Murielle Rivenet
email: murielle.rivenet@ensc-lille.fr

Nicolas Vigier

Pascal Roussel

Francis Abraham

Hydrothermal synthesis

Keywords: Diamine uranyl vanadates, Crystal structure, Hydrothermal synthesis, layered structure

come

Introduction

The solid state chemistry of uranyl -containing inorganic compounds is very rich in diversity. In particular, the association of hexavalent uranium polyhedra and oxoanion such as silicate, phosphate, vanadate, molybdate, tungstate. . . constitutes a true building set leading to structures with varied architectures and dimensionalities, although there is a tendency to form layered structure due to the presence of uranyl bonds that preclude the connection in a third dimension. The basic building units are the uranium polyhedron, which can be hexagonal bipyramid, pentagonal bipyramid or distorted octahedron and the oxoanion which can be tetrahedron, square bipyramid or octahedron. The structural arrangement depends on many factors, such as the U/oxoanion ratio, that influence the degree of polymerization between the uranium polyhedra which can be connected directly through equatorial oxygen atoms or through the oxo -anions polyhedra.

Our group of research mainly focuses on the solid state chemistry of uranyl-vanadates compounds [1][2][3][4][5][6][7]. In almost the studied oxides, the association of uranyl polyhedra through vanadate oxoanions leads to bi-dimensional anionic sheets with contercation in the interlayer space. However in a recent paper, we described a novel three-dimensional open -framework uranyl-vanadate obtained using small monovalent ions (Li and Na) [8].

Owing to the potential applications for this type of materials (radioactive waste management, uranium geochemistry, ion-exchange and catalysis), a series of reactions was conducted in order to obtain new open-framework uranyl-vanadates. One approach for designing novel layered and microporous uranium-bearing materials is to exert influence over structural features through the introduction of templating agents. Amines have historically been used as charge balancing, space filling and templating agents in various systems including molybdate [9][10][11][12][13], phosphate [14][15][16][17][18], phosphite [19], selenate [20], selenite [21], arsenate [17] sulfate [22][23][24][25][26][27][28][29][30][31] and fluoride containing compounds [18,[31][32][33][34][35][36][37][38][39] but organically templated uranyl-vanadates systems remained unexplored so far.

The materials described in the present paper were obtained by means of reactions of inorganic species and U/V ratio equal 1 with excess of amines under hydrothermal conditions. Two linear diamines of different length (1,2-ethylenediamine (en), 1,3-diaminopropane (dap)) and three cyclical diamines based upon piperazine (piperazine (pip), 1-methylpiperazine (mpip) and 1,4-diazabicyclo [2,2,2]octane (dabco)) were employed so as to vary the structures of the organic templates from a reaction to an other. Syntheses led to six new layered uranylvanadates which crystal structure and thermal behavior are reported herein.

Experimental

Synthesis

Uranyl nitrate (UO 2 (NO 3 ) 2 •6H 2 O -Prolabo, R.P. normapur), vanadium oxide (V 2 O 5 -Merck, Extra pur), concentrated chlorydric acid (Carlo Erba, 37%, d=1.186) and amines (listed in Table 1) were used as received. For each synthesis the solutions were heated to 180°C in Teflon -lined 23mL digestion bombs for a time varying from 1 to 30 days. The resulting powders were collected after cooling to ambient temperature, filtration and washing with deionized water. Reaction yields were not quantitatively determined.

(NH 4 ) 2 [(UO 2 ) 2 V 2 O 8 ] (1) was synthesised using UO 2 (NO 3 ) 2 •6H 2 O (301.3mg, 0.600mmol), V 2 O 5 (54.6mg, 0.300mmol), 1,2 -ethylenediamine (C 2 H 8 N 2 -90.2mg, 1.5mmol) and deionized water (15.1g, 840mmol). The mixture was heated at 180°C for 30 days. Absence of ammonium in the starting reactants indicates that the formation of ammonium ions must involve the in situ decomposition of ethylenediamine. Amines decomposition under hydrothermal conditions has previously been observed using pyridine [40], imidazole [41], and guanidinium amines [42][43][44][45]. It is worth noting that we were enable to obtain (NH 4 ) 2 [(UO 2 ) 2 V 2 O 8 ] using a more direct source of ammonium cations.

(H 2 en)[(UO 2 ) 2 V 2 O 8 ]
(2) was prepared by hydrothermal reaction using a mixture of UO 2 (NO 3 ) 2 •6H 2 O (301.3mg, 0.600mmol), V 2 O 5 (54.6mg, 0.300mmol), 1,2 -ethylenediamine (C 2 H 8 N 2 -90.2mg, 1.5mmol) and deionized water (15.1g, 840mmol) added with chlorydric acid (HCl -109.4mg, 3mmol). Time of synthesis was limited to 1 day in order to avoid the ethylenediamine decomposition.

(H 2 dap)[(UO 2 ) 2 V 2 O 8 ] (3), (H 2 pip)[(UO 2 ) 2 (VO 4 ) 2 ]•0.8H 2 O (4), (H 2 dmpip)[(UO 2 ) 2 V 2 O 8 ]
(5) and (H 2 dabco)[(UO 2 ) 2 (VO 4 ) 2 ] (6) were obtained by conducting hydrothermal reactions at 180°C for 2 days using a mixture of UO 2 (NO 3 ) 2 •6H 2 O (301.3mg, 0.600mmol), V 2 O 5 (54.6mg, 0.300mmol), HCl (109.4mg, 3mmol), deionized water (15.1g, 840mmol) and, respectively, C 3 H 10 N 2 (111.2mg, 1.5mmol), C 4 H 10 N 2 (129.2mg, 1.5mmol), C 5 H 12 N 2 (150.3mg, 1.5mmol) and C 6 H 12 N 2 (168.3mg, 1.5mmol). During the synthesis of 5 under the hydrothermal conditions used, some 1-methylpiperazine (mpip) dismuted in 1,-dimethylpiperazine (dmpip) and piperazine (pip). The single crystals grew from mdpip when the presence of pip in the remaining solution was unambiguously proved by 13 C RMN.

The purity of the compounds was checked using X-ray powder diffraction. The X-ray powder patterns of the bulk samples can be fully indexed on the basis of the theoretical data calculated from the crystal structure results which evidences that pure phases are obtained.

For compounds 1, 4, 5 and 6 crystals suitable for single crystal X-ray diffraction experiments have been isolated under optical microscope.

Crystal structure determination

Selected crystals of 1, 4, 5 and 6 were mounted on a glass fibber and aligned on a Bruker SMART CCD X-ray diffractometer. Intensities were collected at room temperature using MoKα (λ = 0.71073 Å) radiation selected by a graphite monochromator. The individual frames were measured using a ω -scan technique. Omega rotation and acquisition time were fixed at 0.3° and 20 s per frame, respectively. 1800 frames were collected in order to cover the full sphere. The Bruker program SAINT [START_REF]SAINT Plus Version 6.22[END_REF] was used for intensity data integration and correction for Lorentz, polarisation and background effects. After data processing, absorption corrections were performed using a semi-empirical method based on redundancy with the SADABS program [START_REF]SADABS Version 2.03: Bruker Analytical X -ray Systems[END_REF]. Details of the data collection and refinement are given in Table 2.

The crystal structures were determined in the centrosymmetric space groups P2 1 /c for 1, P2 1 /b for 5 and C2/m for both 4 and 6. A non-conventional setting was chosen for 5 so that the layers described hereafter are always in the (1 0 0 ) plane for 1, 5, 2 and 3 compounds.

The heavy atoms (U, V) positions were established by direct methods using SIR97 program [START_REF] Altomare | [END_REF]. The oxygen, nitrogen and carbon atoms were localised from difference Fourier maps.

The last cycles of refinement included atomic positions and anisotropic displacement parameters ADP for all atoms, excepted for partially occupied sites. Full-matrix least-squares structure refinements against F were carried out using the JANA2000 program [49]. The atomic positional parameters and displacement parameters are given in Table 3 -compound 1, Table 4 -compound 5 and Table 5 -compounds 4 and 6. Some selected interatomic distances are reported in Table 6 -compounds 1 and 5, and Table 7 -compounds 4 and 6.

As no suitable crystals could be found for the 2 and 3 samples, crystal structure models were checked using X-ray powder diffraction data. The data were collected by means of a Huber G670 diffractometer using an asymmetric Guinier flat sample transmission geometry, equipped with a 2D detector (Image Plate) covering the 2θ range [ 8.

For compound 2, the b and c parameters correspond to a carnotite -type layer, however the space group symmetry operations are incompatible with such a structure and as we could not find any structurally related compound, we attempted to solve the structure of compound 2 by ab initio procedures. The pattern decomposition option of the JANA2000 package [49] was used to extract corrected structure factors from a limited region of the diffractogram (6<2θ<60°). The pattern was fit without any structural model by refining the overall parameters: background, unit-cell parameters, zero-point error, peak shape (pseudo-Voigt). The refinement converged to Rwp = xx and Rp = xx%. A total of xxx reflections were used as input to the direct-methods SIR97 program [START_REF] Altomare | [END_REF]. The positions of two U atoms were derived from this method. At this stage, the positions of the V atoms were deduced from a difference-Fourier map. Then, after refining the heavy atoms position and ADP, another difference-Fourier map revealed O atoms. At this stage we used soft constraints in the U -O and V -O bonds to avoid the structure blow up and to keep a reasonable geometry for the anionic sheets. Unfortunately, probably due to the high contrast between the U and (C -N -H) atoms of the amines, we were not able to see these last types of atomic species in a last difference-Fourier map. Without these "light" atoms and refining all positional parameters Rwp dropped to 1.80 %. Refining isotropic temperature factors freely resulted in some negative values. It is well known that temperature factors, for complex structures with heavy cations and medium resolution X-ray powder data, are quite unreliable. Hence, we decided to refine an overall isotropic temperature factor for U and V atoms and to fixe an overall temperature factor for the oxygen atoms. The refined positional parameters are reported in Table 9.

For the compound 3, the values of the cell parameters and the XRD pattern are very similar to those of uranyl vanadates of divalent A cations built from carnotite type layers. The space group deduced from the X -ray powder pattern indexation, Pmcn, is adopted for A = Ca [50], Mn, Co [START_REF] Saadi | El Jadida[END_REF], Ni, Cd, Zn [START_REF] Cesbron | [END_REF]. The precise lattice parameters were obtained from profile matching and, using the structural model of the divalent carnotite-type compounds layers, the reliability factors reported in Table 8 are obtained.

High temperature X -ray diffraction

The high temperature X-ray powder diffraction patterns were recorded using a Guinier Lenné moving film camera. The samples were deposited on the sample holder (gold grid) using an ethanol slurry which yielded, upon evaporation, a regular layer of powdered compound. The high temperature X-ray diffraction patterns were recorded in the temperature range from 20 to 600°C with a heating rate in the range [12 -15° h -1 ].

Thermal analysis

The thermal analyses were performed on a Setaram TG-DT 2-16.18 apparatus.

Analyses were undertaken in air, in the temperature range from 20 to 600°C, with a heating rate of 300° h -1 , in platinum crucibles.

Results

Cation coordination polyhedra

For all the studied compounds, each of the uranium atom is strongly bonded to two oxygen atoms forming a nearly linear uranyl cation (UO 2 ) 2+ with a O -U -O bond angle ranging from 178.6(4) to 179.5(3)° and U -O bond lengths ranging from 1.775(8) to 1.816 (6) Å (average value of 1.79(2) Å). The uranyl cations are coordinated by five oxygen atoms located in the equatorial plane which forms [UO 7 ] pentagonal bipyramids. The equatorial oxygen ligands show significant variations with U -O distances ranging from 2.295(6) to 2.458(3) Å. However, the average value, 2.36(6) Å, is in good agreement with the average bond length of 2.37(9) Å calculated for uranyl polyhedra of numerous well -refined structures [53].

