Min Yao
email: min.yao@insa-lyon.fr

Philippe Duvauchelle
email: philippe.duvauchelle@insa-lyon.fr

Valérie Kaftandjian
email: valerie.kaftandjian@insa-lyon.fr

Angéla Peterzol- Parmentier
email: angela.peterzol-parmentier@areva.com

Andreas Schumm
email: andreas.schumm@edf.fr

Simulation of Computed Radiography X-ray Imaging Chain Dedicated to Complex Shape Objects

Keywords: Radiographic testing (RT), modeling and simulation, nuclear, Monte Carlo-Deterministic simulation, computed radiography

Computed Radiography (CR) based on photostimulable imaging plate (IP) is increasingly used in the field of non-destructive testing. For the inspections of high attenuation specimens employing high energy sources, the CR performance is poor because of the response of the IP. Simulation is a very useful tool to predict experimental outcomes and determine the optimal operating conditions. We propose a hybrid simulation approach which combines the use of both deterministic and MC codes for simulating rapidly a complex geometry imaging set-up. This approach allows us to take into account the degradation effect introduced by Xray scattering and fluorescence transport as well as the optical photon scattering taking place in IP during X-ray exposure and optical readout processes. The results of different simulation configurations are compared.

Introduction

Computed radiography (CR), as a digital replacement of the traditional film radiography, is increasingly used in the field of non-destructive testing. As CR uses equipment very similar to the film radiography [START_REF] Sonoda | Computed radiography utilizing scanning laser stimulated luminescence[END_REF], the users do not need to modify their existing exposure routine. Moreover, the digital detector used, imaging plate (IP), is flexible (can be bent) and re-useful [START_REF] Seggern | Photostimulable x-ray storage phosphors: a review of present understanding[END_REF]. It is therefore easy to implement and cost-efficient. On the other hand, CR also has its limitations which hold back the industry from completely replacing the traditional radiography with such a technique. Today's available commercial IPs are mostly based on the high Z phosphor, which makes IP energy-dependent: sensitive to low energy and nearly transparent to high energy [START_REF] Rowlands | The physics of computed radiography[END_REF]. To overcome this problem, metallic screens are recommended, by current international standards [START_REF]Non-destructive testing -Industrial computed radiography with storage phosphor imaging plates -Part 2: General principles for testing of metallic materials using X-rays and gamma rays[END_REF], [START_REF]Non-destructive testing of welds --Radiographic testing --Part 2: X-and gamma-ray techniques with digital detectors[END_REF], to be used together with IP to ensure a good image quality. However, the type and thickness of such a screen are not clearly defined and a large panel of possible configurations does exist.

Simulation is a very useful tool to predict experimental outcomes and determine the optimal operating conditions. It makes it possible to study how the relevant operating parameters affect the x-ray image without actually testing it in real life. In the previous works, Vedantham and Karellas have developed a complete (from X-ray exposure to digital readout) analytic CR model to analyze the propagation of system performance factor, such as detective quantum efficiency (DQE) and modulation transfer function (MTF), during image formation process [START_REF] Vedantham | Modeling the performance characteristics of computed radiography (CR) systems[END_REF]. In their approach, the X-ray scattering effect is considered negligible. However, for high energy CR, where the scattering effect becomes dominant, this assumption is no longer appropriate. Souza et al. have proposed a methodology of computed radiography simulation for industrial applications [START_REF] Souza | Methodology for digital radiography simulation using the Monte Carlo code MCNPX for industrial applications[END_REF], [START_REF] Correa | Computed radiography simulation using the Monte Carlo code MCNPX[END_REF]. The IP dose response has been obtained using Monte Carlo method; scattering effect has been accounted. Whereas, the unsharpness introduced by energy deposition process has not been integrated, neither does the effect of metallic screen.

We propose a hybrid simulation approach which combines the use of both deterministic and MC codes. This approach allows us to simulate rapidly a complex geometry set-up and still take into account the physical phenomena (e.g. effect of metallic screens and x-ray or light scattering) during energy deposition and optical readout process. The results of different simulation configurations are compared.

Simulation method

In CR, the imaging plate is used to detect the transmitted radiation emerging from the object. The received radiation interacts with imaging plate resulting in a latent image, which is later read by an optical scanner. Accordingly, the simulation of the CR imaging chain follows three successive stages (Figure 1): i. X-or gamma rays attenuation by object. At this stage, both Monte Carlo and deterministic methods can be used. ii. Latent image generation. This stage consists of two steps: a) interaction of radiation with detector (IP alone or IP with screens); and b) latent image generation. For the former, the detector (IP alone or IP with screens) is modeled by a transfer function which is obtained through a parametric study using Monte Carlo simulation. The latter is simply modeled by an amplification factor, as the latent imaging forming mechanism is still not clearly understood, and furthermore, depending on the materials, the mechanisms are different. iii. Digital image generation. There are also two steps in this stage: optical readout, and the collection, amplification and digitization of the emitted signal (i.e. photostimulated luminescence, PSL). The former is modeled by a transfer function (obtained with a Monte Carlo code) of the IP optical response; the latter is simply modeled as a factor without further blurring the output image.

Different simulation methods have been applied to the three stages, and will be discussed in sections 2.1, 2.2 and 2.3.

Generating object image

This stage simulates the interaction of radiation with object, and outputs an object image obj(E, x, y). Different methods (Monte Carlo or deterministic) could be employed depending on needs. To simulate a simple object, the Monte Carlo methods can deal with very detailed physical phenomena and provide precise results. However, modeling of complex objects with MC method is very time consuming; it would take months or even more. Moreover the manual design of complex geometry is quite frustrating and error-prone. In this case, deterministic methods can simulate rapidly complex set-ups under reasonable approximations; coupling with Computer Aided Design (CAD), the geometry designing can be relatively easy and robust.

In order to generate obj(E, x, y) (whichever method is chosen), a virtual detector is used and should be placed at the actual detector plane (see the green plane in Figure 2 (a)). This virtual detector is divided into M×N pixels to record the spatial distribution of the incident photons. Each pixel patch is also a spectrum counter with a channel width of E width keV. In this way, the transmitted radiation is stored (Figure 2 (b)). The object image is in unit of photon number per pixel area per energy channel.

Generating latent image: CR detector model

The interaction of X-ray with detector is modeled by a transfer function H1 (Figure 3 We generate a set of impulse responses by varying the detector configuration det (i.e. IP/screens combination) and the incident energy E. With R x-ray (det,E,x,y,z), we can have the spectral response (Figure 4 (a)) by summing it over x, y and z; we can also calculate the spatial response (via the Fourier transform of the impulse responses) for different energies (Figure 4 (b)). The latent image is obtained through a convolution (H1) of the object image with the detector model

( ) , ) , , , , (det ) , , ( ) , , , , (det ) , , ( ) , ( H1 ) , , ( , 0 0
∫ ∫∫ ∫         - - ⋅ = * = = - - - E v u ray x E ray x ray x dE dudv z v y u x E R y x E obj dE z y x E R y x E obj R obj z y x Limg (1)
where Limg(x,y,z) is the latent image and det 0 is the given detector.

Generating digital image: optical readout model

The optical readout process is viewed as a transfer function H2 (Figure 5 Excitation energy (keV) Absorption reader: a finely focused laser is used to scan and release, line by line, the latent image; the latent image is modified while the laser spot traverses the IP [START_REF] Leblans | Storage phosphors for medical imaging[END_REF]. Thus different from the previous operator H1, H2 is a modified convolution operation. The final digital image is computed using

[ ] { } , ) , , ( exp 1 ) , , ( ) ( )
where Limg(x,y,z) is the latent image, I(x,y,z) is the IP model (impulse response to a laser beam), P(z) is the probability that a photon (emitted at z) could escape from the front side of IP, σ is the optical cross section of photo-stimulation and t scan is the dwell time of laser spot at (x 0 ,y 0 ).

The IP model I(x,y,z) is obtained through the Monte Carlo method. A Monte Carlo code has been programmed in Matlab to simulate the light propagation problem in IP. Certain physical models of light/IP interaction adopted in the code are based on [START_REF] Wang | MCML-Monte Carlo modeling of light transport in multi-layered tissues[END_REF] and [START_REF] Fasbender | Monte Carlo modeling of storage phosphor plate readouts[END_REF]. Figure 5(b) shows us an example of IP response to a normal incident laser beam, to obtain which 2×10 6 photons has been generated to strike the imagine plate.

(a) (b)

Results

The geometric set-up in Figure 2 has been adopted for our simulation. The object material was set steel, and it was irradiated by a monoenergetic point source (100 keV). A step/hole type image quality indicator (IQI) was placed on the object facing the source. Note that only the central part (where IQI located) has been exposed.

The detector was a combination of IP with metallic screens, where IP was sandwiched between the screens. The IP was modeled as a multi-layered structure which consists of, in sequence, a 6 µm protective layer, a 150 µm phosphor layer, a 254 µm support layer and a 25.4 µm backing layer. The materials of these four layers were respectively Mylar for the protective and support layers, BaFBr:Eu 2+ with a packing factor of 60% for the phosphor layer, and polycarbonate for the backing layer. We have simulated the dose response of the following two detector configurations: a) IP alone ('HRIP') and b) IP with two 0.3 mm lead screens on both sides (0.3Pb+HRIP+0.3Pb).

Latent image

As a first step, we have computed the latent images generated by the two different detectors (see the upper two images in Figure 6). In order to compare the contrast, the two images have been normalized by their maximum gray values. We plot the normalized profiles along the red lines below. In the image, the detector efficiency is also presented, denoted 'AE' (total energy absorbed in the phosphor layer over the total incident energy). With HRIP, about 4.92% of the object image has been detected. With lead screens, the efficiency decreases to 1.96% and we lose the contrast.

Final output of simulation: digital image

In this part, the latent image generated with HRIP has been chosen as input. It has been read using different laser powers (represented by a power factor): 10 10 , 10 13 , 10 14 , 10 15 and 10 16 . This factor is directly proportional to the laser power. In the image obtained with '10 16 ', it is hard to identify the smallest hole.

