as in [Ladner, 1977]. 1 Let ϕ a satisfiable formula: there exists a product STIT-model M = (W, V ) and w such that M, w |= ϕ. First, we construct product STIT-model M satisfying ϕ with selected points of the initial model M:

• In the construction, we take care to create a new point in M for each subformula ∅ ψ of ϕ true in M.

• We also take care to construct enough points so that all subformulas ∅ ψ and [J]χ of ϕ false in M are false in M as well.

Secondly we make sure that M is polynomial sized and that there is a point w so that M , w |= ϕ.

Counterfactual statements in STIT

The following counterfactual statement is a fundamental constituent of an analysis of counterfactual emotions: (*) group J (or agent i) could have prevented a certain state of affairs χ to be true now.

Our STIT fragment enables a formal translation of it. We note CHP J χ this translation, where CHP J χ is defined as follows:

CHP J χ def = χ ∧ ¬[AGT \ J]χ.
The expression ¬[AGT \ J]χ just means that: the complement of J with respect to AGT (i.e. AGT \ J) does not see to it that χ (no matter what the agents in J have chosen to do). This is the same thing as saying that: given what the agents in AGT \ J have chosen, there exists an alternative joint action of the agents in J such that, if the agents in J did choose this action, χ would be false now. Thus, χ and ¬[AGT \ J]χ together correctly translate the previous counterfactual statement (*).

EXAMPLE. Imagine a typical coordination scenario with two agents AGT = {1, 2}. Agents 1 and 2 have to take care of a plant. Each agent has only two actions available: water the plant (water ) or do nothing (skip). If either both agents water the plant or both agents do nothing, the plant will die (dead ). In the former case the plant will die since it does not tolerate too much water. In the latter case it will die since it lacks water. If one agent waters the plant and the other does nothing, the plant will survive (¬dead ). The scenario is represented in the STIT model in Fig. 1. For instance both at 1 Space restrictions prevent from giving the extensive proof of the theorem. The interested reader might read it at the anonymous URL http://rapidshare.com/files/182384345/proofdfstitNP.pdf The following are some interesting properties of the operator CHP J . If J 1 ⊆ J 2 then:

|= STIT (CHP J1 χ 1 ∨ CHP J1 χ 2 ) ↔ CHP J1 (χ 1 ∨ χ 2 ) (1) |= STIT CHP J1 χ → CHP J2 χ (2) |= STIT CHP J1 (χ 1 ∧ χ 2 ) → (CHP J1 χ 1 ∧ CHP J1 χ 2 ) (3)
Proof. We give the proof of Validity 2 as an example. Let M be a STIT-model and w ∈ W such that M, w |= CHP J1 χ. We have M, w |= χ and M,

w |= ¬[AGT \ J 1 ]χ. As R AGT\J1 ⊆ R AGT\J2 , it implies that M, w |= ¬[AGT \ J 2 ]χ. That is why we have M, w |= CHP J2 χ.
According to Validity 1 J 1 could have prevented χ 1 or χ 2 to be true if and only if, J 1 could have prevented χ 1 or could have prevented χ 2 . Validity 2 expresses a monotonicity property: if J 1 is a subset of J 2 and J 1 could have prevented χ then, J 2 could have prevented χ as well. Finally Validity 3 shows how the operator CHP J behaves over conjunction: if J 1 could have prevented χ 1 and χ 2 then, J 1 could have prevented χ 1 and could have prevented χ 2 separately. REMARK. It is worth noting that counterfactual statements of the form "group J (or agent i) could have prevented χ to be true", which are expressible in STIT, are not expressible in other well-known logics of multi-agent interaction such as Alternating-time temporal logic (ATL) [Alur and Henzinger, 2002] and Coalition Logic (CL) [Pauly, 2002]. This is due to the fact that STIT is more expressive than ATL and CL (this is formally proved in [START_REF] Broersen | [END_REF]) so that there are STIT formulas such as [J]χ and ¬[J]χ that cannot be translated into ATL and CL.

A STIT extension with knowledge

This section presents an extension of the fragment df STIT of STIT logic presented in section 2 with standard operators for knowledge of the form K i , where K i ϕ means "agent i knows that ϕ is true". This is a necessary step for the formalization of regret and relief that will be presented in section 5.