The vanadium atoms of structures 1, 2, 3 and 5 are pentacoordinated by five oxygen atoms in a square pyramidal arrangement. Two VO 5 square pyramids related by an inversion centre share an O -O edge to form a V 2 O 8 dimeric unit. Within the VO 5 square pyramids, the apical V -O bond is shorter than the vanadium -oxygen distances of the square base. This vanadyl V -O bond distance is close to that calculated in various carnotite-type compounds [54] and V 2 O 5 [55].

In structures 4 and 6, the vanadium atoms occupy one crystallographic site with tetrahedral environment. The tetrahedra are slightly distorted with V -O distances in the range from 1.721(7) to 1.763(5) Å when the oxygen atoms are shared with a UO 7 polyhedron and shorter distances with O(5) atom not involved in uranium coordination, 1.62(1) and 1.611(6) Å for 4 and 6, respectively. Bond -valence sums were calculated using parameters given by Burns et al. [53] for U -O bonds and by Brese and O'Keeffe [56] for V -O bonds. Calculations resulted in values ranging from 6.00(2) to 6.19(3) v.u. for U 6+ and from 5.08(5) to 5.14(4) v.u. for V 5+ with oxygen valences ranging from 1.57(2) to 2.18(2) v.u.. The lowest sums correspond to O atoms not shared between UO 7 and VO 4 or VO 5 polyhedra.

Structural connectivity

The structural building unit block, with labelled scheme, constituted from two edgeshared UO 7 pentagonal bipyramids and two edge-shared VO 5 square pyramids is shown on Fig. 1 for compounds 1, 2 and 5.

Compound

1 is isotypic with the mineral carnotite K 2 [(UO 2 ) 2 V 2 O 8 ] and other A 2 [(UO 2 ) 2 V 2 O 8 ]•nH 2 O compounds
where A is a monovalent ion [45], CsUNbO 6 [57] and [58]. The structural arrangement between the edgeshared dimers, V 2 O 8 , and the edge-and corner-shared pentagonal bipyramids, UO 7 , further linked by edge-sharing, forms sheets of composition [(UO 2 ) 2 V 2 O 8 ] 2 -parallel to (1 0 0) (Fig. 2b). The ammonium anions are located in the interlayer space and insure the cohesion of the structure.

A 2 [(UO 2 ) 2 Cr 2 O 8 ]•nH 2 O (A=K, Rb Cs)
Compound 3 is built from the same layers. The layers packing along a-axis depends upon the interleaving cation. In 1, as in almost the monovalent containing

A 2 [(UO 2 ) 2 V 2 O 8 ]•xH 2 O
compounds, adjacent layers noted P are deduced by a translation, resulting in the simple PPPP sequence (Fig. 3a). In compound 3 there is a second layer, labelled b, deduced from P by a (1 0 0) mirror plane. P and b layers alternate to yield a PbPb sequence (Fig. 3b). Such a sequence was previously evidenced in

A[(UO 2 ) 2 V 2 O 8 ]•xH 2 O
compounds containing a divalent A cation [50][START_REF] Saadi | El Jadida[END_REF][START_REF] Cesbron | [END_REF].

Using the description developed by Burns et al. [59], the francevillite anion topology of the [(UO 2 ) 2 V 2 O 8 ] 2-sheets is represented in Fig. 2a. Compounds 2 and 5 are built from sheets with the same anion topology and with the same occupation of pentagons by U, squares by V and empty triangles, but, in contrast with all the carnotite-type layer containing compounds described up today (Fig. 2b), half of the V 2 O 8 units are reversed compared to the P sheets (Fig. 2c). A V 2 O 8 dimmer can be referenced as ud with a tetragonal pyramid that point up and one down. In P layers the dimers alternate, along [0 1 0], to form the isomer ud/du, in opposite, the new anionic layer, labelled P' hereafter, represents the ud/ud geometrical isomer. In compound 5, a layer b' is deduced from P' by a two fold axis running along c axis so as the stacking sequence is P'b'P'b' (Fig. 3d) while the stacking sequence in structure 2 can be described as P'P'P'P' (Fig. 3c). Along [0 0 1] the V 2 O 8 dimmers are parallel.

Compounds 4 and 6 are isotypic and contain the same [(UO 2 ) 2 (VO 4 ) 2 ] 2-layer built from (UO 5 ) ∞ zig-zag chains of edge shared UO 7 pentagonal bipyramids running down the baxis further connected by VO 4 tetrahedra. The uranyl vanadate layer has the uranophane sheets anion topology (Fig. 4) adopted by many mineral or synthetic inorganic or hybrid organic-inorganic uranyl compounds. In 4 and 6, all the tetrahedra that share edges with one side of a uranyl chain point down (d), and all the tetrahedra along the other side point up (u) which corresponds to the geometrical isomer du/du as defined by Locock et al. [60]. The three dimensional structure results from the alternate stacking of inorganic layers and sheets of protonated amines and occluded water molecules.

It should be noticed that with triethylamine, Locock and Burns [17] obtained the same structural arrangement but with another geometrical isomer of the uranyl arseniate layer whereas with dabco molecules an autunite-type uranyl arseniate layer is formed.

Interlayer space occupation

In order to study the role of amine on the structural arrangement of the uranylvanadates, the location and connectivity of amines in the interlayer space was systematically investigated for the single crystal studied compounds.

Projected along [1 0 0], the interleaving NH 4 + cations in 1 appears at the centre of the inoccupied triangles of the francevillite anion -type topology (Fig. 5). The with the ionic radii of the 8 -coordinated A + ions [61]. Using the value of ionic radius reported by Khan and Bauer [62] for the 8-coordinated NH 4 + (1.66 Å), the corresponding point does not fit with this straightforward variation, so according to Shannon [61] for the 6coordination, we should conclude that NH 4 + is not different in size from Rb + .

Although in 2 and 3 the diprotonated amines could not be localised one can imagine that they separate the inorganic layers from one another creating spacing of approximately 6.98 and 7.37Å between uranium atom planes and stabilize the structures, both through balancing the negative charge of the inorganic layer, and donating hydrogen bonds.

In observed in diamine templated uranium sulfates [26].

Thermal behaviour

For compound 1, a one-step decomposition from

(NH 4 ) 2 [(UO 2 ) 2 V 2 O 8 ] to (UO 2 ) 2 V 2 O 7
[63] corresponding to 6.1% weight loss in accordance with the 6.5% theoretical loss is observed between 375 and 470°C on the TGA curve. The thermal behavior is confirmed by high temperature X-ray diffraction experiment.

For all the amine bearing materials (2 to 6), the decomposition of the amine and the modification of the uranyl -vanadate arrangement occur in several steps between 280 and 550°C. On the high temperature X-ray diffraction patterns a non -crystalline zone is observed between the starting uranyl -vanadate and the final product (UO 2 ) 2 V 2 O 7 [61]. The total weight losses are in agreement with the calculated values for the transformation from (Diamine)[uranyl-vanadate] to (UO 2 ) 2 V 2 O 7 (exp. / theor. (%): 9.0/9.4 -11.3/10.9 -11.5/12.0 -14.8/14.9 -14.0/14.6 for structures 2 -6 respectively).

Conclusion.

For the five studied structure directing diamines, uranyl vanadates layers with formula (UVO 6 ) 2-are formed. Two types of layers were obtained. For compounds 1, 2, 3 and 5, the [(UO 2 ) 2 V 2 O 8 ] 2-layers are similar to that of the mineral carnotite with two geometrical isomers distinguished by the orientations of the V 2 O 8 units. For compounds 4 and 6, the uranophanetype layers are built from chains of edge-shared UO 7 pentagonal bipyramids connected through VO 4 tetrahedra. In all the compounds, the diprotonated amines reside between the inorganic layers, balancing charge and donating hydrogen bonds to the layers. For all compounds, thermal decomposition led to the divanadate (UO 2 ) 2 V 2 O 7 . Further experiments using other amine structure directing agents to built three-dimensional uranyl-vanadate frameworks are in progress. Chemical formula

Piperazine (pip) ACROS, 99% 3
1 -Methylpiperazine (mpip) ACROS, 99% 4
1,4 -Diazabicyclo[2,2 5
(NH 4 ) 2 [(UO 2 ) 2 V 2 O 8 ] (H 2 pip)[(UO 2 ) 2 (VO 4 ) 2 ].0,8H 2 O (H 2 dmpip)[(UO 2 ) 2 V 2 O 8 ] (H 2 dabco)[(UO 2 ) 2 (VO 4 ) 2 ]

Crystallographic data

Formula weight (g.mol -

9 ≤ h ≤ 9 -17 ≤ h ≤ 17 -12 ≤ h ≤ 12 -23 ≤ h ≤ 23 -11 ≤ k ≤ 11 -7 ≤ k ≤ 7 -13 ≤ k ≤ 13 -9 ≤ k ≤ 9 -13 ≤ l ≤ 13 -7 ≤ l ≤ 7 -10 ≤ l ≤ 10 -9 ≤ l ≤ 9 No. of

  hour and half an hour to a monochromatized Cu-Kα 1 radiation obtained with a Germanium Johanson monochromator. For the ethylenediamine containing compound 2, the powder was dehydrated before measurement by heating the sample at 200°C for 1 hour. For the two compounds, structural models containing only the uranium -vanadium -oxygen sheets with distances constrained to the values calculated for the compounds 5 and 1, respectively are used and led to the results reported in Table

  Fig. 6 shows the variation of the inter-layer distance asinβ for the A 2 [(UO 2 ) 2 V 2 O 8 ] compounds

5 ,

 5 the [H 2 dmpip] 2+ cations, with a chair conformation, are aligned along [0 0 1] in the space between the [(UO 2 ) 2 V 2 O 8 ] 2-layers. The mean plane of the diamines is inclined of about 45° from the layer. As evidenced on Fig. 5, the arrangement of the diamines in the interlayer space is incompatible with the ud/du isomer. The two nitrogen atoms of the [H 2 dmpip] 2+ cation donate their hydrogen bonds to the apical oxygen atoms O(2) of two parallel uranyl-vanadate layers with a N1 ---O2 distance of 2.98(2) Å. In 4 and 6, the [H 2 pip] 2+ and [H 2 dabco] 2+ cations reside between the inorganic layers and are located above the VO 4 tetrahedra of the [(UO 2 ) 2 (VO 4 ) 2 ] 2 -uranophane layers that points down. In 4, the interleaving [H 2 pip] 2+ molecules lie almost parallel to (1 0 0) plane with no strong hydrogen bonds with the uranyl vanadate layer : instead, the shortest N --O distances, 2.71(2) Å involves an oxygen of the interleaving water molecules. The [H 2 pip] 2+ molecules adopt a boat conformation which can be present in two possible orientations related by a mirror parallel to (0 0 1). Thus, the site occupancy factors for the C(1), C(2) and N(1) atoms, located in a general position, were fixed at 0.5. In 6, the pseudo-trigonal [H 2 dabco] 2+ cations (Fig. 7) can be found with two possible orientations modelled by half occupying the general positions of C(1), C(2) and N(1) atoms. Moreover, a disorder presented by the C(3) -C(4) group, was taken in account by fixing at 0.25 the site occupancy for C(3) and C(4) atoms, located in a general position. The [H 2 dabco] 2+ molecule are slightly distorted with C(1) -C(2) and C(3) -C(4) bond lengths respectively equivalent to 1.67(2) and 1.64(5)Å and N(1) -C bond lengths varying within the range [1.40 -1.55] Å. The cations are oriented such as their ammonium moities are directed toward the corner sharing oxygen atoms O(3) of the VO 4 -UO 7 units with a strong N ---O(3) hydrogen bond lying along the N -N axis of 2.78(1) Å. That indicates the presence of a three-dimensional hydrogen bonds network that constrains the [H 2 dabco] 2+ cations to lie almost perpendicular to the layers as previously

Symmetry codes for 1

 1 (i) -x, -y,1 -z ; (ii) -x,1 -y,1 -z ; (iii) x,3/2 -y,1/2+z ; (iv) x,1/2 -y,1/2+z(v) ; -x,1 -y, -z ; (vi) -1+x,1/2 -y,3/2+z ;(vii) -1 -x, -y,1 -z ; for 5 (i) 1 -x,1/2 -y,1/2+z; (ii) x,1/2+y, -1/2 -z; (iii) 1 -x,1 -y, -1 -z ; (iv) 1 -x, -y, -1 -z ; (v) 1 -x,1/2 -y, -1/2+z ; (vi) -x,1 -y, -1 -z

  Figure caption.

