Yong-Feng Deng

Anh-Minh Tang

Prof Yu-Jun Cui
email: yujun.cui@enpc.fr

Xiang-Ling Li

Study on the hydraulic conductivity of Boom clay

Keywords: Boom clay, consolidation, constant-head percolation, hydraulic conductivity, void ratio, correlation

The hydraulic conductivity is a key parameter in geotechnical engineering practice for the seepage and consolidation analysis. Experimental results show that the hydraulic conductivity is mainly governed by the soil porosity, and the correlations with void ratio are usually proposed. The validity of these correlations has been verified for soft clays and sands.

However, few studies were involved in stiff clays. In this work, the hydraulic conductivity of Boom clay, a stiff clay taken from the Essen site in Belgium, was determined from both consolidation and constant-head percolation tests. The data obtained was then analyzed to evaluate the existing correlations to predict the hydraulic conductivity. In addition, as these correlations usually require a referred hydraulic conductivity at a known void ratio, it is often difficult to be used in practice. Thus, a new method was developed allowing the prediction of hydraulic conductivity without the referred value, which was based on two existing correlations and involved the void ratio and the liquid limit. The proposed correlation was verified using the experimental results obtained from Boom clay samples which were collected from various locations in Belgium.

Introduction

Boom clay is a thick deposit of over-consolidated marine clay, of Oligocene age. It can be found in the north-east region of Belgium [START_REF] Bouazza | Some mechanical properties of reconstituted Boom[END_REF]. Its hydraulic conductivity has been investigated for its performance assessment for the deep geological disposal of high-level radioactive waste. Recently, [START_REF] Wemaere | Hydraulic conductivity variability of the Boom Clay in north-east Belgium based on four core drilled boreholes[END_REF] has worked on Boom clay cores taken from four distant boreholes at various depths. The results showed a variability of hydraulic conductivity ranging from 3×10 -12 to 10×10 -12 m/s.

In the laboratory, hydraulic conductivity is usually determined using Darcy's law. The constant-head method is usually applied for high permeability soils (sandy soils for example) and the variable-head method is usually used for low permeability soils (clayey soils for example). For stiff clays or rocks with extremely low hydraulic conductivity (sometimes lower than 10 -12 m/s), pulse tests are usually used [START_REF] Zhang | Evaluation and application of the transient-pulse technique for determining the hydraulic properties of low permeability rocks-Part 1: Theoretical evaluation[END_REF]. In addition to these direct methods, Terzaghi's consolidation theory can also be used to determine the hydraulic conductivity by back analyzing the consolidation results [START_REF] Delage | On the thermal consolidation of Boom clay[END_REF].

Experimental results usually show that hydraulic conductivity (k) mainly depends on soil porosity and various correlations are proposed between hydraulic conductivity (k) and void ratio (e). Some of these correlations are presented in Table 1. The correlation by [START_REF] Taylor | Fundamentals of soil mechanics[END_REF] (equation No. 1) was modified by [START_REF] Samarasinghe | Permeability and consolidation of normally consolidated soils[END_REF] for sandy soils where C and m are two constants (after [START_REF] Indraratna | Laboratory Determination of Efficiency of Prefabricated Vertical Drains Incorporating Vacuum Preloading[END_REF]. [START_REF] Aubertin | Hydraulic conductivity of homogenized tailings from hard rock mines[END_REF] and Sridharan and Nagaraj (2005) analyzed the effects of m and proposed m = 5 for clays. [START_REF] Kozeny | Ueber kapillare Leitung des Wassers im Boden[END_REF] and [START_REF] Carman | The determination of the specific surface of powders[END_REF][START_REF] Carman | Flow of gases through porous media[END_REF] proposed another correlation (i.e. KC function) for porous materials between void ratio and hydraulic conductivity as equation No. 2, where γ is the unit weight of the fluid involved; µ is the viscosity of the fluid, C K C -is the Kozeny-Carman empirical coefficient and 0 S is the specific surface area per unit volume of particles (after [START_REF] David | Goodbye, Hazen; Hello, Kozeny-Carman[END_REF]. Note that if the value of m in equation No. 1 is set as 3, the relation between hydraulic conductivity and void ratio is the same as k ∝ e 3 /(1+e). [START_REF] Aubertin | Hydraulic conductivity of homogenized tailings from hard rock mines[END_REF], Sridharan and [START_REF] Shridharan | Hydraulic conductivity of remolded fine-grained soils versus index properties[END_REF] found that the value of m is generally close to 5 for clays; hence, KC function would over-evaluate the hydraulic conductivity when the void ratio is less than 1.0.

For clayey soils, [START_REF] Taylor | Fundamentals of soil mechanics[END_REF] hydraulic conductivity and void ratio respectively (after [START_REF] Indraratna | Laboratory Determination of Efficiency of Prefabricated Vertical Drains Incorporating Vacuum Preloading[END_REF]. [START_REF] Tavenas | The permeability of natural soft clays Part II: permeability characteristics[END_REF] and Leroueil et al. (1990aLeroueil et al. ( , 1990b) ) validated for soft clays whose in-situ void ratio (e 0 ) is larger than 1.0. The validation of these equations for stiff clays with a void ratio often lower than 1.0, remains to be verified.

In the present work, the hydraulic conductivity of Boom clay cores of low in-situ void ratio (e 0 < 0.79) taken from a borehole drilled in Essen (Belgium) was determined from oedometer and isotropic consolidation tests as well as constant-head percolation tests. Four depths in the range of 220 -250 m were investigated. The obtained results were used to evaluate the existing correlations listed in Table 1. A new method was developed allowing prediction of the hydraulic conductivity of stiff clays using the void ratio and the liquid limit.

This method was verified using the experimental results from Boom clay samples taken from various locations.