We then plot reading efficiency (output signal over input signal) versus the laser power ( Figure 8). The efficiency increases slowly at low laser powers, then we see a significant increase between 10 13 and 10 15 ; and at 10 16 the curve starts to reach its maximum. One may notice that the maximum efficiency does not equal to one. Indeed, a high power increases the photoluminescence, but the photons are emitted isotropically and only a small fraction can escape from the front surface of IP and contribute to the final image. In order to compare the image contraste, the images have been nomalized by their maximum values. In Figure 9, we present the profiles along the IQI. The red curve is the latent image profile. The curves of the first 2 powers overlap each other, then we lose contrast by increasing the power. Comparing the profiles, we also see an obvious shift between the black and the red curves in the IP translation direction.

Finally, we show the evaluation of the profile after passing different stages in Figure 10. In this example, we took the optimal conditions for each stage: to detect the latent image with HRIP then read it with a laser power of '10 15 '. We lose contrast at each stage.

Discussion

In our case study, we have compared the image quality of different detection configurations (i.e. detector and laser power). The optimal imaging condition is obtained by using HRIP alone to receive the object image and reading the latent image with the laser power '10 15 '.

For the optical readout, with the increase of the laser power, the reading efficiency increases (i.e. the fraction of the electrons released by the laser); however we loss the contrast. The reason is that when reading the current point, part of the trapped electrons in the neighborhood are also released and contribute to the signal of the current pixel resulting in blurring. The more we increase the laser power, the more the surrounding pixels are affected. This is also the reason that we observe a shift while using a very high laser power. To optimize the efficiency and to optimize the contrast are two conflicting goals. Compromises should be made for different applications.

Conclusion

We have presented an approach to simulating the computed radiography imaging chain. Different simulation methods have been adopted: the Monte Carlo and deterministic methods.

A simulation tool has been developed based on this approach. We have showed the use of our tool via a case study. With the tool we can simulate the influence of different parameters such as detector configuration and laser power. One can use it to study the CR system performance, and optimize the image quality.

The comparison between simulation and experimental results is in progress.

Figure 1 :

 1 Figure 1: Computed radiography modeling. The CR imaging chain is modeled as three successive stages: x-ray attenuation by an object, latent image generation in IP and optical readout. The output of the previous stage is the input of the current stage. Certain sub-steps are treated as an amplification factor.

Figure 2 :

 2 Figure 2: Generation of the object image with a deterministic simulation tool VXI[9], [10]: (a) geometrical representation of imaging setup and (b) illustration of an output object image.

Figure 3 :

 3 Figure 3: Latent image generation: (a) detector transfer function H1; and b) impulse response of detector (3D map of energy deposition).

Figure 4 :

 4 Figure 4: Spectral and spatial responses of IP (obtained with R x-ray (det,E,x,y,z)).

  (a)), which also requires two inputs: latent image and IP model. Flying spot scanner is the most common CR

Figure 5 :

 5 Figure 5: a) Generation of the digital image using the optical readout transfer function H2; b) an example of IP model (impulse response of IP to laser light).

Figure 6 :

 6 Figure 6: Simulation results: the latent images obtained with two different detector configurations.

Figure 7 Figure 7

 77 Figure 7 compares the object image, the latent image and two final images read with two different laser powers 10 16 and 10 10 . We lose the signal (gray level) and contrast after passing each stage. A great laser power leads a high gray level of the output image; however a slight shift upwards is observed (comparing Figure 7 (c) with (b)). In the image obtained with '10 16 ', it is hard to identify the smallest hole.

Figure 7 :

 7 Figure 7: Simulation results: (a) object image, (b) latent image obtained with HRIP, (c) digital image output (reading factor 10 16 ) et (d) digital image (reading factor 10 10 ).

Figure 9 :

 9 Figure 9: Normalized profiles for different laser power.

Figure 10 :

 10 Figure 10: The object profile propagation. The red line is the object profile before entering the detector; the green line is the latent image detected by 'HRIP'; and the blue line is the output digital image read by the laser power '10 15 '.

).

		0.018 0.018 0.018 0.018
			Efficiency Efficiency Efficiency Efficiency
		0.016 0.016 0.016 0.016	Fit Fit Fit Fit
		0.014 0.014 0.014 0.014
		0.012 0.012 0.012 0.012
	Efficiency	0.008 0.008 0.008 0.008 0.01 0.01 0.01 0.01
		0.006 0.006 0.006 0.006
		0.004 0.004 0.004 0.004
		0.002 0.002 0.002 0.002
		0 0 0 0	1E10 1E10 1E10 1E10	1E13 1E13 1E13 1E13 1E14 1E14 1E14 1E14 1E15 1E15 1E15 1E15 1E16 1E16 1E16 1E16
			Laser power ( a.u.)

Figure 8: Readout efficiency versus laser power.

		1 1 1 1
		0.99 0.99 0.99 0.99
		0.98 0.98 0.98 0.98
	Normalized profile	0.95 0.95 0.95 0.95 0.96 0.96 0.96 0.96 0.97 0.97 0.97 0.97	Image latente Image latente Image latente Image latente 1e10 1e10 1e10 1e10
		0.94 0.94 0.94 0.94	1e13 1e13 1e13 1e13
			1e14 1e14 1e14 1e14
		0.93 0.93 0.93 0.93	1e15 1e15 1e15 1e15
			1e16 1e16 1e16 1e16
		-1 -1 -1 -1 0.92 0.92 0.92 0.92	-0.8 -0.8 -0.8 -0.8	-0.6 -0.6 -0.6 -0.6	-0.4 -0.4 -0.4 -0.4	-0.2 -0.2 -0.2 -0.2	0 0 0 0	0.2 0.2 0.2 0.2	0.4 0.4 0.4 0.4	0.6 0.6 0.6 0.6	0.8 0.8 0.8 0.8	1 1 1 1
							y (mm)

11th European Conference on Non-Destructive Testing (ECNDT 2014), October 6-10, 2014, Prague, Czech Republic


Cyril Uhry
email: cyril.uhry@cea.fr

François Guillet
email: françois.guillet@cea.fr

Philippe Duvauchelle
email: philippe.duvauchelle@insa-lyon.fr

Valérie Kaftandjian
email: valerie.kaftandjian@insa-lyon.fr

Optimisation of the process of X-ray tomography applied to the detection of defects in composites materials

Keywords: computed tomography, X-ray detectors and sources, defect detection, scattering, simulation, X-ray imaging, non-destructive testing, virtual X-ray imaging

Computed tomography (CT) is a relevant nondestructive inspection method for composite material. However, X-ray attenuation contrast difference between fibers and matrix can be low, which makes identification of defects difficult. Optimization of the tomographic process has then to be considered, such as studying the acquisition parameters and phenomena that decrease the contrast. Simulation of the CT method is considered in order to optimize contrast between fibers and matrix. Simulation allows to study phenomena independently to the others. Phantom was built in order to check the accuracy of the computation. Even though results show discrepancies between computation and experimental measurements, these discrepancies have been found to be due to an experimental artefact. When corrected, simulation fits very well with experimental results and this procedure allows to seek an optimal setup for composite material inspection.

Introduction

Because of their outstanding properties [START_REF] Ozcan | Microstructure and elastic properties of individual components of C/C composites[END_REF], carbon fiber-reinforced polymers (CFRP) are the favorite candidate for lightweight thermo-structural applications, like in the aerospace industry. In this study, we will focus on carbon fiber reinforcement composites materials with epoxy resin matrix such as hydrogen reservoirs. These reservoirs are composed of filament of carbon fibers impregnated with epoxy resin. The architecture of the composite material is obtained by filament winding around a cylindrical polyamide liner. Architecture plays an important part for obtaining good properties. However, defects in composites cannot be avoided and occur either during the manufacturing process, or during usage (impact, fatigue). Depending on their nature and size, they can decrease the thermo-mechanical properties of the material. Defects can be porosities in the resin, delamination between woven tissue, or cracks in fibers [START_REF] Yan Zhu | Influence of voids on interlaminar shear strength of carbon/epoxy fabric laminates[END_REF] [START_REF] Shigang | Effect of manufacturing defects on mechanical properties and failure features of 3D orthogonal woven C/C composites[END_REF]. Thus, accurate non-destructive inspections are mandatory, such as X-ray computed tomography. CT provides three-dimensional images of internal structure of material, allowing detection of defects, as porosities or delamination [START_REF] Centea | Measuring the impregnation of an out-of-autoclave prepreg by micro-CT[END_REF] [START_REF] Schilling | Xray computed microtomography of internal damage in fiber reinforced polymer matrix composites[END_REF]. Distinction between porosities and composite material (carbon fiber or epoxy resin) in CFRP is possible due to the relative high density contrast and chemical composition difference between porosities (air or trapped gas) and carbon or epoxy resin. The difference of linear attenuation coefficient is consequently enough for observing this type of defects. However, the distinction between carbon reinforcement and epoxy resin to identify architecture defects is much more complex. A typical epoxide polymer is composed of 76 wt% (weight percentage) of carbon, 17 wt% of oxygen and 7 wt% of hydrogen. The densities of epoxy polymer and carbon fiber are close (typically 1.8 for carbon fiber and 1.2 for epoxy resin). This leads to a low attenuation coefficient difference at voltages required for inspecting objects and consequently to a low contrast in CT scan. X-ray computed tomography allows observation of the internal structure of hydrogen reservoir. Figure 1 display a photo of a hydrogen reservoir (a), a slice perpendicularly to its revolution axis (b) and a cylindrical slice along the blue line in (b). Identification of the filament orientation is difficult. However, a high proportion of porosities make their identification easier.

As the contrast difference between epoxy resin and fiber is weak, optimization of the tomographic process should be considered, beginning with experimental parameters such as voltage. Phenomena decreasing contrast in CT scans such as scattered radiation has also to be studied. A usual approach to carry out optimization is computer modelling and simulation. It allows to study independently the influences of the phenomena [START_REF] Jarry | Characterization of scattered radiation in kV CBCT images using Monte Carlo simulations[END_REF] [7], or the optimal set-up parameters [START_REF] Tigkos | Simulation study for optimization of X-ray inspection setup applied to SFRP aerostructures[END_REF] [9], with the advantage of reducing the experimental acquisition as much as possible. In this study, computation of projections is carried out using the deterministic software VXI (Virtual X-ray Imaging) [START_REF] Duvauchelle | A computer code to simulate X-ray imaging techniques[END_REF]. It allows to compute a set of projection that is reconstructed using the software DigiCT (Digisens). Because of the weak contrast between carbon and epoxy resin, it is very important that the computation are as much accurate as possible. In order to check the accuracy of results, a simple phantom object is considered. Experimental projections and CT scans of this object are compared to the one computed using VXI and the differences are discussed.

