D Bémer

Y Morele

Filtration of ultrafine metallic particles in industry

Keywords: ultrafine particles, thermal spraying, filtration, dust collector

Thermal metal spraying, metal cutting and arc welding processes generate large quantities of ultrafine particles that cause the irreversible clogging of industrial filters. The aim of the study performed was to identify the causes of the clogging of cartridge filters and investigate other paths for cleaning them. This study required the development of a test bench capable of reproducing a thermal spraying process to test the performances of different filtration techniques. This test instrument first, permitted the precise characterisation of the aerosol generated by the process and, second, defined the clogging and cleaning conditions for filters. Several parameters were tested: the type of filter, on-line and off-line cleaning, pre-coating, cleaning by jets of high speed compressed air via a probe.

Introduction

Many industrial processes give rise to the formation of aerosols composed of ultrafine particles (UFP, particle diameter < 100 nm) which, in addition to their specific toxicity due to their size and chemical composition, lead to the clogging of the industrial filters used to eliminate them. Clogged filters result in increased resistance to the flow of air through the filter, leading to a deterioration of the performances of the ventilation installed (booth, suction table, etc.).

The primary particles thus formed by condensation of the metallic vapour agglomerate into bigger particles, mainly by the action of the Brownian diffusion mechanism, up to 300-500 nm [START_REF] Maynard | Is engineered nanomaterial exposure a myth ?[END_REF]. Particles greater than 500 nm are eliminated by deposition on the walls of the different structures (ducts, ventilated booth, etc.) by sedimentation or inertial deposition. Three hundred nanometers corresponds precisely to the MPPS (Most Penetrating Particle Size) i.e. the particle size the most difficult to be retained by media filters, electrofilters, wet scrubbers, cyclones, etc. [START_REF] Bicocchi | Les techniques de dépoussiérage des fumées industrielles[END_REF][START_REF] Chong | Effects of the diffuser and of the particle interception on the penetration of a venture scrubber[END_REF] Actually, filtration using fibrous media presents the best compromise performances/costs and is, for this reason, the separating technique the most widely used in industry for particle removal.

Thermal metal spraying (flame, electric arc, etc.), metal cutting (arc/air process) and arc welding processes generate large quantities of UFP (concentration in number > 10 8 /cm 3 , mass concentration > 100 mg.m -3 , [START_REF] Bémer | Ultrafine particles emitted by flame and electric arc guns for thermal spraying of metals[END_REF]), that are often highly toxic, and the cause of irreversible clogging of the filters used in dust extraction installations. The size of the primary particles composing these aerosols and their aggregates is sometimes very small (9 nm for the primary particles of thermal spraying, 70-100 nm for the aggregates), due to the almost instantaneous dilution caused by the injection of compressed air at the point of metal fusion. The particles generated by arc welding, for example, which can also be considered as ultrafine, present larger primary particles (40-90 nm) [START_REF] Witschger | Les Nanoparticules. Avis d'experts[END_REF], due to the low dilution at the point of fusion. However, the larger the specific surface of the particles (ratio of the surface developed by the particles to the volume), the higher the clogging capacity of the layer of particles [START_REF] Thomas | Clogging of fibrous filters by solid aerosol particles, Experimental and modeling study[END_REF].

The most pollutant operations like thermal metal spraying are performed in ventilated booths. The air contaminated by the particles is directed to the dust extraction installation whose air flow is determined by the section of the booth to be ventilated, i.e. 1400 m 3 /h/m 2 [START_REF]Guide de ventilation n°14 -Décapage, dessablage, dépolissage au jet libre en cabine[END_REF], which leads to high flow rates, generally between 20000 and 25000 m 3 /h. Filtration using pleated filter cartridges is by far the most widely used air treatment technique in industry due to the large filter surface presented by the filters. However the clogging of the filters subject to metallic fumes is currently a source of problems. Indeed, observations in the field have shown that in many cases, clogged filters (Figure 1) are either maintained in the dust collector, in which case the ventilation flows are no longer ensured, resulting in poorer indoor air quality, or simply extracted, leading to the direct discharge of pollutants into the external environment.

The evolution of the filter pressure drop over filtration/cleaning cycles depends on many parameters [START_REF] Lo | Experimental study of pleated fabric cartridges in a pulse-jet cleaned dust collector[END_REF]: the nature of the medium (elasticity, fibre diameter, etc.), the filter (shape, filtration surface, etc.), the pressure of the compressed air used for cleaning, etc. At present, the manufacturers of industrial dust collectors used for metallic UFP filtration, often have no solution other than increase the filtration surfaces in order to decrease filtration velocity and thus limiting further pressure drop and the frequency of filter replacement. Furthermore, increasing filtration surfaces leads to increased investment and operating costs.

The aim of the study performed by the INRS was to identify the causes of cartridge filter clogging and investigate other paths for cleaning filters. This study required developing a test bench capable of reproducing a thermal metal spraying operation by electric arc to test the performances of different filtration techniques. This test resource first permitted the precise characterisation of the aerosol generated with respect to the granulometry and morphology of the materials. Second, the clogging and above all the cleaning conditions of the filters, that is to say their capacity to eliminate the layer of metallic particles causing the increased pressure drop, were studied.

Methodology

The aerosol was generated by a thermal metal spraying process using a Margarido® M25 electric arc gun fed by wires made of Zn/Al (85/15%). The molten metal is sprayed by compressed air onto a rotating metal cylinder. The slag from the melting process is recovered in a tank located in the lower part of the chamber. The fumes and fine particles are extracted by a fan located at the end of the main channel (200 mm in diameter) and directed to a cyclone that eliminates the micronic fraction of the particles but conserves the ultrafine fraction corresponding to the fumes. A fraction of the flow is thus directed to the filtration system studied. The remaining fumes are conveyed to the main dust separator (Figure 2).

The aerosol produced was characterised using different analysers. A Scanning Mobility Particle Sizer SMPS (Grimm 5.403), after dilution at 1/100 (Palas, VKL 100), was used to measure the number particle size distribution. The mass concentration was determined by collection on a filter (filter: GLA 5000 D 47 mm) and by gravimetry. A Tapered Element Oscillating Microbalance TEOM analyser (1400a, R&P) was also used to monitor the evolution of the downstream mass concentration through time The variation in pressure drop (P) of a flat filter (GLA 5000) was measured during clogging. This measure permitted determining the specific resistance coefficient of the particle layer k 2 , according to Darcy's law:

s f 2 M . U . k P    (1)
With , the dynamic viscosity of the gas, U f , the filtration velocity and M s the mass of particles per unit of filter surface. Knowing the coefficient k 2 and the filtration conditions (air flow rate, particle mass concentration), it is theoretically possible to determine the kinetics of increasing P during clogging. Eq.1 also shows that for a given particulate pollutant, the only way to limit the P increase is to reduce the filtration velocity as far as possible.

Two types of pleated cartridges were studied. They are manufactured using the same media (cellulose covered by a PTFE membrane) but present different filter surface areas and fold widths (Table 1). The cartridge was placed horizontally in a closed chamber supplied by air recovered from the test bench. Cleaning was ensured by injecting compressed air in reverse direction flow via a nozzle placed at the inlet of the cartridge. The pressure of the compressed air was 4 bar and the flow rate of the test air was set at 340 m 3 .h -1 , resulting in filtration velocities for cartridges A and B of 0.55 cm/s and 1.1 cm/s, respectively.

A second cleaning device, developed by the INRS, was tested. It comprised a probe supplied with compressed air placed inside the cartridge. The compressed air (at a pressure of 7 bar), discharged through several nozzles, swept the entire internal surface of the cartridge by rotating about its own axis (Figure 3). The cartridge was completely unclogged by four rotations of the probe.

A dust separator composed of four panel type filters, made of a quite rigid polyester covered with a PTFE membrane and placed vertically, was also tested. The filtration surface provided by the filters was 14 m 2 . Cleaning was performed by injecting compressed air in reverse direction flow.

The pressure of the compressed air for cleaning was 6 bar and the air flow rate was set at 320 m 3 .h -1 (filtration velocity 0.6 cm.s -1 ).

Pre-coating tests were performed in the cases of cartridges A and B. This operation consisted in depositing a layer of quite coarse and non toxic particles beforehand on the filter medium, the layer acting like a filter for the fume particles [START_REF] Ravert | Precoating new filters for better airflow, longer filter life[END_REF]. This layer above all presents the advantage of being easy to dislodge. Limestone dust (mass median diameter 3.6 µm) was used for this purpose. A mass of 200 g was injected over a period of ten minutes into the upstream air circuit (Figure 1) before starting metallic fume filtration. The mass per unit area deposited on filter can be determined from the filter surface of the filters. Tests were also performed with coarser dolomite dust (mass median diameter 9.5 µm) and a higher generated mass of 350 g.

The conditions for initiating cleaning were also studied: cleaning was initiated with the air flow rate maintained (V-on) or not (V-off), and with the generation of metallic fumes activated (G-on) or not (G-off). There was obviously only one configuration in V-off mode, since the thermal spraying operation was, of course, stopped in the absence of ventilation.

Results and discussion

Characteristics of metallic fume particles (downstream of the cyclone).

Figure 4 shows the particle size distribution in number downstream of the cyclone, measured using the SMPS analyser, after dilution at 1/100. The mean electrical mobility diameter in number is from 70 to 80 nm for a concentration of 2.10 8 cm -3 , the mass mean diameter is from 100 and 150 µm for a concentration of 85 mg.m -3 . Figure 5 shows a photograph of particles taken by transmission electron microscopy. The mean diameter of the primary particles is 9 nm. The porosity of the strongly aggregated particles causes their density to be much lower than the density of the constitutive material (metal). For example, in the case of Zn/Al particles, the apparent mean density is 1 g/cm 3 for a material density equal to 5.74 g/cm 3 . The apparent density of Zn/Al particles was determined using a Differential Mobility Analyser DMA (Grimm 5.403) and an Aerosol Particle Mass analyser APM (Kanomax, 3601) in series [10]

Coefficient k 2

The measures performed at a flow rate of 4 l/min, give a value of coefficient k 2 = 8.10 11 m/kg  1.6.10 11 m/kg according to the operating conditions. These values appear to be quite stable for most thermal spraying processes, providing that the aerosol fraction > 1 µm is eliminated.