Soils studied

The soil cores studied were taken from the borehole drilled in Essen (Belgium). The Four cores with 1 m in length and 100 mm in diameter were studied. After being taken from the borehole, the cores were sealed in plastic tubes with ends closed and transported to the laboratory for testing. The details of these cores are shown in Table 2. Two cores were taken from the Putte member (Ess75 and Ess83) and two others from the Terhagen members (Ess96 and Ess104). The geotechnical identification parameters of these cores are similar: specific gravity, G s = 2.64 -2.68; liquid limit, w L = 68 -78%; plastic limit, w P = 29 -33%; plastic index, I P = 36 -45. The water content (w) ranges between 26.5 and 29.7 %, and the void ratio (e) between 0.700 and 0.785. The values of degree of saturation (S r ) determined in the laboratory on the cores are equal or close to 100%. The Blue Methylene value (VBS) is equally similar, VBS = 6.20 -6.67. According to the relationship between the specific surface area (SSA) and VBS [START_REF] Yukselen | Suitability of the methylene blue test for surface area, cation exchange capacity and swell potential determination of clayey soils[END_REF], the SSA of Boom clay ranges from 152 to 163 m 2 /g. Nevertheless, the carbonate content of core Ess104 (43.6 g/kg) is significantly higher than that of other cores (lower than 10 g/kg). The particle size distribution determined is shown in Figure 2. The curves obtained for the four depths are similar and the clay content (< 2 µm) is quite high (more than 40%). Note that the particle size distribution of Boom clay at Essen is very similar to that presented by [START_REF] Wemaere | Hydraulic conductivity variability of the Boom Clay in north-east Belgium based on four core drilled boreholes[END_REF] for Boom clay taken from other regions in Belgium (Mol, Doel, Zoersel and Weelde).

Experimental techniques

Three methods were used to study the hydraulic conductivity of the soil cores: (1) constant-head method using an oedometer cell;

(2) back-analysis of consolidation tests in an oedometer cell;

(3) Back-analysis of consolidation tests in an isotropic cell. A synthetic solution having a similar chemical composition as the in-situ pore-water was used to perform the tests. In Table 3, the composition of salts used for preparing the synthetic solution is presented for all depths. A high concentration of NaCl (15 -20 g/L) can be observed. More details on the pore-water chemistry of Boom clay at Essen can be found in De [START_REF] De Craen | Geochemical analyses of Boom Clay pore water and underlying aquifers in the Essen-1 borehole[END_REF].

For the tests in the oedometer cell, a 40 mm long section was cut from the soil core using a metal saw. The confining ring of the oedometer cell having a sharp edge was then pushed into the soil sample. The surfaces of the soil specimen were finished using a steel knife with sharp edge. The final dimension of the soil specimen is 20 mm high and 50 mm in diameter.

Note that special attention was paid to sampling direction: the axis of specimens is perpendicular to the bedding plan. The confining ring having the soil specimen inside was then installed in the oedometer cell. For tests using the isotropic cell, the soil specimen was carefully hand-trimmed to have a final dimension of 38 mm in diameter and 76 mm high. The axis of the specimen is also perpendicular to the bedding plane.

When applying the constant-head test in the oedometer cell, a vertical stress equal to the in-situ vertical stress estimated from the soil densities and depths was applied after installation of the soil specimen in the cell. The specimen was fully saturated and this was checked by considering the initial degree of saturation (see Table 2) and the volume changes due to the in-situ stress application. After stabilization of the soil deformation under this initial load, the porous stone in the base of the cell was fully saturated by the synthetic solution flushing. It was observed that this operation did not induce any volume change of the soil. A water pressure of 1.0 MPa was then applied using a controller of pressure/volume (CPV) from the lower base of the cell; the upper base was kept at atmospheric pressure. The hydraulic conductivity of the soil was calculated from the water volume change recorded by the CPV with Darcy's law. Two tests were performed using this method on core Ess83 (227 m depth) and Ess96 (239 m depth). The in-situ vertical stress was estimated to be 2.27 MPa, based on a mean value of the unit weight estimated at 20 kN/m 3 and a ground water level considered at the ground surface. In the oedometer test, a value of 2.40 MPa was chosen to represent the in-situ vertical stress.

To apply the second method, one oedometer consolidation test was carried out on each of the four cores of Table 2. A vertical stress equal to the in-situ stress was applied to the specimen after its installation inside the cell. After stabilization of the volume change, the subsequent saturation of the drainage system was performed without inducing any volume change of soil. The soil was then subjected to several unloading/loading paths in steps with vertical stresses ranging from 0.05 to 32 MPa. The hydraulic conductivity was finally determined following the Casagrande's method (AFNOR, 2005a).

To apply the third method, one isotropic consolidation test was carried out on each of the four cores of Table 2. The high-pressure isotropic cell described by [START_REF] Cui | Investigating the time dependent behavior of Boom clay under thermomechanical loading[END_REF] was used for this purpose. After installation of the soil sample on the pedestal with dry porous stones, a confining pressure equal to the in-situ stress was applied. After stabilization of the soil volume change, the drainage system was fully saturated by the synthetic solution. As in the case of oedometer tests, no volume change was observed during this operation. Later, the confining pressure and the back pressure were then increased at the same time by a same pressure increment. The final back pressure was equal to 1.0 MPa and the effective pressure of the soil was still equal to the in situ stress. After [START_REF] Delage | Suction effects in deep Boom clay samples[END_REF], this procedure allows a satisfactory saturation of the soil sample without disturbing its initial state (in terms of microstructure and stress state). Finally, the confining pressure was incrementally increased, allowing the determination of the hydraulic conductivity at various void ratios following Casagrande's method (AFNOR, 2005a).

Experimental results

In Figure 3, the results obtained from the test using the constant-head method are shown.

The volume of solution passing through the soil specimen (20 mm high) under a pressure gradient of 1.0 MPa is plotted versus time. For Ess83, a seepage velocity of Using the results from the oedometer test on Ess83, the relationship between the calculated void ratio and the corresponding vertical stress can be obtained as shown in

Erreur ! Source du renvoi introuvable.. The initial loading to the in-situ stress decreased the void ratio from 0.730 to 0.651 (point A). The saturation of the drainage system did not induce significant change in void ratio. The subsequent unloading in steps to 0.125 MPa increased the void ratio up to 0.774 (point B). After the loading in steps to 16 MPa, the void ratio decreased to 0.361 (point C). For the subsequent paths, the void ratios were equal to 0.620 for σ v = 0.125 MPa (point D), 0.270 for σ v = 30 MPa (point E), and 0.569 for σ v = 0.125 MPa (point F). Note that the volume change was considered as stabilized when the deformation rate was less than 0.00125 mm/h following the French standard [START_REF] Afnor | Sols : reconnaissance et essais: essai de gonflement à l'oedomètre, détermination des déformations par chargement de plusieurs éprouvettes[END_REF]. It can be observed that the time needed for the stabilization during the unloading steps was generally longer than during the loading steps. A total of about 80 days was needed to finish the oedometer test.

The hydraulic conductivity using the second method (back analysis from consolidation tests using the oedometer cell) is presented in Figure 5 versus the corresponding void ratio. It can be observed that the relationship between void ratio and hydraulic conductivity can be described with a linear correlation in the semi-logarithmic coordinate. Furthermore, this relationship seems to be independent of the loading history; a unique function can be proposed for all the loading and unloading steps as ) / 10 ( 10 015 . 0

12 1 . 3 s m k e - × =
.