Phantom object

The geometry, size and chemical composition of the phantom object were chosen to be close to the characteristics of hydrogen reservoirs, particularly the filament winding around the liner. The object proposed here is a hollow cylinder consisting in a succession of hollow cylinders of pure graphite and epoxy resin corresponding successively to the filament winding of carbon fiber and epoxide polymer (Figure 2) . K10 graphite and RSF 816 epoxy resin were chosen due to availability and good representativeness of actual materials.

The phantom is approximately 8 cm high, with a diameter of 10 cm. It is composed of four graphite cylinders and three RSF 816 epoxy cylinders, each layer being 2 mm thick.

A chemical analysis was performed on both materials so as to compute the most accurate linear attenuation coefficient possible. Analysis shows that the graphite is composed of 93 wt% of carbon, with impurities (1.1 wt% silicon and 1.1 wt% sulphur). This heavier atoms account for a significant part of linear attenuation coefficient. A precise knowledge of chemical composition is therefore very important. The RSF 816 epoxy resin is relatively close to epoxy resin, 71 wt% of carbon, 17 wt% oxygen and 8 wt% of hydrogen and a low percentage of heavy weight atoms (chlorine 0.6 wt%). So as to validate the chemistry of materials homogeneous samples of graphite and RSF 816 epoxy resin were exposed to X-rays. Computed and experimental projections were compared for different high voltages. The difference between measured and computed X-ray intensity after attenuation for both materials is less than 2 % for a range of voltage between 60 kV and 200 kV.

Projection computation

Virtual X-ray Imaging (VXI) is a deterministic simulation code allowing the simulation of X-Ray radiography of three-dimensional objects. This code is based on ray-tracing technique and the attenuation law of photons passing through an object. The description of the principle of the simulation can be found in [START_REF] Duvauchelle | A computer code to simulate X-ray imaging techniques[END_REF] and [START_REF] Freud | Deterministic simulation of first-order scattering in virtual X-ray imaging[END_REF] VXI allows computation of the projections of the primary radiation, defined as photons that do not undergo interaction before reaching the detector. The computation of the scattered radiation is carried out separately. Only the first-order scattering is considered, higher orders of interaction are neglected.

Projections are computed as follows:

1. The primary radiation and scattered projection : primary and scattered radiations are computed on the central line of the cylinder (on the central line of the Figure 3). Given the symmetry of the cylinder, projection is considered to consist of copies of this line along the vertical axis. Thus, the cylinder is considered to be infinite along the revolution axis. 2. Computation of the images without any object: . 3. Normalization of and with :

. and are computed by calculating the energy deposited in the scintillator, in MeV. Normalization with allows obtaining projections free of unit, as obtained experimentally. As the format of the image is 16 bits, the constant is arbitrarily fixed to , corresponding approximately to .

4. The final projection image : .

Measurement system

The set-up consists in a X-Tek Micro 225kV X-ray tube with a reflective tungsten target. The detector is a 4000x4000 pixels Photonic Science Image Star Large Area, with approximately 100 microns square pixels. The scintillator is composed of a 300µm columnar CsI (Caesium Iodine) layer deposited on a 1 mm thick aluminium substrate. Visible photons emitted by the scintillator are reflected on a 45° mirror and focused on a CCD camera by a coupling lens.

The CCD area is 37x37mm with 9 microns square pixels. Images present artefacts due to the detector such as vignetting artefact (intensity is higher in the centre of the image than in the corner) barrelling distortion and dark current of the photodiodes. Images are corrected from gain and offset using equation 1.

(

) 1
With being the as-measured image, is an image without X-ray exposition, an image without object, as previously described.

Objects considered in the study are hollow cylinders. In the simulation, their axis of revolution corresponds to the axis of rotation of the tomographic acquisition. Consequently, only one projection is computed with XVI, and then duplicated as much as required. Noise can be added to the simulated projection during duplication, but this is not in the scope of this article. Whether projections are computed or experimental, CT reconstructions are performed using DigiCT (DigiSens software version 3.2) using FDK algorithm and Hanning filter.

Results

Experimental parameters:

Measurement of the phantom objet is performed for a 90 kV accelerating voltage, with an exposure of 8.9 s and an intensity current of 550 µA. The source-object distance is 300 mm and the object-detector distance is 620 mm. A set of 3600 projections are measured along 360°. The reconstruction volume corresponds to the central part of the phantoms. Its size is 106.6 mm x 106.6 mm x30 mm, with a 100 µm voxel size. Computed and experimental projections are compared in Figure 4. Figure 5 presents the greyscale profile along the region of interest in the Figure 4.

Projections

Computed grey level is close to experimental grey level. However, some discrepancies can be noticed. First, in the region of the highest attenuation (labeled A in the Figure 5), computed profile overestimates grey level by 25%. Second, the right and left maximum value of experimental grey level should be symmetric. However, the right maximum is 6% lower than the left maximum in the experimental projection and is lower than the expected 30.000 value (label B in the Figure 5).

Comparison between computation and experiment was performed for other voltages (140kV, 160kV, 200kV): the same differences were observed.

Computed tomography

Figure 6 shows a planar slice of the volume, perpendicular to the axis of revolution of the cylinders, for both experimental and computed reconstructions. Bright layers correspond to graphite cylinders and dark cylinders to RSF 816 layer, as darker grey level corresponds to weaker values of linear attenuation coefficient. Heterogeneities can be observed in the graphite layers (see arrows in Figure 7). Linear attenuation coefficient is higher in the border than in the center, which would correspond to higher graphite density in the border of the cylinders. Graphite cylinders were cut from blocks of graphite, a process which could account for higher densities close to the cutting zone. layer on the left of the profile (Figure 7). As deeper layers are crossed, linear attenuation coefficient decreases. This is the well-known cupping effect due to the polychromatic nature of the beam. The comparison of profiles shows that the computed linear attenuation coefficient is higher than the experimental linear attenuation coefficient. The overestimation increases as deeper layers are considered (relative difference of 5% for the external graphite layer and 15% for the internal graphite layer). This is consistent with projection comparison (Figure 5): computed attenuation is overestimated where the attenuation is maximal at the internal border of the object. This leads to a higher linear attenuation coefficient.

Discussion

A plausible hypothesis for the discrepancies between computation and experimentation is that the mirror in the camera backscatters X-rays towards the scintillator. X-rays photons that go through the scintillator without being absorbed are backscattered by the mirror (Figure 8). They are then absorbed and contribute to the projection. The mirror is tilted 45° from the direction of the propagation of X-rays. It is closest to scintillator at the right side of the detector. Consequently, backscattered radiation is more important at the right side because of the higher solid angle per detector unit length. Backscattered radiation was experimentally estimated using a lead panel in front of the scintillator covering its entire surface, except for a vertical band along axis of 1.5 cm width, where X-rays can reach the scintillator. Backscattered radiation is then measured on each side of the band, behind the lead panel. Figure 9 display a profile along the axis for the open band at the right side of the detector and left side of the detector. Scattered intensity is higher at the right side, as it is consistent with the hypothesis of backscattered radiation. The scattered intensity drop at the right of the detector (label A on Figure 9) can not be explained considering the mirror alone but can be due to the actual camera setup (e.g. mirror mounting). (

) 2
is proportional to , defined as the intensity corresponding to X-ray photons that are not backscattered.

is considered to be approximately proportional to

. Several approximations are made. First, the backscattered profile is not measured for a unique pixel, but for a band along with a width of 1.5 cm along . This measurement integrates contribution from the whole 1.5 cm width as well as the height of the band. This second point is however not a problem due to the vertical symmetry of the objects considered in this study. Border effects are also neglected as backscattering is less important at the top and bottom of the image. Figure 10 displays projection profiles of phantom and white images compared to computed backscattered intensity. As expected, backscattered radiation is higher at the right side of the projection. It is also higher in the white image than in the phantom projection, since it is proportional to intensity reaching scintillator in the first place. Computed backscattered signal (Equation 2) has been added to computed projections, and were compared to experimental ones (Figure 11). Grey level at the right of the image is now lower than 30000, close to the experimental profile (label B in Figure 11). Grey level of the highest attenuation region (label in Figure 11) is increased and is also closer to experimental values. Computed tomographic reconstruction was performed using projections corrected with the backscattered radiation (Figure 12). The linear attenuation coefficient is much closer to experimental values. The cupping effect is also increased and matches the experimental one.

This coarse computation of backscattering allows to get closer to experimental projections and CT scans. However, it is not fully accurate and further work is necessary.

Conclusion

In order to optimize the tomographic process, simulation is a relevant tool as it allows to study each different phenomenon independently to the others. However, it has to be checked against experiment. To this end, a phantom was built. The materials of the phantom were chosen for best representativeness of carbon fibers and epoxy resin especially with respects to chemistry. Heavy impurities can change significantly the linear attenuation coefficient, and greatly influence contrast between carbon fiber and epoxy resin which is weak. Comparisons of computed and both measured projections and CT scans were performed. Results reveal discrepancies that allowed to identify a backscattering phenomenon. Its computation allowed to better fit the experimental results, even with a rough estimation.

Figure 1 :

 1 Figure 1: a) photo of a hydrogen reservoir, b) CT slice perpendicularly to its revolution axis c) cylindrical slice along the blue line in (b) displaying tow architecture in reservoir

Figure 2 :Figure 3 :

 23 Figure 2: the phantom composed of K10 graphite and RSF 816 epoxy resin hollow cylinders

Figure 4 :Figure 5 :

 45 Figure 4: comparison of experimental and computed projections

Figure 6 :Figure 7 :Figure 7

 677 Figure 6: simulated and experimental simulation

Figure 8 :

 8 Figure 8: backscattering in the detector Figure 9: profile behind lead panel

Figure 10 :

 10 Figure 10: phantom projection and white image and the respective computed contribution of backscattering to experimental image

Figure 11 :

 11 Figure 11: comparison of experimental projections and simulation with backscattering

Figure 12 :

 12 Figure 12: comparison of the reconstructed profile with backscattered radiation

Digital Industrial Radiology and Computed Tomography (DIR 2015) 22-25 June 2015, Belgium, Ghent -www.ndt.net/app.DIR2015 More Info at Open Access Database www.ndt.net/?id=18056


Hamid Chaachoua

Christophe Viudez

Denis Bouhineau

Jean-François Nicaud

GRAPHICAL REPRESENTATIONS ASSOCIATED WITH ALGEBRAIC TASKS IN APLUSIX

Aplusix is a learning environment devoted to the practice of algebra in high school which has chosen a natural representation for algebraic expressions.