Values of the same magnitude have been obtained with metallic fumes generated by air arc cutting. This value of k 2 leads to a pressure drop increase kinetics of 145 Pa/(g/m2/(cm/s). Applied to the case of cartridge B, using a concentration of 85 mg/m 3 , the P increase would therefore be 560 Pa/h.

Filter performances

The filtration flow rate and pressure drop were recorded continuously. The particle mass concentration was also measured upstream and downstream of the filter. Cleaning was initiated when the pressure drop reached a predefined limit. Three configurations were tested: V-on / G-on, V-on / G-off and V-off, with and without pre-coating. Pre-coating was not studied in the case of the panel filter.

Cartridge filter

Figure 6 shows the almost total lack of efficiency of cleaning on cartridge B whether done in V-on mode or V-off mode. During cleaning in V-off mode, the reduction of pressure drop is more marked, but the increase following the resumption of filtration is also very fast, so that the P value returns to its value before cleaning almost immediately. Regarding cartridge A which have a larger filtration surface, the same phenomenon was observed, though it was simply slower. Note that the increase of P of the cartridge corresponds to the data obtained for the flat filter (coefficient k 2 ). The downstream mass concentrations varied from 2.5 mg/m 3 at the start of the cycle (new or completely cleaned filter) to 0.15 mg/m 3 at the end of the cycle. Note that this low value of the mass concentration (0.15 mg/m 3 ) corresponds to a relatively high number concentration of 4.10 5 particles/cm 3 , compared for instance to the IFA (2009) limit value for oxide metallic UFP of 2.10 4 to 4.10 4 particles/cm 3 [START_REF] Van Broekhuizen | Worplace exposure to nanoparticles and the application of provisional nanoreference values in times of uncertain risks[END_REF].

Cleaning using the compressed air probe was tested on cartridge B in V-off and V-on modes. The results (not shown) show complete cleaning of the cartridge with the total elimination of the cake of particles. After cleaning, the pressure drop returned to its initial value and the P at the resumption of filtration was similar to that of a new cartridge. Continuing ventilation during the cleaning operation (V-on) did not appear to have an influence on the device's capacity to eliminate the layer of particles.

Following the pre-coating of cartridge B with limestone dust of particle size 3.6 µm, cleaning was performed when the ventilation was stopped (Figure 7, V-off). At the end of cleaning, the pressure drop of the cartridge returned to values close to the initial P, but the kinetics of increase was higher than that of a new cartridge, indicating partial cleaning. However, in Von / G-on mode cleaning was totally inefficient (similar results were obtained with cartridge A). With dolomite dust of particle size 9.5 µm, partial cleaning was nonetheless obtained in V-on mode, leading to a residual P (after cleaning) higher than the initial P. This can be explained by the lower adhesion forces developed by this layer of particles, which were larger and therefore easier to detach.

Tests were also performed, still using pre-coating with dust of particle size 9.5 µm, in V-on mode, but this time stopping the thermal spraying operation several seconds before initiating the cleaning (G-off). In this configuration, the cleaning appeared to be more efficient and the pressure drop returned to the evolution described in Figure 7. The almost instantaneous filling of the cleaned surfaces by the fine metallic particles may explain the inefficiency of cleaning in V-on / G-on mode.

Panel filters

Figure 8 shows the evolution of the P of the panel filter over several clogging/cleaning cycles in V-off mode. This time, the cleaning allowed eliminating a sufficient fraction of the filter cake to lower the P durably. Cake detachment occurs by a process of "patchy cleaning", where some areas of cake are completely removed and others not at all (Figure 9) [START_REF] Koch | Dust cake detachment from gas filters[END_REF]. The small reductions (a hundred Pa) observed for the P were caused by interruptions of the tests from one day to another. The test was interrupted for a much longer period of several weeks (indicated by the arrow in Figure 8). The kinetics of increasing P on resuming filtration was therefore much slower. Then the system returned to the behaviour exhibited before the long interruption. The average efficiency of the filter determined over a clogging/cleaning cycle, based on the upstream/downstream particle mass concentrations, was 99.3% with a mean downstream concentration of 0.6 mg/m 3 . During the filter clogging, the formation of the layer of particles on the surface of the medium contributes to increasing the filtration efficiency. Thus the downstream concentration, measured with a TEOM analyser, regularly decreases during clogging as shown in Figure 10. A quite high concentration was recorded at the beginning of the cycle ( 1.8 mg/m 3 ).

Contrary to the cartridge filter, the cleaning performances of the panel filter did not appear to be affected by maintaining ventilation and generation (V-on / G-on). With this mode of operation, cleaning made it possible to obtain a P very close to the initial value and the kinetics of increased pressure drop when re-starting filtration was similar to that of a new filter. But the overly short tests performed, did not allow reaching a definitive conclusion.

Nonetheless, it appears that the slackness conferred by the panel shape of the filter contributes to the improved cleaning observed. This phenomenon was highlighted by the works of Dennis [START_REF] Dennis | Predicting pressure drop for pulse-jet filters[END_REF] in a study in which he effectively showed that optimal cleaning of the layer of particles is obtained for slackness filters with very low intrinsic elasticity. This slackness of the medium provides it with a considerable freedom of movement during pressure cleaning pulse, generating considerable detachment forces during the deceleration phase of the medium, due to its low elasticity.

Conclusion

Observations in the field showing that dust collectors with pleated cartridge filters were irreversibly clogged, were confirmed by our laboratory results. Pulse-jet cleaning of these cartridges by compressed air injected in reverse direction flow appeared to be very inefficient in both on-line (with ventilation on) and off-line (ventilation off) modes. But this solution can nevertheless be retained if the user agrees to frequently change the cartridges, which will result in additional costs.

Use of the pre-coating technique permitted cleaning these cartridges, but only in off-line mode. In on-line mode, once again using pre-coating, cleaning could nonetheless be performed after stopping the thermal spraying operation. However, this solution should be tested on an industrial installation operating over a longer period. But one disadvantage of the pre-coating solution is that it generates large quantities of waste, made of the pre-coating powder contaminated but the metallic UFP.

Cleaning performed using probes introduced inside the cartridge and supplied with compressed air was the most efficient of the cleaning methods tested. Cleaning performance did not depend in this case on whether ventilation was kept on or off. Solutions are now being sought to optimise this technique.

The panel filters tested in this study appeared to be much more efficient regarding the elimination of the particle layer. The greater flexibility of the filter provided by its shape was certainly the reason for the improved performance. Similar results (not given in this report) were obtained with filter bags made of polyester covered with a microporous layer. These tests confirmed that the best cleaning performances were obtained with slackness media, with low elasticity. However, the disadvantage of these filters (panels, pockets, bags) is that they also have small filter surfaces, contrary to the pleated filters used in cartridges and therefore the installations using them are much larger and more expensive.

Regarding the performances in terms of filtration efficiency and downstream particle concentration, the measurements show that they are quite efficient if mass is the reference parameter used, but are much worse if the particle number is considered. Consequently, the air recycling technique cannot be used, under any circumstances, for these particulate pollutants.

Figure 1 .

 1 Figure 1. Clogged pleated cartridge filterdust collector installed on a booth used for thermal spraying with an electric arc gun.

Figure 2 .

 2 Figure 2. Experimental set-up for measuring the performances of filters subjected to metallic metal spraying fumes (electric arc thermal spraying).

Figure 3 .

 3 Figure 3. Photograph of the cleaning probe used for filter cartridges by blowing compressed air.

Figure 4 .

 4 Figure 4. Particle size distribution in number of metallic particles -SMPS measurementdilution of the sample 1/100.

Figure 5 .Figure 6 .Figure 8 .

 568 Figure 5. Transmission electron microscope photograph of the metallic particles generated by thermal spraying (electric arc gun).

Figure 9 .Figure 10 .

 910 Figure 9. View of the panel filter after cleaning (off-line) showing the partially detached filter cake.

Table 1 .

 1 Characteristics of pleated cartridges.

	Medium	Cartridge A	Cartridge B
	Filter surface, S (m 2 )	17.7	8.5
	Fold height, H (mm)	38.1	38.1
	Fold width, L (mm)	2.84	6.03
	Ratio H/L	13.4	6.3

Pre-coating

Pre-coating


Emmanuel Chasseigne

ON THE REGULARIZING EFFECT FOR UNBOUNDED SOLUTIONS OF FIRST-ORDER HAMILTON-JACOBI EQUATIONS

Keywords: 1991 Mathematics Subject Classification. 35F21, 35D35, 35D40 First-order Hamilton-Jacobi Equations, viscosity solutions, regularizing effects. 1

We give a simplified proof of regularizing effects for first-order Hamilton-Jacobi Equations of the form ut +H(x, t, Du) = 0 in R N ×(0, +∞) in the case where the idea is to first estimate ut. As a consequence, we have a Lipschitz regularity in space and time for coercive Hamiltonians and, for hypo-elliptic Hamiltonians, we also have an Hölder regularizing effect in space following a result of L. C. Evans and M. R. James.

Introduction

In this short paper we give a new proof of regularizing effects for Hamilton-Jacobi Equations

u t + H(x, t, Du) = 0 in R N × (0, ∞) , (1.1)
in the case when the aim is to estimate u t first. This new proof is inspired by ideas introduced in [START_REF] Barles | On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations[END_REF] and then simplified in [START_REF] Barles | A new PDE approach to the large time asymptotics of solutions of Hamilton-Jacobi equations[END_REF]; a precise comparison between the results and ideas of [START_REF] Barles | On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations[END_REF][START_REF] Barles | A new PDE approach to the large time asymptotics of solutions of Hamilton-Jacobi equations[END_REF] and ours is provided just after the statement of the main results of this article, at the end of Section 2. More classical proofs can be found in [START_REF] Barles | Regularity results for first-order Hamilton-Jacobi Equations[END_REF] but with stronger assumptions and more tedious proof.

The model equations we have in mind are

u t + |A(x, t)Du| m = f (x, t) in R N × (0, ∞) , (1.2)
where f is a continuous (typically bounded from below) function and A takes values, in the set of N × N symmetric matrices. For such equations, we consider two cases: the coercive case for which A is invertible and, as a consequence |A(x, t)Du| m → +∞ as |p| → +∞, and the non-coercive case where A may be degenerate. In both cases, we provide regularizing effects for bounded from below solutions. The main improvement in the assumptions is easy to describe in the coercive case since we just require that A, f are continuous in x (no uniform continuity assumptions) and, in particular, f may have some growth at infinity. In the non-coercive case, analogous results hold except that we have to impose far more restrictive assumptions on the t-dependence of the equation.