The results of the isotropic consolidation test on Ess83 are also shown in Figure 4, in terms of variations of void ratio as a function of effective pressure. It can be observed that from the in-situ stress of 2.40 MPa (initial void ratio of 0.725), loading in steps up to 20 MPa effective pressure decreased the void ratio to 0.426. Moreover, the relationship between void ratio and the logarithm of (p-p u ) can be described with a linear correlation. This corresponds to the compression curve of a normally consolidated soil with a compression index C c equal to 0.31. It should be noted that in this test each loading step was maintained until stabilization of the volumetric strain according to the French standard (AFNOR, 2005b): the volumetric strain rate should be less than 6×10 -5 /h. The time needed for each step varied from 100 h to 250 h. A total of 40 days was needed to finish the test.

The hydraulic conductivity using the third method (back analysis from the consolidation tests using the high-pressure isotropic cell) was also presented in Figure 5, versus void ratio.

As observed in the case of the consolidation test in the oedometer cell, a linear function can be proposed to correlate the relationship between void ratio and the logarithm of hydraulic conductivity: In order to evaluate the three methods used for the hydraulic conductivity determination, the result of Ess83 obtained by the first method is also presented in Figure 5. It can be observed that similar results were obtained by the three methods: the hydraulic conductivity at a void ratio of 0.64 is between 1.5 x 10 -12 m/s and 5 × 10 -12 m/s, with a value of 2.5 × 10 -12 m/s obtained by the constant-head method. Note that for Ess96 the three methods gave also similar results: the hydraulic conductivity at a void ratio of 0.59 is between 0.9 × 10 -12 m/s and 2.1× 10 -12 m/s, with a value of 2.0 × 10 -12 m/s obtained by the constant-head method. The difference between the results by the three methods is due to both the experimental error and the heterogeneity of natural Boom clay.

All the results obtained from Ess83 core are gathered in Figure 6b. As mentioned before, for Ess96, all the three methods (constant-head method, consolidation using the oedometer and the isotropic cell, respectively) were applied. Only the second and third methods were applied for Ess75 Ess104. The results are plotted in Figure 6c for Ess96, in Figure 6a for Ess75 and in Figure 6d for Ess104. It can be observed that, on the whole, the relationship between void ratio and the logarithm of hydraulic conductivity can be satisfactorily correlated with a linear function. Furthermore, the functions obtained for Ess75, Ess83, Ess96 and Ess104 are similar (see Table 4 for all fitting equations).

Prediction of hydraulic conductivity

The existing correlations allowing the prediction of hydraulic conductivity (in [START_REF] Shridharan | Hydraulic conductivity of remolded fine-grained soils versus index properties[END_REF] showed that m is equal to 5 for clays.

In the following sections, the experimental data obtained on Boom clay from Essen and that from Mol [START_REF] Delage | On the thermal consolidation of Boom clay[END_REF][START_REF] Aertsens | Spatial variability of transport parameters in the Boom Clay[END_REF][START_REF] Coll | Endommagement des roches argileuses et perméabilité induite au voisinage d'ouvrage souterrains[END_REF][START_REF] Le | Comportement thermo-hydro-mécanique de l'argile de Boom[END_REF][START_REF] Wemaere | Hydraulic conductivity variability of the Boom Clay in north-east Belgium based on four core drilled boreholes[END_REF]) are used to assess the various models presented in Table 1. The models requiring the initial hydraulic conductivity ( 0 k ) and the initial void ratio ( 0 e ) are firstly evaluated. The two models that do not require 0 k and 0 e are considered afterwards. Finally, a new correlation is proposed for Boom clay.

Evaluation of existing models requiring k 0 and e 0

To further evaluate the correlation between hydraulic conductivity and void ratio, ) (e f , the equations of other forms in Table 1 are In order to evaluate the predicting models, the two following parameters were calculated:

(1) a is the mean value of R ( 1); where R is defined as the ratio of the predicted value predicted k to the calculated experiment value calculated k and N is the statistical number [START_REF] Tang | A study on the thermal conductivity of compacted bentonites[END_REF] .

∑ = = N i i R N a 1 1 ); (2) b is the root mean square error of R ( ∑ = - = N i i R N b 1 2 ) 1 (
In Figure 7

Models without k 0 and e 0

For Boom clay in particular or stiff clays in general, the hydraulic conductivity is low and thus difficult to be determined. Hence, from a practical point of view, it is essential to predict the hydraulic conductivity without measuring 0 k . For this purpose, equation No. 2 can be developed as follows (after [START_REF] David | Goodbye, Hazen; Hello, Kozeny-Carman[END_REF]; at a temperature of 20 °C):

[1] .While the data of the specific surface area 0 S of Boom clay calculated using VBS (in Table 2) according to [START_REF] Yukselen | Suitability of the methylene blue test for surface area, cation exchange capacity and swell potential determination of clayey soils[END_REF] and liquid limit w L were used to verify the correlation by [START_REF] Mbonimpa | Practical pedotransfer functions for estimating the saturated hydraulic conductivity[END_REF].

With this correlation, the hydraulic conductivity expression can be rewritten as From Figure 8a, it can be observed that the mean value a ( calculated predicted k k / ) ranges from 1.9 to 2.8 and the error b ranges from 1.1 to 2.2. When the void ratio is close to the in-situ value, the predicted hydraulic conductivity fits well with the calculated one; but if the soils are compressed and the void ratio is decreased, the hydraulic conductivity will be over-evaluated. For equation No. 4 (Figure 8b), the mean value a ranges from 5.6 to 11. As a conclusion, when the measurement of k 0 is not available, equation No. 2 can be used by combining the correlation between S 0 and w L [START_REF] Mbonimpa | Practical pedotransfer functions for estimating the saturated hydraulic conductivity[END_REF]. A satisfactory agreement between prediction and calculation ca be obtained, especially when the void ratio is close to the in-situ value.

A correlation for Boom clay

The above evaluation shows that equation No. 3 in Table 1 is suitable for describing the relationship between k and e and has been used in various works (Leroueil et al., 1990a(Leroueil et al., , 1990b;;[START_REF] Nagaraj | Soft clay behaviour -analysis and assessment[END_REF] In Figure 9, the hydraulic conductivity calculated from the proposed correlation is presented versus the measured values. As the parameters were fitted using the data by back analysis, the value of a is equal to 1.0; the error b obtained is equally small, b = 0.4. Note that there are more data available for higher hydraulic conductivity than for lower conductivity, and therefore the proposed correlation is more suitable for the range of high hydraulic conductivity.