We will first present the didactical and technological choices for the development of the graphical representations. Then we will study the contributions of this didactical representation on various algebraic tasks such as factoring, solving equations. Finally, we will describe a first experiment.

INTRODUCTION

In Mathematics, notions can only be handled through semiotic representations such as the natural language register, numerical writing and graphical representations. As specified by [START_REF] Duval | Registres de représentation sémiotique et fonctionnement cognitif de la pensée[END_REF], "semiotic representations are productions made of the use of signs belonging to a semiotic system of representation which has its own meaning and functioning constraints". In this approach, the conceptualisation of mathematical notions needs the handling of several registers for a same mathematical object thus enabling to separate it from its representations. Duval (ibid) distinguishes three types of activities: the construction of an identifiable representation responding to some given rules, the processing of a representation in its own register and the conversion of a representation from a register to another one.

Only the first two activities are generally considered by teaching. But the third one conversion plays an essential role in the conceptualisation. The understanding of a conceptual content relies on the coordination of at least two registers of representation, and this coordination manifests itself by the rapidity and the spontaneity of the cognitive activity of conversion Duval (ibid).

We rely on this didactical hypothesis to make the Aplusix software application evolve. Indeed, in the first version of the application, only one register was implemented: the usual register. This register enables the forming of algebraic expressions and their processing. As part of the European project ReMath, we made the application evolve by adding two other registers of representations: trees [START_REF] Bouhineau | Introduction de nouvelles représentations dans le micromonde[END_REF] and graphs.

With each algebraic task, we associate a graphical representation to the usual representation which is the denotation of the mathematical object which is represented. The purpose is to develop the understanding that correct calculations lead to identical denotations.

1-2

ICTMT 9 -2009

REGISTERS OF REPRESENTATION IN APLUSIX

The Aplusix application

The Aplusix environment [START_REF] Nicaud | Mixing Microworld and CAS Features in Building Computer Systems that Help Students Learn Algebra[END_REF]) is an ILE (Interactive Learning Environment) to practice elementary algebra: transformations of algebraic expressions, solving of equations, inequations and systems, at a high school level.

The student's goal in Aplusix is to solve, as on paper, algebra problems by performing the calculation steps of his/her algebraic reasoning. The mathematical frame offered for this work is solving by equivalence. During training mode activities, feedback is given: the algebraic equivalence between steps is constantly calculated and displayed. There is also a test mode where this feedback is not available.

During the design of Aplusix, the authors made the screen representation of algebraic expressions very close to the usual representation, the one everybody uses on paper and blackboard. So, the first versions of Aplusix implement the usual algebraic register of mathematical expressions.

The graphical representation of the expressions

Mathematical objects available for a graphical representation in Aplusix are the algebraic expressions of one variable, equations and inequations with one unknown and linear systems with two unknowns. A « graphical representation » menu allows the user to get the graphical representation of the object contained in the current step. This representation is added to the graphical representations already displayed in a dedicated window. This is a graphical denotation of the reasoning step it comes from; it dynamically evolves according to the changes of the step. Changes can be performed on the graphical appearance of the curve: It is possible to choose the colour and the thickness for each curve.

More precisely, the graphical representations chosen for the different sorts of expressions are:

For an algebraic expression of one variable, Expr(x): the graph of the function xExpr(x) is drawn in the plane. See figure 1.

For an equation of one unknown, G(x) = R(x): the graphs of the functions xG(x) and xR(x) are drawn in the plane (this is an intermediary graphical representation and not a denotation of the equation) and the solution set of the equation is drawn on the real line (this is the denotation of the equation). See figure 2.

For a system with two unknowns, the lines representing the equations are drawn (intermediary representations which are not the denotation of the system) and the points which are common to all the straight lines (denotation of the system).

For an inequation of one variable, e.g., G(x) < R(x): the graphs of the functions xG(x) and xR(x) are drawn in the plane and the solution set is drawn on the real line. See figure 2.

ICTMT 9 -2009 1-3

The main challenge for designing interesting graphical representations is in the context of Aplusix (students work by equivalence) consisted of drawing the curves of objects having the same denotation, i.e. curves which coincide. We chose to use different thicknesses and colours. The new/last objects are drawn with the thinnest stroke above the others (their stroke thickness is increased if necessary), varying the colours. With the help of a palette, the user can change these settings to better observe each object.

EXPERIMENT WITH THE GRAPHICAL REGISTER

We experimented the use of this new representation with a pair of 10 th grade students (15-16 years old). These students had already used Aplusix at school. Two sessions of 50 minutes were conducted by a researcher and the mathematics teacher of the class.

During the first phase, the researcher presented the graphical representation based on two examples: "expand x(x+2)" and "solving 2x +1= x-4". The goal was to allow them to interpret algebraic calculations in the usual register with the help of the graphical register. Regarding the equation, we distinguished two levels of interpretation. The first level consisted of interpreting the equation P(x) = Q(x) as an equality between two functions f and g defined by f(x) = P(x) and g(x) = Q(x). That is to say, we searched x belonging to the intersection of the definition domain of f and g, we noted it D fg . Then, we interpreted the search of the equality between these two functions in the D fg set as the search of the abscissa of the intersection points of the representative curves of f and g. The second level concerned the processing of the equation which had to be interpreted in the graphical register. Thus the student had to interpret the equivalence between two equations as the invariance of the intersection points of the curves associated with the two equations.

In order to accomplish this goal, we proposed to the pair to solve 4 exercises in the test mode (no display of the equivalence) with the possibility to display the graph. This way used the graph as a control tool for the algebraic solving by mobilising the two control levels developed before.

The pair made a relevant use of the graph as a control tool for the three first exercises: "Expand an simplify (2x+3)(3-x)", "Expand and simplify (x-3)²" and "Factor x(x+3)+(x+3)". For example, in the first exercise, the pair made an error in step 3 (see figure 1), noticed the non superposition of the curves in the graph, then modified step 3 and obtained the superposition of the curves.

However, the fourth exercise "Solving x(x-1)) -x²+1" the students had difficulties regarding the interpretation, because they tried to interpret the equivalence of the equations as a superposition of the curves. The situation required an intervention of the researcher. It was an opportunity to go back on the notion of equivalence between equations and equality between functions (see figure 2).

1-4 ICTMT 9 -2009 Figure 1. Solving an "expand and simplify" exercise. The non coincidence of the curves is a consequence of the non equality between the expressions of the two steps.

Figure 2 .

 2 Figure 2. Solving an equation. The solutions, obtained from the abscissa of the intersection points of the curves are represented on the line under the plan. The non equivalence of the equations is represented by different set of solutions (associated to each equation).


Lionel Clément

Acknowledgment

I would like to thank the following people for their valuable contributions to different parts of XLFG (documentation, graphic design, some ideas on implementation of the front-end, grammar development, samples, feedbacks, etc.) Olivier Bonami (Univ. Paris-Sorbonne) Kim Gerdes (Sorbonne Nouvelle, Paris) Sekou Diao (Univ. Bordeaux) Hélios Hildt (Univ. Bordeaux Montaigne) Fiammetta Namer (Univ. Lorraine) Hugo Nioteau (Univ. Bordeaux) Pape Ousmane Sow (Univ. Bordeaux)

Disclaimer

This document is neither an introduction to Lexical-Functional Grammar (LFG) nor a manual on formal linguistics. It is intended as a resource for people who want to use XLFG for a class, for research purposes, or out of mere curiosity. Readers who wish to learn more about LFG are kindly asked to consult the reference section at the end of this document.

XLFG is constantly evolving; thus, parts of this guide may well become obsolete in the future. This document was meant to be used with XLFG version 9.8.0 It will be updated and made available online each time XLFG is modified.

XLFG has been designed to allow the conservation of works for multiple users (students, teachers, lecturers, researchers, or anybody else). Nevertheless, it is not designed to backup data. We disclaim any liability for the loss of work.

To make sure work and data is properly saved, a variety of alternative methods are readily available.

XLFG is often used for exams and tutorials. While we endeavor to do the utmost to prevent server interruptions, we disclaim any liability for problems resulting from the use of this Web site during exam or tutorial sessions.

XLFG has been designed to with relevance to the field of linguistics. The comments and opinions in the sentences or comments are those of the author alone and do not necessarily reflect those of the XLFG operator.

We reserve the right to terminate any account and remove all contents associated with it, at any moment.

Commitments

We commit to not communicate, for any purpose whatsoever, the email addresses used to register with XLFG, or any personal data.

We commit to removing any account at the request by email of the person concerned and do not keep personal data and records.

1 What is XLFG?

XLFG is a fast, accurate deep parser for LFG grammar. These outputs are phrase structures, predicate-argument structures and predicate-thematic relations. It allows any user to parse an extensive (in fact unlimited) set of phrases in a long sentence with a realistic lexicon in just a few hundredths of a second.

XLFG is divided into two separate parts: back-end and front-end.

For those interested in the back-end part as yet unavailable on the Webportal, design for Natural Language Processing applications, please contact the author Lionel Clément.

This present documentation concerns more particularly the XLFG's frontend, designed to experiment LFG grammar online with small lexicons.

The XLFG's front-end will hereafter be referred to simply as "XLFG".

2 Installing XLFG XLFG does not require software installation. XLFG may be used as an online service using a web browser from any computer connected to the Internet.

One only has to go to the following page: http://www.xlfg.org/ with the Browser one prefers.

Nevertheless, XLFG is compatible with web browsers that can display MathML, SVG et XHTML. It has been tested on Internet Explorer, Firefox, Google Chrome, Safari and Opera.

Accounts

Checking in is not mandatory for trying XLFG. However, registration allows users to share their work and save it on the server.

Creating an account is fast, free and easy. Just sign-up by completing a form, including a username, an email address and a password.

To complete the signup process, we need to confirm that the email address you're using is your own: a confirmation by email will be sent to you to confirm your registration for the website.

Workshops

Any recorded user may create workshops. Workshops are designed for organizing classes. A teacher or a lecturer may create a workshop, projects related to it and may then assign rights for these projects according to the individual member whose part of the project is editable or not.

The workshop owner adds members that are XLFG recorded users identified by their logins.

Projects

A project is a form with a title, an abstract, input and output parameters, grammar and lexicon.