Of course, for (1.2), the equation implies that u t ≤ f (x, t) in R N × (0, ∞) and thefore we just need an estimate from below for u t .

To do so, our approach consists in using the exponential transform, v := -exp(-u). Notice that provided u is bounded below (then we can always assume that u is nonnegative), we get that v is bounded since -1 ≤ v ≤ 0. Moreover, v solves a new Hamilton-Jacobi equation

v t + G(x, t, v, Dv) = 0 , with G(x, t, v, p) := -vH x, t, - p v . (1.3)
In order to estimate v t , a key property (as in all the regularizing effects proofs) is to have a large enough, positive G v when v t = -G is large (but negative) i.e. when G is large and positive. This leads to an assumption on (H p • p -H)(x, t, p) which is classical except that, here, this quantity has to be large when H(x, t, p) is large, and not when |p| is large as it is classical for the estimate on Du.

The paper is organized as follows : in Section 2, we state our main results, one in the case where H is assumed to be coercice, one in the degenerate case and we give and we deduce full regularizing effects, i.e. in space and time. The proofs of the main theorems are are given in Section 3. Then, in Section 4, we treat several explicit examples. We have put some technical results about Hamiltonian G in an appendix.

Acknowledgement -Both authors were partially supported by the french ANR project WKBHJ (Weak KAM beyond Hamilton-Jacobi), ANR-12-BS01-0020.

Assumptions and main results

2.1. Assumptions. In order to state and prove our results, we use several structure conditions, which all rely on the following fundamental hypothesis We first point out that, for the model equation (1.2), (H0) is satisfied with c 0 = 0 and φ(s) = (m -1)s by homogeneity, both in the coercive and non-coercive cases.

(H0) The function H is continuous in R N × [0, T ] × R N and there exists c 0 = c 0 (H) ≥ 0 such that • H is locally Lipschitz in the p-variable, in a neighborhood of the set {(x, t, p); H(x, t, p) ≥ c 0 }; • there exists a continuous, increasing function φ : [c 0 , ∞[→ [0, ∞[ such that for some A > c 0 , +∞ A 1 sφ(s) ds < +∞ (2.
Remark 2.1. Roughly speaking, (H0) is a superlinearity condition on H either in the p-variable or a linear function of p (possibly depending on x, t and possibly degenerate). Indeed, it is easy to show that if H(x, t, p) satisfies (H0) then H(x, t, a(x, t)p) also satisfies (H0) for any bounded matrix-valued function a. We also refer to Section 4 for various examples. Even if we were not able to prove it in full generality, Condition (2.1) should be automatically satisfied for convex and superlinear Hamiltonians.

The first main consequence of (H0) is the following: if we set

∀τ ∈ (c 0 , ∞) , F (τ ) := 2 ∞ τ dσ σφ(σ)
and ∀s ∈ (0, F (c 0 )) , η(s) = F -1 (s) , then η ′ (s) = -η(s)φ(η(s))/2, s → η(s) is decreasing and η(0 + ) = +∞. Notice that if c 0 = 0, since φ(s) ∼ φ(0)s near s = 0, the integral blows-up and

F (0) = ∞. Similarly, if φ(c 0 ) = 0 we have F (c 0 ) = +∞.
The next assumption concerns the dependence of H in t. In order to formulate it, we notice that η → 2η + φ(2η) is increasing from (c 0 , ∞) to (2c 0 + φ(2c 0 ), ∞). Hence, its inverse, denoted by ψ : (2c 0 + φ(2c 0 ), ∞) → (c 0 , ∞) is well-defined and also increasing. Notice that ψ is sublinear but ψ(τ ) → ∞ as τ → ∞.

(H1) There exists c 1 > 0 such that the function H is locally Lipschitz in t in a neighborhood of the set {(x, t, p); H(x, t, p) ≥ c 1 } and for any x, p ∈ R N , t ∈ [0, T ],

|H t (x, t, p)| ≤ ψ H(x, t, p) (H p • p -H)(x, t, p) a.e. in {(x, t, p); H(x, t, p) ≥ c 1 } .
For the model equation (1.2), (H1) is satisfied in the coercive case as soon as A is Lipschitz continuous in t, while, in the non-coercive case, it imposes rather restrictive (or at least non-natural assumptions) since A should satisfy |A t (x, t)p| ≤ k|A(x, t)p|.(|A(x, t)p| m ) for any x, t, p such that |A(x, t)p| m ≥ c 1 , a property which imposes particular types of degeneracies (see Section 4 for a study of the scalar case). However (H1) is satisfied with c 1 = 0 if A does not depend on t.

We then define precisely the coercive case:

H is coercive if (H2) H(x, t, p) → +∞ when |p| → +∞, uniformly for (x, t) in a compact subset of R N × (0, T ].
And we have to impose some additional restrictions on the behavior of H in x in the non-coercive case i.e. when (H2) does not hold.

(H3) There exists c 2 > 0 such that the function H is locally Lipschitz in x in a neighborhood of the set {(x, t, p); H(x, t, p) ≥ c 2 } and there exists a constant γ > 0 such that, for any

x, p ∈ R N , t ∈ [0, T ], |H x (x, t, p)| ≤ γ(|p| + 1)(H p • p -H)(x, t, p) a.e. in {(x, t, p); H(x, t, p) ≥ c 2 } .
For the model equation (1.2), (H2) is satisfied (in the non-coercive case) as soon as A is Lipschitz continuous in x.

2.2.

The main theorem. In order to state our main results, we first introduce

c * := max(c 0 , c 1 ) if (H0)-(H1)-(H2) holds , max(c 0 , c 1 , c 2 ) if (H0)-(H1)-(H3) holds .
and we set

t * = t * (H) := η -1 (c * ) = F (c * ) if c * > 0 , +∞ if c * = 0 .
Notice that this is consistent with the fact that F (0 + ) = +∞.

Theorem 2.2. We assume that either (H0)-(H1)-(H2) or (H0)-(H1)-(H3) holds and let u be a continuous, bounded from below solution of (1.1). Then for any t ∈ (0, t * ) and x ∈ R N , the function

v := -exp(-u) satisfies inf 0≤s≤t v(x, t) -v(x, s) + (t -s)η(s) ≥ 0 . (2.2)
Notice that since u and v are continuous functions, u t and v t are defined in the sense of distributions. A distributional formulation of (2.2) is the following: v t ≥ -η(t) in R N × (0, t * ). Similarly, we have u t ≥ -η(t)u on the same set.

2.3. Consequences. The general statement of this theorem may hide some non-trivial consequences concerning global estimates, at least in some particular cases. First, if c * > 0 and we want to get an estimate for t > t * , we can apply the estimate of the theorem to w(x, s) := u(x, s + t -t * /2) which is a solution of an analogous (just shifted in time) pde for s ∈ (0, T -t + t * /2). This interval corresponds to the interval (t -t * /2, T ) for u. Hence the

Corollary 2.3. If c * > 0, setting C = η(t * /2) we have v t ≥ -max(η(t), C) and u t ≥ -max η(t), C u a.e. in R N × (0, T ) .
If T = +∞, the previous estimate gives the lower estimate u t ≥ -Cu for t big enough. However, if c * = 0 then t * = ∞ and we get a much better asymptotic estimate Corollary 2.4. If c * = 0 and T = +∞,

v t ≥ -η(t) and u t ≥ -η(t)u a.e. in R N × (0, ∞) .
In particular, (u t ) -→ 0 locally uniformly in R N as t → ∞.

We first point out, as we already do it after the statement of (H1), that c * = 0 implies in particular c 1 = 0, which is a rather restrictive assumption in the non-coercive case. We come back on this point in Section 4.

This asymptotic result leads us to the comparison with the results of [START_REF] Barles | On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations[END_REF][START_REF] Barles | A new PDE approach to the large time asymptotics of solutions of Hamilton-Jacobi equations[END_REF] which concern cases where H does not depend on t. The main issue in these papers was the asymptotic behavior of u which is expected to be of ergodic type, namely

u(x, t) = -λt + φ(x) + o t (1) when t → +∞ ,
where λ is the ergodic constant and φ a solution of the ergodic problem

H(x, Dφ) = λ in R N .
In [START_REF] Barles | On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations[END_REF][START_REF] Barles | A new PDE approach to the large time asymptotics of solutions of Hamilton-Jacobi equations[END_REF], the key idea is to reduce to the case when λ = 0 and then to show that (u t ) -→ 0 (or (u t ) + → 0) locally uniformly in R N as t → ∞. To do so, one has to take into account the fact that λ = 0 and φ plays a key role since it interferes in the analogue of (H0). With all these ingredients, the proofs of [START_REF] Barles | On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations[END_REF][START_REF] Barles | A new PDE approach to the large time asymptotics of solutions of Hamilton-Jacobi equations[END_REF] are slightly more sophisticated that ours. But we point out that, remarking that an analogous proof could provide regularizing effects is (to the best of our knowledge) completely new.

This analysis through the ergodic behavior of u also shows that Corollary 2.4 can only hold in particular situations.

2.4. Gradient estimates and spatial regularity. In the coercive case, we can deduce a gradient estimate from (2.2). Indeed, Theorem 2.2 implies

H(x, t, Du) ≤ -u t ≤ η(t)u ,
and standard arguments in viscosity solutions' theory show that u is locally Lipschitz continuous. Moreover, a local estimate of Du follows from the above inequality.

In the non-coercive case we studied above (Example 3), the bound on u t implies a local estimate on |A(x, t)Du| and using a result of Evans and James [START_REF] Evans | The Hamilton-Jacobi-Bellman equation for time-optimal control[END_REF], we can deduce a local C 0,α bound when A satisfies some "hypoelliptic" type assumptions.

