J Pellegrino

R Bourgain

S Jennewein

Y R P Sortais

A Browaeys

S D Jenkins

J Ruostekoski

Observation of suppression of light scattering induced by dipole-dipole interactions in a cold atomic ensemble

Keywords: numbers: 42.50.Ct, 42.50.Nn, 42.25.Fx, 32.80.Qk, 03.65.Nk

We study the emergence of collective scattering in the presence of dipole-dipole interactions when we illuminate a cold cloud of rubidium atoms with a near-resonant and weak intensity laser. The size of the atomic sample is comparable to the wavelength of light. When we gradually increase the atom number from 1 to ∼ 450, we observe a broadening of the line, a small red shift and, consistently with these, a strong suppression of the scattered light with respect to the noninteracting atom case. We compare our data to numerical simulations of the optical response, which include the internal level structure of the atoms.

When resonant emitters, such as atoms, molecules, quantum dots, or meta-material circuits, with a transition at a wavelength λ, are confined inside a volume smaller than λ 3 , they are coupled via strong dipole-dipole interactions. In this situation, the response of the ensemble to near-resonant light is collective and originates from the excitation of collective eigenstates of the system, such as super-and sub-radiant modes [1][2][3]. Dipole-dipole interactions affect the response of the system and the collective scattering of near-resonant light differs from the case of an assembly of noninteracting emitters [4]. It has even been predicted to be suppressed for a dense gas of cold two-level atoms [5].

Following the recent measurement of the collective Lamb shift [6] in a Fe layer [7], in a hot thermal vapor [8] and in arrays of trapped ions [9], it was pointed out [10] that the collective response of interacting emitters is different between ensembles exhibiting inhomogeneous broadening, such as solid state systems or thermal vapors, and those free of it, such as cold atomic clouds. In particular, inhomogeneous broadening suppresses the correlations induced by the interactions between dipoles, leading to the textbook theory of the optical response of continuous media [10,[START_REF] Born | Principle of Optics[END_REF]. In the absence of broadening, however, this theory fails and should be revisited to include the light-induced correlations [START_REF] Morice | [END_REF][13][14][15][16][17][18][19]. Several recent experiments aiming at studying collective scattering with identical emitters used large and optically thick ensembles of cold atoms [20][21][22][23]. However, the case of a cold atomic ensemble with a size comparable to the optical wavelength has not been studied experimentally, nor has the transition between the well-understood case of scattering by an individual atom [24] to collective scattering. In particular, the suppression of light scattering when the number of atoms increases in a regime of collective scattering has never been directly observed.

Here, we study -both experimentally and theoretically -the emergence of collective effects in the optical response of a cold atomic sample due to dipole-dipole interactions, as we gradually increase the number of atoms. To do so, we send low-intensity near-resonant laser light onto a cloud containing from 1 to ∼ 450 cold 87 Rb atoms, with a size comparable to the wavelength of the optical transition at λ = 780 nm. Starting from one atom, we observe a broadening of the line as the number of atoms increases, as well as a small red shift and a strong suppression of the amount of scattered light with respect to the case of noninteracting atoms. We show that this suppression is consistent with the measured broadening and shift. We finally compare our measurements to a numerical simulation of the response of the system in the low excitation limit, including the internal level structure of the atoms.

The suppression of light scattering by resonant dipoledipole interactions can be understood qualitatively as follows. Consider a laser radiation with frequency ω impinging on an ensemble of classical radiating dipoles with resonance frequency ω 0 = 2πc/λ (see Fig. 1). When the dipoles interact through the dipole-dipole potential

V lβ jα = -V dd p αβ (ikr -1) + q αβ (kr) 2 e ikr , (1)
the system features collective modes with various eigenfrequencies and decay rates. Here, j and l denote two dipoles separated by a distance r, V dd = 3Γ/4(kr) 3 , k = 2π/λ, Γ is the radiative decay rate in the absence of interactions, and the angular functions p αβ and q αβ depend on the polarizations α and β and the relative orientations of the dipoles j and l [25] [START_REF]Here p αβ = δ αβ -3(ê * the interatomic separation and êσ the unit polarization vector[END_REF]. In our experiment, the geometry of the atomic system and its orientation with respect to the excitation are such that the incident laser best couples to only a few modes with decay rates Γ c larger than Γ (super-radiant modes), leading to a broader excitation spectrum. The excitation rate, and therefore the amount of scattered light, should thus be reduced by a factor (Γ/Γ c ) 2 . The effect is stronger when the average distance between dipoles r is smaller than λ/2π.

To study the collective scattering by an ensemble of atoms coupled via resonant dipole-dipole interactions we use the setup depicted in Fig. 1(a). We prepare small clouds containing up to 450 atoms at a temperature ∼ 100 µK, confined in a microscopic dipole trap [START_REF] Bourgain | [END_REF], and illuminate them with laser light nearly resonant with the atomic transition at λ = 780 nm. The Doppler width of the sample (150 kHz) is much smaller than the atomic linewidth Γ/2π = 6 MHz, making inhomogeneous broadening negligible. The anisotropy of the trap results in an elongated cloud with calculated root-mean-square thermal sizes σ ρ = 0.3λ and σ z = 2.4λ. The maximal density is ρ = 2.5 × 10 14 at/cm 3 and the minimal average interatomic distance r = ρ -1 3 = 0.2λ (Fig. 1b). In this regime, k r ∼ 1, leading to V dd ∼ Γ, and the resonant dipole-dipole interaction will therefore have an effect on the scattering.

Experimentally, we prepare the trapped atoms in the F = 2 hyperfine manifold with an efficiency better than 95%. We then release them in free space by switching off the trapping light while exciting them with σ + polarized light at a frequency ω = ω 0 + ∆ tuned near the (5S 1/2 , F = 2) to (5P 3/2 , F ′ = 3) transition (see Fig. 1).

In this way we avoid extra light-shifts induced by the trapping beam that would obscure the measurement of small collective shifts and broadening. Also, we choose the intensity saturation I/I sat = 0.1 to be in the low excitation limit (I sat = 1.6 mW/cm 2 ). We interleave excitation pulses with duration 125 ns and recapture periods in the dipole trap with duration 1 µs. This sequence is repeated 200 times using the same cloud of atoms, in order to improve the duty cycle of the experiment. Finally, we prepare a new atomic sample and repeat the set of excitation pulses a few hundred times. The scattered light that we collect in the z direction is therefore the result of an average over many spatial configurations of the atoms. The choice of the number of pulses ( 200) is a trade-off between getting a good signal-to-noise ratio and avoiding light-assisted losses [27] or heating of the cloud, both of which would lower the density. We checked that both effects do not exceed 5% over the entire set of pulses and that less than 5% of the atoms are depumped in the (5S 1/2 , F = 1) hyperfine level during the excitation.

Figure 2 shows the number of photons n z (N, ∆) detected by the I-CCD as a function of the detuning ∆ of the excitation laser, for various atom numbers N . A Lorentzian fit agrees well with the data for the range of atom numbers explored here. As expected from the qualitative argument described above, we observe that the full-width-at-half maximum (FWHM) increases with the number of atoms (see Fig. 3a), since the interatomic distance then decreases, leading to stronger dipole-dipole interactions. We also measure a small red shift δω of the center frequency (Fig. 3b). For N = 1 atom, the FWHM is 1.35 ± 0.15Γ, in agreement with the short duration of the excitation pulses (125 ns), which broadens slightly the resonance. Figure 2 also shows that the amount of light scattered in the z direction at resonance does not increase linearly with the number of atoms as one would expect for noninteracting atoms, but actually increases more slowly. Fig. 4(a) indicates that this is also the case off resonance, where we plot n z (N, ∆)/n z (N = 1, ∆) for different atom numbers and detunings. For noninteracting atoms this ratio is equal to the number of atoms N (and is thus independent of the detuning ∆), as we verified by collecting the scattered light after letting the atomic cloud expand in free space for a sufficiently long time [28]. By contrast, here we observe that the amount of scattered light is strongly suppressed on resonance as the number of atoms increases, and that we gradually recover the behavior of noninteracting atoms as we detune the laser away from resonance.