With the proposed correlation and the values of w L (59% -83% ) and e 0 (0.56 -0.68)

reported

Discussion

From the correlation between hydraulic conductivity and void ratio in Figure 6, it can be seen that all the correlations in the coordinate k log -e show a linear relationship. However, when the void ratio is equal to 0.77, the predicted hydraulic conductivity by both equation No. 2 and correlation [3] is close to the measured one.

Conclusions

There have been various works in the literatures focusing on the measurement and the prediction of hydraulic conductivity of soft clays. Satisfactory agreement is often obtained.

For stiff clays as Boom clay, the existing correlations are however not verified. In the present work, the hydraulic conductivity of Boom clay taken from the Essen site in Belgium was first measured using various techniques (constant-head method or back-analysis from consolidation tests). The results show a strong correlation between the hydraulic conductivity and the void ratio. Secondly, the obtained results and the results collected from other works on Boom clay were used to evaluate some existing correlations. The following conclusions can be drawn:

(1) When the in-situ hydraulic conductivity 0 k and the in situ void ratio 0 e are available, the two following equations can give satisfactory predictions: (2) When the in-situ hydraulic conductivity is not available, the following equation can be used for stiff clay especially when the void ratio is close to the in-situ value: Table 1. Correlations between hydraulic conductivity and void ratio Table 2. Details of soil cores studied Table 3. Salts used to prepare the synthetic pore-water (concentration g/l) Table 4 . Mean value of R (a) and the root mean square error of R (b) Boom clay at Mol [START_REF] Le | Comportement thermo-hydro-mécanique de l'argile de Boom[END_REF] Boom clay at Mol [START_REF] Delage | On the thermal consolidation of Boom clay[END_REF] Boom clay at Mol [START_REF] Coll | Endommagement des roches argileuses et perméabilité induite au voisinage d'ouvrage souterrains[END_REF] Singapore clay [START_REF] Arulrajah | Characteristics of Singapore marine clay at Changi[END_REF] Figure 10. Relationship between k C and initial void ratio 0 e

List of Tables

List of Figures

  also proposed equation No. 3 where k C is a permeability change index, 0 k and 0 e are referred values (usually, the in-situ values) of

  Essen site situates in the north east of Belgium, 60 km far from the underground research laboratory (URL) at the Mol site (Figure 1). The Boom clay formation ranged from 153 to 280-m in depth. It can be sub-divided into four zones: Transition zone (153 -200 m); Putte member (200 -238 m); Terhagen Member (238 -260 m); and Belsele-Waas Member (260 -280 m).

  is similar to that obtained from the oedometer test.

  , the hydraulic conductivity values predicted by the above equations are plotted versus the calculated values for the four soil cores from Essen (a, b, c, d) and Boom clay from Mol (e). In Figure 7f, all the data are put together for the analysis. The mean value of R and the root mean square error of R are shown in Table 5. It can be observed that equation No. 1 underestimates the hydraulic conductivity for almost all cores (except Ess83) with a relative large scatter (b = 0.5). On the contrary, equation No. 2 overestimates the hydraulic conductivity for all cores with a large scatter (b = 0.8). Equation No. 3 and No. 4 can predict the hydraulic conductivity satisfactorily for all cores. As a conclusion, when k 0 and e 0 are available, equation No. 3 and No. 4 are suitable for predicting the hydraulic conductivity of Boom clay by modifying these equations

  specific surface area.The correlation by[START_REF] Mbonimpa | Practical pedotransfer functions for estimating the saturated hydraulic conductivity[END_REF] was applied to estimate 0 S from the liquid limit w L of Boom clay:

γ

  is the unit weight of soil particles. The above expression [2] and equation No. 4 (in Table 1) were used to predict the hydraulic conductivity of Boom clay from Essen and Mol without using k 0 . The predicted values are plotted versus the calculated (or measured) values in Figure 8a (equation No. 2) and Figure 8b (equation No. 4). The parameters a and b are also presented to evaluate these models.

  2 and the error b ranges from 5.2 to 11.1. That means this equation overestimates by about 10 times the hydraulic conductivity of Boom clay.

  , while equation No. 2 presents the advantage of estimating parameter C without measuring k 0 . As C K-C in equation No. 2 is aWhere A and B are two constants. In the case of Boom clay, the back analysis gives:

  by François et al. (2009), the predicted hydraulic conductivity using expression [3] (0.9×10 -12 -4.4×10 -12 m/s) covers the range of the hydraulic conductivity of Boom clay (1.2×10 -12 -4.2×10 -12 m/s) measured by Aertsen et al. (2004) at Mol and by Wemaere et al. (2008) at Mol, Doel, Zoerel and Weedle.

.

  and Mol (collected from the present work and others) and Singapore clay (after[START_REF] Arulrajah | Characteristics of Singapore marine clay at Changi[END_REF] are calculated and plotted in Figure10, versus e 0 . According to the analysis on the data presented by[START_REF] Tavenas | The permeability of natural soft clays Part II: permeability characteristics[END_REF]results of Singapore clay are plotted for its low value of e 0 (< 1.0), comparable with the initial void ratio of Boom clay. Figure10shows that the values of C k for Boom clay and Singapore clay lie generally below the line of 56. Actually, from equation No. 3 (Table 1), if k C is taken as the mean value of k C for Boom clay ( k C = 0.3), the B value is equal to 3.33. This value is similar to that obtained for the proposed correlation. On the other hand, the value of A can be also calculated from equation No. 2. Indeed, taking the unit weight s γ of Boom clay equal to 2This value is 2200 times larger than that obtained from the correlation [3].

  This equation was verified using the experimental data on Boom clay.(3) In the case of Boom clay, the following equation was proposed:This equation was developed based on the linear correlation observed on the experimental data between the logarithm of hydraulic conductivity and void ratio. The liquid limit w L is introduced to take into account the variable characterization of Boom clay at various locations.

Figure 1 .

 1 Figure 1. Location of the Essen site and Mol site (De Craen et al., 2006) Figure 2. Particle size distribution of Boom clay at Essen Figure 3. Relationship between water volume change and elapsed time by the constant-head method for Ess83 and Ess96 Figure 4. Void ratio versus effective vertical stress for oedometer and isotropic consolidation tests onEss83 Figure 5. Hydraulic conductivity versus void ratio obtained with three methods for Ess83 Figure 6. Relationships between hydraulic conductivity and void ratio; (a) Ess75, (b) Ess83, (c) Ess96, (d) Ess104 Figure 7. Predicted hydraulic conductivity versus experimental one -models with known k 0 ; (a) Ess75, (b) Ess83, (c) Ess96, (d) Ess104, (e) Mol, (f) all data Figure 8. Predicted hydraulic conductivity versus experimental one -models without k 0 ; (a) Equation No. 2, (b) Equation No. 4 Figure 9. Hydraulic conductivity predicted by the proposed correlation versus the experimental one Figure 10. Relationship between k C and initial void ratio 0 e

Figure 1 .Figure 3 .Figure 4 .