Proof of Theorem 2.2

In all the proof, we fix 0 < t < t * where we recall that t * = η -1 (c * ). Assuming that for some x 0 ∈ R N , inequality (2.2) does not hold, we deduce that for α, β, ǫ > 0 small enough with ǫ ≪ α, β, we have also

min t≥s≥0, x,y∈R N v(x, t) -v(y, s) + (t -s)η(s) + |x -y| 2 ǫ 2 + α|y| 2 + βη(s) ≤ -δ , (3.1)
for some δ > 0 which does not depend on α, β, ǫ. Notice first that this min is attained because of the α-term, and it cannot be attained at s = 0 because of the β-term. For simplicity, we still denote by x, y, s, t the variables for which the min is attained. Since v is bounded, we have the usual estimates

|x -y| ≤ O(ǫ), |y| = o(α -1/2 ).
Next, the equation for v given in (1.3) involves a new Hamiltonian G whose properties follow from the assumptions on H (see the proof in Appendix): if G(x, t, v, p) ≥ c * , (H0) implies (G0) :

G v (x, t, v, p) ≥ φ(G(x, t, v, p)) and |G p (x, t, v, p)| ≤ κ G v (x, t, v, p) , (H1) implies (G1) : |G t (x, t, v, p)| ≤ ψ(G(x, t, v, p)) G v (x, t, v, p) , (H3) implies (G3) : |G x (x, t, v, p)| ≤ γ(|p| + 1)G v (x, t, v, p) .
Using the supersolution inequality for v(x, t) in the viscosity sense, and the subsolution for v(y, s) we get, with

p ǫ = -2(x -y)/ǫ 2 , -η(s) + G x, t, v(x, t), p ǫ ≥ 0 , -η(s) + (t -s + β)η ′ (s) + G y, s, v(y, s), p ǫ + 2αy ≤ 0 .
We begin with the coercive case, assuming (H0)-(H1)-(H2). We remark that, if α and β remain fixed, then x, y stay in a compact subset of R N and t, s in a compact subset of (0, t * ) and we can let ǫ tend to zero. By the uniform continuity of H and v on such compact sets, we end up with

-η(s) + G x, t, v(x, t), p ≥ 0 , -η(s) + (t -s + β)η ′ (s) + G x, s, v(x, s), p + 2αx ≤ 0 , (3.2)
for some p ∈ R N . In order to estimate the difference of the hamiltonians, we denote by v 1 = v(x, t), v 2 = v(x, s), q α = 2αx and ξ σ := (x, t, v 1 , p) + σ(0, s -t, v 2 -v 1 , q α ).

The following arguments rely on the next two lemmas, where s and t are fixed, and σ is the running variable.

Lemma 3.1. Let 0 ≤ σ 1 < σ 2 ≤ 1 such that η(s) ≤ G(ξ σ ) ≤ η(s) + φ(η(s)) for any σ ∈ [σ 1 , σ 2 ] . (3.3) Then, the function σ → G(ξ σ ) is increasing on [σ 1 , σ 2 ].
Proof -We argue as if G were C 1 (otherwise a standard mollification argument allows to reduce to this case) and compute

d dσ G(ξ σ ) = G t (ξ σ )(s -t) + G v (ξ σ )(v 2 -v 1 ) + G p (ξ σ ) • q α .
Inequality (3.1) yields (recall that we have let ǫ tend to 0)

v 2 -v 1 ≥ (t -s + β)η(s) + α|x| 2 + δ , (3.4)
and therefore, as long as G(ξ σ ) ≥ η(s), we have G v (ξ σ ) ≥ 0 and

d dσ G(ξ σ ) ≥ G t (ξ σ )(s -t) + G v (ξ σ ) (t -s + β)η(s) + α|x| 2 + δ + G p (ξ σ ) • q α .
Denoting

I 1 := G t (ξ σ )(s -t) + 1 2 G v (ξ σ )(t -s)η(s) , I 2 := 1 2 G v (ξ σ )α|x| 2 -G p (ξ σ ) • q α , I 3 := 1 2 G v (ξ σ )δ ,
we have to estimate these three terms, as long as η(s) ≤ G(ξ σ ) ≤ η(s) + φ(η(s)). Recall that since η(s) ≥ η(t * ) = c * , as long as G(ξ σ ) ≥ η(s), we have G(ξ σ ) ≥ c * and we can apply the hypotheses on G.

First,

I 1 ≥ 1 2 (t -s)G v (ξ σ ) η(s) -2 |G t (ξ σ )| G v (ξ σ ) ,
and by (G1),

|G t (ξ σ )|/G v (ξ σ ) ≤ ψ(G(ξ σ )) ≤ ψ η(s) + φ(η(s)) = η(s)/2
, the last equality coming from the definition of ψ. Hence

I 1 ≥ 0 on [σ 1 , σ 2 ].
Concerning I 2 , we have

I 2 ≥ 1 2 αG v (ξ σ ) |x| 2 -2 G p (ξ σ ) G v (ξ σ ) |x| ≥ 1 2 αG v (ξ σ ) |x| 2 -2κ|x| .
Using (G0) we remark that the quantity in the bracket is positive if |x| > 2κ and therefore we are left with considering the case when |x| ≤ 2κ. But in this case, I 2 is estimated by 4ακ 2 G v (ξ σ ) and I 3 controls this term since δ ≥ 4ακ 2 for α small enough, so that I 3 + I 2 > 0.

In any case, I 1 + I 2 + I 3 > 0 on [σ 1 , σ 2 ] and we deduce that, if t * is chosen small enough and α is small enough,

d dσ G(ξ σ ) > 0 for σ ∈ [σ 1 , σ 2 ]. Q.E.D.
The next step shows that we can apply Lemma 3.1 on the whole interval σ 1 = 0 and σ 2 = 1:

Lemma 3.2. For any σ ∈ [0, 1] η(s) ≤ G(ξ σ ) ≤ η(s) + φ(η(s)) .
Proof -We recall that in this proof s is fixed. We first notice that viscosity inequalities (3.2) yield G(ξ 0 ) ≥ η(s) and

G(ξ 1 ) ≤ η(s) -(t -s + β)η ′ (s) = η(s) + (t -s + β)η(s) φ(η(s)) 2 .
From (3.4) we deduce that (t

-s + β)η(s) ≤ v 2 -v 1 -δ ≤ 1 -δ so that G(ξ 1 ) ≤ η(s) + (1 -δ)φ(η(s)).
Now assume by contradiction that G(ξ σ ) > η(s) + φ(η(s)) for some σ ∈ [0, 1) and denote by σ the supremum of such σ. Then, since G(ξ 1 ) ≤ η(s)

+ (1 -δ)φ(η(s)), necessarily σ < 1 and G(ξ σ ) = η(s) + φ(η(s)). Since G(ξ 1 ) < G(ξ σ ) = η(s) + φ(η(s)), by continuity of σ → G(ξ σ ), there exists a subinterval [σ 1 , σ 2 ] ⊂ [σ, 1], such that G(ξ σ1 ) > G(ξ σ2 ) and for any σ ∈ [σ 1 , σ 2 ], η(s) ≤ G(ξ σ ) ≤ η(s) + φ(η(s)) .
But this yields a contradiction with Lemma 3.1. Hence for any σ ∈ [0, 1], G(ξ σ ) ≤ η(s) + φ(η(s)).

For the other inequality, the argument is the same: if there exists σ such that G(ξ σ ) < η(s) (using this time G(ξ 0 ) and the interval [0, σ] where σ is the infimum of the above set of σ), we get a similar contradiction with Lemma 3.1. Hence the proof is complete.

Q.E.D.

In order to conclude the proof of the first part of Theorem 2.2, we subtract the viscosity inequalities, we obtain

(t -s + β)η ′ (s) + G(ξ 1 ) -G(ξ 0 ) = (t -s + β)η ′ (s) + G x, s, v 2 , p + q α -G(x, t, v 1 , p) ≤ 0 ,
and we can perform almost the same computations to compute the left hand side, except that we write

G v (ξ σ ) = 1 2 G v (ξ σ ) + 1 2 G v (ξ σ ) ≥ 1 2 G v (ξ σ ) + 1 2 φ(η(s)) ,
and we obtain

(t -s + β) η ′ (s) + 1 2 φ(η(s))η(s) + 1 * G t (ξ σ )(s -t) + 1 2 G v (ξ σ ) (t -s + β)η(s) + α|x| 2 + δ + G p (ξ σ ) • q α dσ ≤ 0 .
By the choice of η(•), the first term is 0 and the integral is strictly positive for t * and α small enough. Therefore we have a contradiction, proving the claim in the coercive case.

Now we turn to the non-coercive case, assuming (H3) instead of (H2), where the main difference is that we are not able to let ǫ tend to 0. We define G(ξ σ ) in a similar way and prove that G(ξ σ ) ≥ η(s) (we omit the proof here since it follows, as in Lemma 3.1, from the same argument as below). Computing the difference of G in the same way, we can write the difference of the viscosity inequalities as

1 * Q 1 (σ) + Q 2 (σ) + Q 3 (σ) + Q 4 (σ) dσ ≤ 0 , where Q 1 (σ) := (t -s + β)η ′ (s) + 1 2 G v (ξ σ ) (t -s + β)η(s) + δ ,
and by the same arguments as above Q

1 (σ) ≥ 1 2 G v (ξ σ )δ. Q 2 (σ) := G x (ξ σ ) • (x -y) + 1 2 G v (ξ σ ) |x -y| 2 ǫ 2 ,
which can be rewritten as

Q 2 (σ) ≥ 1 2 ǫ 2 G v (ξ σ ) - G x (ξ σ ) G v (ξ σ ) |p ǫ | + 1 4 |p ǫ | 2 ≥ 1 2 ǫ 2 G v (ξ σ ) -γ|p ǫ |(|p ǫ | + 1) + 1 4 |p ǫ | 2 . But ǫ 2 |p ǫ | 2 = |x -y| 2 ǫ 2 = o ǫ (1) if α, β are fixed and therefore Q 2 ≥ o ǫ (1)G v (ξ σ ) is controlled by Q 1 for ǫ small enough. Q 3 (σ) := (t -s) 1 2 G v (ξ σ )η(s) -G t (ξ σ )
is as I 1 above and clearly the quantity in the bracket is positive if η(s) is large enough, i.e. t * small enough. Finally

Q 4 (σ) := 1 2 αG v (ξ σ ) |y| 2 -2 G p (ξ σ ) G v (ξ σ ) |y| ,
is treated as I 2 above. And the proof is complete.

Examples and applications

Example 1 -We first consider the model equation (1.2) where, for m > 1

H(x, t, p) = |A(x, t)p| m -f (x, t) ,
the function f being bounded from below. In this case

(H p • p -H)(x, t, p) = (m -1)|A(x, t)p| m + f (x, t) = (m -1)H(x, t, p) + mf (x, t) .
We can assume without loss of generality that f (x, t) ≥ 1 (which can be obtained by changing u into u(x, t) + Ct), so that (H0) is satisfied with φ(s) = (m -1)s for s ≥ 0 and c 0 = 0. Moreover,

|H p (x, t, p)| = m|A(x, t)p| m-1 and if H(x, t, p) ≥ 0 then |A(x, t)p| ≥ f (x, t) 1/m ≥ 1. Therefore |H p (x, t, p)| ≤ m|A(x, t)p| m ≤ m m -1 (H p • p -H)(x, t, p) .
A direct computation shows that η(s) = (m -1)s -1 which yields the estimate u t ≥ -u/ (m -1)t .