Fig. 1 :Fig. 2 :

 12 Fig. 1: The structure building unit block formed of two edge-shared UO 7 pentagonal bipyramids and two edge-shared VO 5 square pyramids further connected by edge with the labelled scheme for a) compounds 1 and 5, b) compound 2. Fig. 2: The francevillite anion topology (a) and the uranyl vanadate layers in carnotite -type compounds: (b) ud/du isomer layers P in M + -, M 2+ -or 1, 3 -diaminopropane -containing compounds; (c) ud/ud isomer layers P' in ethylenediamine -and 1,4 -dimethylpiperazinecontaining compounds.

Fig. 3 :

 3 Fig. 3: Stacking of the P and P' layers in carnotite type compounds containing monovalent ions and 1 (a), divalent inorganic ions and 3 (b), 1,4 -dimethylpiperazine 5 (c) and ethylenediamine (not localized from X -ray powder diffraction data) 6 (d).

Fig. 4 :

 4 Fig. 4: The uranophane type uranyl vanadate sheet in the structure of 4 and 6.

Fig. 5 :

 5 Fig. 5: Localisation of the ammonium cation above the ud/du geometrical isomer layer in 1 (a), dimethylpiperazine ions above the ud/ud geometrical isomer layer in 5 (b), piperazine and dabco above the uranophane-type layer in 4 and 6, (c) and (d), respectively.

Fig. 6 :

 6 Fig. 6: Variation of the interlayer distance versus the ionic radii of 8-coordinated monovalent ion A 2 [(UO 2 ) 2 V 2 O 8 ] compounds built on carnotite-type layers.

Fig. 7 :

 7 Fig. 7: One orientation of the pseudo-trigonal [H 2 dabco] 2+ cations showing the disorder presented by the C(3) -C(4) group. The general positions of C(1), C(2) and N(1) atoms are half occupied.

Table 1 :

 1 Amine molecules of the studied UO 3 -V 2 O 5 -amine systems.

	Name	Formula	Origin	Compound
	1,2 -Ethylenediamine (en)	H 2 N (CH 2 ) 2 NH 2	ACROS, 99%, d=0.899	1 and 2
	1,3 -Diaminopropane (dap)	H 2 N (CH 2 ) 3 NH 2	ACROS, 99%, d=0.870

Table 2 :

 2 Crystal data, intensity collection and structure refinement parameters for (NH 4 ) 2 [(UO 2 ) 2 V 2 O 8 ] (1), (H 2 pip)[(UO 2 ) 2 (VO 4 ) 2 ].0,8H 2 O (4), (H 2 dmpip)[(UO 2 ) 2 V 2 O 8 ] (5) and (H 2 dabco)[(UO 2 ) 2 (VO 4 ) 2 ] (6).

Table 3 .

 3 Atomic coordinates and isotropic displacement parameters (in Å 2 ) for (NH 4 ) 2 [(UO 2 ) 2 V 2 O 8 ] (1).

	Atom	Wyck.	x	y	z	U eq
	U1	4e	0.01539(4)	0.47727(3)	0.81974(2)	0.0110(2)
	V1	4e	0.1147(2)	0.8516(2)	0.0557(2)	0.0122(4)
	O1	4e	0.2776(8)	0.4265(7)	0.8795(5)	0.020(2)
	O2	4e	-0.2457(9)	0.5260(6)	0.7598(5)	0.019(2)
	O3	4e	0.0406(8)	0.0607(6)	0.1079(5)	0.016(2)
	O4	4e	0.0418(8)	0.6553(6)	-0.0044(4)	0.016(2)
	O5	4e	0.3484(9)	0.8708(7)	0.0648(5)	0.025(2)
	O6	4e	0.0930(8)	0.7909(6)	0.2192(5)	0.016(2)
	N1	4e	-0.458(2)	0.7697(9)	0.8626(7)	0.029(3)

Table 4 .

 4 Atomic coordinates and isotropic displacement parameters (in Å 2 ) for (H 2 dmpip)[(UO 2 ) 2 V 2 O 8 ] (5)

	Atom	Wyck.	x	y	z	U eq
	U1	4e	0.48421(4)	0.47738(4)	-0.31760(3)	0.0125(2)
	V1	4e	0.4327(2)	0.1248(2)	-0.4674(2)	0.0149(7)
	O1	4e	0.6969(8)	0.5718(8)	-0.3076(7)	0.024(3)
	O2	4e	0.2702(7)	0.3856(8)	-0.3308(7)	0.020(3)
	O3	4e	0.5371(8)	0.0699(8)	-0.5981(6)	0.017(3)
	O4	4e	0.5028(8)	0.3475(8)	-0.5114(6)	0.016(3)
	O5	4e	0.2473(7)	0.0417(8)	-0.4949(7)	0.022(3)
	O6	4e	0.4657(8)	0.1971(7)	-0.2998(6)	0.019(3)
	N1	4e	0.004(2)	0.488(2)	-0.3662(9)	0.061(6)
	C1	4e	0.095(2)	0.648(2)	-0.424(2)	0.041(5)
	C2	4e	-0.113(2)	0.350(2)	-0.436(2)	0.043(5)
	C3	4e	-0.015(2)	0.487(2)	-0.222(2)	0.062(8)

Table 5 .

 5 Atomic coordinates and isotropic displacement parameters (in Å 2 ) for (H 2 pip)[(UO 2 ) 2 (VO 4 ) 2 ].0,8H 2 O (4) (bold type) and (H 2 dabco)[(UO 2 ) 2 (VO 4 ) 2 ] (6) (italic type).

	Atom	Site	Occ.	x	y	z	U eq/ iso*
	U1	4i		0.75640(3)	0	-0.11646(6) 0.0118(2)
	U1	4i		0.75602(2)	0	-0.11813(3) 0.0138(2)
	V1	4i		0.7204(2)	0	-0.6690(2)	0.0108(6)
	V1	4i		0.71988(7)	0	-0.6700(2)	0.0138(3)
	O1	4i		0.8722(5)	0	-0.036(1)	0.020(3)
	O1	4i		0.8588(3)	0	-0.0535(8)	0.027(2)
	O2	4i		0.6409(5)	0	-0.191(2)	0.026(3)
	O2	4i		0.6528(3)	0	-0.1811(8)	0.027(2)
	O3	4i		0.7838(5)	0	-0.435(1)	0.020(3)
	O3	4i		0.7763(3)	0	-0.4412(7)	0.020(2)
	O4	8j		0.7433(4)	-0.1807(7)	-0.8236(7)	0.021(2)
	O4	8j		0.7428(2)	-0.1799(5)	-0.8221(5)	0.024(2)
	O5	4i		0.6179(6)	0	-0.659(2)	0.032(3)
	O5	4i		0.6289(4)	0	-0.6494(9)	0.038(2)
	O6w	4g	0.42(2)	0.5	-0.184(2)	0	0.014(6)*
	N1	8j	0.5	0.504(2)	-0.386(2)	-0.667(2)	0.025(4)*
	C1	8j	0.5	0.532(2)	-0.432(2)	-0.673(2)	0.022(5)*
	C2	8j	0.5	0.482(2)	-0.304(2)	-0.543(2)	0.026(5)*
	N1	8j	0.5	0.0660(5)	-0.037(2)	0.453(2)	0.033(3)*
	C1	8j	0.5	0.9325(9)	0.153(3)	0.371(2)	0.059(7)
	C2	8j	0.5	0.0182(8)	0.118(2)	0.296(3)	0.068(7)
	C3	8j	0.25	0.040(2)	0.163(4)	0.447(6)	0.054(9)*
	C4	8j	0.25	0.946(2)	0.154(5)	0.451(7)	0.047(9)*

Table 6 .

 6 Principal interatomic distances (Å) for 1 and 5.

		Compound 1		Compound 5
	U1 -O1	1.793(5)	U1 -O1	1.802(7)
	U1 -O2	1.782(6)	U1 -O2	1.816(6)
	U1 -O3 iv	2.295(6)	U1 -O3 i	2.340(6)
	U1 -O4 i	2.338(5)	U1 -O4	2.368(7)
	U1 -O4 ii	2.356(5)	U1 -O4 iii	2.319(7)
	U1 -O6 i	2.369(5)	U1 -O6	2.356(7)
	U1 -O6 iii	2.342(5)	U1 -O6 ii	2.327(7)
	V1 -O3 v	1.946(5)	V1 -O3	1.857(8)
	V1 -O3 v	1.899(5)	V1 -O3 iv	1.940(8)
	V1 -O4	1.784(5)	V1 -O4	1.809(7)
	V1 -O5	1.596(6)	V1 -O5	1.594(6)
	V1 -O6	1.832(6)	V1 -O6	1.853(6)
			V1 -V1 iv	2.988(3)
			N1 -O2	2.98(2)
			N1 -C1	1.42(2)
			N1 -C2	1.43(2)
			N1 -C3	1.53(2)
			C1 -C2 vi	1.48(2)

Table 7 .

 7 Principal interatomic distances (Å) for 4 and 6.

	Compound 4		Compound 6
	U1 -O1	1.775(8)	U1 -O1	1.784(5)
	U1 -O2	1.783(8)	U1 -O2	1.791(5)
	U1 -O3	2.326(7)	U1 -O3	2.306(5)
	U1 -O4 i	2.448(5)	U1 -O4 i	2.456(4)
	U1 -O4 ii	2.330(5)	U1 -O4 ii	2.339(4)
	U1 -O4 iii	2.330(5)	U1 -O4 iii	2.339(4)
	U1 -O4 iv	2.448(5)	U1 -O4 iv	2.456(4)
	V1 -O3	1.721(7)	V1 -O3	1.737(5)
	V1 -O4	1.763(5)	V1 -O4	1.747(4)
	V1 -O4 v	1.763(5)	V1 -O4 v	1.747(4)
	V1 -O5	1.62(1)	V1 -O5	1.613(7)
	C1 -C2	1.59(2)	C1 -C2 vii	1.67(2)
			C3 -C4 xiv	1.64(5)
	N1 xiv -C1	1.38(2)	N1 xii -C1	1.47(2)
	N1 xii -C2	1.54(2)	N1 v -C2	1.40(2)
			N1 -C3	1.51(3)
			N1 -C4 x	1.55(4)
	N1 i -O6w	2.71(2)	N1 vii -O3	2.78(1)
	N1 xii -O6w	2.71(2)	N1 viii -O3	2.78(1)
	Symmetry codes (i) -x,-y,1-z ; (ii) 3/2-x,-1/2-y,-1-z ; (iii) 3/2+x,1/2-y,-1+z ; (iv) 1/2+x,1/2-y,1+z ; (v)
	1/2+x,1/2-y,z ; (vi) 3/2-x,-1/2-y,-2-

z ; (vii) 1-x,-y,-z ; (viii) 1+x,-y,z ; (ix) 2+x,-y,1+z ; (x) 1+x,-y,1+z ; (xi) 3/2+x,1/2-y,z ; (xii) 1-x,-y,1-z ; (xiii) x,-y,1+z ; (xiv) -1-x,-y,-z ; (xv) -1/2+x,1/2-y,z ;(xvi) 2-x,-y,1-z

Table 9 .