 134 Figure 1. Location of the Essen site and Mol site[START_REF] De Craen | Geochemical analyses of Boom Clay pore water and underlying aquifers in the Essen-1 borehole[END_REF]

Figure 5 .Figure 6 .Figure 7 .

 567 Figure 5. Hydraulic conductivity versus void ratio obtained with three methods for Ess83

Figure 8 .

 8 Figure 8. Predicted hydraulic conductivity versus experimental one -models without k 0 ; (a) Equation No. 2, (b) Equation No. 4

  applied equation No. 3 to soft clays with a wide range of void ratios, and proposed that k C is a function of the in-situ void = .[START_REF] Mesri | Settlement of embankment on soft clays[END_REF] analyzed the data of soft clays and proposed equation No. 4 where CF is clay fraction and A c is soil activity. Although there are various correlations proposed, equation No. 3 is usually used to describe the variation of hydraulic conductivity of clays versus void ratio changes. Note that these equations have been only

	ratio 0 e :	C k	5 0 e .	0

Table 1

 1 No. 1 and No. 3 in Table1. In other cases, C is correlated with other soil basic parameters (equation No. 2 and No. 4). In equation No. 1, m is a parameter related to the curve shape. As mentioned before, the results of[START_REF] Aubertin | Hydraulic conductivity of homogenized tailings from hard rock mines[END_REF] and Sridharan and

	)

Table 1 .

 1 Correlations between hydraulic conductivity and void ratio

	Eq. No.	Proponent	Correlations	Soil type	References
	1	Taylor (1948)

Table 2 .

 2 Details of soil cores studied

	Core No.	Depth (m)	Member	G s	w L (%)	w p (%)	I p	w	e 0	S r (%)	VBS	carbonate content (g/kg)
	Ess75	218.91-219.91	Putte	2.65 78 33 45 29.7 0.785 100 6.47	9.1
	Ess83	226.65-227.65	Putte	2.64 70 33 37 27.2 0.730 98 6.67	7.6
	Ess96	239.62-240.62 Terhagen 2.68 69 33 36 26.5 0.715 99 6.20	2.4
	Ess104 247.90-248.91 Terhagen 2.68 68 29 39 27.7 0.700 100 6.67	43.6

Table 3 .

 3 Salts used to prepare the synthetic pore-water (concentration g/L)

	Salt	Ess75	Ess83	Ess96	Ess104
	NaHCO 3	3.009	3.009	3.009	3.009
	Na 2 SO 4	3.460	3.712	4.126	4.392
	KCl	0.229	0.229	0.229	0.229
	CaCl 2 .2H 2 O	0.367	0.385	0.422	0.459
	MgCl 2 .6H 2 O	1.381	1.464	1.632	1.757
	NaCl	14.542	15.976	18.002	19.287

Table 4

 4 Fitting expressions from test results

	No.	Core	Fitting expressions
	1	Ess75	k=0.015×10 3.2e (10 -12 m/s)
	2	Ess83	k=0.013×10 3.3e (10 -12 m/s)
	3	Ess96	k=0.020×10 3.2e (10 -12 m/s)
	4	Ess104	k=0.038×10 2.9e (10 -12 m/s)

Table 5 .

 5 Mean value of R (a) and the root mean square error of R (b)

			Equation No.1	Equation No.2	Equation No.3	Equation No.4
		Ess75	0.6	1.8	1.3	1.1
		Ess83	1.1	1.7	1.3	1.1
	a	Ess96 Ess104	0.8 0.7	1.4 1.3	1.1 1.0	1.0 0.8
		Mol	0.7	1.1	0.9	0.8
		All data	0.8	1.4	1.1	0.9
		Ess75	0.5	1.2	0.7	0.4
		Ess83	0.4	0.9	0.6	0.4
	b	Ess96 Ess104	0.5 0.5	0.8 0.5	0.4 0.3	0.4 0.4
		Mol	0.5	0.4	0.3	0.5
		All data	0.5	0.8	0.4	0.4

Acknowledgements

ONDRAF/NIRAS (The Belgian Agency for Radioactive Waste and Enriched Fissile Materials) is gratefully acknowledged for its financial support. The first author is grateful to the National Science Foundation of China for its supports (No 50908049).


Claire Gardent¹
email: claire.gardent@loria.fr-yannick.parmentier@univ-orleans.fr-guy.perrier@loria.fr-sylvain.schmitz@lsv.ens-cachan.fr

Yannick Parmentier²

Guy Perrier³

Sylvain Schmitz⁴

Lexical Disambiguation in LTAG using Left Context

Keywords: parsing, lexical disambiguation, lexicalized tree adjoining grammar

, and extends it by computing a polarity-based abstraction, which contains information about left context. This extension allows for a faster lexical disambiguation by reducing the filtering automaton.

Introduction

When deep-parsing natural language with LTAG, one needs to first retrieve, for each word of the sentence to parse, the grammatical structures (i.e., LTAG trees) this word is associated with in the lexicon. This operation is called lexical selection. 1 Unfortunately, in large lexicalized grammars, a given word may be associated with hundreds of grammatical structures. This high lexical ambiguity causes a combinatorial explosion, which highly impacts parsing times. Indeed, lexical selection may yield an exponential number of sequences of structures (in the worst case, one needs to retrieve the cartesian product of the lexical entries of each word of the sentence to parse). In order to reduce lexical ambiguity, one wants to compute, for a given input sentence, a subset of the grammatical structures, where useless structures with respect to the input sentence are discarded. [START_REF] Bangalore | Supertagging: An Approach to Almost Parsing[END_REF] proposed to use ngrams and probabilities to compute the set of most probable grammatical structures. This probability-based lexical selection is called supertagging. A major drawback of this approach is that it heavily relies on the training corpus used for assigning lexical probabilities. Supertagging may thus ignore valid structures (i.e., structures that are actually needed to parse the input sentence), which in turn can degrade parsing accuracy.