Example 2 -Let us now adress a non-homogeneous situation where, again for m > 1

H(x, t, p) = |A(x, t)p| m -b(x, t) • Du -f (x, t) . (4.1)
Here, the function b is continuous and there exists C b > 0 such that for any (x, t), |b(x, t)| ≤ C b . We are also considering the coercive case where A is an invertible matrix and we assume that there exists C A > 0 such that for any (x, t), A(x, t) , A -1 (x, t) ≤ C A .

In this case, as for the homogeneous case we have

  Example 3 -In the non-coercive case for (4.1), i.e. if A is not invertible, the same computations for (H0) can be done if the Hamiltonian H has the formH(x, t, p) = |A(x, t)p| m -c(x, t) • A(x, t)Du -f (x, t) , where c : R N × [0, T ] → R N . This means that b has to be of the form b(x, t) = A t (x, t) • c(x, t).Notice that in this degenerate case, at least for large H, (H3) is automatically satisfied if A, c and f are Lipschitz continuous in x, uniformly in t.Example 4 -In the non-coercive case it is also interesting to examine the simple example whereH(x, t, p) := a(x, t)|p| m ,The proof of (G3) is similar:|G x (x, t, v, p)| ≤ |v||H x (x,t, -p/v)|. Then, using (H3), |v| ≤ 1, we get |G x (x, t, v, p)| ≤ γ p v + 1 |v|G v (x, t, v, p) ≤ γ(|p| + 1)G v (x, t, v, p) , which gives the result. Q.E.D.

(H p • p -H)(x, t, p) = (m -1)|A(x, t)p| m + f (x, t) .

By Young's inequality

|b(x, t) • p| = |A -t (x, t)b(x, t) • A(x, t)p| ≤ 1 m |A(x, t)p| m + (m -1) m |A -t (x, t)b(x, t)| m/(m-1)

where A -t is the inverse of the transpose matrix of A. Therefore, by our hypotheses on b and A there exists two constants

C 1 , C 2 > 0 such that (1 -1 m )|A(x, t)p| m -C 1 -f (x, t) ≤ H(x, t, p) ≤ (1 + 1 m )|A(x, t)p| m + C 2 -f (x, t) ,

and from there we deduce that (H0) is satisfied with φ(s) = m(m -1)/(m + 1) s -C for some constant C > 0 and for all s > 0.

where m > 1 and a : R N × [0, T ] → [0, +∞) is Lipschitz continuous in t. Taking into account that φ(s) = (m -1)s and ψ(s) = s/(2m) , assumption (H1) imposes |a t (x, t)| ≤ m -1 2m a(x, t) a(x, t)|p| m for a(x, t)|p| m ≥ c 1 , for some constant c 1 ≥ 0. Clearly this implies |a t (x, t)| ≤ c • a(x, t) for some c > 0, which implies some particular degeneracy: for any x either a(x, t) = 0 for any t or a(x, t) > 0 for any t. An -admittedly-not very natural hypothesis. Moreover if we want c 1 = 0 then necessarily the above constant c has to be 0 and therefore a cannot depend on t. (ii) We can consider other types of growths: if for instance H(x, t, p) = e |p| we have φ(s) = s ln(s -1), defined for s > 2 = c 0 . Hypotheses (H0)-(H1)-(H2) are satisfied with c 1 = 2 and in this case, η(s) ≫ s -1 as s → 0.

Remarks

(iii) If on most examples the function φ grows at least linearly, this is not always true. For instance, the case H(p) = |p| ln(1 + |p|) leads to H p • p -H ≃ |p| for |p| big, which leads to a function φ which is not superlinear, only asymptotically linear at infinity. Indeed, for any ǫ > 0, one can check that for σ > 0 big enough, σ ≤ φ(σ) ≤ σ 1-ǫ so that the function [σφ(σ)] -1 is still integrable near infinity. Hence, (H0) is satisfied and our results apply.

Appendix

Lemma 4.1. Assumptions (H0),(H1),(H3) imply that for any (x, t, v, p) such that G(x, t, v, p) ≥ c * ,

since φ is increasing. The second part of (G0) is obvious.

To get (G1) we start from (H1):

Then we notice that ψ satisfies the following sublinear property: for any λ ∈ [0, 1], λψ(H) ≤ ψ(λH). Indeed, since φ is increasin, for any λ ∈ [0, 1] and η > c * , (λ2η)φ(λ2η) ≤ λ(2ηφ(2η)). Taking the inverse yields the result. Thus, taking λ


Bruno Mery
email: bruno.mery@me.com

Richard Moot
email: richard.moot@labri.fr

Christian Retoré
email: christian.retore@lirmm.fr

Typed Hilbert Operators for the Lexical Semantics of Singular and Plural Determiner Phrases -Short Abstract

.

Overview

The original idea beneath the selection operators is Russell's Iota, which is ι x .F: the unique individual x such that F(x). It was later amended by von Heusinger: a contextually salient x such that F(x) (asserting unicity is notoriously problematic for logics that include negation). It is intuitively suitable for modelling the semantics of definite determiners such as the.

Hilbert provided two related operators. The first is Epsilon, in which ε x .F provides an existential quantification for x, which exists whenever F is satisfiable. This operator is suitable to model the semantics of indefinite determiners such as a. The second is Epsilon's dual Tau, written τ x .F, a form of universal quantification for x in F (it is true that F(τ x .F) when all possible individuals x satisfy F).

They denote the selected individuals : ∃x.F(x) is F(ε x .F), ∀x.F(x) gets written as F(τ x .F), the cat as ι x .λ x.(cat x) (and cat(ι x .λ x.(cat x)) is true).

The operators ε and τ being dual (ε x .F = τ x .¬F), ε is sufficient and can be used to (with standard application and reduction) to form a logical calculus. In its original formulation, the Epsilon calculus is a conservative extension to first-order logic, as the classical ∀, ∃ quantifications can be translated straightforwardly using ε, τ (sometimes yielding overly complicated formulae). However, some formulae of the Epsilon calculus are not equivalent to any first-order formula (such as P(ε x .Q)).

In [Egli and von Heusinger, 1995], von Heusinger uses ε and η to model determiners such as the, or a. He insists that there are no fundamental differences in semantics between the operators, η being used to introduce new referents. In this work, the quantification does not have the same strength as classical existential quantification (ε x .F is a term, but does not necessarily entail F(ε x .F)). He demonstrated the pertinence of Hilbert's operators for the semantics of natural language, as a possible alternative to choice (Skolem) functions (see also [von Heusinger, 2004]).

Types for Determiners in NTT

Lexical semantics imply a system of types with many sorts and second-order operators, in the view of so-called New Type Theories (NTT). There have been several efforts to model the theories introduced in [Pustejovsky, 1995], including [Asher, 2011] and our own ΛTY n (details in [Retoré, 2014]). As our goal is to implement the resulting system in an operational syntax/semantics parser (Grail, see [Moot, 2010]), we need to integrate many complex phenomena in a single, sound logical framework and keep computer tractability. We have thus been interested in variations of the Epsilon calculus for determiners. See [Retoré, 2014].

There are many different opinions, in a post-Montagovian, New Type Theory world, as to what constitutes a correct semantics for determiners. The full scope of those issues is far too broad for a single article, but they include co-variance and contra-variance in sub-typing (see e.g. [START_REF] Luo | Coercive subtyping: Theory and implementation[END_REF]). For those reasons, in [Luo, 2012], Zhaohui Luo's option is to have common noun as types. Then determiners should be typed with Πα.α → α : the constant representing a is the term Λαλ x α .x that, given a nominal type, yields an individual of that type (this amounts to a semantic identity, but can be interpreted in different ways).

We, however, adhere to the Montagovian view of common noun as predicates (modelling the property of being an individual of such denomination); this is much more convenient for quantificational purposes. Then determiners should be typed with Πα.(α → t) → α. The corresponding term would yield an individual (different according to the context and the interpretation of the specific determiner) that enjoys the property of the predicate passed as argument. This view is motivated by our approach of the granularity of the type system: we favour a system with many sorts corresponding to salient cognitive distinctions. Human should be a distinct type, Child could be a distinct type, but Broccoliliking-children should most probably not be a type, yet we need to be able to parse DPs such as the children who like broccoli. We argue that the easiest way to do so is to have common nouns be predicates over a finite (and relatively small) number of basic types, and that children who like broccoli is the predicate λ x H .children H→t (x) ∧ like broccoli H→t over the type H of human agents. We develop this typing for determiners in Section 3.

Note that, if need be, we can also model the alternate view with this typing by having α a predicate of type e → t for any type α. This fits neatly with von Heusinger's vision of Hilbert's operators.

Typing the Operators for Use in our System

We use ε, η mostly as von Heusinger does, using ε to signify "a (new) individual in the context of enunciation that satisfies the property" and to model the indefinite determiner a (as Hilbert's original notation and von Heusinger's η) and ε 1 to signify "the most salient/interesting existing individual in context that satisfies the property" and to model the "definite" determiner the (as von Heusinger's η, and Russel's ι minus the unicity condition). These operators have identical operational semantics, different interpretations in context ; we add an existential presupposition (F(ε x .F), F(ε 1

x .F) respectively) in every case, as speaking about something asserts its essence. Those presuppositions, similar to Asher's in [Asher, 2011] can reside with typing judgements and contextually salient entities and facts in a dedicated area that can be implemented via Cooper storage, Luo's signature structure, multi-dimensional analysis, or simply conjoined to the main formula -for the sake of simplicity, we will not include the resulting presuppositions in the typing of the operators, as it is pretty straightforward but dependent on the chosen paradigm. As before, τ is the dual of ε (and also entails F(ε x .F)).

Our principal contribution is to provide a typing in ΛTY n that enables the operators to be used in our framework, and to have a polysemous interpretation depending on context. That typing is ε : Πα.(α → t) → α, a single polymorphic operator (again, ε 1 has the same type and semantic term with a different interpretation). When applied to a predicate F(x) t = λ x a .(F x) for some type a, the operator is specialised for a.