All the observations reported above can be reproduced by a single functional form:

n z (N, ∆) = C N Γ c (N ) 2 + 4[∆ -δω c (N )] 2 , (2)
where C includes the detection efficiency of the imaging system. This is illustrated in Fig. 4(b): we find that the quantity

R(N, ∆)/R(N = 1, ∆), where R(N, ∆) = n z (N, ∆) × [Γ 2 c + 4(∆ -δω c ) 2
] and Γ c and δω c are respectively the phenomenological fits of FWHM and the shift (see Fig. 3), collapses on a single curve whatever the detuning. For N 300, this curve is linear with N with a slope of 1, in agreement with Eq. ( 2). It emphasizes that in this regime, the scattered intensity is suppressed by a factor (Γ/Γ c ) 2 at resonance, as expected from the qualitative discussion earlier. We note that this scaling cannot be explained by a model where the suppression would come from an incoherent superposition of the intensities scattered by each atom with resonant frequencies inhomogeneously distributed over a distribution of width FWHM: that would lead to a suppression that would scale as Γ/Γ c near resonance, instead of the (Γ/Γ c ) 2 scaling observed here. For N > 300, the departure from the linear law indicates that C depends on the atom number in this regime, and that the simple Lorentzian form (2) becomes inaccurate, as also found in the simulation (see below).

We have performed numerical simulations of the collective dynamic response of the atomic sample to nearresonant pulsed light in the low excitation limit. In this model, each atom, located at position r j (j = 1, . . . , N ) and with dipole d j , is driven by the incident laser field and by the fields scattered by all the N -1 other atoms, i.e. each dipole is coupled to the N -1 other dipoles via the resonant interaction of Eq. ( 1). This classical electrodynamics simulation incorporates all the interactions between an ensemble of non saturated discrete dipoles. This approach has been used to study dielectric media comprising two-level or spatially averaged isotropic electric dipoles [10,14,16,18,29,30] as well as magneto- dielectric circuit resonator systems [31]. Here, we also incorporate the Zeeman level structure of the atoms [13] and the shifts associated to the presence of the magnetic field. To calculate the dipoles d j in our experimental con-figuration, we stochastically sample the positions of the atoms according to a 3-dimensional Gaussian density distribution with root-mean-square sizes given by the thermal sizes of the cloud along and perpendicular to the trap propagation axis; each atomic position is treated as an independent and identically distributed random variable. At each realization the N atoms are fixed at positions r j (j = 1, . . . , N ) and we stochastically sample the magnetic quantum number of the Zeeman states m j of each atom j. The probability of atom j being in state |g, m (m = ±2, ±1, 0) is the initial population of that Zeeman state p m (0 < p m < 1; m p m = 1). The optical pumping used in the preparation step before the excitation sequence, skews the initial populations; here we use the values p 0 = p 1 = p 2 = 1/3 and p -1 = p -2 = 0. We write the positive frequency component of the dipole produced by each atom j that oscillates at the laser frequency as d j = D σ êσ C (σ) mj P jσ , where the sum runs over the unit spherical polarization vectors σ = ±1, 0. The amplitude of the atomic dipole j associated to the optical transition |g, m j → |e, m j + σ is proportional to the reduced dipole matrix element D, the atomic coherence P jσ , and the corresponding Clebsch-Gordan coefficient C (σ) mj . The temporal evolution of the coherences is given by the set of coupled equations

Ṗjα -i (∆ jα + iΓ/2) P jα = -iΩ jα (t) -i l =j β C (β) m l C (α) mj V lβ jα (r) P lβ , (3)
where Ω jα (t) and ∆ jα = ω -ω jα are respectively the time-dependent Rabi frequency and the detuning of the driving laser with respect to the Zeeman shifted transition of the α-polarized atom j with frequency ω jα , and β = ±1, 0. Here, we deduce Ω jα (t) from the experimentally measured temporal profile of the excitation pulse. The last term in Eq. ( 3) couples the α-polarized dipole j to the β-polarized dipole l separated by r = r jr l according to Eq. ( 1). We have solved Eqs. (3) numerically in the presence of a 1G magnetic field to calculate the light field amplitude that is scattered into the solid angle encompassed by the aspherical lens in the far field. Finally, accounting for the polarization-sensitive detection scheme, we calculated the measured light intensity. The simulation predicts that the spectra n z (N, ∆) should present an increasing broadening and asymmetry (Fig. 5), a negligible shift (Fig. 3b), as well as a suppression of the scattered light (Fig. 4a) when the number of atoms increases. These features are in good agreement with our data for N 50. In this range, the simulated spectra are well fitted by a Lorentzian for N 50, thus justifying our fitting of the data by Eq. ( 2) and the collapse of the data shown in Fig. 4(b). For N 50, the agreement is only qualitative, as the effects are found to be less pronounced experimentally. We attribute these discrepancies to two possible reasons. Firstly, forces in- duced by the dipole-dipole interactions may expel atom pairs with shortest inter-atomic distances, thus breaking down the assumption that atoms have frozen positions during the sequence of pulses excitations. This is all the more likely as the number of atoms is large, and could explain the evolution of the FWHM in Fig. 3(a). This effect is hard to check experimentally since the sample is smaller than the diffraction limit of our imaging system. We found numerically, however, that an increase by a factor 2 in the widths σ ρ and σ z already restores a nearly Lorentzian profile close to the measured spectra (see Fig. 5), and yields the observed suppression of light scattering (see Fig. 4a). Secondly, the simulation predicts that the number of detected photons increases by a factor 2 when the initial distribution of Zeeman state populations varies from p 0 = p 1 = p 2 = 1/3 to p 2 = 1.

For large atom numbers, optical pumping during the set of excitation pulses may change the distribution of populations, an effect not accounted for in our model.

In conclusion, we have directly measured the suppression of light scattering induced by dipole-dipole interactions in an ensemble of cold atoms driven by a nearresonant weak laser field and compared it with a timedependent model of coupled dipoles. The model reproduces the observed trends. In the future, we plan to investigate to what extent the observed collective scattering involves beyond-mean-field scattering processes, i.e. is cooperative in nature. Experimental investigations of the temporal response of the system, and comparisons to the case of a single atom [24], should also provide insight into the interplay between dipole-dipole interactions and collective scattering.

We acknowledge support from the E.U. through the ERC Starting Grant ARENA, from the Triangle de la Physique, EPSRC, and Leverhulme Trust. We thank P. Pillet, J.-J. Greffet, and J. Javanainen for fruitful discussions.

FIG. 1 :

 1 FIG.1:(a) Experimental setup. The atoms are initially confined in a microscopic single-beam dipole trap (not shown) (wavelength 957 nm, depth 1 mK, and a waist 1.6 µm, oscillation frequencies ωx = ωy = 2π×62 kHz and ωz = 2π×8 kHz). The excitation laser propagates along the quantization axis x, set by a B ∼ 1 G magnetic field. We collect the scattered light along z, after a polarizer P oriented at an angle of 55 • with respect to x, using a lens L with a large numerical aperture (NA= 0.5) and an image intensifier followed by a CCD camera (I-CCD). (b) Simulation of the distribution of nearest neighbors for a single stochastic realization of a cloud of N = 450 atoms. (c) Structure of 87 Rb atoms relevant to this work. The excitation light at frequency ω is near-resonant with the transition at λ = 2πc/ω0 = 780 nm.

FIG. 2 :

 2 FIG.2: Amount of scattered light detected nz(N, ∆), versus the detuning ∆ of the excitation light for numbers of atoms N = 1, 5, 20, 50, 200, 325, 450 (from bottom to top). The amplitudes of the curves are normalized to the amount of light detected at resonance for a single atom, nz(N = 1, ∆ = 0). Solid lines : Lorentzian fits to the data. Typical uncertainties: 10% (vertically) and 20% (horizontally).

FIG. 3 :FIG. 4 :

 34 FIG. 3: (a) FWHM and (b) line shift δω with respect to the atomic frequency for N = 1 atom, in units of Γ. Filled data extracted from the Lorentzian fits in Fig. 2, versus the number of atoms. Dashed lines : phenomenological fits of the FWHM and shift by, respectively, Γc/Γ = 1.49(6) × N 0.08(1) , and δωc/Γ = 47(9) × 10 -5 N . The error bars are from the fits of Fig 2. Green solid line: results of the simulation (see text).

FIG. 5 :

 5 FIG.5: Comparison between experiment and theory for the number of detected photons nz(N, ∆) (normalized to the single atom case at resonance) for N = 450. The red, blue and green lines correspond to samples with widths σρ and σz multiplied by a factor 1, 1.44, and 2 respectively.


Philippe De Groote

Yoad Winter

A type-logical account of quantification in event semantics

.