 9 Atomic coordinates and isotropic displacement parameters (in Å 2 ) for (H 2 en)[(UO 2 ) 2 V 2 O 8 ] (2).

	Atom	Wyck.	x	y	z	U iso/eq
	U1	4e	0.2367(4)	-0.2646(9)	0.2617(3)	0.009(2)
	U2	4e	0.2576(5)	-0.236(1)	0.6282(3)	0.009(2)
	V1	4e	0.276(2)	0.107(2)	0.470(2)	0.009(2)
	V2	4e	0.679(2)	0.386(3)	0.092(2)	0.009(2)
	O1	4e	0.3849(6)	-0.232(3)	0.608(3)	0.006
	O2	4e	0.1306(4)	-0.242(1)	0.650(2)	0.006
	O3	4e	0.3649(3)	-0.259(3)	0.257(4)	0.006
	O4	4e	0.1086(4)	-0.2723(9)	0.265(2)	0.006
	O5	4e	0.241(2)	-0.0941(9)	0.4378(3)	0.006
	O6	4e	0.387(2)	0.144(6)	0.441(5)	0.006
	O7	4e	0.738(2)	0.5924(9)	0.0464(3)	0.006
	O8	4e	0.570(2)	0.402(5)	0.053(5)	0.006
	O9	4e	0.2202(8)	0.198(7)	0.3497(9)	0.006
	O10	4e	0.7336(3)	0.463(1)	0.248(2)	0.006
	O11	4e	0.267(2)	0.321(5)	0.5411(9)	0.006
	O12	4e	0.2627(5)	0.036(2)	0.638(2)	0.006

To whom correspondence should be addressed; E-mail: Murielle.rivenet@ensc-lille.fr

Acknowledgment.

The authors gratefully thank Catherine Méliet (UCCS, USTL) and Marc Bria (SCM RMN-RPE, USTL) for their assistance in collecting and interpreting the 13 C NMR spectra. The "Fonds Européen de Développement Régional (FEDER)", "CNRS", "Région Nord Pas-de-Calais" and "Ministère de l'Education Nationale de l'Enseignement Supérieur et de la Recherche" are acknowledged for fundings of X-ray diffractometers.

Chemical formula

Crystallographic data

Formula weight (g.mol -1 ) 832 846.


Jacques Fages

Didier Mathon

Béatrice Poirier

André Autefage

Daniel Larzul

Eliane Jean

Patrick Frayssinet

Supercritical

Supercritical processing enhances viral safety and functionality of bone allografts


François Coste
email: francois.coste@inria.fr

Jacques Nicolas
email: jacques.nicolas@inria.fr

Olgierd Unold

Witold Dyrka

Wojciech Wieczorek

International Conference on Grammatical Inference Learning local substitutable context-free languages from positive examples in polynomial time and data by reduction

Keywords: Learnability, substitutable languages, context-free grammar, normal form

To study more formally the approach by reduction initiated by ReGLiS, we propose a formal characterization of the grammars in reduced normal form (RNF) which can be learned by this approach. A modification of the core of ReGLiS is then proposed to ensure returning RNF grammars in polynomial time. This enables us to show that local substitutable languages represented by RNF context-free grammars are identifiable in polynomial time and thick data (IPTtD) from positive examples.

Introduction

The idea that strings occurring in the same contexts are substitutable, an early intuition from linguistics [START_REF] Harris | Distributional structure[END_REF], has been particularly fruitful in grammatical inference for learning context-free or mildly context-sensitive languages from positive examples. It has been the root of unsupervised empirical grammatical inference approaches targeting natural languages: seminal algorithms of the field such as EMILE [START_REF] Adriaans | Language Learning from a Categorial Perspective[END_REF][START_REF] Adriaans | The EMILE 4.1 grammar induction toolbox[END_REF], ABL (van Zaanen, 2000) or ADIOS [START_REF] Solan | Unsupervised learning of natural languages[END_REF] basically detect strings sharing some contexts (either on the basis of clustering, alignment or fixed-length window along paths in a pseudograph) and group them in substitutability classes to build accordingly a context-free (or context-sensitive) grammar. More fundamental studies have been later initiated by [START_REF] Clark | Polynomial identification in the limit of substitutable context-free languages[END_REF] to better understand these approaches. They focused on the simplest case, when it is sufficient for two strings to share one context to be substitutable everywhere, thereby defining the class of substitutable languages. They proposed SGL, a simple polynomial learning algorithm enabling to prove a first polynomial identification in the limit result for substitutable context-free languages.

While substitutability paved the way to learnability results on more and more expressive languages [START_REF] Yoshinaka | Identification in the limit of k, l-substitutable context-free languages[END_REF][START_REF] Clark | Distributional learning of some context-free languages with a minimally adequate teacher[END_REF][START_REF] Yoshinaka | Efficient learning of multiple context-free languages with multidimensional substitutability from positive data[END_REF][START_REF] Clark | Distributional learning of parallel multiple contextfree grammars[END_REF], [START_REF] Clark | Polynomial identification in the limit of substitutable context-free languages[END_REF] stated themselves that their polynomial identification in the limit result was unsatisfactory since it was obtained "by relaxing the constraint for polynomial size of the characteristic set, to merely requiring polynomial cardinality", and would "allow algorithms to use exponential amounts of computation, by specifying an exponentially long string in the characteristic set". The problem was left open for further study, asking for a consensual theoretical setting able to deal with context-free grammars.

The learnability result targeted by [START_REF] Clark | Polynomial identification in the limit of substitutable context-free languages[END_REF] is the polynomial time and data identification in the limit, defined by de la Higuera (1997), which provides a realistic learnability criterion for tractable grammatical inference from given data by bounding polynomially 1) the running time of the learning algorithm and also 2) the size of the data required in the training sample to ensure identification: Definition 1 (Polynomial time and data identification in the limit) A representation class R is identifiable in the limit from polynomial time and data (IPTD) iff there exist two polynomials p() and q() and an algorithm A such that:

• Given a training sample S of size S , A returns a representation R in R consistent with S in O (p( S ));

• For each representation R in R of size R , there exists a sample CS (named characteristic sample) of size at most O (q( R )), such that A returns a representation R equivalent with R for any sample S containing CS.

The first results by [START_REF] De | Characteristic sets for polynomial grammatical inference[END_REF] show the difficulty of learning languages in this setting: context-free grammars, linear grammars, simple deterministic grammars, and nondeterministic finite automata are not IPTD while, in contrast, deterministic finite automata are IPTD. This later result illustrates the importance of the choice of the representation for being IPTD. Deterministic automata are IPTD while non-deterministic automata are not IPTD, although they represent the same class of languages. The point is that nondeterministic finite automata can be exponentially smaller than the deterministic ones for the same language. A characteristic sample of polynomial size with respect to the deterministic target can then be too big with respect to an exponentially smaller non-deterministic automaton representing the same language, while no smaller training set may ensure the convergence towards an equivalent automaton. Although one may argue about the importance of the representation choice (see for instance [START_REF] Eyraud | Efficiency in the identification in the limit learning paradigm[END_REF], IPTD has remained the classical learnability criterion to meet for practical and non-statistical grammatical inference from given data. The most expressive grammars that have been shown to be IPTD so far are essentially subclasses of linear grammars: the even linear grammars and hierarchies built upon them, as a consequence of the positive result for deterministic automata [START_REF] Takada | Grammatical interface for even linear languages based on control sets[END_REF][START_REF] José | A characterization of even linear languages and its application to the learning problem[END_REF][START_REF] Takada | A hierarchy of language families learnable by regular language learners[END_REF], and more recently deterministic linear grammars (de la [START_REF] De | Inferring deterministic linear languages[END_REF][START_REF] Calera | Identifying left-right deterministic linear languages[END_REF].

These results have been obtained for complete samples, i.e. from positive and negative examples, but IPTD criterion is also interesting for learning from text, i.e. from positive examples only (see [START_REF] Yoshinaka | Learning efficiency of very simple grammars from positive data[END_REF], for a discussion). In that setting, the substitutabilitybased learning approach seems well suited to get IPTD results for expressive classes of grammars. The polynomial time part of IPTD has already been proven for the inference of substitutable and k, l-substitutable context-free languages [START_REF] Clark | Polynomial identification in the limit of substitutable context-free languages[END_REF][START_REF] Yoshinaka | Identification in the limit of k, l-substitutable context-free languages[END_REF]. The problem is that the characteristic sample size cannot be bounded polynomially with respect to the size of any context-free grammar representing substitutable languages. Indeed, as pointed out by [START_REF] Clark | Polynomial identification in the limit of substitutable context-free languages[END_REF], if we consider the languages consisting of the single string a 2 n (the symbol a repeated 2 n times) represented by the context-free grammars (see notations in section 2)

G n = {a}, {N i : 1 ≤ i ≤ n}, {N i → N i-1 N i-1 : 1 < i ≤ n} ∪ {N 1 → aa}, {N n } ,
the size of the unique training sample that can be used is always exponentially bigger than the grammar's size.

A solution proposed by [START_REF] Yoshinaka | Identification in the limit of k, l-substitutable context-free languages[END_REF] is to change the theoretical setting to consider not only the representation's size, but also its thickness τ R which is the maximum length of the smallest word generated from each rule (see [START_REF] Yoshinaka | Learning mildly context-sensitive languages with multidimensional substitutability from positive data[END_REF]. Requiring the size of the characteristic sample to be at most O (q( R τ R )) instead of O (q( R )) defines the identification in polynomial time and thick data (IPTtD) learnability criterion [START_REF] Yoshinaka | Learning mildly context-sensitive languages with multidimensional substitutability from positive data[END_REF][START_REF] Eyraud | Efficiency in the identification in the limit learning paradigm[END_REF].

The work presented here is based on previous research motivated by the practical application of substitutability-based learning framework to proteins. It led us to introduce a hierarchy of subclasses of substitutable languages, based on local rather than global contexts substitutability criterion, and to propose ReGLiS, an efficient algorithm learning contextfree grammars in reduced normal form for these classes directly (Coste et al., 2012a(Coste et al., ,b, 2014)). Designed to make the most from incomplete training sets, ReGLiS is not guaranteed to run in polynomial time when the training set is not characteristic. Moving from a practical to a theoretical perspective, we focus here on the core of the approach by reduction of ReGLiS. Providing a new formal characterization of grammars in reduced normal form (RNF), we propose a modification of the reduction algorithm, limiting its generalization capacities to ensure returning a RNF grammar in polynomial time, enabling us to show that local substitutable languages represented by context-free grammars in reduced normal form are identifiable in polynomial time and thick data (IPTtD) from text.

Substitutable languages

First, we present the definitions and notations that will be used.

Formal languages. Let Σ be a non-empty finite set of atomic symbols. Σ * is the set of all finite strings over the alphabet Σ. We denote the length of a string x by |x|, the empty string by , the set of non-empty strings Σ * \ { } by Σ + , the set of strings of length k {x : |x| = k} by Σ k and the set of strings of length smaller than or equal to k {x : 0 ≤ |x| ≤ k} by Σ ≤k . We will assume an order ≺ on Σ and its standard order extension on strings of Σ * (comparing the first different symbols with ≺ if they are of same length, or their lengths with < otherwise). The concatenation of strings is denoted x 1 • x 2 or simply x 1 x 2 . A string or language x concatenated k times is denoted x k . Given a string x, we denote by x[i], with 1 ≤ i ≤ |x|, the symbol at the position i in x and by x[i, j], with 1 ≤ i ≤ j ≤ |x|, the substring x[i] . . . x[j] of x from positions i to j.