(For the sake of simplicity of the resulting formulae, we will use the usual ∀, ∃ quantification for variable-binding along with ε, ε 1 ).

Groups and Sets: Using Hilbert's Operators with Predicate Variables

We model groups and massive entities using specific operators, with predicates denoting sets, and an operator | | giving the cardinality of the different individuals satisfying a predicate (we hold that utterances only predicate on a finite number of distinct individuals) . We have specific sorts for group individuals and mass individuals, and lexical transformations ensure that everything goes smoothly; see [START_REF] Mery | Computing the Semantics of Plurals and Massive Entities using Many-Sorted Types[END_REF] for details. A team is ε x .λ x g H .(team x). Someone in the team is ε x .λ x H .member of(x, (ε 1 y .λ y g H .(team y))). This follows Link's classical account of plurals and masses found in [START_REF] Link | The logical analysis of plurals and mass terms: A lattice-theoretic approach[END_REF], adapted for multiple sorts.

The "easy" way to model Some people in the team would simply to select all pertinent individuals and aggregate them together in a group. We have detailed a straightforward way to account for settheoretic union in [START_REF] Mery | Plurals: individuals and sets in a richly typed semantics[END_REF] based on Link's account, constructing a predicate representing the set of all individuals. However, as constructing a predicate to represent the set of individuals implies quantification over predicates, we could directly apply the ε operator to the target predicate and have much more control over the result. We can simply use ε P .λ P H→t .(R (H→t)→t P), with the Epsilon operator quantifying over the set of individuals, (represented as their characteristic predicate) and applied to the desired property. The full semantics of Some people in the team would then be ε P .λ P H→t .(|P| > 1 ∧ (∀x H .P(x) ⇒ member of(x, ε 1 y .(λ y g H .(team y)))))

Allowing ε and ε 1 to be specialised for any type gives us sufficient flexibility to quantify over predicates, relations and sets. As we already use quantification over predicates (and types) as features of our system (John and Mary is modelled as λ x H .(x = j) ∨ (x = m)), we should not refrain from doing so with determiner quantification.

Conclusion

We believe that typed, polymorphic Hilbert-derived operators are suitable tools to model the various polysemous usages of determiners. Those operators avoid the standard pitfalls of simplistic quantification such as symmetry (some A are B) and incomplete (nominal) sentences. We have shown that they can be adapted for recent frameworks including lexical semantics and contextual adaptation by the means of an higher-order typing system. The handling of their interpretation, and the correct usage of presuppositions and their inferences/implicatures remains to be investigated, for instance when combining quantification over a set of individuals and distinct anaphoric references to some of those individuals. We are confident that those issues can be resolved within our system.


Bruno Mery
email: bruno.mery@me.com

Lexical Semantics with Linear Types

We have proposed a framework based upon the λ -calculus with higher-order Intuitionist types for the symbolic computation of the semantic analysis, integrating lexical data. This proposal is sufficient for many phenomena and accurately incorporates lexical semantics by the means of type theory, but some issues linger in the linguistic data. In the present paper, we revisit this proposal with a version of the λ -calculus based upon higher-order linear types, that aims to resolve those issues and present an integrated framework for meaning assembly. 1

Introduction

In the computation of Natural Language Semantics for use in NLP applications, there are two main approaches being pursued : the symbolic, compositional approach that is derived from the Montagovian program (from [Montague, 1974]), in which the meaning of the utterance is derived from the combination of single meaningful postulates associated with each word (most often by the means of β -reduction in a version of the λ -calculus), and the numeric, statistical approach that use contextual information and rules acquired from corpora (using strategies similar to hidden Markov models). It is generally held that the former is closer to the structure of the sentence and able to process correctly complex issues of binding, anaphora, and quantification scope, and that the latter is more informative as it is able to derive precise senses for the words, disambiguating between different meanings that can be conveyed by the same lexical item. There are also many hybrid approaches that try to combine the two, but statistical methods and heuristics are prevalent in most of the real-world applications.

Another approach that has gained traction in the theoretical circles is the integration of complex lexical data and compositional rules in the symbolic analysis. The principle is that words are no longer atomic lexical items, but provide enough information in order to influence their context, and be influenced by other lexical items in return. The use of co-compositionality means that the assembly of the meaning of an utterance from its constituting lexical items is more complex than simple application, but, in many cases, the precise determination of the sense attached to polysemic words can be done in a straightforward, deterministic way. Statistical methods, and specifically semi-automated learning, are still useful, but are restricted to the construction of such a complex lexicon.

This alternative approach, pioneered by Nunberg and Sag (see eg. [Nunberg, 1993]) and Pustejovsky (in [Pustejovsky, 1991]), has come to be known as the Generative Lexicon Theory. Thoroughly developed by Pustejosvky in [Pustejovsky, 1995], criticised and enriched by many linguists since then, it has become a reference in linguistic theories. There have been many early attempts to use the principles of the generative lexicon in symbolic semantic analyses for NLP applications (such as 1 The present paper presents work in progress by the author, currently employed at a full-time teaching position in higher education, and thus still preliminary. This work relates to ΛTY n , a framework developed at LaBRI and LIRMM by many different people including Christian Bassac, Richard Moot and Christian Retoré from 2007 on. As a research team, we would like to thank Nicholas Asher, Daisuke Bekki, Stergios Chatzikyriakidis, Robin Cooper, Thomas Icard, Luo Zhaohui, Koji Mineshima and many others from long and helpful discussions and collaborations. We are indebted to Frédéric Lang, Anaïs Lefeuvre and Livi Real for working with us on this project. The author would like to extend his gratitude to the NLCS organising committee and the reviewers of the present publication for their kind and helpful comments and the opportunity to return to NLCS. [START_REF] Pustejovsky | The Metaphysics of Words in Context. Objectual attitudes[END_REF], [START_REF] Gupta | Interpreting Events Using Generative Sublanguage Ontologies[END_REF] and [START_REF] Pinkal | Feature Logic for Dotted Types: A Formalism for Complex Word Meanings[END_REF]), but the sheer complexity of the concepts involved prevented really satisfactory analyses for a long while.

More recently, there have been several proposals that use "New" Type Theories (logical frameworks of higher order, in the tradition of Curry) in order to accurately integrate lexical semantics (and some additional phenomena) in their analysis. They comprise the Type Theory with Records ( [Cooper, 2007]), Dependant Type Theory ( [START_REF] Bekki | Logical polysemy and subtyping[END_REF]), Type Presuppositions ( [Asher, 2011]), Unified Type Theory ( [START_REF] Luo | Coercive subtyping: Theory and implementation[END_REF]), and our very own λ -calculus with higher-order types and many sorts ( [Retoré, 2014]). All these proposals use a form of higher-order typing, binding or quantification in order to provide mechanisms for co-composition, sub-typing and type coercions.

Concerns of complexity and decidability have been raised over the use of higher-order logics. Partial implementations and illustrations of the constraints in the use of the expressive power of such systems have alleviated some of those; however, the formal characterisation of the fragment actually used remains a work in progress. Other problematic points include the coverage of such systems (the treatment of additional phenomena is detailed on a regular basis), and the use of features outside of the formal systems in order to patch in some specific idiosyncrasies.

The present paper aims at addressing some of these concerns by proposing an alternative to ΛTY n that uses a λ -calculus with higher-order Intuitionist linear types. We will demonstrate that this approach is adequate to model the variations in the felicity of co-predicative utterances, and does not imply the use of external features. It also provides a single, well-known formal setting for meaning assembly.

Our Type-Theoretical Framework for Lexical Semantics

We will first detail ΛTY n as a theory of meaning composition, in the tradition of Montague.

Principles of the Approach

In constructing ΛTY n , we aimed at producing a system that integrates as many features of lexical semantics as possible while remaining useable, i. e. sufficiently similar to recent re-interpretations of Montagovian semantics to be plugged into our text analysis workflow. Such a system should be sufficiently flexible to keep up to date with the evolution of linguistic studies, and to add support for various phenomena.

We thus adopted a compositional paradigm in which meaning is computed along the lines of the syntactic structure of the utterance. More precisely, composition in Montague-style semantics proceeds as follows : words are given syntactic types (according to syntactic and grammatical categories), and are assembled using a grammar-like formalism such as A/B grammars. This is the first analysis where typed terms are combined in order to form well-typed sentences that yield the syntactic structure. The "meaning" derived from this first step of calculus is that structure, generally depicted as a tree (or sometimes forest).

The second step is meaning assembly, wherein the semantics is calculated. The lexicon provides semantic types and terms associated to words (or lexemes), and the Curry-Howard isomorphism states that there is a correspondence between syntactic and semantic types, up to a point (and up to the integration of designs between the formalisms used). The syntactic structure is used to seed the basic layout of the semantics terms in a compositional way, i. e. every term is the argument or predicate of a term that corresponds to the words that shared the corresponding syntactic relationship in the actual utterance.

Meaning assembly consists of the computation, by means of function application, of the formulae given by that transcription. Semantic types are (propositional, Intuitionist) formulae, terms being proofs of those formulae, and the Curry-Howard isomorphism ensuring that everything follows smoothly. At the point of meaning assembly then, there are two notions of formulae : the propositional ones corresponding to the types of (individual or composed) terms, and the result of the combination of the terms themselves.

The latter formulae, when in normal closed form, as a result of β -reduction (or cut-elimination at the proof-theoretic level) can be interpreted as the meaning as the semantics analysis, usually in its firstorder predicate logic form. That representation can, in its turn, be interpreted in a model-theoretic way if needed.

The Montagovian analysis uses e and t as base types. In order to account for the many distinctions driving selection restrictions, we adopt a many-sorted logic, i. e. a logic with many base types in addition to t that are effectively used to represent different kinds of e (entities). Our principles are, as Pustejovsky suggested, that the types given to lexemes in the lexicon guide the compositional process, and that some creative use of words is used to create new meaning in event of a type clash.

Main Features

Work on this framework is continuing (see e. g. [START_REF] Bassac | Towards a Type-Theoretical Account of Lexical Semantics[END_REF], [Mery, 2011], [START_REF] Mery | Recent advances in the logical representation of lexical semantics[END_REF] and most recently [Retoré, 2014] and [Mery et al., 2015a]).