Introduction

It is well known that combining Davidsonian event semantics [START_REF] Davidson | The logical form of action sentences[END_REF] with Montague's treatment of quantification [START_REF] Montague | The proper treatment of quantification in ordinary english[END_REF] may give rise to unexpected semantic interpretations. To understand the potential problem, consider the standard interpretations of proper names and transitive verbs, which allow one to give a semantic interpretation to simple sentences like [START_REF] Champollion | Quantification and negation in event semantics[END_REF].

(1) a. John = λp. p j : (e → t) → t b. Mary = λp. p m : (e → t) → t c. kissed = λpx. p (λy. kissed x y) : ((e → t) → t) → e → t

(2) John kissed Mary.

Adapting lexical entry (1-c) to the Davidsonian approach consists in providing the binary relation kissed with an additional event argument of type v, 3 which results in lexical entry (3-a). This allows adverbial modifiers to parallel adnominal modifiers (see lexical entry (3-b)). Then, in order to interpret a sentence as a truth value rather than as a set of events, one has to apply an existential closure operator (3-c).

(3) a. kissed = λpxe. p (λy. kissed e x y) :

((e → t) → t) → e → v → t b. passionately = λpe. (p e)∧(passionate e) : (v → t) → v → t c. e-clos = λp. ∃e. p e : (v → t) → t
Using the above apparatus, we obtain the semantic interpretation of sentence (4) by computing the value of expression [START_REF] Davidson | The logical form of action sentences[END_REF], which results in formula [START_REF] Davidson | Essays on Actions and Events[END_REF]. [START_REF] Curry | Some logical aspects of grammatical structure[END_REF] John kissed Mary passionately.

(5) John (λx. e-clos ( passionately ( kissed Mary x)))

(kissed e j m) ∧ (passionate e)

Consider now sentence [START_REF] Ph | Towards abstract categorial grammars[END_REF], which includes a quantified noun phrase.

John kissed every girl.

The standard interpretation of this quantified noun phrase is based on the following lexical interpretations:

(8) a. girl = λx. girl x : e → t b. every = λpq. ∀x. (p x) → (q x) : (e → t) → (e → t) → t

Then, using expression [START_REF] Landman | Events and Plurality: The Jerusalem Lectures[END_REF] to compute the semantic interpretation of sentence [START_REF] Ph | Towards abstract categorial grammars[END_REF] results in a counterintuitive interpretation (formula [START_REF] Montague | The proper treatment of quantification in ordinary english[END_REF]).

(9) John (λx. e-clos ( kissed ( every girl ) x))

(10) ∃e. ∀x. (girl x) → (kissed e j x)

According to formula [START_REF] Montague | The proper treatment of quantification in ordinary english[END_REF], there should be a single kissing event involving John and every girl. This requirement appears because the existential closure operator takes wide scope over the universally quantified noun phrase. The problem with this analysis becomes more apparent when we consider the interaction of events with quantifiers that are not upwardmonotone, as in the following sentences.

(11) a. John kissed no girl. b. John kissed less than five girls. c. John kissed exactly one girl.

Consider the standard lexical interpretation of the no quantifier, together with a semantic analysis akin to expression [START_REF] Landman | Events and Plurality: The Jerusalem Lectures[END_REF].

(

) a. no = λpq. ∀x. (p x) → ¬(q x) : (e → t) → (e → t) → t b. 12
John (λx. e-clos ( kissed ( no girl ) x))

This leads to the following problematic interpretation:

(13) ∃e. ∀x. (girl x) → ¬(kissed e j x)
According to formula [START_REF] Winter | Event semantics and Abstract Categorial Grammar[END_REF], sentence (11-a) would be true if John kissed a girl, but there is another event where John did not kiss any girl. Indeed, since in the latter event John kissed no girl, the sentence might be incorrectly analyzed as true in this situation. The same kind of problem can also be shown with sentences (11-b) Using a standard interpretation of negation (lexical entry (14-b)), expression (14-c) leads to a correct semantic interpretation of sentence (14-a), namely, formula (14-d). There is, however, another possible analysis of sentence (14-a), which is expressed by expression (14-e). This leads to formula (14-f), where the existential quantifier over events takes scope over negation. According to this fallacious interpretation, sentence (14-a) might be true in a situation where John kissed Mary, provided there is another event where he did not kissed Mary.

Besides quantification and negation, conjunction is also known to interact badly with event existential closure. Consider the following sentence, whose interpretation is meant to be distributive:

(15)
John kissed Mary and [then] Sue.

The standard interpretation of distributive coordination is given by the following lexical entry:

(16) and = λpqr. (p r) ∧ (q r) :

((e → t) → t) → ((e → t) → t) → (e → t) → t
Nevertheless, the semantic analysis given by expression (17-a) yields an interpretation (formula (17-b)) akin to a collective reading. According to this problematic interpretation, Mary and Sue would be the patients of the same kissing event.

(17) a. John (λx. e-clos ( kissed ( and Mary Sue ) x)) b. ∃e. (kissed e j m) ∧ (kissed e j s)

To obtain the right interpretations of sentences like ( 7), (11-a)-(11-c), (14-a), and (15), the event existential closure operator need to take narrow scope. For sentence [START_REF] Ph | Towards abstract categorial grammars[END_REF], this is illustrated by expression (18), which is interpreted as in formula (19). ( 18) John (λx. every girl (λy. e-clos ( kissed (λp. p y) x)))

(19) ∀x. (girl x) → (∃e. kissed e j x)
In many works, expressions such as (18) result from some covert movement operation. This explains why most approaches that combine event semantics and quantification either rely on syntactic devices that control the scope of the quantifiers [START_REF] Landman | Events and Plurality: The Jerusalem Lectures[END_REF], or depart significantly from standard assumptions in compositional semantics [START_REF] Beaver | On the logic of verbal modification[END_REF][START_REF] Krifka | Nominal reference, temporal constitution and quantification in event semantics[END_REF]. Following Champollion [START_REF] Champollion | Quantification and negation in event semantics[END_REF][START_REF] Champollion | The interaction of compositional semantics and event semantics[END_REF] and Winter and Zwarts [START_REF] Winter | Event semantics and Abstract Categorial Grammar[END_REF], we see this enrichment of standard systems as problematic. Accordingly, we refer to the problem of combining standard compositional semantics and event semantics as the event quantification problem.

Two solutions to the event quantification problem

The recent literature provides two solutions to the event quantification problem. The first one is due to Champollion [START_REF] Champollion | Quantification and negation in event semantics[END_REF][START_REF] Champollion | The interaction of compositional semantics and event semantics[END_REF]. It consists in interpreting sentences as generalized quantifiers over events ((v → t) → t) rather than as sets of events (v → t). This allows the existential closure to occur at the lexical level. Accordingly, the lexical entries in (3) are adapted as follows.

(20) a. kissed = λpxf. p (λy. ∃e. (kissed e x y) ∧ (f e)) :

((e → t) → t) → e → (v → t) → t b. passionately = λpf. p (λe. (f e) ∧ (passionate e)) : ((v → t) → t) → (v → t) → t c. clos = λp. p (λe. true) : ((v → t) → t) → t
The semantic interpretation of sentence ( 7) is then obtained by computing the value of expression (21). This expression is akin to expression [START_REF] Landman | Events and Plurality: The Jerusalem Lectures[END_REF]. Nevertheless, this time, the computation yields the correct interpretation (formula (19)). The second solution, which is due to Winter and Zwarts [START_REF] Winter | Event semantics and Abstract Categorial Grammar[END_REF], is expressed in the framework of Abstract Categorial Grammar [START_REF] Ph | Towards abstract categorial grammars[END_REF]. It takes advantage of the so-called tectogrammatic level [START_REF] Curry | Some logical aspects of grammatical structure[END_REF] for the treatment of scope interactions. Following the type-logical tradition, Winter and Zwarts distinguish, at the abstract syntactic level, the category of noun phrases, NP , from the category of quantified noun phrases, (NP → S) → S. They also distinguish the category S of sentences interpreted as truth values (t) from the category V of sentences interpreted as sets of events (v → t). 4This results in abstract syntactic structures specified by a signature akin to the following one. In this setting, the only abstract structure corresponding to sentence [START_REF] Ph | Towards abstract categorial grammars[END_REF] is the following well-typed expression (whose derivation and surface realization are given in Appendix A).

(24) every girl (λx. e-clos (kissed x john))

Then, using the semantic interpretation given here below, the evaluation of expression (24) yields the expected result, i.e., formula (19).