To prevent lexical selection from ignoring valid structures, [START_REF] Boullier | Supertagging: a Non-Statistical Parsing-Based Approach[END_REF] proposed to rather define a grammatical abstraction (i.e., a morphism), which produces an abstract grammar from the input one, and to use this abstract grammar to parse the input sentence. For each set of abstract structures that succeed in parsing the input sentence, one selects the original structures associated with these by the morphism. This technique improves parsing efficiency only if parsing with the abstract grammar is less complex than with the input grammar. In his experiments, [START_REF] Boullier | Supertagging: a Non-Statistical Parsing-Based Approach[END_REF] abstracted a Context-Free Grammar from an LTAG. Parsing with LTAG has a polynomial complexity of O(n ) ⁶ , n being the 1 Lexical selection is related to part-of-speech tagging.

While the latter assigns to a word its possible syntactic categories in a given context, lexical selection assigns to a word its possible grammatical structures.

length of the sentence to parse, while parsing with CFG has a complexity of O(n³). The main drawback of this approach is that one needs to first parse with the abstract grammar, which may still be time-consuming. Following [START_REF] Boullier | Supertagging: a Non-Statistical Parsing-Based Approach[END_REF], [START_REF] Bonfante | Polarization and abstraction of grammatical formalisms as methods for lexical disambiguation[END_REF] proposed a non-probabilistic lexical selection, where one uses a polarity-based abstraction. This abstraction is inspired by Interaction Grammar [START_REF] Perrier | Interaction Grammars. 18 Th Int. Conference on Computational Linguistics, COLING 2000[END_REF]. 2Interaction Grammar can be seen as an extension of Categorial Unification Grammar where attribute-value pairs are replaced with attribute-polarity-value triples that also encode valency constraints. [START_REF] Bonfante | Polarization and abstraction of grammatical formalisms as methods for lexical disambiguation[END_REF] thus aimed at reducing the combinatorial explosions of a polarized grammar by preliminary filtering with the polarity constraints. Concretely, they projected sets of polarities from the input grammatical structures. These sets were then combined according to the sentence to parse, in order to compute neutral sets referring to valid selections (we will elaborate on this in Section 3). In this paper, we build on the work of [START_REF] Bonfante | Polarization and abstraction of grammatical formalisms as methods for lexical disambiguation[END_REF], and propose to compute a polarity-based abstraction, which also takes into account the linear order of the words of the sentence to parse to speed up filtering.

This paper is organized as follows. In Section 2, we briefly introduce LTAG. Then, in Section 3, we give a detailed presentation of the polarity-based lexical selection of [START_REF] Bonfante | Polarization and abstraction of grammatical formalisms as methods for lexical disambiguation[END_REF] and show how we extend it to reduce the cost of lexical selection. In Section 4, we present an implementation of this extended lexical selection within an LTAG parser, and an evaluation of this approach. Finally, in Section 5, we conclude.

Lexicalized Tree-Adjoining Grammar

Definition

A Lexicalized Tree-Adjoining Grammar [START_REF] Joshi | Tree-Adjoining Grammar. In: Handbook of Formal Languages[END_REF]) is a tree rewriting system, where elementary trees can be rewritten using one of the two following operations: substitution and adjunction.

Substitution consists in replacing a leaf node (marked with a ↓) of an elementary or derived tree with an elementary tree whose root is labeled with the same syntactic category as this leaf node. Adjunction consists in replacing an internal node of an elementary or derived tree with an elementary tree having both a root node and a leaf node (marked with *, called foot node) labeled with the same category as this internal node. As an illustration, see Fig. 1 below. It shows the substitution (resp. adjunction) of the tree associated with John (resp. deeply) into the tree associated with sleeps.

Furthermore, in a Lexicalized TAG, every elementary tree is associated with at least one lexical item (called anchor). If a tree is associated with several lexical items, there is one anchor, and several co-anchors.

Parsing with LTAG

A large LTAG is made of thousands of elementary trees. To reduce redundancy within this huge set of trees, a three-layer lexicon architecture has been proposed by the XTAG Group (2001). It consists in dividing the LTAG lexicon in three sub-lexicons:  a morphological lexicon associating inflected words with lemmas,  a syntactic lexicon associating lemmas with tree schemas (i.e., unanchored LTAG trees),  and a grammatical lexicon containing tree schemas to be anchored. These tree schemas are gathered into families according to the sub-categorization frame they describe. When parsing with LTAG, (i) the input sentence is first tokenized, then (ii) each token is associated with the corresponding entries from the morphological lexicon. For each selected morphological entry, the syntactic lexicon is used to retrieve corresponding lemmas. For each pair (word, lemma), corresponding tree schemas are selected and anchored. (iii) In a final step, parsing is done using the selected sub-grammar.

As mentioned above, step (ii) may yield an exponential number of sequences of anchored grammatical structures, while step (iii), i.e., LTAG parsing, has a polynomial complexity in the length of the input sentence. The global complexity of (ii)+(iii) can be evaluated as O(|G|.A ) ⁶ where |G| is the size (i.e., number of structures) of the input grammar and A the number of transitions of the automaton that describes the sequences of useful anchored grammatical structures [START_REF] Lang | Recognition can be harder than parsing[END_REF].

In our work, we aim at reducing the practical cost of parsing, by discarding as many useless grammatical structures as possible during lexical selection (step (ii)). The next section shows how this can be done using a polarity-based grammar abstraction.

Lexical disambiguation

Following [START_REF] Bonfante | Polarization and abstraction of grammatical formalisms as methods for lexical disambiguation[END_REF][START_REF] Bonfante | Lexical disambiguation with polarities and automata[END_REF], our approach for lexical disambiguation uses a polarity-based abstraction over the input grammar. Let us first see how this abstraction is built by [START_REF] Bonfante | Polarization and abstraction of grammatical formalisms as methods for lexical disambiguation[END_REF] and applied to LTAG. We shall then show how it can be extended to improve disambiguation times. [START_REF] Bonfante | Polarization and abstraction of grammatical formalisms as methods for lexical disambiguation[END_REF] define a general layout for lexical disambiguation, that fits a large number of lexicalized grammar formalisms. They apply this layout to Lambek Grammar [START_REF] Lambek | The Mathematics of Sentence Structure[END_REF] and LTAG. Their lexical disambiguation is a three-step process:

Bonfante et al. (2004)'s lexical disambiguation

1. (grammar polarization) First, the input grammar is polarized. Every grammatical structure g is associated with a set of polarities Sg made of polarities of the form (f,v,p) where f is a feature (constant), v a value (constant), and p a (possibly negative) integer. Since the grammar is lexicalized, we can associate each word w of the lexicon with its corresponding multiset of polarities M (recall that a word can anchor several grammatical structures, each associated with a possibly non-unique set of polarities).