For a language L ⊆ Σ * , its set of substrings is Sub

(L) = {y ∈ Σ * : x, z ∈ Σ * , xyz ∈ L} and its set of contexts is Con(L) = { x, z ∈ Σ * × Σ * : y ∈ Σ * , xyz ∈ L}. The empty context is , . The distribution D L (y) of a string y ∈ Σ * for a language L is the set of contexts surrounding it in L: D L (y) = { x, z ∈ Σ * × Σ * : xyz ∈ L}.
2 by L 1 L 2 = {y 1 y 2 : y 1 ∈ L 1 , y 2 ∈ L 2 }, let us remark that for any strings y 1 , y 2 in Σ * we have [y 1 y 2 ] L ⊇ [y 1 ] L [y 2 ] L .
We will also consider the local distribution D k,l L (y) of a string y in a language L, where k and l define respectively the size of left and right contexts. To handle short strings and their borders in a smooth way as in (Luque and López, 2010), we will consider in the sequel that the strings w of a language are virtually extended to k w l , where and are specific start and end terminal symbols (not in Σ), and that local contexts can start and end with these symbols, i.e. that left and right contexts can be taken respectively in

Σ k = (Σ∪ ) k ∩( * .Σ * ) and Σ l = (Σ ∪ ) l ∩ (Σ * . * ). The set of k, l-local extended contexts of y in L is then E k,l L (y) = { u , v ∈ Σ k × Σ l : u yv ∈ Sub( k L l )} and the corresponding set of k, l-local contexts is D k,l L (y) = { u, v ∈ Σ ≤k × Σ ≤l : ∃ su, ve ∈ E k,l L (y) with s ∈ * , e ∈ * }.
Classes of substitutable languages. Interested by studying learnability of natural languages in the substitutability framework, [START_REF] Clark | Polynomial identification in the limit of substitutable context-free languages[END_REF] introduced the class of substitutable languages on the basis of the very simple criterion that strings sharing one common context are substitutable. More formally, we define two non-empty strings y 1 and y 2 to be weakly substitutable in a language L, denoted y [START_REF] Clark | Polynomial identification in the limit of substitutable context-free languages[END_REF]

1 . = L y 2 , if D L (y 1 ) ∩ D L (y 2 ) = ∅. A language L is substitutable
iff for any y 1 , y 2 ∈ Σ + : y 1 . = L y 2 =⇒ y 1 ≡ L y 2 .
The preceding definition of weak substitutability is based on global contexts. Motivated by the application to proteins, we came up in (Coste et al., 2012a) with a less demanding criterion based on local contexts. Introducing parameters k and l specifying the sufficient left and right lengths of the local contexts used as evidence for strings substitutability, we define two non-empty strings to be weakly k, l-local substitutable, denoted by y

1 . = k,l L y 2 if D k,l L (y 1 ) ∩ D k,l L (y 2 ) = ∅. A language L is said k, l-local substitutable (Coste et al., 2012a) iff for any y 1 , y 2 ∈ Σ + : y 1 . = k,l
L y 2 =⇒ y 1 ≡ L y 2 . We denote by L S the class of substitutable languages, by L k,l-LS the class of k, l-local substitutable languages for given k and l and by L -LS the whole class of local substitutable languages (defined as the union of the languages L k,l-LS for finite k and l). From the definition, any language in L k,l-LS is also in L k ,l -LS for any k ≥ k and l ≥ l (since

y 1 . = k,l L y 2 =⇒ y 1 ≡ L y 2 =⇒ y 1 . = k ,l L y 2 )
. Each class of L k,l-LS is strictly included in the class L k+1,l-LS and L k,l+1-LS , forming a double hierarchy of included classes that tends in the limit towards the class of substitutable languages L S that can be considered as the L ∞,∞-LS class of languages.

Context-free grammars. In next sections, we consider the problem of learning substitutable languages represented by context-free grammars. A context-free grammar (CFG) is defined as a 4-tuple G = Σ, N, P, N 0 where Σ is a finite alphabet of terminal symbols, N is a finite alphabet of non-terminal symbols, N 0 ∈ N is the start symbol (also named axiom) and P is a finite set of production rules of the form A → α where the left-hand side (lhs) A belongs to N and the right-hand side (rhs) α belongs to (N ∪ Σ) * . We say that a sentential form δAγ from (N ∪ Σ) + can be derived into δαγ, denoted δAγ ⇒ G δαγ, if there exists a production rule A → α in P . The transitive closure of ⇒ G is denoted ⇒ + G and its reflexive transitive closure ⇒ * G . Given a sentential form α in (N ∪ Σ) + , we denote by L(α) the set of sentential forms {β ∈ (N ∪ Σ) * : α ⇒ * G β} and L(α) the set of strings {w ∈ Σ * : α ⇒ * G w} that can be derived from α. The context-free language defined by a context-free grammar G is L(G) = L(N 0 ). We will assume that target grammars contain no useless symbols or rules [START_REF] Hopcroft | Introduction to automata theory, languages, and computation -international edition[END_REF].

Learning grammars in reduced normal form

To study the approach by reduction from a theoretical point of view

C S ← (C S \ {[y 1 ], [y 2 ]}) ∪ {[y 1 ] ∪ [y 2 ]} /* Keep only C S -prime substitutability classes */ P S ← {C ∈ C S : ∀C 1 , C 2 ∈ C S , C C 1 C 2 } /* Build bottom grammar */ N ← ∅, P ← ∅ foreach C ∈ (P S ∪ {[S]}) do N ← N ∪ {N C } foreach y ∈ C do P ← P ∪ {N C → y}

Learning by reduction

The general sketch of the approach is to start from a bottom grammar, representing the training sample S and the classes of substitutable substrings from S that are implied by weak k, l-local substitutability evidence in S.

More precisely, the bottom grammar for S contains a non-terminal N C for each class C of substitutable substrings from S and a rule N C → y to derive directly from N C each substring y in C. The language generated from each non-terminal N C is then the set of strings from C. Its start symbol being the non-terminal for the class including S (N [S] in the pseudo-code), the bottom grammar recognizes only the training set S.

Generalization is then achieved by reduction of the right-hand sides. The idea is to replace rules of the form B → δαγ such that α can be derived from non-terminal A, by rules B → δAγ. Such a reduction shortens the right-hand side of the rule and generalizes the derived language, from L(δ)L(α)L(γ) to L(δ)L(A)L(γ) with L(α) ⊆ L(A). Remark that if L(A) = L(α) this reduction, as well as the non-terminal A, is useless from the generalization point of view and introduces unnecessary additional derivation steps for parsing. Otherwise, the generalization ensures the required substitutability of strings derived from this occurrence of α by all the strings in their substitutability class, which will be represented by A. Also note that reductions can compete on the right-hand side of a rule, resulting in potentially incomparable generalizations. These will require the replacement of one rule by several reduced alternatives to ensure required substitutability.

Combined with the removal of the non-axiom non-terminals that are head of exactly one derivation rule (replacing their occurrences by the right-hand side of the rule) and with the unification of non-terminals that can be derived into the same right-hand side, reductions of bottom's grammar rules will converge towards a grammar with fully reduced rules, said in reduced normal form (RNF). Instead of simply defining RNF as the result of the reduction, we propose here a formal characterization of RNF which does not rely on any predefined class of languages:

Definition 2 (RNF) A context-free grammar Σ, N, P, N 0 is in reduced normal form if:

1. ∀A ∈ N, L(A) = [L(A)] L
(a non-terminal represents exactly one congruence class)

2. ∀A ∈ N, A = N 0 : |{(A → α) ∈ P }| > 1 (each non-terminal, other than the axiom, has alternative derivations)

3. ∀A, B ∈ N : (B ⇒ * G δαγ ∧ A ⇒ * G α) =⇒ (∃(B → δ Aγ ) ∈ P : δ ⇒ * G δ ∧ γ ⇒ * G γ) (existence of reduced rules) 4. ∀B ∈ N, ∀(B → β 1 ), (B → β 2 ) ∈ P : L(β 1 ) ⊆ L(β 2 ) =⇒ β 1 = β 2
(only fully reduced rules)

RNF is a canonical form for local substitutable languages [START_REF] Coste | A bottom-up efficient algorithm learning substitutable languages from positive examples[END_REF]. The first condition implies that the class of RNF-grammars is a subclass of strongly congruential context-free grammars (SC-CFG) [START_REF] Scicluna | Grammatical inference of probalistic context-free grammars[END_REF]. We conjecture that RNF could also be a canonical form for the class of languages represented by SC-CFG.

The second condition discards non-terminals giving rise to vacuous local derivation trees [START_REF] Clark | A language theoretic approach to syntactic structure[END_REF]. The corresponding classes of congruence are said to be composite since they can be factorized into the concatenation of several congruence classes. Intuitively, if A → α 1 α 2 . . . α |α| is the unique derivation of A, then

L(A) = [L(A)] = [L(α 1 )][L(α 2 )] . . . [L(α |α| )] = [L(α 1 )L(α 2 ) . . . L(α |α| ]. Formally:
Definition 3 (Composite class) Let L be a language whose set of non-zero and non-unit congruence classes is

C L . A class C ∈ C L is composite for L iff: ∃C 1 , C 2 ∈ C L , C = C 1 C 2
We say that a congruence class is prime if it is not composite. From the two first conditions, RNF grammars have non-terminals only for prime congruence classes. Non-terminal alphabet being finite by definition, reduced normal form can represent only languages with a finite set of primes, excluding thus substitutable context-free languages with an infinite number of primes [START_REF] Clark | Learning trees from strings: a strong learning algorithm for some contextfree grammars[END_REF]. In contrast, this does not restrict the class of local substitutable languages that can be represented, since their number of congruence classes, and thus of prime congruence classes, is always finite (the number of contexts of fixed size k, l on a finite alphabet characterizing them being finite). Our intuition is that the class of substitutable context-free languages with a finite number of primes actually corresponds to the class of local substitutable languages.

The last two conditions ensure to have only fully reduced rules. The premise of the third condition is the same as in the definition of Non-Terminal Separable (NTS) languages [START_REF] Boasson | NTS languages are deterministic and congruential[END_REF]. This condition implies that RNF grammars are NTS, but the goal is here to ensure the existence of rules in (fully) reduced form. Introducing such rules does not change the language represented by the grammar because of the first condition. Keeping only right-hand sides generating most general languages, as implied by the fourth condition, will ensure to keep only fully reduced rules since they subsume the other rules.

RNF grammars are compact NTS and SC-CFG representations of languages without useless non-terminals and with smallest non-redundant right-hand-sides. As for deterministic versus non-deterministic representations of automata, RNF grammars can yet be exponentially bigger than non-RNF context-free grammars: in the example of languages {a 2 n }, RNF grammars are {a}, {N 0 }, {N 0 → a 2 n }, N 0 (with only one non-terminal since congruence classes are unique) and are exponentially bigger than the context-free grammars G n presented in the introduction. Note that for any other finite language F , the RNF grammar of F looks alike and is: Σ, {N 0 }, {N 0 → w : w ∈ F }, N 0 (but represents a substitutable language only if F is already susbstitutable).

Efficient reduction

We detail here how k, l-local substitutable languages are learnt efficiently by reduction from a training sample S in Algorithm 1.

Bottom grammar on prime substitutability classes. The set C S , partitioning substrings from S into classes of substitutable substrings, is built classically by initializing the partition to singleton sets and merging the classes of each pair of weakly k, l-local substitutable substrings in S.

(A → α[i]) ∈ R then E ← E ∪ {(i, i + 1, A)} else E ← E ∪ {(i, i + 1, α[i])} /* Trivial irreducible path to next position */ IPaths[i, i + 1] ← {(i, i +
∈ IPaths[1, |α| + 1]} 1 2 3 4 5 1 2 3 4 N 1 N 2 N 3 Figure 1: Parsing graph of α = α 1 α 2 α 3 α 4 with N 1 → α 1 α 2 , N 2 → N 1 α 3 , N 3 → α 2 α 3 α 4 .
Full reductions of α are then N2α4 and α1N3, corresponding respectively to irreducible paths

(1, 4, 5) and (1, 2, 5), while N1α3α4 is not fully reduced as witnessed by (1, 3, 4, 5) r (1, 4, 5).