( 22) compels the existential closure operator (e-clos) to take scope below the quantified noun phrase (every girl). This is because the abstract syntactic category assigned to every (N → (NP → S ) → S ) is given in standard terms of the abstract category S (interpreted as truthvalues) rather than V (interpreted as sets of events). Consequently, each of the derived sets of events (i.e., each expression of type V ) must first be "closed" (i.e., turned into an expression of type S) before quantification can apply. This solution is easily transferable to the cases of negation and conjunction. It suffices to express their abstract categories in term of S. Unlike Champollion, Winter and Zwarts do not consider the case of quantified adverbial modifiers such as everyday or everywhere. The present paper aims to fill this gap.

  . clos ( kissed ( every girl ) x))

  : NP b. girl : N c. kissed : NP → NP → V d. every : N → (NP → S ) → S e. passionately : V → V f. e-clos : V → S This signature comes with a lexicon that specifies the surface realization of the abstract syntactic structures: (23) a. john := John b. girl := girl c. kissed := λxy. y + kissed + x d. every := λxf. f (every + x) e. passionately := λx. x + passionately f. e-clos := λx. x

5

 5

  (26) a. not : (NP → S • ) → (NP → S ) b. and : NP → NP → (NP → S ) → S ) Thus, boolean negation and conjunction are still treated by using the boolean S-based types, rather than the Davidsonian V -based types.

We follow a Davidsonian approach as opposed to a neo-Davidsonian approach. We also distinguish the type of events (v) from the type of entities (e). These choices, which are rather arbitrary, will not affect our purpose.

Following Montague's homomorphism requirement, these two abstract categories should indeed be distinguished since they correspond to different semantic types. The fact that they share the same surface realizations may be considered as a mere contingence. Alternatively, we may relax the homomorphism requirement, e.g. as in[START_REF] Partee | Noun phrase interpretation and type shifting principles[END_REF], who treats indefinite NPs as ambiguous between predicates and quantifiers.

The reason for distinguishing between type S and type S•, which is merely syntactic, is explained in Appendix B.

Acknowledgement

The authors would like to thank Lucas Champollion for fruitful discussions, and the two anonymous referees. The work of the first author was supported by the French agency Agence Nationale de la Recherche (ANR-12-CORD-0004). The work of the second author was supported by a VICI grant 277-80-002 of the Netherlands Organisation for Scientific Research (NWO).

Consider sentence (27) together with a plausible semantic interpretation (formula (28)). (27) John kissed Mary everyday.

(28) ∀x. (day x) → (∃e. (kissed e j m) ∧ (time e x))

In formula (28), the underlined subformulas are derived from the semantic interpretation of everyday. In a compositional setting, this makes it necessary that the lexical semantics of everyday acts both inside and outside of the scope of the existential closure. The solution to this puzzle is in accordance with the type-logical tradition. It consists in distinguishing the category of adverbial modifiers from the category of quantified adverbial modifiers. Similarly to the treatment of noun phrases, the category of quantified adverbial modifiers is obtained from the category of non-quantified adverbial modifiers by type-shifting. This results in the following type assignment.

(29) everyday :

Then, the abstract structure corresponding to sentence (27) may be expressed as follows.

(30) everyday (λq. e-clos (q (kissed mary john)))

Finally, using lexical entry (31), one may compute compositionally the interpretation of sentence (27).

(31) everyday := λq. ∀x. (day x) → (q (λpe. (p e) ∧ (time e x))) :

It is to be noted that our approach is consistent with the treatment of quantified adverbial prepositional phrases such as in every room. Consider the category and the semantic interpretation assigned to the preposition in.

(32) a. in :

We may then derive an abstract structure of the appropriate type, corresponding to the prepositional phrase in every room.

(33) λq. every room (λx. q (in x)) :

We give a toy-grammar that summarizes our approach in appendix B. This grammar allows one to handle examples such as sentence (34).

(34)

John did not kiss Mary for one hour.

Sentences such as (34) are known to exhibit a scope ambiguity that yields two different semantic interpretations. In the present case, these two interpretations may be paraphrased as follows.

(35) a. For one hour, it was not the case that John kissed Mary. b. It was not the case that John kissed Mary for one hour.

Using our toy-grammar, these two interpretations may be obtained by computing the semantic interpretations of the two following abstract structures whose surface realization is sentence (34).

(36) a. for

The abstract constants marked with small circles (for • , e-clos • , kissed • , etc.) involve a syntactic treatment of negation, which avoids ill-formed strings like * John did not kissed Mary. For more details see Appendix B.

We conclude that the ambiguity of sentence (34), which is treated in Champollion's system as an ordinary case of scope ambiguity, is similarly treated here, by following an adaptation of the system proposed by Winter and Zwarts.

A final remark has to do with sentences like the following.

(37) John did not kiss Mary deliberately.

This sentence presents an ambiguity similar to the one of sentence (34):

(38) a. It was deliberate that John did not kiss Mary. b. It was not deliberate that John kissed Mary.

Nevertheless, sentences like (37) cannot be treated like sentence (34) because the adverb is not quantificational. Following Davidson, we assume that adverbs like deliberately, unlike standard manner adverbs, have a modal element to their meaning, and accordingly modify full propositions rather than events. Like Champollion, we believe that the treatment of modal adverbs and other modal operators is orthogonal to the main tenets of event semantics.

Appendix A

Derivation of expression (24):

x : NP ⊢ kissed x john : V x : NP ⊢ e-clos (kissed x john) : S ⊢ λx. e-clos (kissed x john) : NP → S

. . . .

Appendix B

This appendix presents a toy grammar that covers the several examples that are under discussion in the course of the paper. It mainly consists of three parts:

• a set of abstract syntactic structures, specified by means of a higherorder signature; • a surface realization of the abstract structures, specified by means of a homomorphic translation of the signature; • a semantic interpretation of the abstract structures, specified by means of another homomorphic translation;

B.1 Abstract Syntax

The signature specifying the abstract syntactic structures is given in Table 1. It uses a type system built upon the following set of atomic syntactic categories:

• N , the category of nouns;

• N u , the category of nouns that name units of measurement;

• NP , the category of noun phrases;

• NP τ , the category of noun phrases that denote time intervals;

• S and S • , the category of sentences (positive and negative);

• V and V • , the category of "open" sentences (positive and negative).

The reason for distinguishing between the categories of positive and negative (open) sentences is merely syntactic. Without such a distinction, the surface realization of a negative expression such as:

not (kissed mary) john would be: * John did not kissed Mary Without this distinction, it would also be possible to iterate negation. This would allow the following ungrammatical sentences to be generated:

B.2 Surface Realization

The surface realization of the abstract syntactic structures is given in table 2. This realization is such that every abstract term of an atomic type is interpreted as a string. Accordingly, abstract terms of a functional type are interpreted as functions acting on strings.

Surface

B.2 Semantic interpretation

The semantic interpretation of the abstract syntactic categories is given in Table 3. Besides the usual semantic types e and t, we also use v, i, and n. These stand for the semantic types of events, time intervals, and scalar quantities, respectively.

The semantic interpretation of the abstract constants is then given in Table 5. This interpretation makes use of the non-logical constants given in Table 4.

Semantic Types


Samuel Nowakowski

Ivana Ognjanović

Monique Grandbastien

Jelena Jovanovic

Ramo Šendelj

Two Recommending Strategies to enhance Online Presence in Personal Learning Environments

Keywords: Web-based learning, social presence, online presence, ontology based resource recommendation, Kalman filter, learning trajectories, AHP, CS-AHP 1

Aiming to facilitate and support online learning practices, TEL researchers and practitioners have been increasingly focused on the design and use of Webbased Personal Learning Environments (PLE). A PLE is a set of services selected and customized by students. Among these services, resource (either digital or human) recommendation is a crucial one. Accordingly, this chapter describes a novel approach to supporting PLEs through recommendation services. The proposed approach makes extensive use of ontologies to formally represent learning context that, among other components, includes students' presence in the online world, i.e., their online presence. This approach has been implemented in and evaluated with the OP4L (Online Presence for Learning) prototype. In this chapter, we expose recommendation strategies devised for OP4L. One is already implemented in OP4L, it is based on the well-known Analytical Hierarchical Process (AHP) method. The other one which has been tested on data coming from the prototype is based on the active user's navigation stream and used a Kalman filter approach.