2. (abstraction) Second, an abstract lexical selection is operated using the previously computed polarized grammar. Concretely, this abstract lexical selection amounts to first computing the cartesian product P of the multisets associated with the words w1 … wn of the input sentence : P = M1 × … × Mn. Then, a binary rewriting rule (called neutralization rule) is applied on the elements E = (S1, … , Sn) of this product P as follows.

Let (S, S') be a couple of polarity sets in E, these are replaced with a set S'' such that: ▪ if a feature f with value v is present in both S and S' as (f,v,p) and (f,v,p') respectively, then the polarity (f,v,p + p') is added to S'' . ▪ any polarity (f,v,p) in (S U S') \ (S ∩ S') is copied into S''. This rewriting goes on (in an arbitrary order) until all elements of E have been consumed, thus producing one polarity set per element E.

(filtering)

In the end, the elements E of P are filtered to keep only those whose abstraction after neutralization is a set made of exactly one polarity (cat, axiom, +1) and an arbitrary number of polarities of the form (f,v,0), and nothing else. For each such element, the associated grammatical structures in the input grammar are lexically selected.

Note that this general layout relies on the computation of a huge number of sequences of structures (i.e., the elements of the product P). As pointed out by [START_REF] Perrier | Désambiguisation lexicale à l'aide d'automates de polarités[END_REF], among possible implementations of this lexical selection, using automata helps in sharing information (i.e., merging sequences with identical sets of polarities). Also, a crucial point of this approach lies in the definition of polarization. Polarization should include enough information (i.e., enough features) to distinguish between useful grammatical structures in the context of the sentence to parse, and useless ones, and yet it should not lead to a complex, time-consuming abstraction (i.e., applying neutralization to polarized structures should stay computable in reasonable times).

Application to LTAG

Let us now see how this lexical selection is applied to LTAG, where grammatical structures are trees.

1. The first step (polarization) consists in associating each tree t of the input LTAG with a set of polarities of the form (cat, x, n), where : ▪ x is either the category of the root, then n is 1 minus the number of substitution or foot nodes labeled with the category x, or ▪ x is not the category of the root, and then n is -1 times the number of substitution or foot nodes labeled with the category x, or ▪ x is a lexical item (i.e., a word) labeling a coanchor (e.g., a preposition), and then n is -1 times the number of such co-anchors.3 That is, we consider only one type of feature (category). The trees from Fig. 1 Neutralization consists in summing, for each element E of P, the polarities for compatible (f,v) pairs. In our toy example, we obtain:

neutralization(E)={(cat, S, +1) (cat, NP, 0) (cat, VP, 0)} 3. The third step (filtering) keeps among neutralized elements of P, those which are wellformed. In our case, there is only one element in P and its neutralization is well-formed for an axiom S (i.e., if we are parsing a sentence).

Note that lexical selection based on polarization and abstraction as presented here does not rely on any particular word order. One can compute well-formed elements of P (i.e., apply neutralization) following or not the word order defined by the input sentence. Also, as mentioned above, using deterministic automata to implement this lexical selection allows to factorize information, which is very important to deal with the huge number of sequences of structures. The idea of this automaton-based implementation is the following. States are associated with sets of polarities. We create an empty initial state. We create as many edges from the initial state as candidate polarized grammatical structures for the first word of the input sentence. These edges lead to new states whose set of polarities is the sum of the polarities of the original state and those of the structure labeling the edge. We merge states with similar sets of polarities. We go on with the second word, that is, we create as many edges from these new states as candidate polarized structures for this word, leading to new states, and so on. As an illustration, consider Fig. 2 below. In this figure, one can see the lexical selection for the sentence "John eats a cake". For the word John, there is only one associated grammatical structure in the lexicon, namely that for proper nouns. It is polarized as (cat, NP, +1). For eats, there is an ambiguity between intransitive and transitive verbs. There are thus two new states, depending on which tree is used. When the sentence is fully processed, we obtain two final states, among which only one is well-formed (state number 7).

Lexical disambiguation using word order

As mentioned above, the polarity-based lexical selection introduced in Section 3.1 does not rely on a specific order. In particular, polarization and abstraction do not use any information about the word order within the input sentence. In order to enhance lexical selection, we propose to use such information by extending [START_REF] Bonfante | Polarization and abstraction of grammatical formalisms as methods for lexical disambiguation[END_REF]'s lexical selection as follows.

The polarization of input grammar (step 1) is extended to produce pairs of sets of polarities. That is, polarization generates for each input grammatical structure, a pair of sets of polarities, where the first element is the classical polarization of [START_REF] Bonfante | Polarization and abstraction of grammatical formalisms as methods for lexical disambiguation[END_REF], and the second element a polarization related to left context. This second polarization is similar to that presented in Section 3.2 except that it only considers the root node and substitution nodes that are on the left of the anchor. For instance, the word eats in Fig. 2 is now polarized as: polarities(eats) = polarities(ttrans) U polarities(tintrans) = { ({(cat,S,+1) (cat, NP, -1)},{(cat, S, +1) (cat,NP,-1)}) ({(cat,S,+1) (cat, NP, -2)},{(cat, S, +1) (cat,NP,-1)}) } Abstraction (step 2) is then applied to these pairs of sets of polarities. The definition of the neutralization rule extends that of [START_REF] Bonfante | Polarization and abstraction of grammatical formalisms as methods for lexical disambiguation[END_REF], by applying on pairs of sets, while following the word order of the input sentence. Let (a1, b1) and (a2,b2) be two such pairs: neutralization((a1,b1), (a2,b2)) = (neutralization(a1,a2), neutralization (b1,b2))

Furthermore, b1 and neutralization(b1,b2) are constrained to only contain polarities of the form (f,v,i) where i ≥ 0.

Finally, filtering (step 3) remains identical to that of [START_REF] Bonfante | Polarization and abstraction of grammatical formalisms as methods for lexical disambiguation[END_REF], and applies on the first element of the neutralized pairs.

Let us now see how this extension impacts the automaton-based implementation of lexical selection. Consider the sentence "say it to her", and a lexicon that associates the verb say with an imperative, a transitive and a ditransitive grammatical structure. We can build an automaton for lexical selection as previously (see Fig. 3).

Fig. 3: Lexical selection using left context

We notice that only the imperative lexical entry for say prevents the left context abstraction from having negative polarities (i.e., transitive and ditransitive grammatical structures have "uncombinable" substitution nodes on the left of the anchor). Once an abstraction has a non-empty set of left polarities, there cannot be any other abstraction that could combine to produce a well-formed polarity set. In Fig. 3, we can thus discard states 2 and 3, and we do not need to build states 5, 6, 8, 9, 11 and 12.