The owner may assign each project to one of their workshops and then accord privileges for each part of the project: two levels of privilege are available for any workshop member: Read-only access, Read-write access. Obviously, any user may build a project without attaching it to his or her workshops. In any case, XLFG grants privileged access to the user's own projects.

When the same project is shared by a workshop, only copies of it are editable by each member. Thus, the workshop owner may supervise a project and read or correct the different changes introduced by each member to the original.

A practical example of using projects and workshops would be to organise a tutorial:

The teacher creates a workshop and invites the students to enroll. Then he produces project including a grammar with read-only privilege and a lexicon with read-write privilege for our example.

Each student make a personal copy of the project and may change the lexicon entries by following the teacher's recommendations without any possibility of changing anything else.

The teacher may change from read-write to read-only mode the lexicon after a deadline, and read and correct each result. The easiest way to get to know XLFG is to use one of the sample grammars.

Choose English with functional-structure in the Grammars menu. A page opens with a number of pre-filled fields.

Declaration This part will contain some useful declarations. It's not helpful for us in this first example.

Lexicon contains the lexicon, written as a sequence of words followed by the relevant terminal symbol from the grammar and a functional structure associated with it.

Grammar contains the syntactic rules, written as context-free rules allowing to build phrase structure trees, plus functional constraints allowing to build functional-structures.

Parsing contains the input sequence that will be parsed: John sees the man with a telescope in our first example. It also contains a list of buttons to choose the display mode you prefer. For now keep this standard configuration unchanged.

When clicking on the Parse button, a new window (or a new tab depending on the configuration of your browser) opens, which contains:

• A graph encoding the lattice of words sequence and all the constituents found (fig. 2).

• One of the two possible constituent-structures for the sentence (fig. 3).

• The corresponding functional-structure for the sentence (fig. 4).

• An argument structure for the sentence (fig. 5).

If this does not work, perhaps the browser used has been configured to disallow popup windows or new tabs. Uncheck this option.

As this number can be quite large or unlimited, XLFG displays only one tree at a time or a shared representation, namely a forest-representation of the constituent-structures. In our exemple, to see the other constituent-structures and the related functional-structures, click on the link labelled next to the VP node. This link brings the other possible analyses of the verb phrase (in which the phrase with a telescope is attached to the man and is not a complement of the verb sees).

Click on the other links of the graphes to see the related functional-structures and argument-structures. Notice that capitalization and punctuation are taken into account; one should be careful about this when writing a grammar.

Error messages

If you make a mistake, you will get a (hopefully helpful) error message which provides a line number and the project part (declaration, grammar, or lexicon).

Modifying the grammar

You may write a new grammar from scratch or modify one of the given projects. In both cases just add rules conforming to the XLFG conventions.

For instance, adding the following rule to our example:

N → Adj N { ↑ = ↓1; ↓2 ∈ (↑ MOD); };
and the lexical entry: little Adj[PRED:'little'];

will allow you to parse the sentence John sees the little man with a telescope. At this stage we need to give a more precise description of XLFG grammars, although the sample grammars should provide an excellent idea of what they look like and allow data to be manipulated. The following sections describe the makeup of an XLFG grammar in detail. We start by explaining how grammars found in LFG books can be implemented into XLFG.

LFG theory and XLFG parsing

Solving equations

XLFG implements a parser for LFG grammars whose main feature is to rely on a shared representation for different parses of the same sentence. For instance, the sentence John sees the man with a telescope has two parses, where the phrase with a telescope either modifies the noun man or is a complement of the verb sees. In both cases, the phrase with a telescope is the same. It is thus convenient to share this part of the two parses. This sharing avoids a lot of redundant computations and makes XLFG quite efficient, even when used with realistic grammars and lexicons based on real data.

Whereas the PP is the same in both parses of the sentence, its function in the sentence is not the same. The structures it attaches to are different and cannot be shared between the two parses.

In LFG, the rule describing PPs modifying a noun might be written as:

NP → Det N [PP] ↑ = ↓ ↑ = ↓ ↓ ∈ (↑ ADJ)
In this rule, the f-structure of the noun and noun phrase are identified through the use of equality. This structure also contains a set-valued feature ADJ whose value must contain the f-structure of the modifier (in our example, with a telescope). It is evident that such a statement cannot be verified when the noun has no complement, because the PP attaches outside of the NP. Thus, in LFG, a separate computation must be made for each parse: each phrase structure tree requires its own f-structure computation. Without such a condition, the system of equations would have no globally coherent solution.

Obviously, an exponential number of trees are produced in case of structural ambiguity, unlimited in the case of cyclic derivation (in some grammars with empty categories for exemple). In XLFG, we decided not to limit our analysis to a small number of trees on the one hand, and we would like to remain the semantic of equality on the other. Thus, we chose to have shared phrase structure representations, in order to be able to parse highly ambiguous sentences. For this reason a different solution for computing f-structures needed to be built which did not affect either the LFG linguistic theory or the associated formal model too greatly.

XLFG models the Φ projection (from constituent structures to functional structures) using copy operations rather than strict unification for the solution of equations.

This design strategy explains why some LFG grammars cannot be used directly in XLFG, but must be adapted to meet conventions and constraints that are particular to XLFG. This is the XLFG version of the preceding rule:

NP → Det N [PP] { ↑ = ↓1; ↑ = ↓2; ↓3 ∈ (↑ ADJ); };
In this declaration, the values ↑, ↓1 and ↓2, which correspond respectively to the f-structure of the NP, the determiner and the noun, are not identical during parsing. The = symbol which denotes an equality of its operands is implemented in XLFG as a bottom-up copy operation.

The declaration ↓3 ∈ (↑ ADJ) modifies the f-structure of the NP, but it must not modify the f-structure of the noun, because that f-structure is shared in the representation of both parses of the sentence.

In our example, the value of the f-structure ↑ is the unification of ↓1 and ↓2 to which the set-valued feature ADJ is added; but the values of ↓1 and ↓2 are not modified.

Parsing in LFG thus relies on a general rule that f-structures propagate bottom-up, from the most embedded to the least embedded, through copy operations.

As this exemple attests, XLFG notations are close to those commonly used in LFG literature and a typical user should not be too greatly perturbed.

Furthermore, XLFG allow us to adapt the main principles agreed in the LFG literature in a simple and elegant manner.

Formal definition of = symbol in XLFG

Given a shared forest where the node N i dominates N j , an equality declaration x = y where x is embedded in a projection of N i and y is embedded in a projection of N j .

Given φ = ⊥, the most general unifier (mgu) for x and y. x is assigned with φ(x).

Hereafter, the term assignment after unification will be used to refer to this implementation.

6 Using the XLFG Web-portal interface XLFG Web-portal interface is pleasant to use and extremely intuitive and simple.

All you need to do is connect onto the interface from any computer using any browser. Once you have accessed the XLFG software, just let yourself be guided by the XLFG interface.

If you want to try XLFG with samples only, you do not need to create an account. If, however, you do wish to create new projects and save your work on our server, you must create an account by clicking on Sign-up in the top right corner. To log into an existing account, click on Login in the top right corner.

Once logged in, the XLFG Web-portal interface looks like this:

Grammars

The Grammars menu contains some XLFG grammars in further languages with the aim of facilitating XLFG development.

These freely grammars was developed by XLFG contributors and XLFG developer. Thanks a lot to all contributors who are helping us to improve XLFG data!

Projects

Each user may edit and save projects involving a grammar, a lexicon and output parameters. Any member enrolled in a workshop can copy and then edit a new version of a shared project. Only the read-write parts can be then edited, and the workshop owner may read and correct this version. It is very efficient for organize tutorial. Of course, in addition, any user can make a personal copy of any readable project. This personal copy is not readable by the workshop owner by default.

Workshops

If you are the administrator of a workshop, you can click on the button Edit to add users and projects. You will then be able to create and access the various projects of the workshop to which you belong. Otherwise, you will only be able to see information concerning the workshop and access shared projects in workshops of which you are a member.

Users

Any registered user is listed here with workshops and projects he or she manages. This interface allows anyone to send a notification to a user and to ask to join his workshops. We hide all emails. The communications between users can only be with XLFG notifications.

Notifications

Any recorded user may send notifications to the workshop administrators, or to the members of a given workshop he or she administrates. The unread notifications icon then appears at the top of the XLFG home-page. By clicking on it, new notifications may be read.

At the request of XLFG users, XLFG can automatically send an email to the users when a new notification arrives.

XLFG grammar

An XLFG grammar consists of a set of phrase structure rules.

Phrase structure rules describe clause structure, or more precisely regularities in clause structure. Constituents of the clause differ in category (noun phrase, verb phrase, etc.), in position (pre-verbal, post-verbal, etc.) and in grammatical relations (or syntactic functions) they enter into (subject, object, etc.).

Categories for phrases are defined by the projection of a lexical element (a noun, verb, etc.) and a constitution.

Syntactic positions are determined by word order and the structural arrangement of words.

Syntactic functions may be expressed by case morphology (case endings, typically in "free word order" languages), or by lexically-registered valence requirements.

The aim of an XLFG grammar is to use theoretical constructs from LFG to describe the syntactic regularities of a language. XLFG will produce a parse for grammatical sentences. The analysis of ungrammatical sentences will display information allowing the user to determine which constraints were violated. These constraints are always typeset in red.

The following examples are not meant to be part of a realistic grammar fragment, but to illustrate various theoretical concepts.

XLFG Notations

The content of a XLFG grammar and lexicon is made with UTF-8 encoding characters. Capitalizations are taken into account to distinguish notations. Blank characters and line feeds are not taken into account, with the exception of strings.

Comments

Comments are destined for the reader only and are skipped by XLFG. A one line comment start with the // symbol, the text is skipped up to the end of the line. A multi-line comment starts with /* and ends with */. The enclosed text is skipped.

Identifiers

Whitout any quotes, all the strings beginning with a latin character, or "_" followed by alpha-numeric characters are identifiers, when they are not a keyword.

• The latin characters are a . . . z, A . . . Z, à, á, â, ã, ä, å, ae, ç, è, é, ê, ë, ì, í, î, ï, ð, ñ, ò, ó, ô, õ, ö, ø, ù, ú, û, ü, ý, ÿ

, θ, À, Á, Â, Ã, Ä, Å, AE, Ç, È, É, Ë, Ì, Í, Î, Ï, Ð, Ñ, Ò, Ó, Ô, Õ, Ö, Ø, Ù, Ú, Û, Ü, Ý, Ÿ, Þ, ß
• The keywords are the following:

-PRED -LEXEME -SUBCAT -case -else -functions -if -in -not -start -switch -symbol -with
Identifiers are used on different occasions:

• As a terminal symbol in the lexicon and grammatical rules (common_noun, verb, preposition, . . . ).