Introduction

Web-based Personal Learning Environments (PLEs) have been increasing adopted by the TEL research community as a mean to support and facilitate online learning practices [START_REF] Atwell | Personal Learning Environments -the future of eLearning? eLearning Papers, www.elearningpapers[END_REF]. From the technical perspective, a PLE is a customizable set of tools and services aimed at enhancing students' learning experiences and learning outcomes. Among these services, resource (either digital or human) recommendation is a crucial one, given the number and the diversity of available resources on the Web. Various approaches have been proposed to improve the recommendation of resources and adapt them to the learners' needs [START_REF] Vuokari | Special Issue on Social Information Retrieval for Technology Enhanced Learning[END_REF], [START_REF] Manouselis | Recommender Systems in Technology Enhanced Learning[END_REF], [START_REF] Santos | Educational Recommender Systems and Technologies[END_REF]. They all rely on a learner profile and include a more or less rich description of the learning context, often based on ontologies.

In the last few years, we are witnessing a steady increase in the students' use of Web-based social software tools. This has lead to the emergence of novel forms of social presence in online learning environments, PLEs being no exception. Hence, the dominant forms of establishing and maintaining social presence become online status updates, online visibility, availability for online communication and the like. Semantic Web technologies, ontologies in particular, allow for taking these forms of social presence into account when generating recommendations for students.

This chapter reports on the recommendation strategies that have been implemented in the OP4L (Online Presence For Learning) [START_REF]OP4L project[END_REF] framework and evaluated using the data coming from the students' use of the framework. The first approach is based on the well-known Analytical Hierarchical Process (AHP) method and its adoption for handling conditionally defined preferences, named Conditional Stratified AHP (CS-AHP) method. The PeerRec service is developed based on the adoption of prioritization algorithms in a PLE. Based on the students' preferences about some important features of the learning process, the service offers recommendation of peers to communicate and/or collaborate with.

The second approach is based on the active user's navigation stream: we consider that users browsing the Web can be seen as objects moving along trajectories in the Web space. Having this assumption, we derive the appropriate description of the so-called recommender space to propose a mathematical model and state estimate based on a Kalman filter describing the behaviour of the users along the trajectories of the recommender space.

We present the theoretical background, and report on the obtained results and performances. The chapter concludes with perspectives for further developments and prospective evolution studies.

Background

In her "vision" paper on the design of social learning environments [START_REF] Vassileva | Towards Social Learning Environments[END_REF], Vassileva identifies three main roles that should be performed by PLEs: [START_REF] Atwell | Personal Learning Environments -the future of eLearning? eLearning Papers, www.elearningpapers[END_REF] support the learner in finding the right content (right for the context, particular learner, specific purpose of the learner and pedagogically), [START_REF] Vassileva | Towards Social Learning Environments[END_REF] support learner to connect with the right people (…) and (3) motivate/incentivize people to learn. To devise PLEs with such features, TEL researchers and developers rely on the body of knowledge and experiences originating from several interrelated research domains. The discovery and retrieval of learning resources is one of those domain, and has been widely investigated, beginning with the work on metadata interoperability, then going on with the use of ontologies to better match the learners' needs and context. As social web applications, such as collaborative tagging, became available, solutions mixing both ontology-and folksonomy-based approaches were proposed. Meanwhile, the recommender systems community developed powerful algorithms for the e-commerce sector, and PLE developers tried to adapt them to e-learning purposes [START_REF] Vuokari | Special Issue on Social Information Retrieval for Technology Enhanced Learning[END_REF], [START_REF] Manouselis | Recommender Systems in Technology Enhanced Learning[END_REF], [START_REF] Santos | Educational Recommender Systems and Technologies[END_REF].

Social presence is another relevant research field. It has been identified as a crucial success factor in e-learning for many years [START_REF] Aragon | Creating social presence in online environments[END_REF], [START_REF] Cob | Social Presence and Online Learning: A current view from a Research Perspective[END_REF], [START_REF] Lowenthal | Social Presence[END_REF]. At the beginning of e-learning practices, social presence was mostly implemented through online forums and Instant Messaging tools that allowed for establishing and maintaining social presence in online learning settings. The wide adoption of social web applications, such as online social networks, resulted in the inclusion of these applications and connections that students had established in them into online learning environments. Though in theory students can interact with their entire social network, in practice they do not get any indicator about who is really available in the given moment and who is really capable of helping in the current task. Although recommending knowledgeable people for performing a given task is not new, it has been mostly investigated in company settings such as reported, for instance, in [START_REF] Beham | Recommending knowledgeable people in a work-integrated learning system, 1 st RecSysTEL workshop[END_REF]. The OP4L framework brings solutions for the two aforementioned challenges in the manner described in the following sections.

OP4L framework

Background and objectives

The OP4L project was running between 2010 and 2012. Its aim was to explore the use of Web-based tools and services for supporting social presence in online learning environments, and thus lead to an improvement in the students' learning experience. In this chapter, we use the term OP4L to name both the project and the developed prototype.

OP4L defines online presence as a temporary description of a user's presence in the online world. It can be considered as an image that a person projects about him/herself into the online world. We explored online presence in the context of DEPTHS [START_REF] Jovanovic | Ontologies for Effective Use of Context in e-Learning Settings[END_REF], a PLE customized for the domain of Software Design Patterns. It makes use of ontologies as a common foundation for the integration of different systems, services and tools in a common environment for collaborative learning of software design patterns. OP4L extends the set of services offered by DEPTHS, by processing online presence data at the semantic level.

Functional description

A complete technical description of the OP4L framework can be found in [START_REF] Jeremic | OP4L : Online Presence Enabled Personal Learning Environments[END_REF] and [START_REF] Milikic | Infrastructure for Exchanging Online Presence Data in Learning Applications[END_REF], as well as in deliverables available on the project's web site1. Therefore, in the following we give just a brief overview of the OP4L framework in order to draw an overall picture of the services it offers and the ontologies that make possible the provided services.

From a functional perspective, the primary goal behind the OP4L framework was the development of a context-aware PLE through integration of learning context data from different learning systems/tools/services, using a flexible ontology-based model [START_REF]OP4L Models[END_REF]. We define learning context, i.e., the context of a given learning situation as an interplay of the following main components:

• the learning activity that was performed or the learning-related even that occurred,

• the content that was used and/or produced during the learning activity, • the individual(s) involved (e.g., learners, teachers, experts) and their respective on line presence statuses, • the (online) environment where the learning activity took place,

• the time when the learning activity took place.

The notion of learning context is formally modeled through an interlinked set of ontologies collectively named LOCO (Learning Object Context Ontologies) framework [START_REF] Jovanovic | Ontologies for Effective Use of Context in e-Learning Settings[END_REF]. Within the OP4L project, the notion of learning context is extended to include the notion of Online Presence. Accordingly, links have been established between the existing LOCO ontologies and the Online Presence Ontology (OPO) [START_REF] Jovanović | Online Presence in Adaptive Learning on the Social Semantic Web[END_REF] to allow for explicitly defining the semantic of this extended notion of learning context. These ontologies served as the foundation for the development of the OP4L prototype with the following main features:

• Integration of data and resources from diverse learning applications that students interact with; • Context-aware recommendation of resources on software design patterns from online repositories, learning artifacts produced and shared by peers, software projects, discussion threads, chats, etc.; • Context-aware recommendation of other students, experts and/or teachers to offer help in the given situation.

These services make use of learners' overall learning context, including his/her online presence data, when providing them with recommendations about whom to ask for help or collaborative work. These data are periodically "pulled in" the OP4L system by specific software modules developed for that purpose. Within the online presence data, a key indicator is the "online status" [START_REF] Stankevic | Modeling Online Presence[END_REF] as declared by the user. For instance, a peer whose online status indicates that he/she is busy in the given moment will not be recommended; on the other hand, the system would recommend a face-to-face study session with a peer who has just checked in the same building and whose status indicates that he/she can be freely contacted.

For course designers, one of the main challenges is to adapt interactions to the students' state of presence and to provide services so that interactions can be established smoothly among the participants.

Main features of OP4L prototype

OP4L services are accessible through a dedicated Moodle platform. The services become available after a student selects a course to study (e.g., the Design Pattern course) and a learning activity (e.g., updating patient's data problem using the UML modeling tool). Fig. 1 shows the OP4L online presence services as presented to the student in the user interface. Specifically, they appear in the form of an online presence box in the upper left side of the screen. The box indicates who is competent for the given problem and available online for help or collaboration. It also indicates how to contact potential helper(s)/collaborator(s), either on the Moodle platform itself, or via Facebook or Twitter.