Implementation and evaluation

In order to evaluate the benefits brought by left context, we implemented polarity-based lexical selection using left context within the TuLiPA parser [START_REF] Parmentier | A Syntax-Semantics Parsing Environment for Mildly Context-Sensitive Formalisms[END_REF]. 5 TuLiPA is an open source parser for mildly context-sensitive formalisms which supports LTAG and already includes the automaton-based lexical selection of [START_REF] Bonfante | Polarization and abstraction of grammatical formalisms as methods for lexical disambiguation[END_REF]. This makes it easier to compare the two approaches. To exhibit the benefits of left-context disambiguation, we used two types of resources, a toy LTAG and a large one. These are described below. The question of which metrics to use to evaluate the gain of polarity filtering is not trivial. We chose to compare the sizes of automata when using left context or not. Another interesting metrics would be total parsing times, even if the additional cost of automaton-based filtering is expected to be small compared with the cost of parsing with highly ambiguous grammars. Due to time and space restrictions, this is left for future work.

Qualitative study

In a first experiment, we performed polarity-based lexical selection using a toy LTAG made of 12 tree schemas (intransitive verb with clitic / canonical/ extracted subject, active transitive verb with clitic / canonical / extracted subject, passive transitive verb with clitic / canonical / extracted subject, proper nouns, clitics and auxiliaries), 9 lemmas, and 30 morphological entries (note that there is no part-of-speech tagging ambiguity in our toy lexicon, i.e., lexical ambiguity here is purely grammatical). The results are given in Table 1 1, the column labelled #lex gives the general lexical ambiguity of the sentence (i.e., cardinality of the cartesian product of the lexical entries selected by each word). The column #pol gives the remaining lexical ambiguity once polarity filtering has been performed (in this toy example, the remaining ambiguity is the same using left-context optimization or not). The column #st.1 gives the size (i.e., number of states) of the polarity-based automaton of [START_REF] Bonfante | Polarization and abstraction of grammatical formalisms as methods for lexical disambiguation[END_REF], and #st.2 the size of the automaton that uses left context. One can notice that using left context significantly reduce the size of the polarity-based automaton, even with a small grammar and short sentences.

Quantitative study

In a second experiment, we used the French LTAG of [START_REF] Crabbé | Représentation informatique de grammaires fortement lexicalisées : application à la grammaire d'arbres adjoints[END_REF]. This grammar is compiled from a metagrammar developed in the XMG language (Duchier et al., 2004). 6 It is made of 6,080 tree schemas, and focusses on verbal, adjectival and nominal predicatives.

We used this grammar to evaluate polarity-based lexical disambiguation on a subset of the Test Suite for Natural Language Processing [START_REF] Lehmann | TSNLP -Test Suites for Natural Language Processing[END_REF]. This subset contains 90 sentences whose length ranges from 2 to 12 words. Fig. 4 displays the average number of states of the automaton (y-axis) depending on the length of the input sentence (x-axis). One can notice the combinatorial explosion of the size of the automaton of [START_REF] Bonfante | Polarization and abstraction of grammatical formalisms as methods for lexical disambiguation[END_REF] with sentences having more than 7 words. As suggested by Fig. 4, for polarity filtering to stay computationally tractable when parsing real data, optimizations are needed. From these first experiments, using information about left context seems to be adequate in practice. This still needs to be confirmed with other evaluation resources and metrics.

Conclusion

In this paper, we presented an extension of the technique of [START_REF] Bonfante | Polarization and abstraction of grammatical formalisms as methods for lexical disambiguation[END_REF] for lexical disambiguation. This extension consists in defining a grammar abstraction containing information about left context. In LTAG, this information corresponds to substitution sites located on the left of the anchor. This abstraction makes it possible to reduce on-line the size of the automaton used for lexical selection, thus improving parsing time efficiency.

As advocated by [START_REF] Bonfante | Dependency Constraints for Lexical Disambiguation[END_REF], using information about context (e.g., dependency constraints) for improving lexical disambiguation may be applied to several types of lexicalized grammar through the concept of companionship. Similarly, it would be interesting to apply left context abstraction to other formalisms.

Another interesting question concerns the definition of polarization. How to polarize a grammar so that invalid grammatical structures can be filtered out as soon as possible while keeping neutralization computationally tractable ? For LTAG, we chose to use only polarities related to syntactic categories. For Interaction Grammar, other types of features are used (e.g., morpho-syntactic features), leading to a complex abstraction because of neutralization cost. There is a trade-off to be found.

Fig. 1 :

 1 Fig. 1: Tree rewriting in LTAG

  are polarized as follows: polarities(tJohn ) = {(cat, NP, +1)} polarities(tsleeps) = {(cat, S, +1) (cat, NP, -1)} polarities(tdeeply) = {(cat, VP, 0)} 2. The second step (abstraction) consists in computing the cartesian product of the set of polarities according to the sentence to parse: 4 P= {polarities(tJohn )}×{polarities(tsleeps)}×{polarities(tdeeply)}

Fig. 2 :

 2 Fig. 2: Automaton-based lexical selection

Fig. 4 :

 4 Fig. 4: Distribution of the size of polarity automata

Table 1 .

 1 . Polarity-based lexical disambiguationIn Table

	Sentence	# lex #pol #st.1 #st.2
	Jean aime Marie (John loves Mary)	12 2 26 4
	Marie dort (Mary sleeps)	3	1	5	3
	Ils dorment (They sleep)	3	1	5	3
	Jean qui dort aime Marie (John who sleeps loves Mary)	36 8 78 7
	Marie est appelée par Jean (Mary is called by John)	6	1 21 9
	Jean qui dort est aimé par Marie (John who sleeps is loved by Mary)	18 4 63 13

In this formalism, grammatical structures are tree descriptions, where nodes are labeled with polarized feature-structures and parsing corresponds to computing syntactic tree models where polarities are neutralized.

In step 2, input words which can be coanchors in the lexicon, are associated with trees made of a single root node. These trees bring the +1 polarity needed to neutralize the coanchor in a valid lexical selection.

Note that in our toy example, there is only one tree per word of the input sentence (i.e., there is no lexical ambiguity), and thus P has a single element. The purpose here is merely to illustate the lexical selection process.

See http://sourcesup.cru.fr/tulipa.

Crabbé's metagrammar is available at https://sourcesup.cru.fr/scm/viewvc.php/trunk/METAGR AMMARS/FrenchTAG/?root=xmg and its French documentation at http://www.linguist.univ-parisdiderot.fr/~bcrabbe/frenchgrammar/index.html

Acknowledgement

The implementation of the left-context automaton has been inspired by Johannes Dellert's code within TuLiPA.