As it stands now, the logic used is the second order Intuitionist logic based on the logic of types with n sorts described in [Muskens, 1996]. The"second-order" aspect symbolised by the Λ represents the ability to use type-level abstraction, application and quantification, and is based on Jean-Yves Girard's System-F (see [Girard, 1972]). Our calculus for meaning assembly is the λ -calculus with Intuitionist, second-order types of many sorts, while our final logic used for representations is the many-sorted higher-order predicate calculus.

The difference with basic semantic analysis that encompasses the main contribution of lexical semantics to the construction of meaning can be summarised as follows: to lexemes are associated as the meaning-assembly start not a single typed term, but a collection of such: a single "main" term with associated type, and zero to finitely many additional ones. We use these additional terms in case of a type mismatch to provide transformations to the terms in the meaning assembly phase, and the resulting logical representation incorporates those transformations as modifiers to the meaning of the terms.

More specifically, the transformations are used to resolve type mismatch situations such as the following:

(1) The book is heavy.

(2) The book is interesting.

No matter what the type for book is, heavy is a predicate restricted to arguments that are physical objects while interesting is restricted to information (or facts). Lexemes such as those that have multiple facets are called dot objects in the Pustejovskian tradition, and have provided some of the most difficult phenomena to model in a logically sound way. We simply have a type Book, a type ϕ and a type In f o that are three different sorts, and the lexical entry for book provides, in addition to the expected main λ -term, the following two additional ones: f ϕ Book→ϕ , that turns terms of type Book into terms of type ϕ, and f Info Book→Info , that turns terms of type Book into terms of type In f o.

In the above examples, composition proceeds as normal until a type clash occurs. In example (1), it occurs between (the book) Book and is heavy ϕ→t . However, the optional term f ϕ , made available by the lexical entry for book, can be used to resolve the type mismatch in the following way : (is heavy ( f ϕ (the book))), which has a felicitous typing and yields semantics consistant with our expectations2 .

This very simple mechanism allows us to take into account a number of different linguistic phenomena.

The main difference between our approach and those of Asher, Bekki, Luo. . . is that the typechanges transformations that are called coercions in most of the literature are, almost always, given as relations between types. Our transformations are optional terms given in the lexicon, and thus are introduced by individual words. This is a system guided by types, but enabled by terms at the lexical level, which allows us to model specific idiosyncrasies and hypostasis, among other advantages.

However, a case where type-driven systematic coercions is useful is the derivation of ontological relations such as hyperonymy. Consider the following:

(3) The Honda moved.

(4) The car moved.

(5) The vehicle moved.

(6) The object moved.

(7) The thing moved.

Clearly, the meaning of ( 7) is implied in (6), while ( 5) is more precise than (6), etc. Lexical semantics is often based upon an ontological hierarchy of types, a tree, forest or lattice consisting of sub-typing relations. (There is some disagreement between different type-theoretical views here, depending on the typing of the argument and of the instantiation strategy used in the semantics of the determiner; see further on.)

Pustejovsky calls examples (3-7) instances of type accomodation: Honda is a subtype of Car which is a subtype of Vehicle which is a subtype of ϕ which is a subtype of e, and any subtype may be used anywhere the original type can be used (again, with subtle issues depending on determiner and DP semantics). This is formalised as a coercive mechanism, as rules at the logical level, in [START_REF] Soloviev | Coercion completion and conservativity in coercive subtyping[END_REF] and [START_REF] Luo | Coercive subtyping: Theory and implementation[END_REF]. Briefly, having a type A subtype of B is written A c B and means that there is some coercion c A→B turning objects of type A into objects of type B. As detailed in [Retoré, 2014], we can use the same mechanism to introduce type-driven coercions for ontological sub-typing, while keeping our term-driven framework for all other transformations.

A specific issue being disputed in recent type-theoretical developments is that of the typing of common nouns. In the classical Montagovian analysis, nouns are predicates, i. e. typed as cat e→t . Nouns represent the property of being an individual represented by such a predicate, and determiners select the appropriate individual, with a typing such as a (e→t)→t . The combination is a determiner phrase of type e.

Luo, among others, departs from this view by having common nouns typed with atomic types. Thus he has cat Animal and a α→α for any sort α; the common noun represents a generic individual, which is determined by the determiner.

We argue that having nouns typed as predicates is very useful for determination and quantification purposes, but that having access to the individual is also necessary in other cases (adjectives, adverbs. . . ). We use operators derived from Hilbert's ε for that purpose. We associate a characteristic predicate to all sorts, α e→t where α is the sort, and axiomatise for all such predicates that α(ι( α)). In the end, determiner phrases are typed by a single sort. We have presented a complete account of this system founded on type-theoretic principles in [START_REF] Mery | Typed Hilbert Operators for the Lexical Semantics of Singular and Plural Determiner Phrases[END_REF].

Compositionality

The compositional approach to formal semantics, in agreement with cognitive evidence, advocates that the meaning of the discourse is derived from the composition of the meaning of its parts. Crucially, at each composition step from the elementary lexical term (lexeme) to the final formula in normal form (logical representation of the utterance / discourse), each term can still be interpreted and convey constructive meaning. While this is transparent in classical Montague semantics, it is also the case with our additional mechanisms. Here, we will only present an intuitive interpretation of the denotation of adaptative predications.

An intransitive verb such as to fall is modelled as a predicate over a sort (physical objects) that can be adapted to other argument types: Λαλ x α f α→ϕ .(P ( f x)). The meaning associated to P is that of asserting that x, if it can be coerced into a physical object by some f , enjoys P. Such predicates always provide the identity morphism for the appropriate type, here id ϖ = λ x ϕ .x, and so the "usual" meaning of the verb can be inferred as λ x ϕ .(P x). In The book fell, the argument will provide the correct transformation in order to infer correctly that the verb applies to the physical aspect of the book. Thus, compositionality and intermediate meanings are preserved through the process of meaning assembly.

This approach is also compatible with accounts of semantics that are centred on context, as with dynamic or continuation semantics. The "slots" provided for transformations allow the context to alter the felicity of predications such as * The idea fell, depending if some morphism of type Abstract → ϕ or similar has been introduced in the discourse.

However, the above mechanisms are not sufficient in order to treat all phenomena, as issues with co-predicative utterances illustrate.

Co-Predicative Utterances

Co-predicative utterances, or co-predications for short, are phrases in which a polysemous word is used for more that one of its meanings at once. They are among the more difficult problems treated with the help of lexical semantics, and have been used as a test case for such theories. All these sentences contain a co-predication: the book in ( 8) is used by a predicate restricted to physical and another restricted to informative objects, London in (9) has its facets of a group of people and a place with geographical limits being used, while the salmon in ( 10) is used infelicitously at the same time as an animal and cooked food.

The Problem of Co-Predication

In classical Montagovian analysis, there are no ways to distinguish co-predications. Everything being typed as e, there are no selection restrictions. Even when a separate mechanism is used for selection restriction, when the individual predications are fine, their conjunction is also.

In a many-sorted framework, without a specific attention to co-predications, every co-predication is infelicitous, as the two predicates use different types as their arguments.

The difficulty lies in producing a mechanism that allows (8-9), but is able to block (10).

Co-Predications in our Current System

We treat the co-predication issue using a specific operator for conjunction, the polymorphic "and" defined as follows:

Λαβ τλ P α→t Q β →t x τ f τ→α g τ→β .(∧ t→t→t (P ( f x)) (Q (g x)))
This operator allows the conjunction of predicates bearing on different sorts to the same argument, if there are transformations that can accommodate both predicates.

Example ( 8) is thus computed as AND is heavy ϕ→t is interesting Info→t (the book) Book f ϕ Book→ϕ f Info Book→Info which reduces to

(∧ t→t→t (is heavy ( f ϕ (the book))) (is interesting ( f Info (the book))))
In addition, in order to deal with incompatible co-predications, we have a mechanism in the lexicon that attaches to every transformation a label (either Flexible or Rigid). Rigid transformations are destructive transformations, such as the ones in (10) that turns a salmon into food through the process of cooking. A simple set of rules ensures that such examples are not computed. However, this is rather unsatisfactory, as those features and rules are outside of the scope of the logical system itself.

Outstanding Issues

This framework is sufficient to deal with most co-predication issues, but not all. First, the use of an external system to reject infelicitous co-predications is unfortunate and inelegant.

More importantly, it also fails to predict felicity in many cases. Sentences such as (10) are clearly infelicitous, but can be rewritten as:

(11) The salmon was lightning-fast. It is delicious.

Thus, it appears that rather simple syntactic means (here, anaphoric reference) can alter the felicity of co-predications. There is also the example of city names, apparent in utterances such as:

(12) London is a huge city. City names have an extremely high degree of polysemy, and can refer to a number of different things depending on context. Co-predications between any two of those different meanings have a felicity which is variable in function of many parameters. A more complete account of the possible combinations is forthcoming (and outside the scope of this article), but this illustrates the difficulty of characterising coherent behaviours for co-predications.

Moreover, those examples necessarily rely upon the subjective appreciation of the authors and the limited data available. They are contestable, and might be idiosyncratic (personal communications from Asher and Luo, as well as anecdotical evidence, suggest significant differences in behaviour between English and Chinese). While a thorough empirical study of those issues, including a survey of native speakers of several languages, is worthy to conduct, we do not need to wait such in order to complete a framework that can provide such behaviours, while being adaptive enough to integrate updated linguistic data.

4 Linear Second-Order Types

Linear Logic

Linear Logic is an alternative to Classic and Intuitionist Logics developed by Jean-Yves Girard in [Girard, 1987], and detailed in [Girard, 1995]. It has garnered much attention in the field of formal logic for its simplicity, its geometrical interpretation and its relevance in category theory.

It differs from other logics in that hypothesises are used exactly once in order to form the conclusion. Where, in Classical Logic, from a proof of A and one of A → B, one can deduce a proof of B and still be able to use the hypothesises for the rest of the deduction, in Linear Logic, if from a proof of A and one of A B, one deduces a proof of B, neither A nor A B can be used afterwards. The possibility of duplicating some elements is constrained to exponent operators such as !, and thus non-linearity is clearly marked in the logic.

In computer science, Linear Logic is interesting because of the intuitive resource interpretation associated: rather than being abstract propositions that have some truth value in an eidetic space, the base elements of the logic can be thought of as resources. An atom A is a single instance of a resource. The linear implication A B represent the ability to convert an instance of A into an instance of B, consuming both the resource and the converter in the process. Abundance of a resource is noted !A, etc. This is obviously well-suited to many real-world applications.