Services linking Moodle to Facebook and Twitter have been developed so that each student can remain using his/her current application, for instance, Moodle for the student looking for help/collaboration and Facebook for the student being contacted. Based on the online statuses declared by the peer students, the learner who is looking for help/collaboration will know in which manner he/she can communicate with the peers. In the case presented on Fig. 1, all of the peers can be contacted on Facebook and by email. The system offers several other services as described below. The system recommends appropriate contents related to the topic of the course (Fig. 2). Its originality is to augment the course digital library with resources brought in and built by the students during the course.

Fig. 2. Interface illustrating recommendation of digital resources

For enhancing collaboration, students are also given a brainstorming tool where ideas can be annotated and rated. Finally, students can upload their work and benefit from peer evaluations, as shown in Fig. 3. They can assess solutions proposed by fellow students only after uploading their own solution.

Recommendations strategies

To support students through recommendation of digital and human resources, the OP4L framework needs to implement recommendation algorithms. In this section, we present the algorithm that has been implemented to recommend peers and digital content. We also present a complementary approach based on the students' learning trajectories.

Recommendation algorithm based on the Conditional Stratified Analytic Hierarchy Process

In a learning environment each individual student has different characteristics, motivation and performance, which all together indubitably should be considered when designing and/or adapting the learning process. It has been well recognized that there is a need to move away from the 'one size fits all' paradigm, and to offer personalized learning experience to learners [START_REF] Dagger | Personalisation for All: Making Adaptive Course Composition Easy[END_REF]. Most of the approaches aimed at adapting the learning process to individual learners or learning groups have been based on the learners' level of knowledge [START_REF] Popescu | Adaptive educational hypermedia systems: A focus on learning styles[END_REF], [START_REF] Stash | Incorporating cognitive styles in AHA! (The Adaptive Hypermedia Architecture)[END_REF]. Other learner features taken into account are background, hyperspace experience [START_REF] Brusilovsky | Adaptive hypermedia. User modeling and user adapted interaction[END_REF], preferences and interests, as well as learning styles and their effect on learning achievements [START_REF] Mustafa | An approach to Adaptive E-Learning Hypermedia System based on Learning Styles (AEHS-LS): Implementation and evaluation[END_REF]. However, despite interests in exploring these diverse learners' features, in recent years, modeling of adaptive systems has still revolved around acquiring and representing learners' knowledge. This orientation does not properly reflect real life situations where each learner has a variety of selection criteria and requirements over them when choosing other learners for collaboration and cooperation [START_REF] Ognjanović | Teachers' requirements in dynamically adaptive e-learning systems[END_REF].

Representation and analysis of preferences have been studied in many fields such as economics, especially in project and risk management, decision theory, social choice theory, with further developments and applications in areas such as operational research, databases, security analysis, and artificial intelligence [START_REF] Ognjanović | Conditional Preferences in Software Stakeholders' Judgments[END_REF]. Modeling of user preferences is a great challenge, as it is difficult to express human opinion in a way that can be easily processed by computers [START_REF] Yu | Toward an Understanding of User-Defined Conditional Preferences[END_REF]. The adoption of user preferences in the design of a PLE brings in additional challenges in coordination of the flow of information among the learners involved in a learning process, and in encouragement of interactions across learning systems/tools/services. In particular, communication between peers may be induced by different needs and expectations (e.g., general questions about the course organization, help needed in learning and understanding of some course topics, etc.). In some cases, urgent response is needed, while in others, only answers of good learners or good senior learners are useful. The following problems emerge from this observation: (i) characterization of peers for communication related to the selection criteria, and (ii) definition of specific requirements and preferences over them. Furthermore, it is reasonable to expect that appropriate peers are not available all the time for each learning topic, or the most appropriate peers are not there when a specific learner needs their assistance. Having all this in mind, the PeerRec service, developed as a part of the OP4L framework, integrates: i) the aforementioned semantic representation of online presence data (see Section 3.2); ii) two-level hierarchical structure of concerns and qualifier tags for semantic representation of selection criteria proposed in [START_REF] Ognjanović | A Stratified Framework for Handling Conditional Preferences: an Extension of the Analytic Hierarchy Process[END_REF]; and iii) the CS-AHP prioritization algorithm for presentation and ranking of users' preferences [START_REF] Ognjanović | A Stratified Framework for Handling Conditional Preferences: an Extension of the Analytic Hierarchy Process[END_REF]. [START_REF] Saaty | The Analytic Hierarchy Process[END_REF] for different kinds of preferences using a two-level hierarchical structure of concerns and qualifier tags. Concerns are a set of quality characteristics that represent important matters of interest for learners such as fields of professional specialization, spoken languages or preferred message response time. Qualifier tags represent possible values for each concern (e.g., qualifier tags for spoken languages could be goodLevel, mediumLevel, lowLevel, unknownLanguage).

CS-AHP algorithm CS-AHP (Conditional Stratified Analytic Hierarchy Process) adopts Analytical Hierarchy Process (AHP) technique

Following the well-known AHP framework for expressing and ranking user requirements, the PeerRec service enables learners to express their requirements by defining relative importance between concerns, and between qualifier tags of each concern. Relative importance is typically defined with odd numbers ranging from 1 (equal importance) to 9 (extreme importance of one concern over the other). The options available to learners in respect to their online presence (i.e. peers for collaboration) are also associated with qualifier tags. Once the relative importance is set between all pairs of concerns, the AHP algorithm performs a tuned pair-wise comparison of the learners' requirements. The outcome of this process are ranks , which provide values from the [0,1] interval over the set of available options. The process is done in two main steps: i) the set of concerns and their qualifier tags are locally rankedwith ii) rank of each available option is calculated based on the ranks of the qualifier tags associated with that option.

CS-AHP also allows for setting conditional preferences. For example, learners are often aware that requirement of expertise is hard to meet, so they may define a compromise: they are only prepared to wait for response from a learner with expert knowledge, if the waiting time is kept at a very minimum; otherwise, they are willing to contact learners with lower level of expertise but with medium delays in response. CS-AHP is simple to perform, and requires quadratic number of comparisons, which brings linear time complexity to the number of available options [START_REF] Ognjanović | A Stratified Framework for Handling Conditional Preferences: an Extension of the Analytic Hierarchy Process[END_REF].

Applying the CS-AHP Algorithm in a PLE

In order to develop a user-friendly service for the implementation of CS-AHP in a PLE, a structure of concerns and qualifier tags is built on the Preview framework related to learning environments [START_REF] Ognjanović | Teachers' requirements in dynamically adaptive e-learning systems[END_REF]. The following three categories of concerns are recognized from the aspect of definition and management issues [START_REF] Ognjanović | Making judgments and decisions about relevant learning resources[END_REF]: statically defined concerns, concerns dynamically defined for each conversation, and concerns with dynamic updates.

Statically defined concerns present general information about a learner who is looking for another peer appropriate for conversation; examples include the known language(s), preferred subject area, and availability for F2F contact. The recognized concerns are defined as attributes in the learner's profile developed for the PeerRec service. Information in the profile may be changed upon a learner's request; otherwise, it remains unchanged, representing the learner's characteristics that are used in each call of the service.

On the other side, some concerns are directly related to the context of the upcoming conversation (i.e. concerns dynamically defined for each conversation), and, thus learners are enabled to define them explicitly. For example, a learner may define the type of conversation (e.g. the help in understanding, etc.) and/or the urgency level (e.g. extra urgent) (see Fig. 4). Contrary to the concerns defined in the learner's profile (with qualifier tags statically defined and changeable only on the learner's explicit request), concern defined as ConversationRate should be updated for each completed conversation. To this end, the PeerRec service asks learner to rate each peer after communication (see button Rate conversation on Fig. 5), and based on the learner's feedback updates the values to be used in further communications with the same peer (see attributes Response speed and Response relevancy on Fig. 6). In cases when learner is not interested in setting rates (from different personal reasons ranging from current disengagement, lack of time or interests), he/she may select the N/A option meaning that experience from the latest conversation does not bring any change to the aggregated rates. Furthermore, if no rate is previously aggregated, initial selection of N/A represents indifferent and undeclared learner (neither positive nor negative rate is specified). After invoking the PeerRec service, all currently available peers are ranked in decreasing order (based on the results obtained from applying the CS-AHP algorithm), and the learner can decide with whom to start a conversation (see Fig. 7). Previous analyses from [START_REF] Ognjanović | A Stratified Framework for Handling Conditional Preferences: an Extension of the Analytic Hierarchy Process[END_REF] give a frame for addressing different preference structures with scales of input and output information with different semantics [START_REF] Boutilier | CP-nets: a tool for representing and reasoning with conditional ceteris paribus preference statements[END_REF], [START_REF] Brafman | Introducing variable importance tradeoffs into CP-nets[END_REF], [START_REF] Ognjanović | A Stratified Framework for Handling Conditional Preferences: an Extension of the Analytic Hierarchy Process[END_REF], [START_REF] Berander | Requirements Prioritization[END_REF], [START_REF] Zavadskas | Multi-attribute assessment of road design solutions by using the COPRAS Method[END_REF], [START_REF] Chen | Reasoning with Conditional Preferences across Attributes[END_REF], [START_REF] Wilson | Computational techniques for a simple theory of conditional preferences[END_REF]. Unique representational and reasoning technique that can effectively address different kinds of preferences and reasoning queries does not exist. Therefore, selection of the most appropriate method should be done depending on the characteristics of the problem at hand in the given field [START_REF] Zavadskas | Multi-attribute assessment of road design solutions by using the COPRAS Method[END_REF].