The language L df KSTIT of logic df KSTIT is defined by the following BNF:

χ ::= ⊥ | p | χ ∧ χ | ¬χ (propositional formulas) ψ ::= [J]χ | ψ ∧ ψ (see-to-it formulas) ϕ ::= χ | ψ | ϕ ∧ ϕ | ¬ϕ | ∅ ψ | K i ϕ
(see-to-it, "can", knowledge formulas ) where p ranges over ATM , i ranges over AGT and J over

2 AGT \ ∅. Definition 3. A KSTIT-model is a tuple M = (W, {R J } J⊆AGT , {E i } i∈AGT , V ) where: • (W, {R J } J⊆AGT , V ) is a STIT-model;
• For all i ∈ AGT, E i is an equivalence relation.

Truth conditions for atomic formulas and the boolean operators are again entirely standard. Truth conditions for the STIT operators [J] are given in section 2. Truth conditions for knowledge operators are defined in the standard way:

M, w |= K i ϕ iff M, v |= ϕ for all v ∈ W such that wE i v.
That is, agent i knows that ϕ at world w in model M if and only if ϕ is true at all worlds that are indistinguishable for agent i at world w.

A formula ϕ is KSTIT-valid (noted |= KSTIT ϕ) iff ϕ is true in every world of every KSTIT-model. Theorem 3. The satisfiability problem of df KSTIT is NP-complete if card(AGT) = 1 and PSPACE-complete if card(AGT) ≥ 2. Sketch of proof. If card(AGT) = 1, we can notice that there are only three operators: [∅], [{1}], and K 1 . Nevertheless, the operator [{1}] can be removed because we force R AGT = id W in our models. As no K 1 -operator can appear after a [∅]-operator, we can prove by a selected points argument (in [Ladner, 1977], it is done for S5) that if a df KSTIT -formula is KSTIT-satisfiable, then it is in a polynomial sized model.

If card(AGT) ≥ 2, we can create a tableau method to see if a df KSTIT -formula ϕ is KSTIT-satisfiable. The idea consists in using a classical tableau algorithm for the knowledge part considering the STIT-subformulas as propositions (see tableau method for S5 n in [START_REF] Halpern | [END_REF]). At each step, we take care of STIT-subformulas by choosing non-deterministically a polynomial sized STIT-model. We can prove correctness and soundness of this tableau method. This algorithm runs using a polynomial space memory so the satisfiability problem of df KSTIT is NPSPACE.

As NPSPACE = PSPACE (Savitch's theorem [Papadimitriou, 1994]), it is PSPACE. It is PSPACE-hard because the logic S5 n is embedded into df KSTIT .

Regret and rejoicing: a formalization

In order to provide a logical characterization of counterfactual emotions such as regret, we need to introduce a concept of agent's preference. Modal operators for desires and goals have been widely studied (see e.g. [Cohen and Levesque, 1990;[START_REF] Meyer | [END_REF]). The disadvantage of such approaches is that they complicate the underlying logical framework. An alternative, which we adopt in this paper is to label states with atoms that capture the "goodness" of these states for an agent. Our approach supposes a binary relation of preference between worlds.

Let us introduce a special atom good i for every agent i ∈ AGT . These atoms are used to specify those worlds which are positive for an agent.

We say that χ is good for agent i if and only if, necessarily if the current state is a good state for agent i then, χ is true in that state. Formally:

GOOD i χ def = [∅](good i → χ).
Now, we are in a position to define the concept of desirable state of affairs. We say that χ is desirable for agent i if and only if, i knows that χ is something good for him:

DES i χ def = K i GOOD i χ.
As the following valid formulas highlight, every operator DES i satisfies the principle K of normal modal logic, and the properties of positive and negative introspection: χ is (resp. is not) desirable for i if and only if i knows this.

|= KSTIT (DES i χ 1 ∧ DES i (χ 1 → χ 2 )) → DES i χ 2 (4) |= KSTIT DES i χ ↔ K i DES i χ (5) |= KSTIT ¬DES i χ ↔ K i ¬DES i χ (6)
We have now all necessary and sufficient ingredients to define the cognitive structure of regret and to specify its counterfactual dimension. Such a dimension has been widely studied in the psychological literature on regret (see [START_REF] Kahneman | [END_REF]Kahneman, 1995] for instance). Our aim here is to capture it formally. We say that an agent i regrets for χ if and only if ¬χ is desirable for i and i knows that it could have prevented χ to be true now. Formally:

REGRET i χ def = DES i ¬χ ∧ K i CHP i χ.
The following example is given in order to better clarify this definition.