Each singleton class C in C S contains a unique string y that has no evidence in S for being substitutable with any other string. It would introduce a unique and useless derivation rule of N C in the bottom grammar, in contradiction with RNF condition 2. Such classes, as well as other classes without evidence in S for being prime, are discarded by ReGLiS and ReGLiS core . To carry out this early detection in ReGLiS core , a class C in C S is said C S -prime when it is not included in the concatenation of any two other classes of

C S , i.e. ∀C 1 , C 2 ∈ C S , C C 1 C 2 . A C S -prime class C contains thus at least two sequences y 1 , y 2 in C, such that for any of their decomposition u 1 v 1 and u 2 v 2 , we have [u 1 ] = [u 2 ] and [v 1 ] = [v 2 ].
Assuming that the congruence partition of the substrings of S remains unchanged during generalization, C will require at least two rules to produce sequences y 1 and y 2 . It satisfies then RNF condition 2 and is kept by ReGLiS core 1 to build the bottom grammar.

Full reduction with parsing graph. Grammar generalization in ReGLiS core is a simple pass full reduction of the rules ordered by their length. This ordering enables us to use the result of the reduction of each substring of the rule's right-hand side, to reduce it. As introduced by ReGLiS, full reduction of the rule is performed efficiently by dynamic programming on the parsing graph of the right-hand side of the rule. An example of parsing graph is shown in Figure 1. The vertices are the positions (numbered from 1 to |α| + 1) between the symbols of the string α to be reduced. Edges (i, j, l) indicate that α[i, j -1] can be derived from symbol l. To have at most one edge between two positions, edges between consecutive positions i, i + 1 are labeled by the non-terminal generating α[i] if it exists and by α[i] otherwise. A path π in the parsing graph can then be represented as a sequence of positions, which defines implicitely a reduction β of α (returned by labeling(π, E), with E being the set of labeled edges) and can easily be compared to other reductions of α. A path π 1 is reducible in another path π 2 , denoted π 1 r π 2 , if π 2 is a subsequence of π 1 with same starts and ends. The property π 1 r π 2 holds iff there exists a series of reductions of α from the one defined by π 1 to the one defined by π 2 . A path π is irreducible if it is minimal with respect to partial order r . The set of irreducible paths in the parsing graph represent then all full reductions of α.

1. CS-prime test is a simplified version of ReGLiS' S-Prime test, which can be written ∀D ∈ CS : C DΣ + ∧ C Σ + D. In CS-prime test, C1 and C2 need to be in CS. In S-Prime test, only D needs to be in CS: the other class E, such that C = DE or C = ED, is assumed to exist from substitutable languages properties, even if it is not in CS. This enables ReGLiS to better discard composite classes in the absence of characteristic sample, when ReGLiScore will stick to RNF condition 2. Finding all irreducible paths can be achieved efficiently by dynamic programming. Pseudo-code of function reduce rhs (Algorithm 2), shows how to do it simultaneously with a bottom-up chart parsing construction of the parsing graph. Irreducible paths between pairs of positions are stored in a triangular matrix IPaths. It can be initialized trivially, as well as the parsing graph, for successive positions. Parsing graph and IPaths are then filled incrementally up to increasing positions j, by detecting new edges from positions i < j to j (in increasing span order to ensure bottom up completion of parsing) and by computing IPaths[i, j] accordingly. This requires an additional loop to test for each position mid between i and j, if the concatenation of the labels along the irreducible path traversing mid matches a rule's right-hand side. In such a case, an edge from i to j labelled by the left-hand side of the rule is added to the parsing graph. The set IPaths[i, j] is computed during the same loop. If an edge is detected, it is (i, j). Otherwise it is the set of irreducible paths among the paths chaining paths of IPaths [i, mid] and IPaths [mid, j]. Chaining paths π l and π r without duplicating position mid is written π l + π r in pseudo-code

Analysis of learning complexity

Polynomial time. A key to control the complexity of the algorithm is to limit the number of irreducible paths. While running in polynomial-time when a characteristic sample is included in the training set, ReGLiS can nevertheless generate a non-polynomial number of irreducible paths otherwise. An example of parsing graph with a non-polynomial number of irreducible paths is given in Figure 2. Analyzing this example, it can be noted however that some edges are missing. For instance there should be an edge from position 2 to position 5 if N 2 and N 3 represent prime congruence classes. Indeed, the class represented by the reduction of α 2 α 3 α 4 is represented by two irreducible right-hand-side N 2 α 4 and α 2 N 3 and is thus prime. If there was a non-terminal N 5 for that class and the corresponding edge (2, 5), the substring α 2 α 3 α 4 could be fully reduced in a unique right-hand side N 5 . On the other hand, if N 2 (or N 3 ) does not represent a prime congruence class, the edge between 2 and 4 (or 3 and 5) should be removed, and α 2 α 3 α 4 would be reduced in a unique α 2 N 3 (or N 2 α 4 ). By generalizing this reasoning, we say that a parsing graph is nice (for learning local substitutable languages) if there is only one irreducible path between its pairs of positions, except between start and end positions (the "missing" edge would then be the edge for the class itself). This assertion is violated by pairs (1, 4), (2, 5), (3, 6), (1, 5), (2, 6) in Figure 2 and is satisfied in Figure 1. When a graph is not nice, one can try to fix it by adding the missing edges, and eventually the missing classes. The solution adopted in ReGLiS core is simply to raise an error and wait/ask for a more complete training sample, when the test at the end of the reduce rhs() iterations, except the last one, fails. It ensures also the uniqueness of irreducible path between pairs of positions before last iteration. Parsing is then simplified, since π l , π r , β l , β r are unique, and reduce rhs() is then guaranteed to return at most |α| -2 reduced rules from α, ensuring so ReGLiS core to run in polynomial time.

To establish more precisely time complexity, we recall that the maximal number of substrings of a sequence of length m is m(m+1)

2

. For a training sample S with n sequences of maximal length m, the total number of substring occurrences (and thus of classes) is O(nm 2 ). Complexities for retrieving substitutability classes and building the bottom grammar are thus also O(nm 2 ), while complexity for keeping only C S -prime classes remains the same as for S-Prime test: O(nm 3 ) [START_REF] Coste | A bottom-up efficient algorithm learning substitutable languages from positive examples[END_REF]. The main generalization loop over right-hand sides |α| is done O(nm 2 ) times. The complexity of one iteration is that of reduce rhs(): O(m 3 ), thanks to the integration of irreducible paths computation in the three nested loops of bottom-up parsing, dominating O(m 2 ) generation of reduced righthand sides. Overall, the complexity of ReGLiS core is O(nm 5 ).

The strategy of raising an error is also used to cope with the detection of two different non-terminals for the same class of substitutability (due to the presence in R of two S-Prime classes for one class) in contradiction with RNF condition 1 and the detection of non-terminals with only one production rule (due to a S-Prime class that is not prime) in contradiction with RNF condition 2. These guards, combined with the full reduction by reduce rhs(), ensuring RNF conditions 3 and 4, enable ReGLiS core to return a RNF grammar in the absence of exception. When an error is raised, ReGLiS core returns simply the RNF grammar of S. The first condition of IPTtD is then satisfied: ReGLiS core returns a RNF grammar consistent with S, in polynomial time with respect to the size of S.

Polynomial data. [START_REF] Clark | Polynomial identification in the limit of substitutable context-free languages[END_REF] defines the characteristic sample of a grammar G = Σ, N, P, N 0 by CS(G) = {uwv ∈ Σ * : (A → α) ∈ P, (u, v) = c(A), w = w(α)} where G is not assumed to be in any special form, with w(α) and c(N ) denoting respectively the smallest string generated from a sentential form α and the smallest context enabling to reach a non-terminal N from N 0 . Since each local substitutable language L can be represented by its canonical grammar in RNF G R (L), we can be more precise here and define the characteristic sample of a language L in L k,l-LS by CS(L) = CS(G R (L)). This characteristic set ensures the existence of a non-terminal for each prime class and the possibility of building by reduction the target grammar from the smallest substitutable strings of each class. Other sequences in the training set are then correctly handled (parsing and irreducible paths) by this grammar, allowing to discard eventual redundant classes in P S by the test β ∈ N of ReGLiS core . The number of strings is linear in the number of rules by definition. From the definition of the thickness of a grammar G (τ G = max{|w(α)| : (A → α) ∈ P }), the length of the string generated for a rule is trivially linearly bounded by the thickness. The second condition of IPTtD is then satisfied: for each k, l-local substitutable language L, there is a characteristic sample of polynomial size with respect to the size of its canonical grammar in RNF G R (L) and its thickness, ensuring ReGLiS core to return G R (L) when it is included in the training set and k, l parameters are given. This shows that k, l-local substi-tutable languages represented by RNF grammars are IPTtD. Remark that convergence will also be ensured if bigger values are chosen for k and l, from the inclusion of L k,l-LS classes of languages. This allows a certain independence with regard to the choice of parameters. At the extreme, since training samples are always finite in practice, choosing for k and l the maximum length of words in S would emulate contexts of infinite lengths for learning any local substitutable language.

Discussion and perspectives

We proposed a formal characterization of RNF grammars, which are the grammars obtained by the reduction approach introduced by ReGLiS. This characterization is independent from the class of language represented and we think that RNF could also be useful for the inference of languages represented by strongly congruential grammars. We presented also a simplified version of ReGLiS enabling easier study of the core of the approach by reduction. Introducing the requirement of only one irreducible path between pairs of positions of the parsing graph, except between the first and last positions, provides a way to detect insufficient training samples. An efficient algorithm performing this detection during construction of parsing graph was presented. In practice, this detection could be exploited to add new substitutability classes or initiate interactions with the user. It is also a first step to test if a grammar represents a substitutable language. In this work, it ensures to return a consistent RNF grammar in polynomial time. Combined with the polynomial size of the characteristic sample with respect to the size of the representation and its thickness, this enables to show that k, l local substitutable languages represented by RNF context-free grammars are IPDtT with this algorithm.

  Two strings y 1 and y 2 in Σ * are syntactically congruent for a language L, denoted y 1 ≡ L y 2 , iff D L (y 1 ) = D L (y 2 ). The equivalence relation ≡ L defines a congruence on the monoid Σ * since y 1 ≡ L y 2 implies ∀x, z in Σ * , xy 1 z ≡ L xy 2 z. We denote the congruence class of y by [y] L = {y ∈ Σ * : y ≡ L y } and, by extension, of a set of congruent strings Y by [Y ] L . The congruence class [ ] L is called the unit congruence class. The set {y : D L (y) = ∅} = Σ * \Sub(L), when non-empty, is called the zero congruence class. A congruence class is non-zero if it is a subset of Sub(L). Defining classically the concatenation of languages L 1 and L

/

  * Generalize grammar by reduction */ try R ← ∅; foreach (N C → α) ∈ P , ordered by increasing |α| do /* Reduce rule (see Algorithm 2) */ foreach β ∈ reduce rhs(α, R) do 14 if β ∈ N then 15 R ← R ∪ {N C → β} /* Check RNF condition 2 */ if ∃N C = N [S] : |{(N C → β) ∈ R}| = 1 then raise exception FAIL (Non-terminal for composite class) catch exception (FAIL) return Σ, {N 0 }, {N 0 → w : w ∈ S}, N 0 /* Trivial RNF grammar of S */ /* Success */ return Σ, N, R, N [S]

Algorithm 2 :

 2 reduce rhs Input: String α ∈ (Σ ∪ N ) + , Set R of rewriting rules from N to (Σ ∪ N ) + Output: Set of irreducible reductions of α or FAIL exception /* Initialization */ V ← {i ∈ [1, |α| + 1]} /* set of vertices (parsing positions) */ E ← ∅ /* set of labeled directed edges in parsing graph */ for i ← 1 to |α| do /* Labeled edge to next position */ if ∃A :

  1)} /* Incremental bottom up computation by dynamic programming */ for j ← 3 to |α| + 1 dofor span ← 2 to j -1 do i ← j -span Paths ← ∅ /* set of paths from i to j */ for mid ← i + 1 to j -1 do 14 {π l } ← IPaths[i, mid] ; β l ← labeling(π l , E) {π r } ← IPaths[mid, j] ; β r ← labeling(π r , E) 16 if ∃A : (A → β l β r ) ∈ R then 17 if ∃(i, j, B) ∈ E, B = A then18 raise exception FAIL (Two non-terminals for same class) /* New edge in parsing graph */ E ← E ∪ {(i, j, A)} /* There is thus a direct path from i to j */ 20 Paths ← {(i, j)} 21 else if Paths = {(i, j)} then /* Path from i to j through mid is π l chained with π r */ 22 Paths ← Paths ∪ {π l + π r } /* Keep only irreducible paths */ IPaths[i, j] ← {π ∈ Paths : ∀π ∈ Paths, π r π =⇒ π = π , } if (i = 1 and j = |α| + 1) and |IPaths[i, j]| > 1 then raise exception FAIL (Two irreducible paths from/to internal position) /* Return labeling of irreducible paths from first to last positions */ return {labeling(π, E) : π

Figure 2 :

 2 Figure 2: Parsing graphs with F |α|-1 irreducible paths. The number of irreducible reduc-tions at vertex i is the sum of the number of irreducible paths reaching i -1 and i -2, and is thus given by Fibonacci number F i-1 with F 0 = 0, F 1 = 1.