In the learning environment, different aspects of recommendations are considered and analyzed, according to their own mission, vision, and objectives [START_REF] Hanna | Data mining in the e-learning domain[END_REF], [START_REF] Merceron | Educational data mining: A case study[END_REF], [START_REF] Baker | The state of educational data mining in 2009: A review and future visions[END_REF],[67] such as, dealing with the assessment of the student's learning performance, providing course adaptation and learning recommendations based on the student's learning behaviour, evaluation of learning material and educational web-based courses, developments for detection of atypical students' learning behaviours, etc. Some of them are focused on creation of Recommendation Systems that link users with items (course, learning material, peers for communication,...) [START_REF] Schafer | The application of data-mining to recommender szstems[END_REF], associating the content of the recommended item or opinion of other individuals with the actions or opinions of the original users of the system.

However, to date, there has been comparatively less progress in direction of focusing to students' requirements and preferences [67], although there is currently an increasing interest in applying techniques and methods primary developed and used in other fields to the educational environment [START_REF] Lazcorreta | Towards personalized recommendation by two-step modified apriori data mining algorithm[END_REF].In this context, recently developed CS-AHP algorithm extends well-known Analytical Hierarchical Process (AHP) proposed by Saaty [START_REF] Saaty | The Analytic Hierarchy Process[END_REF] which is a widely adopted multi-criteria decision making method to make complex decisions [START_REF] Büyüközkan | Strategic analysis of healthcare service quality using fuzzy AHP methodology[END_REF], [START_REF] Chen | The critical factors of success for information service industry in developing international market: Using analytic hierarchy process (AHP) approach[END_REF]. Also, the use of two-layered structure of concerns and qualifier tags has several explanations, sufficient expressiveness (according to [START_REF] Yu | Toward an Understanding of User-Defined Conditional Preferences[END_REF]) and analogous to the concept of attributes in all developed techniques for addressing unconditional and conditional preferences [START_REF] Ognjanovic | Conditional Preferences in Software Stakeholders' Judgments[END_REF].

Learning trajectories algorithm

In this section we present an approach that does not limit itself to the content recommendation, but aims at analyzing the way that students learn. This approach starts from the following hypothesis: learning is linked to the identity; so, when digital learning environments are used, we consider that digital identities have to be studied.

From a philosophical point of view, digital identities can be seen as the "sèmes" which identify an individual in its singularity. We can then speak of "individuation" as a process leading from the undifferentiated to uniquely defined and personalized. Consequently, we can identify someone by his/her personal trajectory in the cyberspace. To validate this approach, we have implemented a service (using Matlab) with input data from the OP4L learning framework. The following sections show the theoretical background of our approach, as well as some numerical results and perspectives.

Input Data

The input data we consider is made up of traces that users have left (log files and users' ratings) when interacting with the system (intranet, Web site, etc.). These data are first used to discover patterns of usage, and then to perform recommendations. In our case, data are coming from the log files of the OP4L prototypes. They show (for each user) the pages visited in the learning environment.

Markov model

One well-known approach to making use of users' online history is to compute predictions by using Markov models. The use of Markov models in the frame of the Web has been first dedicated to the reduction of access time by pre-fetching and caching pages [START_REF] Padmanabhan | Using predictive prefetching to improve World Wide Web Latency[END_REF]. With the same goal, Box and Jenkins in [START_REF] Box | Time series analysis: forecasting and control[END_REF] estimated conditional probabilities of transitioning directly from one page to another within a given time. First order Markov models are not very accurate in predicting the user's browsing behavior since these models do not look far in the past to efficiently discriminate different histories [START_REF] Despande | Selective Markov Models for Predicting Web Pages Accesses[END_REF]. Pirolli and Pitkow in [START_REF] Pirolli | Distribution of Surfer's paths through the World Wide Web: Empirical Characterizations[END_REF] and [START_REF] Pitkow | Mining longest Repeating Subsequences to predict World Wide Web surfing[END_REF] showed that the prediction accuracy is increased when using a longer history. Higher order Markov models, also called k th order Markov models, are used to capture longer histories. Given the navigation history of size k, the probability of each resource is computed, and the resources with the highest conditional probability are recommended. The use of k th order Markov models lead to a high accuracy.

Let us notice that k th order Markov models are similar to frequent contiguous patterns of fixed size k + 1 in the case when support and confidence thresholds are set to 0. One drawback of k th order Markov models is the storage requirements; indeed, in a k th order Markov model a huge number of states are handled (this number increases according to the order of the model) [START_REF] Padmanabhan | Using predictive prefetching to improve World Wide Web Latency[END_REF]. Moreover, as with previous approaches, we are faced to a reduced coverage due to the problem of matching the active history and training data. Many approaches can overcome coverage limitation. For example, we can mention the development of Markov models of orders varying from 1 to k called the all k th order Markov model [START_REF] Nakagawa | Impact of Site Characteristics on Recommendation Models based On association Rules and sequential patterns[END_REF]. However such a model dramatically increases the complexity and storage space drawbacks.

When using Markov models, the order of navigation is taken into account and the sequences are strictly contiguous, hence, these models are not permissive. If a given user performs parallel navigations or goes to an unwanted resource (noise), the model cannot correctly handle such a behaviour and will thus reduce the size of the history considered. Such situations are handled by association rules and sequential patterns as resources are not contiguous. Moreover, when the model does not match the complete history, the most distant consulted resources are discarded for computing predictions. Thus, the most recent resources are always considered while some of them may be not important or may be navigation mistakes and should be discarded.

In the following section, we show how we can derive from Markov model an approach based on Kalman filtering and target tracking. In our approach, we based our recommendation strategy on a transformation of the web space.

Principles

Kalman filter is an optimal state estimator of a linear system [START_REF] Anderson | Optimal filtering[END_REF]. It can estimate the state of the system using a priori knowledge of the evolution of the state and the measurements. Kalman filter has main applications in control systems and target tracking.

Target tracking in the cyberspace

We consider a user who browses Web pages or online resources. Each page/resource belongs to a category (categories are related to the classification of the available resources). We then consider that one category corresponds to one dimension of a space. And the aggregation of all the dimension builds a space having as dimensions as there are categories of resources. Then, all possible categories define the geometrical structure of the space.

Knowing that to see one resource corresponds to a specific position in the space i.e., a specific vector. Successive vectors ( to ) give successive "positions" in the space. In the context of online learning, these "positions -learning positions" define the trajectory of a user in the recommender space. , m is the number of categories; each seen page belongs to one or more categories. In our case, i.e., the analysis of the learning processes on the OP4L platform, , each page coming from the tested site is classified (courses, exercises, …). The structure of each vector will be as follows. Considering that we have m possible categories for the resources, to see one specific resource is to have a vector containing "1" in the corresponding row, and "0" elsewhere.

(1)

Kalman filter: equations Hypothesis

Our main hypothesis is the following: considering that users are moving along a trajectory defined by a set of vectors, we assume that the user can be considered as a target which is described by three components in the state space, i.e., position, speed and acceleration. These three components will completely describe the dynamics of the moving users [START_REF] Anderson | Optimal filtering[END_REF], [START_REF] Box | Time series analysis: forecasting and control[END_REF], [START_REF] Gevers | Processus stochastiques, estimation et prediction[END_REF]. Thus, we choose to represent the state vector by concatenating these three components. The state vector has the following form [START_REF] Nowakowski | Automatic tracking and control for web recommendation. New approaches for web recommendation[END_REF]: [START_REF] Vassileva | Towards Social Learning Environments[END_REF] where:

• contains the components of the position vector, dimensions • contains the components of the speed vector, dimensions • contains the components of the acceleration vector, dimensions .