EXAMPLE. Consider the popular two-person hand game "Rock-paper-scissors" (Roshambo). Each of the two players AGT = {1, 2} has three available actions: play rock, play paper, play scissors. The goal of each player is to select an action which defeats that of the opponent. Combinations of actions are resolved as follows: rock wins against scissors, paper wins against rock; scissors wins against paper. If both players choose the same action, they both lose. The scenario is represented in the STIT model in Fig. 2. It is supposed winning is something good for each agent and each agent has the desire to win the game: GOOD 1 1Win, GOOD 2 2Win, DES 1 1Win and DES 2 2Win are true at worlds w 1 -w 9 . Suppose world w 1 is the actual world in which 1 plays rock and 2 plays paper. In this world 1 loses the game (¬1W in), and 1 knows that (by playing scissors) it could have prevented ¬1Win to be true (i.e. K 1 CHP 1 ¬1Win is true at w 1 ). It follows that at w 1 player 1 regrets for having lost the game, that is, REGRET 1 ¬1Win is true at w 1 .

As the following validity highlights, regret implies the frustration of an agent's desire:

|= KSTIT REGRET i χ → (K i χ ∧ DES i ¬χ) (7)
More precisely, if agent i regrets for χ then, i knows that χ holds and ¬χ is something desirable for i (in this sense i feels frustrated for not having achieved ¬χ). Moreover, regret satisfies the properties of positive and negative introspection:

|= KSTIT REGRET i χ ↔ K i REGRET i χ (8) |= KSTIT ¬REGRET i χ ↔ K i ¬REGRET i χ (9)
That is, i regrets (resp. does not regret) for χ if and only if i knows this.

As emphasized by some psychological theories of counterfactual emotions (see [START_REF] Zeelenberg | [END_REF] for instance), the positive counterpart of regret is rejoicing: while regret has a negative valence (i.e. it is associated with the frustration of an agent's desire), rejoicing has a positive valence (i.e. it is associated with the satisfaction of an agent's desire). According to these theories, a person experiences regret when believing that the foregone outcome would have been better if she did a different action, whilst she rejoices when believing that the foregone outcome would have been worse if she did a different action. More precisely, an agent i rejoices for χ if and only if, χ is desirable for i and, i knows that it could have prevented χ to be true now by doing a different action. Formally:

REJOICE i χ def = DES i χ ∧ K i CHP i χ.
EXAMPLE. Consider again the game "Rock-paper-scissors" represented by the STIT-model in Fig. 2. Suppose world w 2 is the actual world in which player 1 plays rock and player 2 plays scissors. In this world player 1 is the winner (1W in) and it knows that (by playing paper or scissors) it could have prevented 1Win to be true (i.e. K 1 CHP 1 1Win is true at w 2 ). Since DES 1 1Win holds at w 2 , it follows that at w 2 player 1 rejoices for having won the game, that is, REJOICE 1 1Win is true at w 2 .

The following validity highlights that rejoicing implies desire satisfaction:

|= KSTIT REJOICE i χ → (K i χ ∧ DES i χ) (10)
More precisely, if agent i rejoices for χ then, i knows that χ and χ is something desirable for i (in this sense i feels satisfied for having achieved χ). As regret, rejoicing satisfies the properties of positive and negative introspection:

|= KSTIT REJOICE i χ ↔ K i REJOICE i χ (11) |= KSTIT ¬REJOICE i χ ↔ K i ¬REJOICE i χ (12)

Related works

As emphasized in the introduction, there are other researchers who have exploited logical methods in order to build formal models of emotions and affective agents.