  , we introduce here a simplified version of ReGLiS (Reduced Grammar inference by Local Substitutability), modified to ensure returning a RNF grammar in polynomial time. The pseudo-code of this new algorithm, named ReGLiS core , is given in Algorithms 1 and 2, using [x] to denote the class in C S of a substring or set of substrings x. In subsection 3.1, we present first the approach by reduction introduced by ReGLiS from a general point of view and we introduce a formal characterization of the RNF grammars which are targeted by the learning algorithm. The details of the pseudo-code for an efficient reduction based on parsing graphs are then presented in subsection 3.2.

Algorithm 1: ReGLiS core . Input: Set of strings S on alphabet Σ, int k, int l Output: Reduced grammar consistent with S /* Partition substrings into substitutability classes wrt evidence in S */ C S ← {{y} : y ∈ Sub(S) \ { }} /* Partition substrings into different classes */ foreach y 1 , y 2 : y 1 . = k,l S y 2 do /* Merge classes of substitutable substrings */

c 2018 F. Coste & J. Nicolas.

Acknowledgments

FC and JN wish to acknowledge the reviewers for their detailed comments and suggestions to improve the clarity of the manuscript. FC is also grateful to Rémi Eyraud and Ryo Yoshinaka for helpful comments and discussion on an early version of this work in Kyoto.


domain: math shs

text:
Carlos Valenzuela García

Olimpia Figueras
email: figuerao@cinvestav.mx

David Arnau
email: david.arnau@uv.es

Juan Gutiérrez-Soto
email: juan.gutierrez-soto@uv.es

Number line as a conceptual and didactical resource for teaching fractions using applets

Keywords: Fractions, local theoretical models, applets for teaching, number line, improper fractions

Results of secondary school students' performance who participated in a pilot study of a general research project whose purpose is to contribute to building better mental objects for fractions are described. A Local Theoretical Model for fractions is used as a theoretical and methodological framework to design and develop a seven stages teaching sequence based on the use of applets created with GeoGebra and the number line as a conceptual and didactical resource. In this paper details of the design, development, and results of the first two stages are given. Results show children's preferences to represent fractions on the unit segment, that is, they think of proper fractions. A majority of the participants paid more attention to the graphical aspects of the applet.

Introduction

The teaching and learning of fractions and rational numbers have been studied during several decades by researchers such as [START_REF] Freudenthal | Didactical phenomenology of mathematical structures[END_REF], [START_REF] Behr | Rational number concepts[END_REF], [START_REF] Kieren | Personal knowledge of rational numbers. Its intuitive and formal development[END_REF] and [START_REF] Figueras | Dificultades de aprendizaje en dos modelos de enseñanza de los racionales[END_REF]. In the last decade, fractions have been considered as one of the most complex concepts studied in basic education (see for example [START_REF] López-Bustamante | Observación de procesos de aprendizaje de los niños en el aula. Una forma de aprender a enseñar fracciones[END_REF] in Mexico, [START_REF] Contreras | Problemas multiplicativos relacionados con la división de fracciones. Un estudio sobre su enseñanza y aprendizaje[END_REF] in Spain and [START_REF] Petit | A focus on fractions: Bringing research to the classroom[END_REF] in the United States of America).

The construction of better mental objects of fractions during elementary school is considered important because that concept is widely accepted as an integral part of mathematics curriculum. Moreover, [START_REF] Siegler | Early predictors of high school mathematics achievement[END_REF] characterized the knowledge of fractions and division as unique predictors of students' mathematics performance from elementary to high school.

On the other hand, the technological tools as a resource for teaching mathematics have been incorporated into the curriculum because there is evidence that those are cognitive resources (e.g., [START_REF] Kieran | Research on the role of technological environments in algebra learning and teaching[END_REF]. In this sense, an attempt is made to design a teaching model that includes these tools to promote the building up of students' better mental objects of fractions.

Research objectives

The principle aim of the general study is to construct a Local Theoretical Model for fractions in order to enrich the actual teaching model (in Figueras' sense, 1988, pp. 21-22) for Mexican elementar y school. To achieve this aim a seven stage teaching sequence based on the use of applets was designed. The focus of this paper is the design and results of the first two stages of a pilot study. The purposes of those stages are: 1) to characterize the type of fractions students keep in mind to represent them on the number line displayed on the screen, 2) to make an exploration inquiry about students' ideas related to density and order of fractions, and 3) to identify if the students relate numerical aspects of fractions with their graphical representation.

Theoretical framework and related literature

The idea of Local Theoretical Model (LTM) developed by Filloy (see [START_REF] Filloy | Educational algebra. A theoretical and empirical approach[END_REF] is used as a theoretical and methodological framework. From the theoretical point of view, the LTM serves to focus on the object of study through four interrelated components: (1) formal competence, (2) teaching models, (3) models for cognitive processes, and (4) models of communication. The construction of these components allows having an interpretative framework to identify the different aspects of fractions, which according to [START_REF] Freudenthal | Didactical phenomenology of mathematical structures[END_REF] appear as fracturer, comparer, measurer, fraction operator and numbers.

The building up of the teaching models component enabled the detection in Mexican and Spanish curricula, that the study of properties of order, density, the equivalence of fractions, and proper and improper fractions were considered in the last years of primary school. [START_REF] Bright | Identifying fractions on number lines[END_REF] and [START_REF] Saxe | Representing fractions with standard notation: A developmental analysis[END_REF] support the idea that working with the number line gives students an approach to the notion of above concepts. However, in some sixth-grade' textbooks used in Spain (e.g., [START_REF] González | Matemáticas. 6 Primaria[END_REF] continuous and discrete models are more often than not used for teaching fractions, and activities that include the number line are lacking. These are some reasons for using the number line in this research as a didactical resource.

A revisiting of specialized literature was carried out to construct the cognitive processes' models component. Some of the papers analysed report students' difficulties faced when they learn with the number line. Michel and Horne (2008) mentioned three principal misconceptions: (1) Instrumental part-whole knowledge -difficulties with unit-forming, that is, children consider any line segment as a unit-. (2) Counting lines, not spaces -to represent a fraction on the number line some students count the lines or points (considering the zero point) instead of counting intervals-. (3) Decimalising the count -to represent any fraction, some children always divide the unit segment in ten parts-.

Finally, the communication component is formed by the observation of communication processes between student-applet interactions. The building up of the four components grounds the design of the teaching sequence and its trial. The latter is detailed in the following sections.

Methodology and methods

From the methodological point of view the LTM serves to organize the research project in two main parts: (1) the building up of the four components of an initial LTM for fractions as a reference framework of the general research project and (2) an experimentation with students.

Three phases comprise the experimental part. The first one relates to the design of a pre-test, a posttest and a seven-stage teaching sequence. The second phase is the application of the teaching sequence and is structured as shown in Figure 1. Each stage is composed of two parts, one is a GeoGebra applet and the other a series of questions posed with the purpose that students show the ideas they bring into play about fractions when interacting with the applets. Applets and questions are set up on a Webpage that is associated with a database to record students' responses and interactions with the applets. The third phase corresponds to data analysis and characterization of students' performance. The first two stages of the teaching sequence were done in one 45-minute session. Student individ ua l interactions with applets were collected in a non-invasive manner. Answers given were stored in a database and collected with computers provided by researchers. At the end of the experimental phase, stored information was joined and organized to proceed with its analysis. The applets are used as a resource to teach fractions and to collect data.

Setting and participants

The pilot study was carried out with 45 students from 12 to 14 years old in a secondary school located in a troubled urban area of Valencia in Spain. According to their mathematics teacher, participants have a large history of difficulties in mathematics. The students have serious problems of truancy, for this reason, not all of them completed the trial of the teaching sequence. Due to this fact, only data from students that completed sequential steps were considered, that is, 28 students made the first stage and 25 completed the second stage and so on (see Figure 1). The students worked alone during the teaching sequence trial. In this study, the teacher applied the pre-test and post-test.

Applets' design and results

The applets were constructed in a learning environment for fostering the development of conceptual understanding of fractions, taking into account the didactical functions of technology in mathematics education adapted by Drijvers (2013, p. 3). As aforementioned, each stage of the teaching sequence has an applet with an exploration/interaction component (Figure 2 and4) and a list of questions for students to reflect on what they observe during the interaction. To respond, students can turn to applets and observe the animation or representations of fractions.

To characterize the answers given by students, schemes that enable a codification have been constructed (Figure 3 and5). Answers given by pupils were grouped in different types determined by the form in which questions are posed. Type i are answers to questions that are general statements with diverse interpretations. Type ii collects answers that can be classified solely as correct or incorrect. Type iii groups answers to questions where a justification is required and Type iv are answers to questions that requires information students must write on the applets' windows.

Applet design for the first stage. The applet's tasks for the first stage were developed considering two parallel lines of action. One directed towards the student's familiarity with the interactive environment. The other leads the student to represent different aspects of fractions on the number line and to introduce them to a proper use of the fractions' mathematical sign system (Figure 2). Three indications appear on the screen in Figure 2. The first one -'Move the sliders and watch what happens on the number line'-, has the purpose to focus students on the effects of the numerator and denominator sliders that appear at the upper left corner of the screen and to relate those to the graphical and symbolic representations of fractions also shown on the screen. With the second indication -"Represent fractions 1/2, 3/2 and 7/2 by moving the sliders"-, students are asked to represent the first two fractions to see them in the line segment on the screen. The main idea for asking students to represent 7/2 is to promote reflection regarding characteristics of fractions that can be visualized on the screen and of those that cannot.

The third indication -'Represent the fractions 1/3 and 4/3 and observe the blue segment that is drawn on the number line'is provided in order to identify the point representing the fraction or the fraction as number, but also to focus students' attention on the magnitude representing the fraction, that is the length of the segment that represents the fraction. Thus, fraction as a measurer emerges, taking into account the part-whole relationship.

In addition to the above information, seven questions (Figure 3) are posed to make students write their ideas about the observations made during exploration/interaction period. Students can read questions and explore the applet as many times as necessary to answer them.

Data analysis and results of the first stage.

For applet 1 there are only questions of types i and ii. The codification of students' answers is done using the scheme shown in Figure 3.

Questions (1, 2, 4, 5 y 6) Type i Q1. Represent the fraction 1/4. What happens to the fraction if you move the numerator slider and the denominator slider is fixed? Q2. Represent the fraction 7/8. What happens with the fraction if you move the denominator slider and the numerator slider is fixed? Q4. What will happen if the number 25 in shown in the denominator slider? Q5. What happens when the numerator and the denominator are equal? Q6. Why the fraction 7/2 cannot be seen on the screen? Students answers are referred to:

The interpretation is considered:

1. Numerical aspects 2. Graphical aspects 3. Numerical and graphical aspects. 4. Could not be interpreted a. Complete b. Incomplete c. Incorrect or ambiguous

Questions (3 y 7) Type ii

Q3. Represent the fraction 6/7. In how many parts does the line segment that starts at 0 and ends at 1 is divided? How many of those parts are coloured? Q7. Write two fractions that cannot be represented on the line segment shown on the screen.