• As a non-terminal symbol in grammatical rules (S, NP, PP, . . . ).

• As an attribute or atomic value in feature structures (number, singular, . . . ).

• As a grammatical function (SUBJECT, OBJECT, . . . ).

• As argument (Agent, Patient, . . . ).

In order to write identifiers without latin characters in the grammar or lexicon, one can use ' quotes. For exemple the rule S → Subject Object Verb; may be written in Tibetan characters:

The maximal number of different identifiers in the same project is limited to 248, witch is suffisant for building large coverage XLFG grammars for many languages.

Phrase structure rules

This is an exemple of XLFG phrase structure rule:

VP → [aux] V [NP|S];
Brackets indicate optional constituents, the vertical bar indicates an alternative between NP and S .

This rule describes the composition of a phrase of type VP: this phrase consists of a possible constituent aux followed by V and possibly NP or S. These three constituents must be contiguous and in the specified order.

In order to reiterate a constituent in a rule, one has to write a recursive rule instead; because the usual LFG Kleene star notation is not allowed in XLFG. The following LFG rule: A phrase structure rule may have an empty right hand side. This allows for an explicit modeling of empty categories, as are postulated in some syntactic frameworks. This is particularly relevant when such rules are associated with functional descriptions giving rise to constraints on grammatical functions.

VP → V PP *
NPro →;

Start symbol

The start symbol corresponds to the type of phrase XLFG will attempt to parse. It is most often of the type Sentence, but of course XLFG can be used to parse any type of phrase compatible with the grammar.

By default XLFG assumes that the first symbol it encounters in the grammar is the start symbol. If this behavior is not appropriate a different symbol may be specified by writing the given statement using the keywords start symbol in the declaration section of the project. start symbol: NP;

Functional structures

A functional structure (F-structure hereafter) is represented as a feature-structure, namely a set of attribute-value pairs. It is represented with brackets, the features are separated with commas and the attribute-values pairs with the colon character: The number of atomic values is limited to 251.

• A literal value is a symbol marked between simple quotes.

PREPOSITIONAL_FORM: 'into',

The number of literals is not limited.

The feature PRED

While the attribute is PRED, the value is written between two single quotes.

A regular lexeme is noted with a single symbol followed by sub-categorization information.

Following LFG conventions, the grammatical function list is noted between chevrons for those which correspond to a thematic argument, and after for those which do not. Optional grammatical functions are noted with [ ] brackets.

(1) a. The Commission would like to give some of the results that are included in the document. b. The Commission seems to give to the shareholders some of the results.

PRED: 'TO_GIVE < SUBJ, OBJ, [OBLto] >' PRED: 'TO_SEEM < XCOMP > SUBJ'

If a list of grammatical functions is allowed for the same syntactic position, the grammatical functions are separated with | symbol.

(2) a. The conclusions say that a new approach is needed.

b. The Commission is unable to say the degree of progress.

PRED: 'TO_SAY < SUBJ, [ OBJ | SCOMP ] >'
This attribute is central to the analysis of an utterance. The PRED feature of an f-structure is projected from lexical entry -for instance a particular reading of a polysemous lexeme -.

XLFG lexica are simple, and do not allow us to apply lexical rules, nor α projections. We assume the lexica used for XLFG come from time-deferred applications not included in XLFG parser, but in an other software.

However, the user can associate grammatical functions with particular arguments through PRED specifications and assign them thematic roles as requested in LFG theory.

(3) a. The Commission wants to place the shareholder at the center of his commitment.

Here is a sample PRED feature in the XLFG notation:

PRED: 'TO_PLACE < SUBJ . Agent, OBJ . Patient, [OBL . Location] >'

In this example the lexeme is place. It combines with three arguments corresponding to the three functions SUBJ, OBJ, OBL. Each one is associated with a thematic role: respectively Agent, Patient, and Location. In this case, the OBL locative complement is not mandatory.

Functions that do not instantiate a thematic argument of the predicate should be listed outside the angled brackets. This is the case e.g. for the impersonal subject of weather verbs such as rain: PRED: 'TO_RAIN <> SUBJ' or for subjects of raising verbs such as seem:

PRED: TO_SEEM < XCOMP . theme > SUBJ'
In other cases, a single function may correspond to different arguments of the predicate. For instance in the following examples, the subject of crash corresponds either to the agent or the patient argument.

(4) a. The computer crashed.

b. Luke crashed the computer.

In XLFG one may explicitly annotate a function with the name of the role of the argument it realizes. PRED: 'TO_CRASH < SUBJ . patient >' PRED: 'TO_CRASH < SUBJ . agent, OBJ . patient >'

The links constructed by such thematic relations are displayed as a dependency graph. This graph may serve as a first step towards a semantic representation.

Complexe predicates

A feature of LFG is that it is impossible to unify two structures with distinct PRED features. This is the standard way of ensuring that each syntactic function is instantiated no more than once by PRED-bearing constituents, without barring the possibility that distinct constituents contribute to the description of an f-structure.

Let consider these sentences:

(5) a. Mary promised to come. b. Mary promised that she would come c. *Mary i promised [her i to come] This feature of LFG can be put to use, for instance, to ensure that controlled infinitives will not get a c-structure subject. The f-structure of 5a is:

      PRED 'promise<SUBJ, XCOMP>' SUBJ 1 PRED 'Mary' . . . XCOMP PRED 'come < SUBJ > SUBJ ⇒ 1      
Notice that the matrix and embedded SUBJ are identified, thanks to a control specification originating in the lexical entry of promise:

(↑ XCOMP SUJ) ⇒ (↑ SUJ)
By contrast, the f-structure of 5b is:

        PRED 'promise<SUBJ, COMP>' SUBJ PRED 'Mary' . . . COMP   PRED 'come < SUBJ > SUBJ PRED 'PRO' . . .          
Here the two subjects have distinct f-structures corresponding to distinct PRED values (the fact that they might be coindexed semantically is a separate issue that we do not model here). Now let us consider what the f-structure of the agrammatical sentence 5c. Assuming that a well-formed c-structure could be assigned to this sentence, its f-structure would be:

        PRED 'promise<SUBJ, XCOMP>' SUBJ 1 PRED 'Mary' . . . XCOMP   PRED 'come < SUBJ > SUBJ 1 | PRED 'PRO' . . .          
This f-structure is ill-formed, because the PRED values 'PRO' cannot subsume -a fortiori unify -'Mary', despite the fact that the f-structures they occur in are constrained to being identified by the control equation.

This said, it is well known that complex predicate constructions rest on a situation where two distinct constituents contribute to the specification of a PRED value. Particle verbs, support verb constructions, decomposable idioms, and serial verb constructions are examples of cases that may be modeled as complex predicates.

To model such cases, XLFG supports an operator prefix or suffix "-" that derives a PRED value from two other PRED values. The predicate name is the concatenation of the two predicate names, and the set of functions is the union of the sets of functions of the two PRED-bearing constituents. PRED: ' lexeme < . . . > . . . ' PRED: ' prefix -< . . . > . . . ' PRED: ' -suffix < . . . > . . . ' Let us illustrate this situation with a support verb construction: give a lecture. This is a partially grammaticalized construction: both give and lecture seem to have their usual meaning, but (i) something must be said to the effect that give is used rather than other candidate verbs such as make or do, and (ii) the verb seems to inherit something from a valence requirement originating in the noun: in the following sentence, on the subject is a complement of the verb, but it is the noun lecture and not the verb give that is lexically specified for an oblique complement in on1 .

The lecture he gave on the subject in Salzburg was judged as one of the turning points in the evolution of theoretical physics. (A. Calaprice & T. Lipscombe, Albert Einstein: a biography, p. 46, Greenwood Publishing Group 2005)

The lexical entries are as follow.

give V [PRED: 'give_ -<SUBJ>', tense: present]; lecture N [PRED:'lecture<(onOBL)>', number: singular];

The two structures may unify to produce an appropriate f-structure for the sentence above: As one can see, it is always possible to unify more than two PRED, thanks to the idempotence property of unification on prefix and suffix. This possibility may happily be used for serial verb analysis.

      PRED '

The feature LEXEME

In order to describe only the lexeme attribute but not the subcategorization of a predicate, one may use the attribute LEXEME instead of PRED.

This attribute can combine with a PRED to form a complex predicate without changing anything about the lexical valence given by the main predicate.

The lexical entries for a verbal particule look for may be encoded as follow SUBCAT: '<Subj.agent>'

A SUBCAT feature may unify with a PRED. In such a case, the result is a PRED.

Here, in summary, are the various combinations for unification between PRED, LEXEME and SUBCAT attributes in XLFG: PRED:'X<Y>Z' LEXEME:'X' SUBCAT:'<Y>Z' PRED:'A<B>C' PRED:'X U A<B U Y>C U Z' PRED:'X U A<B>C' PRED:'A<Y>Z' LEXEME: 'A' PRED:'X U A<Y>Z' LEXEME:'X U A' PRED:'A<Y>Z' SUBCAT: '<B>C' PRED:'X<B>C' PRED:'X U A<B>C' none

Lexical entries for Complexe Predicates

In many cases where the predicate depends on a combination of two components, the meaning and the argument structure depend on a indecomposable idiom. That is the case for a particule verb for exemple.

The predicate of the particule verb to give up is give_up that don't be the combination between to give something to someone and the preposition up something. Moreover, the subcategorization if particule verb is different from the subcategorization of to give. The feature-structure that has resulted from unification is the following:

• the lexeme is the combination between TO_GIVE and -_UP,

•

Shared functional structures

As XLFG has been designed to share the analysis of ambiguous sentences, multiple F-structures are represented in a unique structure by using distributed features.

Distributed features

The ambiguous analysis for the sentence John sees the man with a telescope corresponds to these two F-structures: An economic way to represent this pair of F-structures is to share the equivalent attributes into a common factor and distribute the differences into a list written with vertical bar characters like this:

Constraints on functional structures

A sentence is considered grammatical if the grammar can assign it with at least one constituent structure, and the Φ projection of that c-structure is a coherent, complete, extended coherent and consistent f-structure. These criteria result from implicit constraints of the theory, to which parochial constraints associated to lexical items or c-structure rules can be added.