In [Meyer, 2006;[START_REF] Steunebrink | [END_REF] a logical approach to emotions based on the modal logical framework KARO [START_REF] Meyer | [END_REF] is proposed. KARO is a framework based on a blend of dynamic logic with epistemic logic, enriched with modal operators for motivational attitudes such as desires and goals. In Meyer et al.'s approach each instance of emotion is represented with a special predicate, or fluent, in the jargon of reasoning about action and change, to indicate that these predicates change over time. For every fluent a set of effects of the corresponding emotion on the agent's planning strategies and decision processes are specified, as well as the preconditions for triggering the emotion. Although Meyer et al. provide a very detailed formal analysis of emotions, they do not take into account counterfactual emotions. This is also due to some intrinsic limitations of the KARO framework in expressing counterfactual reasoning and statements of the form "agent i could have prevented χ to be true" which are fundamental constituents of this kind of emotions. Indeed, standard dynamic logic on the top of which KARO is built, is not suited to express such statements. On the contrary our STIT-based approach overcomes this limitation.

In [El-Nasr et al., 2000] a formal approach to emotions based on fuzzy logic is proposed. The main contribution of this work is a quantification of emotional intensity based on appraisal variables like desirability of an event and its likelihood. For example, following [Ortony et al., 1988], in FLAME the variables affecting the intensity of hope with respect to the occurrence of a certain event are the degree to which the expected event is desirable, and the likelihood of the event. However, in FLAME only basic emotions like joy, sadness, fear and hope are considered and there is no formal analysis of counterfactual emotions as the ones analyzed in our work. Indeed, the formal language exploited in [El-Nasr et al., 2000] is not sufficiently expressive to model counterfactual reasoning about agents' choices and actions.

Conclusion

Directions for our future research are manifold. An analysis of intensity of regret and rejoicing was beyond the objectives of the present work. However, we intend to investigate this issue in the future in order to complement our qualitative analysis of affective phenomena with a quantitative analysis. Moreover, we have focused in this paper on the logical characterization of two counterfactual emotions: regret and rejoicing. We intend to extend our analysis in the future by studying the counterfactual dimension of "moral" emotions such as guilt and shame. Several psychologists (see [Lazarus, 1991] for instance) have stressed that guilt involves the conviction of having injured someone or of having violated some norm or imperative, and the belief that this could have been avoided.

It has been proved in [START_REF] Herzig | Properties of logics of individual and group agency[END_REF]] that the logic STIT with agents and groups proposed by Horty [Horty, 2001] is not only undecidable but also not axiomatizable, i.e. there is no axiomatization Ax such that that for every formula ϕ in L STIT , Ax ϕ if and only if |= STIT ϕ. In this work, we have dealt with the first problem by presenting a decidable fragment of Horty's logic which is sufficiently expressive for our purpose of formalizing counterfactual emotions. It is still an open question whether we can find a finite and complete axiomatization for our fragment df STIT of STIT with agents and groups. Our future work will also be devoted to solve this problem in order to come up with a complete and decidable fragment of STIT with agents and groups.

Figure 2 :

 2 Figure2: Again, vertical circles represent the actions that player 1 can choose, whereas horizontal circles represent the actions that player 2 can choose. For the sake of simplicity, we suppose that players 1 and 2 do not have uncertainty: everywhere in the model players 1 and 2 only consider possible the world in which they are (reflexive arrows represent indistinguishability relations for the two players).

  Figure1: The four worlds w 1 , w 2 , w 3 and w 4 are in the equivalence class determined by R ∅ . Vertical circles represent the actions that agent 1 can choose, whereas horizontal circles represent the actions that agent 2 can choose. For example, w 1 is the world that results from agent 1 choosing the action water and agent 2 choosing the action skip.

world w 2 and w 4 , formulas CHP 1 dead and CHP 2 dead are true: each agent could have prevented the plant to be dead. Indeed, at world w 2 , dead and ¬[2]dead are true: given what agent 2 has chosen (i.e. water), there exists an alternative action of agent 1 (i.e. skip) such that, if 1 did choose this action, dead would be false now. At world w 4 , dead and ¬[2]dead are also true: given what agent 2 has chosen (i.e. skip), there exists an alternative action of agent 1 (i.e. water) such that, if 1 did choose this action, dead would be false now. The case for agent 2 is completely symmetrical.

This works has been supported by the Agence Nationale de la Recherche, contract No. ANR-08-CORD-009.