We will not delve into much detail for the syntax and semantics of Linear Logic (see [Braüner, 1996] for an introduction), but will consider Intuitionist Linear Logic, a fragment thereof that employs the following connectives:

• : linear implication (A B consumes an A and yields a B).

• ⊗: multiplicative conjunction (A ⊗ B means that both A and B have to be used).

• &: additive conjucntion (A&B means that either A or B can be used, the formula that consumes this can "choose" which).

• !: exponential (!A means as many and as little instances of A as needed).

Computational Interpretation

Although Linear Logic is often used for its geometrical or proof-net interpretations, it is possible to define computational interpretations for it. The seminal work of Samson Abramsky ( [Abramsky, 1993]) has illustrated this possibility, detailing terms, operational semantics and evaluation strategies for a computational interpretation of Intuitionist, second-order Intuitionist and classical Linear Logic.

Specifically, it is possible to define a version of λ -calculus where the terms are typed with formulae from Linear Logic rather that Intuitionist Logic.

Standard λ -calculus is the computational interpretation of the Curry-Howard Correspondence for formulae of Intuitionist Logic. Abramsky introduced such a calculus for Linear Logic (and several of its fragments), which was later studied and refined in e. g. [START_REF] Benton | A term calculus for intuitionistic linear logic[END_REF]. The term calculus in itself is quite similar to standard λ -calculus, with abstraction and application unchanged. It presents a direct implementation process for linear functional programming.

Our Proposal for Linear Second-Order Types

The diverse possibilities of compatibilities for co-predicative utterances, explored above in section 3.3, has led us to consider using the improved power provided by Linear Logic, rather than a simple fragment of Intuitionist logic for types.

To be more precise, we want to propose a system based on Linear Second-Order ΛTY n for meaning assembly, i. e. a version of λ -calculus that uses linear, Intuitionist, second-order, many-sorted types. Meaning assembly proceeds as it always has but constrained by linear types, and the features specific to linear logic will disappear after composition so that the (normal, closed) resulting term can be transcribed as a formula in regular predicate logic (with Intuitionist types).

Linear Logic is well-suited to the meaning assembly phase. In particular, the intuitive resource interpretation fits well with the imagined process of composition, where the meaning of arguments and predicates are consumed to form the semantics of the complex predication as a whole. In fact, simple composition for simple predications are easily transcribed using linear types in the usual fashion: With to bark (non-transitive version) being a predicate selecting arguments of type Dog, the dog a determiner phrase of that type, the chair of type ϕ and the sergeant of type Human, the terms for every sentence are:

• (16): Λτλ x τ f τ Dog .(bark Dog t ( f x)) the dog Dog
This reduces to (bark (id Dog Dog Dog the dog)), or more simply (bark the dog).

• (17): Λτλ x τ f τ Dog .(bark Dog t ( f x)) the chair ϕ This has no correct typing, assuming (as in a normal context) that there is no way to turn (generally inanimate) physical objects into dogs. (Dogs can be envisioned as physical objects, not the other way around.)

• (18): Λτλ x τ f τ Dog .(bark Dog t ( f x)) the sergeant Human To reduce this, there has to be a way to consider humans as dogs. In this specific case, the lexical entry for sergeant has such a term available, the term f Order Human Dog that implies that the delivery of a specific kind of speech has a certain dog-like quality. We thus have (bark ( f Order the sergeant)) as a felicitous result. Thus, simple predicates, transformations and coercions all share a type being a single direct linear implication.

In fact, non-linear composition only occurs in cases of co-predication and, in general, of conjunction.

Our polymorphic conjunction has to duplicate the argument. A first step is simply this:

Λαβ τλ P α t Q β t x !τ f τ α g τ β .(copy x as y, z in (∧ t t→t (P ( f y)) (Q (g z))))
This is the original notation proposed by Abramsky. For convenience's sake, this can be abbreviated as:

Λαβ τλ 0 P α t Q β t f τ α g τ β λ x τ .(∧ t t→t (P ( f x)) (Q (g x)))
In this notation, λ 0 introduces linear terms and λ Intuitionist non-linear terms that can be duplicated in the formula.

We therefore use non-linear terms (implemented by !-exponentiation) for the specific purpose of duplicating the argument in conjuncts.

A Proper Treatment of Co-Predications and Constraints

Recall that our goal in proposing an implementation with linear types was to give an account of the compatibilities, or lack thereof, of transformations used in co-predications. For this, we refine the typing of transformations as follows:

f : (τ α)&((τ δ ) t)
The linear operator &, read "with", is the additive conjunction. This means that a transformation f can be viewed as a pair < π 1 f , π 2 f >. (π 1 f ), of type τ α, is the transformation we have been using up to now, turning an argument of a type τ into another of type α which is needed by some predicate. The second member of the pair, (π 2 f ) of type ((τ δ ) t), is a compatibility test for other transformations. In a nutshell, this will be applied to other transformations to check type compatibilities (there have to be transformations available with τ as their source and δ , or a subtype thereof, as their target) and to accept or reject specific transformations (the term itself returns a value of type t that might be true or false).

We modify our polymorphic conjunction as follows:

Λαβ δ γτλ 0 P α t Q β t λ x τ f (τ α)&((τ δ ) t) g (τ β &((τ γ) t) .(∧ t t t t (P ( f x)) (Q (g x)) (comp f g))
(Note that, in order to be tested for compatibility, f and g have to be duplicated and are no longer labelled as "linear".)

The comp operator is simply the following:

comp = def Λαβ γδ τλ f τ α&((τ δ ) t) g τ β &((τ γ) t) .(∧ t t t ((π 2 f )(π 1 g)) ((π 2 g)(π 1 f )))
Or, abbreviating a bit and omitting typing and projections, (∧ ( f g) (g f )). Recall that the linear & reflects a pair of objects, only one of which will be ultimately used; in transformations, the first part (the transformative term) is used for adapting the argument and as argument for compatibility tests, the second (the compatibility test) is used only when it has a transformation as its argument. The typing ensures that there are no other possibilities.

Using linear mechanisms for compositional semantics, as straightforward as it is from a logical and formal point of view, requires a change of perspective in the interpretation of semantical compositionality. As it is apparent that the entities involved in linear applications do not disappear from the mind of the speaker as meaning is constructed, the "meaning assembly" phase presented here should be integrated with additional mechanisms. We always suppose that the context of each utterance is provided in a mechanism similar to Luo's Signatures, Asher's presuppositions, or as Cooper storage or in continuation-style semantics. We have not elaborated on this aspect here, but providing an account in the style of, e. g. Dynamic Semantics is straightforward and require the manipulation and output of a provided complex object recording the context. With linear application, this becomes necessary in order to keep track of the consumed terms.

There are thus two different mechanisms at work in the "meaning assembly" phase:

1. A representation of the entities, relations and facts inferred by the discourse, updated incrementally;

2. A mechanism that derive the predicated facts, from instances of the entities concerned, that are consumed as the composition proceeds.

In each step of this composition, fresh copies of the entities involved are created and combined to provide a set of new entities and facts that are integrated to the context. Simple predication asserts a fact, the use of transformations produce different entities that can be referred to later in the discourse, and copredications duplicate the entity involved in order to assert a number of facts on potentially different facets of the entity.

Reasons of the Upgrade

Linear ΛTY n is a step forward from ΛTY n in several ways. It allows for greater flexibility in the definition of constraints on co-predicative utterances, covering different possible phenomena with a word-level distinction. It is based on a more constrained logic, linear Intuitionist logic second-order logic, which is an interesting fragment in itself (with many operations that can be resolved by lazy evaluation), with constraints on the higher-order quantification; we are much closer to the probable complexity of human compositional semantics than with full System-F. More importantly, this implementation dispenses with external constraints that are stapled to System-F with an ad hoc verification mechanism. The constraints of compatibility between transformations is now integrated to the system, and each transformation now has these constraints as a part of its implied meaning. The result is much more satisfying than ΛTY n , even if it is still far from complete.

Summary

The computation of semantics is a long, segmented process, starting from a syntactic analysis to infer the rough form of a Chomskyan deep structure, integrating terms derived from the lexicon and composing them in the meaning assembly phase, writing the resulting term as a logical representation and interpreting it as a truth-valued statement, a model-checker, a set of possible worlds or a sequence of judgements on entities and facts, before combining any interpretation with others to form a representation of the discourse. The study of lexical semantics tends to add separate mechanisms that allow discourse-level and semantic-level data to influence the lexicon (and reciprocally), resulting in an even more complex picture.

The strength of this proposal is thus to keep things in a single framework for meaning assembly, an usual λ -calculus-based process using terms typed with formulae of second-order Intuitionist linear logic in a many-sorted type set, and to express constraints on the use of terms internally through the typing of the terms. Moreover, our system makes use of a restricted fragment of the wider second-order linear logic, and is a straightforward extension of existing Montagovian frameworks.

Linear ΛTY n should be completed with resolution rules for type mismatches that better embed coercive sub-typing and its formal properties (publication pending, by Frederic Lang). In addition, there are several additional structural constructs to be specified: Determiner Phrases and quantification using Russell/Hilbert operators (work in progress), as well as adjuncts such as adjectives and adverbs. A proper characterisation of sorts, lexical transformation and wide-coverage lexica should also be undertaken (via semi-automated acquisition from corpora).

A capable system would also need extended linguistic investigations. While automated or semiautomated acquisition is possible to build a starting lexicon, specific studies would allow us to define more precisely some aspects of the system, specifically the constraints on transformations used in copredicative utterances. It would also allow us to refine term-specific constraints and transformations according to language-and dialect-specific idiosyncrasies.

For now, we propose this simple implementation of lexical semantics, including the treatment of constraints on co-predications, using the λ -calculus with second-order Intuitionist linear types. We are confident in its ability to cover target phenomena and prove flexible enough to account for idiosyncrasies, dialects and creative uses.

  Consider the following sentences (classical examples used by Asher, Pustejovsky and us in the past): (8) The book is heavy and interesting. (9) London is a lively, sprawling city.(10) (?) The salmon is lightning-fast and delicious.

( 13 )

 13 London has castigated Moscow. (14) (?) London is a huge city. They have castigated Moscow. (15) (*) London is a huge, Moscow-castigating city.

( 16 )

 16 The dog barked. (17) (*) The chair barked. (18) The sergeant barked.

The specific question of quantification and the interpretation of determiners is not detailed here; we have adapted selection operators from Russell and Hilbert. See[Retoré, 2014].