Implicit constraints

Coherence

A functional structure is coherent if the attributes of all the governable functions it includes are specified on the PRED value, and all embedded f-structures are coherent.

Here is an example of an incoherent f-structure: *Mary sleeps her bed.

          PRED 'sleep<SUBJ>' SUBJ PRED 'Mary' . . . OBJ PRED 'bed' . . .          

Completeness

An f-structure is complete if all the attributes specified in its PRED value occur and are instantiated locally, and if all embedded f-structures are complete.

Here is an example of an incomplete f-structure. *Mary precedes.

    PRED 'precede<SUBJ, OBJ>' SUBJ PRED 'Mary' . . .     Extended Coherence
A functional structure is extended coherent if all the governable functions include a PRED value.

Here is an example of an extended incoherent f-structure: *sleeps.

  PRED 'sleep<SUBJ>' SUBJ P ERSON 1|2|4|5|6  

Consistency

An f-structure is consistent if each local attribute has a unique value, and each embedded f-structure is consistent.

Here is an example of an inconsistent structure. *The children sleeps.

      PRED 'sleep<SUBJ>' SUBJ   PRED 'child' NUMBER sing | plur . . .        

Explicit constraints

In addition to these implicit constraints, explicit constraints can be added through the use of several logical operators between local or non local features. In the following chapter we will describe such constraints and how functional structures are made.

Functional descriptions

Functional descriptions are a crucial part of LFG theory. They define the Φ projection allowing one to construct a functional structure from a constituent structure. They also provide explicit constraints on the output functional structures.

Syntactic relations, local and nonlocal agreement, subject control, sub-categorization, etc., can be modeled using functional descriptions. The following examples are only illustrative, and the interested reader is directed to LFG literature for further reading and analyses.

Grammatical Functions

A list of terms which denote argument functions may be defined by the user. It allows them to distinguish them from the other grammatical functions. The argument functions specified in the PRED feature must be present in the local Fstructure and all argument functions must be selected by their local PRED. These conditions correspond to traditional LFG contraints: Completeness Condition, Coherence Condition. In addition, the Extended Coherence Condition stipulates that all F-structures containing at least one argument function will also contain a PRED. The latter requirement is exposed in [START_REF] Bresnan | Topic, pronoun, and agreement in Chiche ŵa[END_REF].

The keyword functions is used to define arguments functions in the declaration section of a project: functions: SUBJ, OBJ, XCOMP;

Without such a declaration of argument functions, the Completeness Condition, Coherence Condition, and Extended Coherence Condition are not verified.

Functional equations

Since the symbol = is implemented in XLFG as an assignment (see above chapt. 5.1), not equality; the only possibility of writing functional equations are the following. ↓1 and ↓2 denote the f-structure of the dominated constituent, in the present instance respectively the NP and the VP, whereas ↑ denotes the functional structure of the dominating category, here S, the full sentence. When a rule introduces a single constituent, ↓ can be used equivalently to ↓1. The equation (↑ SUJ) = ↓1 instantiates a new attribute SUBJ in the f-structure ↑ and assigns the structure denoted by ↓1 as its value. If the attribute already existed in ↑, its value would become the unification of (↑ SUJ) and ↓1.

Let us take an example: the sentence My father came.

The functional structure initially assigned to the VP is:

    PRED 'come < SUBJ>' TENSE past SUBJ NUMBER singular PERSON 3    
The functional structure initially assigned to the NP is:

  PRED 'father' NUMBER singular GENDER masculine   Applying the equation (↑ SUJ) = ↓1 updates ↑ to:         PRED 'come < SUBJ>' TENSE past SUBJ     PRED 'father' NUMBER singular PERSON 3 GENDER masculine            
We see here that the value of SUBJ is the unification of

Set-valued attributes

Attributes corresponding to modifiers have a set of f-structures as their value.

The value of such attributes is constructed using declarations of the form ↓i ∈ (↑ <path>)

(↓i <path>) ∈ (↑ <path>)
where (↓i <path>) (or just ↓i) is the description of an f-structure X and (↑ <path>) an attribute. If the attribute is already present in the structure as a set, X is added to it. If the attribute is present with another type, an error is reported.

As an example, let us consider a noun modified by an adjective and a relative clause: the technical issues that plague the project.

The following rules allow for an adequate analysis:

NP → DET N { ↑ = ↓1 ; ↑ = ↓2 ; }; N → ADJ N { ↓1 ∈ (↑ ADJ); ↑ = ↓2 ; }; N → N REL { ↑ = ↓1 ; ↓2 ∈ (↑ ADJ); };
Here is the resulting f-structure, assuming a simplified lexicon:

                  PRED 'issue' ADJ                            PRED 'technical' . . .         PRED 'plague<SUBJ, OBJ>' SUJ PRED 'pro' . . . OBJ PRED 'project' . . . . . .                                    . . .                  

Links between f-structures

The use of equality in LFG functional equations allows for a single f-structure to be the common value of two attributes modeling distinct syntactic functions. This is how subject control is modelled in LFG: the subject of an XCOMP or XADJ is shared with a function of the matrix sentence. For instance the sentence Mary wants to stay is analyzed as:

        PRED 'want<SUBJ, XCOMP>' SUBJ PRED 1 'Mary' . . . XCOMP   PRED 'stay<SUBJ>' SUBJ 1 . . .          
In this example, the subject of the infinitive is controlled by the matrix verb want, whose lexical entry states that it is shared with the subject of wantbut the name Mary is realized just once in the constituent structure. In LFG, this effect is modeled by putting the following equation in the lexical entry of want:

(↑ XCOMP SUBJ) = (↑ SUBJ)
Remember however that there is no equality in XLFG: equality is replaced by assignment after unification. But copy is not appropriate in the present instance: we do not want to construct a new subject.

In XLFG, subject control is modeled using the operator ⇒:

(↑ XCOMP SUBJ) ⇒ (↑ SUBJ)
This states that the left-hand side subject refers to the second one.

The general form of a link declaration is the following:

(↑ <path>) ⇒ (↑ <path>)
If the attribute corresponding to the left-hand side is already present with a feature-structure value, it must subsume the second one to produce a wellformed F-structure. If it is present, but with another type, an error is reported.

Conditionals

Since functional descriptions are assigned to phrase structure rules rather than constituents, we added the operators if that allows one to turn on or off the functional descriptions associated with optional constituents. Here is an example:

<constant> == <constant> Where <constant> is: (↑ <path>) (↓ <path>) <atom> <literal>
Obviously, a constraint equation that makes reference to a constant which does not exist fails.

Negative constraints

The operator = is the opposite of ==. A constraint such as the following is verified if either there is no CPLZER attribute in the structure, or its value is not THAT.

(↑ CPLZER) = that; <constant> = <constant> This constraint fails if and only if the constants exist with values that matches.

Existential constraint

It is possible for a syntactic rule or a lexical entry to require a feature to be present without a particular value. For exemple, a finite verb is required when the complementizer that is the head of a clause whatever is the tense of such a verb.

(7) a. The Commission is also still debating whether to apply the law.

b. *The Commission is also still debating if to apply the law. c. The Commission is also still debating if one should apply the law.

We use only the functional description to describe such a constraint.

that CPLTZ [] {(↑ TENSE);};

Conversely, a syntactic rule or a lexical entry may require a feature to be not present. For exemple, a verb introduced by the to particule is no finite. We use the ¬ symbol to represent describe constraint.

(8) a. *The Commission is also still debating whether to should apply the law.

to PART [] {¬(↑ TENSE);};

Variable attributes

A single verb may combine with two oblique complements. In such cases, the LFG practice is to index the syntactic function of the complement with the name of the adposition introducing it. This allows for a unique function to be assigned to each complement, in accordance with the unicity requirement on syntactic functions. Thus the PRED value for a verb such as talk is:

PRED: 'TALK<SUBJ, OBL to , OBL about >'

To make sure that the right preposition is used within each complement, it is necessary to constrain the PCASE value associated with the preposition to match the indexed function, as in the following example.

              PRED 'TALK<SUBJ, OBLto, OBLabout >' SUBJ . . . OBLto PCASE to . . . OBLabout PCASE about . . .              
In XLFG, such variable attribute names can be denoted by concatenating a description to the left of an attribute name: OBL -(f PCASE) names an attribute constructed by concatenating OBL with the value of the PCASE attribute of f using the operator "-". The following rule allows one to construct the preceding schematic f-structure from appropriate lexical entries.

VP → V [PP] { ↑ = ↓1; l                       PRED 'ask<SUBJ, (COMP)>' SUBJ x PRED 'John' COMP y               PRED 'should < SUBJ, VCOMP > FOCUS k PRED 'file' SUBJ z PRED 'Luke' VCOMP w   PRED 'open < SUBJ, OBJ > SUBJ ⇒ z OBJ ⇒ k                                       (↑ (COMP SUBJ)) denotes structure [PRED 'Luke']-that is, z. (↑ (COMP | SUBJ)) denotes the structures [PRED 'should'] or [PRED 'John']-
that is, neither exclusive to x or y.

(↑ (COMP | VCOMP) * ) denotes all the structures l, y, w.

XLFG lexica

Although XLFG has been developed to extract syntactic properties from sentences, but not for phonological or morpho-syntactical treatment, it allows us to carry out a basic analysis of compounds or portmanteau forms. Words are written in UTF-8 encoding. The system will accept some accented characters directly, but when using non latin alphabets or less used symbols or keywords, double quotes should be added.

In order to parse words witch correspond to a local grammar (i.e. recognizable with a regular expression) without using the power of the XLFG context-free parser and with a finite lexica, we added special forms: _EMAIL_, _URL_, _INTEGER_, and _REAL_. The first matches a regular email adresse as lionel.clement@xlfg.org, the second matches an url as http://www.xlfg.org, etc.

Here some exemples of accepted forms:

John "emergency exit" Schreibmaschinenpapier _INTEGER_ ";"

An XLFG lexical entry consists of a triplet (category label, functional structure, set of local functional constraints). Functional structures and functional constraints are optional.

A simple form is associated with one triplet, while an homonym form is associated with several ones separated with a | symbol.