The purpose of these questions is that the students observe the graphical representation (line segment). The answers can be: One of the most common mistakes to respond Q3 was that students focused on counting the lines or points considering the zero point instead of counting spaces. 22 students identified at least one fraction greater than three to respond Q7. In the answers to questions type i (for example, Q1, Q2, Q4, Q5 and Q6), pupils orient their attention on what happens in the line segment that appears on the screen (Table 1). Of the 140 responses (28 students x 5 questions), 63 (45%) were classified in this group (code 2, in grey). Eight of these answers are considered complete (code 2a), since a consis tent explanation is offered, 25 incomplete (code 2b) and 30 incorrect or ambiguous (code 2c). Two answers related to these results are included to exemplify the way the coding is done.

(1) The answer given by student S1 to question Q1 is: "the bar is moving to the right". This response is coded as 2b (see Table 1) because the focus is posed on what happens on the line segment. Specifically, it refers to the movement that occurs in the blue segment ("the bar" named by the student) representing the fraction when the slider moves to the right, that is when the value of the fraction is increased. The response is not considered complete because the student centres his attention on the movement to the right and presumably does not move the slider to the left. ( 2) The answer given by S3 to question Q1 is: "there are more points between numbers (denominators), in particular, four points between numbers". In this case, the answer also reveals a focus on the number line; the student observed the partition of the line segment, but the interpretation is incorrect or ambiguous because the student did not explore using the sliders (code 2c).

The focus on numerical aspects of the fraction (code 1, in green) was observed in 38 of the 140 responses to the questions of type i (27.15%); two answers were classified as complete (1a), 14 as incomplete (1b) and 22 as incorrect or ambiguous ( 1c). The answer given by student S17 to question Q6 is: "because the denominator is 2, and this is smaller than 7, so the numerator is bigger, so it is not possible". The student observed the values of the numerator and denominator, makes a comparison between the numerical values and justifies his answer; it was coded as 1c (see Table 1). Only in 17 of the 140 (12.14%) answers to the questions of type i (in purple) the focus on numerica l and graphical aspects is made evident. Five of them were classified as complete (3a) and twelve as incomplete ( 3b). The answer given by student S19 to Q6 is: "because in the line, one can only represent numbers between 0 and 3, and 7/2 is greater than 3". The student observed the structure of the line segment; his focus is posed on the graphical representation but also refers to the fraction as a number. For this reason, the answer was coded as 3a.

Applet design for the second stage. A new form of symbolic representation is introduced in this applet (see Figure 4). In this case, students are asked to write five fractions in a pop-up window when they click the start button (INICIO in Spanish). This button is associated to a JavaScript subroutine that offers feedback and stores the student actions. Fractions are limited to those with denominator between 2 and 10 and numerator between 0 and 40.

If a pupil writes a fraction that does not meet these conditions, a pop-up window appears with a message advising to take into account the characteristics of the numbers involved. When introduc ing a fraction greater than three, another alert window appears to indicate that the fraction cannot be seen on the number line. Fractions that are written by the user leave a trace in the form of red dots and the value of the fraction on the number line (see Figure 4). The visible trace on the screen helps the students answer questions posed in the Webpage in which the applet is embedded.

Data analysis and results of the second stage. In the applet that corresponds to the second stage, there are questions of type i, ii and iv. The codification is done using the scheme in Figure 5.

Questions (1, 3 and 4) Type i Students answers are referred to:

The answer could be: Q1. The 5 fractions you wrote appear on the number line on the screen? Why?

1 Only 12 of 25 students were able to answer correctly Q2 (code 1). This result has an effect on the answers to questions Q3 and Q4, as shown in Table 2. The students who order fractions correctly chose correctly the greater or smallest fraction for questions Q3 and Q4 respectively. To justify the order of those fractions S1 considered the length of the blue segment that represents the fraction on the number line (2a). S2 considered the position of the point representing the fraction on the number line (2a), i.e. graphical aspects of fractions. Students S5 and S18 considered the characteristics of the numerator and denominator of the fraction, that is, numerical aspects of fractions (code 1a).

The justification for most students who do not respond correctly to questions Q3 and Q4 is based on comparing the numerators and denominators of the fractions. Two of these cases are the following:

(1) Student S8 choose 8/3 as the smaller fraction, "because the denominator is the smallest." The comparison made with fractions 4/10, 3/9 and 12/5. He also chose 4/10 as the greater fraction "because the denominator is the largest." (2) Student S7 chose 2/3 as the smaller fraction between 5/4, 3/7, 3/4, "Because that [fraction] has the smaller numbers."

S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S19 S20 S21 S22 S23 S24 S25
Q1 2a 1a 2b 1b 4a 4b 2b 4b 4b 4b 4a 4a 4b 4b 4a 2a 4b 2b 1a 1b 4b 4a 4b 2a 4b 4b 1b 1a 4a 1b 1b 2b 4b 1a 1a 2a 4b 4b 2b 2b 1a 1a 1b 1a 4b 2b 2b 2a Q4 2a 1a 4b 1b 1a 4a 4b 1b 2b 4b 4a 2b 4a 4b 4b 4b 2b 1a 2a 1b 1a 4b 2b 2b Of the 50 answers (25 students x 2 questions) related to the density of fractions (questions Q5 and Q6), 35 were classified with code a, because the answer refers to a finite number of fractions, 6 were blank (code c), and only in 9 answers the density property was mentioned in some sense (code b), because some of the responses were "infinite fractions" or "as much as one wants". Finally, Table 2 shows that students wrote more often proper (56) than improper fractions (38).

Q2 1 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 0 1 0 0 0 1 Q3 2a 2a
3 1 3 0 1 2 3 1 3 3 0 2 3 1 3 1 4 2 56 Im 0 0 0 2 1 3 1 2 2 1 4 2 2 1 4 1 1 2 1 0 3 0 3 1 1 38 U 0 0 0 0 2 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 5

Conclusions

Results of the first two stages described before allow to highlight the fact that few students were able to relate the numerical and graphical representation of fractions. However, students who are able to relate these two representations can give complete and correct answers. To justify questions related to fractions' order, some students relied solely on the numerical aspect, for example, comparing the numerator and the denominator, which led to incorrect or ambiguous answers. Whereas answers in which graphical aspects are used, for example, the position of the point on the number line or the length of the blue segment, led to correct answers. Although most students focused on the latter aspect, they encountered difficulties in representing a fraction on the number line. The most common mistakes are instrumental part-whole knowledge and counting lines, not spaces, also reported by Michel and Horne (2008).

On the other hand, the idea of density that students had seems to be strongly related to the number of fractions they represented during their interactions with the applet, these results are a warning to continue investigating these ideas in later stages and reflect the influence of applets. The students wrote more proper fractions even though the number of proper fractions with small denominators as 2 or 3 are few compared with the number of improper fractions. These results can be related to fact that teaching models favouring the recognition of proper fractions are widely used.

Figure 1 :

 1 Figure 1: Structure of the teaching sequence design and its trial

Figure 2 :

 2 Figure 2: Screenshot of the applet for the first stage

Figure 3 :

 3 Figure 3: Scheme to characterize the answers to questions posed in the first stage

  2c 3b 2b 3b 2c 1c 1c 1c 2c 2b 2b 2b 3b 2c 1c 2b 1b 2c 4c 3b 2c 1c 2b 2b 2c 4c Q2 1b 1c 2c 2b 2b 3b 3b 1c 1c 1c 2c 2b 2c 1c 3b 1b 3a 2c 3b 2b 1b 3b 2c 2c 3a 2b 2c 2c Q4 2b 2c 2c 2b 2b 2c 2c 4c 4c 4c 4c 2c 4c 4c 2b 2b 1b 2b 2a 1c 1c 4c 4c 2c 1c 4c 4c 4c Q5 2a 3a 2a 2c 4c 2b 1b 1c 1c 1b 4c 1a 1b 4c 2a 2a 1b 2a 3a 1b 4c 1b 1a 2b 2b 2a 4c 2a Q6 3b 1c 2c 1c 1b 1c 3b 4c 2c 1c 2c 1c 1c 2b 2b 2b 1c 3b 3a 4c 2c 4c 2c 2c 4c 2b 2c 1b Q3 c a b a b a c b b b a b b b a c c c a c b b c c c b a a Q7 b c a a c c c b b a b b a b b b a a a b a b b b c c b b

Figure 4 :

 4 Figure 4: Screenshot of the applet for the second stage

Figure 5 :

 5 Figure 5: Scheme to characterize the questions of the second stage

Table 1 : Characterizations of students' answers in the first stage

 1

Table 2 : Characterizations of students' answers in the second stage

 2

Proceedings of CERME10

Acknowledgements

This research has been funded from the Research Programmes EDU2015-69731-R (Ministerio de Economía y Competitividad/FEDER), GVPROMETEO2016-143 (Conselleria d'Educació, Investigació, Cultura i Sport) and 230188 (Consejo Nacional de Ciencia y Tecnología, México).


Vincent J Henry

Ivan Moszer

Olivier Dameron

Marie-Claude Potier

Martin Hofmann-Apitius

Olivier Colliot

Martin Hofmann- Apitius

Integrating ontological representation and reasoning into a disease map: application to Alzheimer's disease

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

Background and Objectives

Systems medicine disease maps are ongoing projects that provide fine curated knowledge on pathophysiology at the molecular and phenotypic level (http://disease-maps.org/projects). They are based on a common framework that includes the use of standards and guidelines for defining biological processes and entities. Despite this standardisation effort, there remains some formal and semantics ambiguities: for instance, processes with unexpected participants (e.g. translation of genes instead of transcripts) or phenotypes named with process names (e.g. apoptosis). Ontologies provide a consistent framework to deal with these issues: axiomaticbased definition and logical reasoning properties allow one to underline formal inconsistencies and refine semantic descriptions. In the present work, we propose to use an ontology in order to increase the consistency of the AlzPathway map [1].

Approach

We first built an ontological model based on the definition and axiomatization of high-level classes offered by the CellDesigner diagram [2] including:

-the disjunction of gene, gene product (RNA, protein and complex) and phenotype.

-the definition of processes according to their actions (e.g. transcription is a process that has as input at least one gene and as output at least one RNA).

-the definition of phenotype naming as a state of the system and not as an action. Then, the knowledge contained in AlzPathway was integrated as subclasses of the previously defined high-level classes. This resulted in 2,429 subclasses. We then performed automatic reasoning. It allowed us to identify 285 redundancies, 53 inconsistent processes and 14 inconsistent participants. Moreover, the fine exploration of the ontology led to the identification of 55 phenotypes that refer to biological aggregated processes. We manually fixed these inconsistencies and validated the correction with automatic reasoning. Finally, we validated the ability of this modified map to manage biomedical data using multiomics data from a clinical study.

Conclusions

In conclusion, we integrated ontological properties into AlzPathway. Our model clears out ambiguities in the gene, gene product, metabolite, phenotype and biological process specifications, and thus, facilitates the integration of multiomics data. Furthermore, our work points out the lack of consensual definition of phenotypes and the need to manage process granularity. Interestingly, the integration of ontological standard into AlzPathway opens perspectives to link AlzPathway with the Alzheimer Disease Ontology [3].

Vincent J. Henry, Ivan Moszer, Olivier Dameron, Marie-Claude Potier, Martin Hofmann-Apitius, et al.. Integrating ontological representation and reasoning into a disease map: application to Alzheimer's disease. DMCM 2018 -3rd Disease Maps Community Meeting, Jun 2018, Paris, France. pp.1-2. hal-