Here an example of homonym entries for the form left: past tense of leave or opposite of right.

left commonNoun [PRED: 'LEFT']

| verb [PRED:'LEAVE'];

A poly-categorial word (compound or portmanteau word that must be analyzed according to a morphological theory, or agglutinate word) is represented by a list of triplets separated with the ampersand character &. For example, the French word auquel is the agglutination of the preposition à and the relative pronoun lequel

auquel (prep [PRED: 'à']

& relPro [PRED: 'lequel', GENDER: ms, NUMBER: sg]);

Obviously, one may combine these two possibilities. For example, the little French word du is either a partitive determiner, or a definite article le following the preposition de. The F-structure in an lexicon entry may be followed by local functional constraints. It allows us to give the syntactic property of the word depending on its context. Let us take the example of a subject control verb such as want. An optimal lexical entry will look like this: Information on the nature of the predicate and subject agreement are constant in uses of this entry, so they should be specified in the F-structure. Yet the constraint linking the subject of the infinitive to the local subject depends on the context and should thus be stated separately. Thus the only context anchor for the functional constraints in reference to a lexical entry is ↑, not ↓ i.

Unknown words

When a unknown word is encountered, XLFG assigns it with the special value _UNKNOWN_. One can associate open categories (nouns, verbs, adjectives,) with the unknown words, but not grammatical lexemes such as preposition, particules or determiners.

In this case, and also in the case of regular expressions which we have already discussed, the keyword _THIS_ corresponds to the form encountered in the input. This enables us to rewrite this form in the calculated F-structure.

Macros

As a lot of similar attributes with the same values are used, we added a convenient way of writing this only once using an assigned variable marked with "@": @ms: GENDER: masc, NUMBER: sing;

It is possible to use this variable in the definition section @m: GENDER: masc; @s: NUMBER: sing; @p: NUMBER: plural; @ms: @m, @s; @K: VFORM:participle; @Kms: @K, @ms; @_12: PERSON: 1|2; @P: tense: present, mood: indicative; @V12s_p: SUBJ: [ ( [@_12, @s], [@p] ) ]; @P12s_p: @P, @V12s_p;

Then every macros may be used in the grammar or in the lexicon:

give V [PRED: 'TO_GIVE', @P12s p ];

Compiled lexicon

The back-end of XLFG has been designed to work with large lexica. Thus a compiled lexicon can be given to the parser in addition to the explicit lexicon.

In the case of conflit between the two, the explicit lexicon overrides the other. XLFG has granted the users registered as teachers or XLFG contributors to upload lexicons. Such a lexicon is a simple text file (i.e., no formatting) where each line represents one lexical form in XLFG format.

In order to use a large lexicon into a project, a teacher or an XLFG contributor may just attach the lexicon name withit.

Figure 2 :

 2 Figure 2: Lattice of words and phrases

Figure 4 :

 4 Figure 4: Functional-Structure

Figure 6 :

 6 Figure 6: Home XLFG Web-portal

Figure 7 :

 7 Figure 7: Project page

  is an equivalent to the XLFG one: VP → V PPs; PPs → PP PPs; Rules may be recursive to the left or to the right, immediately or not. This is used to model recursive phrase embedding, as e.g. in NPs occurring inside relative clauses themselves occurring inside NPs. N → N AP; N → AP N; N → N RelVP; RelVP → RelPro S; S → NP VP; NP → Det N [PP]; . . . A given constituent type need not have a unique possible composition. Alternate compositions are described by multiple rules with the same left hand part. VP → [aux] V [NP | S]; VP → VP adv;

[

  PRED: 'snore<SUBJ>', TENSE: present, MOOD: indicative SUBJ: [PRED: 'John', NUMBER: sg, person: 3]]7.3.1 Atomic or literal featureWhile an attribute is an identifier, the possible values for attributes are atoms, literals, embedded F-structure, or a set of embedded F-structures.• An atomic value is a number, an identifier, the symbols + or -, or a list of atomic values separated by the symbol |.

  look V [PRED: 'TO_LOOK < SUBJ, [OBJ] >', tense: present]; for VERB_PART [LEXEME: ' -FOR ']; The unify result is the compositional predicate [PRED: 'TO_LOOK_FOR < SUBJ, [OBJ] >', tense: present] 7.3.5 The feature SUBCAT By contrast, a value for SUBCAT attribute is the subcategory part of a PRED attribute except the lexeme.

  (6) a. He gave a toy to a child. b. He gave me his phone number. c. He gave a concert. d. They gave up their personal possessions. In order to introduce a new lexical entry from such a combination, we use the # symbol followed by the new lexeme as follow: # TO_GIVE_UP [SUBCAT: '< SUBJ, [XCOMP | OBLon] >']; Given the lexical entries for the main verb to give and the particule up encoded as follow: give V [PRED: 'TO_GIVE<SUBJ, OBJ, [OBLto | OBL]>', tense: present]; up VERB_PART [LEXEME:' -_UP'];

  The subcategorization is given only by the # lexical entry, • The others features are given by the unification of the third lexical entries (verb, particule, verbal particule) [PRED: 'TO_GIVE_UP < SUBJ, [XCOMP | OBLon] >', tense: present]; 7.3.7 Embedded feature-structures Feature values for grammatical functions are embedded feature-structures [PRED: 'TO_SNORE<SUBJ>', tense: present, SUBJ: [PRED: 'BOY'] ] 7.3.8 Sets of feature-structures A set of F-structures is written within braces: MOD: {[PRED: 'little'], [PRED: 'big']} Such a set of F-structures is used to describe an unordered list of modifiers.

[

  PRED: 'SEE<SUBJ, (OBJ)>', SUBJ: [PRED:'JOHN'], OBJ: [PRED:'MAN'], MOD: {[PRED: 'TELESCOPE' ]}] [ PRED: 'SEE<SUBJ, (OBJ)>', SUBJ: [PRED:'JOHN'], OBJ: [PRED:'MAN', MOD: {[PRED: 'TELESCOPE' ]}]

[

  PRED: 'SEE<SUBJ, (OBJ)>', SUBJ: [PRED:'JOHN'], OBJ: [PRED:'MAN'], ( [ MOD: {[PRED: 'TELESCOPE' ]}] , [ OBJ: [MOD: {[PRED: 'TELESCOPE' ]}] ] ) ] The OBJ attribute in the second F-structure is the result of unification between [OBJ: [PRED:'MAN']] and [OBJ: [MOD: {[PRED: 'TELESCOPE']}]].An other exemple is the unique lexical entry for a regular English verb without s ending. It's encode both a plural subject and singular first or second person subject.A useful way to encode the two entries is the following: eat verb [ PRED: 'TO_EAT<SUBJ, [OBJ]>', SUBJ: [ ( [ number:singular, person:1|2], [ number:plural ] ) ] ]; eats verb [ PRED: 'TO_EAT<SUBJ, [OBJ]>', SUBJ: [ number:singular, person:3] ];

↑

  = ↓i; (where optional i is a number) ↓i = ↑; (↑ <path>) = ↓i; ↓i = (↑ <path>); (↑ <path>) = (↓i <path>); (↓i <path>) = (↑ <path>); (↑ <path>) = <value>; (where value stands for a functional structure or an atomic value) <value> = (↑ <path>); (↑ PRED) = '<pred>'; (where pred stands for a predicate value) '<pred>' = (↑ PRED) (↑ LEXEME) = '<lexeme>'; (where lexeme stands for a lexeme value) (↑ SUBCAT) = '<subcat>'; (where subcat stands for subcategorization data) '<subcat>' = (↑ SUBCAT);Functional equations are needed to construct f-structures associated with cstructures. One may for instance assume a rule such as the following to associate a preverbal NP with the subject function:

  du det [NUMBER: sg, PARTITIVE: true, DEFINED: false] | (prep [PRED: 'DE', PCASE: DE] & det [GENDER: ms, NUMBER: sg, DEFINED: true]);

  wants v [PRED:'WANT<SUBJ.agent, VCOMP.theme>', TENSE: present, SUBJ: [NUMBER: sg, PERSON: 3]] { (↑ VCOMP SUBJ) ⇒ (↑ SUBJ); };

  _UNKNOWN_ verb[PRED: '_THIS_'] | noun[PRED: '_THIS_'] | adjective[PRED: '_THIS_'] | adverb[PRED: '_THIS_'];

Thanks to Olivier Bonami for his remarks.

VP → AUX [advneg] VP { if ($2) (↑ neg) = true; else (↑ neg) = false; } In this example, the f-structure of the VP will always carry a feature neg, with value true if a negative adverb is present, false otherwise.

Using the operator if is not needed if the functional description includes a reference to the functional structure of the optional term. The rule

The general form for a conditional functional description is the following:

Selection

A compact and economical way to write LFG rules is to use the selection like this example from [Falk, 2001] p.76

The equivalent XLFG notation for term selection, is written by using

In order to associate a function description with each selection, the keyword switch is used.

Here is the complete exemple:

The operator switch concerns DP|NP designated by $2: depending on whether this term is DP or NP, the different statements identified by the keyword case is applied.

The general form for a selection is the following:

Constraining equations

This is the analogue of LFG constraining equations, noted with the operator = c . Such constraints do not build structure, but check that some attribute in a given f-structure has the required value.

As a possible application, notice that in English, finite clauses with the function of complement only optionally begin with a complementizer, whereas finite clauses with the function of a subject need a complementizer. To account for this, we may assume that the complementizer that introduces a feature CPLZER THAT , and that the c-structure rule for clausal subjects checks for the presence of that feature through an equality constraint:

As only an existing constant may be checked without building a structure, all the functional descriptions are accepted as equality operands:

Long distance dependencies

Long distance dependencies are standardly modeled in LFG through functional uncertainty, that is, the use of regular expressions in attribute path descriptions. This is readily implemented in XLFG. For instance, the following is a standard rule for describing wh-questions in English such as Who do you think John saw?

In this description (↑ (COMP | VCOMP)*) denotes a sequence of COMPs and VCOMPs embedded in each other ↑. All values that correspond to this existing sequence are assigned to the variable $x. ($x OBJ) denotes the OBJs embedded in them.

The general structure of such statements is with $<identifier> in (↑ <regexp>) <statement>

where X names the function of the fronted constituent in the embedded clause, and <regexp> is a regular expression over the set of attribute names.

A regular expression denotes a path in a functional structure. The simplest kind of regular expression is just an attribute name. From two regular expressions A and B, one can derive the complex expressions (A B) (A|B) and A*, corresponding respectively to concatenation, disjunction, and iterative closure.

Let