The dynamic of this state vector is modeled by a state space model of the following form:

(3) Matrix A includes the relationship between the position, its first and second derivations will inform us on the geometrical characteristics of the trajectory. This is the matrix form of the cinematic equation linking position to speed and acceleration. T is a parameter that introduces time in the equation. In our case, we consider T equal to 1 because time is fixed each time the user goes to another webpage. The results of the algorithm are not sensitive to T. [START_REF] Aragon | Creating social presence in online environments[END_REF] Where: Many values of parameter α have been tested. The chosen value does not influence our numerical results. and (equation ( 3)) are random noises (their properties will be given in the next section) which take into account unexpected variations in the trajectories.

Matrix H (equation ( 3)), called the measurement matrix, is structured to obtain the values of the positions in the recommender space. Thus, H will have the following structure: [START_REF] Cob | Social Presence and Online Learning: A current view from a Research Perspective[END_REF] Where:

General equations of the filter

Having the state space model (equations 3) and the structure of the state vector, we can derive the equations of the filter. First, we present some important properties of the Kalman filter:

• Information about Z is given as a Markov model i.e., Z is a linear combination of the components of X; • Estimations of X are obtained from any initial instant; • Estimations can be obtained for non-stationary process i.e., timevarying models. • and are uncorrelated white noises where and .

The Kalman filter equations comprise the following equations [START_REF]OP4L project[END_REF]: Prediction: it is the predicted state knowing past values: [START_REF] Lowenthal | Social Presence[END_REF] Kalman gain: it describes the dynamic of the filter. The dynamic takes into account the variations of the moving target. [START_REF] Beham | Recommending knowledgeable people in a work-integrated learning system, 1 st RecSysTEL workshop[END_REF] The evolution of the uncertainty on the estimation is then given by the following Riccati equation: [START_REF]OP4L project[END_REF] where the initial conditions (which initialize the filter) are given by: , [START_REF] Jovanovic | Ontologies for Effective Use of Context in e-Learning Settings[END_REF] and the state prediction is given by: The Kalman predictor will predict the future position in the recommender space i.e. the most possible category knowing the past of the user.

Recommendation strategy

The user profile is built from the list of pages visited on the platform. Each page/resource is defined by a subset of categories such as "modelling", "courses", etc.

Our new recommending strategy is based on the control loop shown in Fig. 9. This control loop will observe the difference between the estimated value of the category and the calculated category, and will integrate the controller/recommender to build the most accurate model of the user. Hence this configuration can predict where the user will "move" in the recommender space. The recommendation strategy will use the predicted position to "suggest" to the user the appropriate category of content.

Conversely to existing methods that recommend specific content items to a given user, this method performs on the macroscopic level, i.e., subspaces of specific categories. The strategy isolates the appropriate subspace and the recommendation is done in the related categories. Then, we can imagine providing a more precise recommendation by doing the second iteration of computing on the subspace (target tracking in the trajectory in the subspace and positions prediction) -a kind of zoom effect. To summarize, the recommendation is based on two arguments:

• the user's actual state of mind • a subset of retained dimensions We then have a set of items to be recommended. Furthermore, according to the pages the user has visited during the day, we can refine our recommendation.

Results

In order to model the trajectories, we have to identify categories of pedagogical resources that define the recommender space. Based on the extensive log analysis, we propose the following basic description:

• assessment (MA) • courses (C) • resources (R) • forum (F) • modelling workgroup (MW) • modelling brainstorm (MB) • modelling (M)
Each position vector is built using relation [START_REF] Atwell | Personal Learning Environments -the future of eLearning? eLearning Papers, www.elearningpapers[END_REF]. We obtain position vectors containing only "0" and "1" on the appropriate dimension.

Thus, we consider that users are moving in a 7 dimensions space. Using our approach and algorithms, we obtain the following results for a specific user (Fig. 10 and Fig. 11). In the following figures, we show one learner's trajectory along specific dimensions of the space (Dimensions MA, C and R). In both figures, X-axis represents time and Y-axis represents the seen item thus it is "0" or "1". Combining all these evolutions, we obtain trajectories in 7dimensions space. Applying our tracking algorithm, we can compute the next viewed category of contents. For example, we can show the results in one dimension, dimension R (Fig. 12). 12 shows the predicted trajectory to the real trajectory relatively to one dimension of the recommender space -dimension R. We can see that the prediction PR follows the real trajectory R. Having this prediction (PR), we can now derive a specific recommender system based on these categories. For example, using the information given by the prediction, the system can recommend specific content items to the student.

Thus, the strength of our approach is in its capability to make recommendations that consider the users' habits, i.e. give the main directions to follow knowing the trajectory in the space and not to suggest specific resources.

Conclusions and perspectives

The OP4L prototype implementing the CS-AHP algorithm has been used with several groups of students in France, Slovenia, Macedonia and Serbia during the academic year 2011-2012. It was mainly evaluated with undergraduate students in Computer Science, within a course on Software Engineering. The first results are described in [START_REF] Grandbastien | Using online presence data for recommending human resources in the OP4L project[END_REF], and the students' responses to the questionnaires used in the study do not mention any mistake in the recommended people. The CS-AHP algorithm is working. The Kalman filter approach differs from other recommendation approaches. Indeed, it focuses on users' behaviour (modeling their path in the space of the resources) to predict the categories of resources that are likely to fit their needs. In this formalism, categories of resources correspond to a subspace of the space of reference, which allows us to identify a set of resources that can meet the needs of the user. In the presented examples, we show that these trajectories can serve as monitoring tools for pedagogues.

We are thus able to describe behaviours by learning trajectories and predict what types of resources a learner is likely to access.

We would like to conclude with the perspectives offered by the learning trajectories algorithm. In particular, the approach based on learning trajectories allows one to:

• model learning process, i.e. to understand how someone learns

• recommend content items more accurately, i.e. knowing about someone's learning process we can recommend exercises, courses, self-assessment, … • develop a recommender system based on different levels of analysis, i.e. to identify the accurate subspace and then re-compute in a zoom effect to identify precisely the concerned dimensions.

Moreover, this geometrical approach opens a new field of research focused on the geometrical description of the recommender space, and how this geometry could lead to better recommendation and dynamics understanding. In the OP4L context, this approach of learning trajectory analysis will enable a better understanding of all the experiments done or to be done with students. Thus, the results obtained in the studies done in France, Macedonia, Serbia and Slovenia could be enforced and enriched by learning trajectories modelling. Moreover, this combined methodology will help us to identify the appropriate functionalities, those which are important, those which are not used, and will lead our research on the integration of others social networks (technically and also by recommending to students, some resources coming from the social network), and the definition of new ontologies related to other disciplines.

Fig. 1 .

 1 Fig. 1. Inside the prototype with recommended peers on the left

Fig. 3 .

 3 Fig. 3. Interface illustrating evaluation of other students' work.

Fig. 4 .

 4 Fig. 4. Definition of the Urgence level and Request Type at the beginning of each conversation

Fig. 5 .Fig. 6 .

 56 Fig. 5. Rating conversationsbased on Response speed and Response relevancy

Fig. 7 .

 7 Fig. 7. The PeerRec service invocation and ranking of all peers available for communication

Fig. 8 .

 8 Fig. 8. Learning trajectory

Fig. 9 :

 9 Fig. 9: Control loop for recommendation

Fig. 10 :Fig. 11 :

 1011 Fig. 10: Evolution in MA, C and R dimensions

Fig. 12 :

 12 Fig. 12: Comparison between Real Dimension R and predicted one PR

Figure

  Figures 10 and 11 show the evolution of the trajectory along the dimensions of the recommender space. Figure12shows the predicted trajectory to the real trajectory relatively to one dimension of the recommender space -dimension R. We can see that the prediction PR follows the real trajectory R. Having this prediction (PR), we can now derive a specific recommender system based on these categories. For example, using the information given by the prediction, the system can recommend specific content items to the student.Thus, the strength of our approach is in its capability to make recommendations that consider the users' habits, i.e. give the main directions to follow knowing the trajectory in the space and not to suggest specific resources.

http://op4l.fon.bg.ac.rs/

Acknowledgements:

This work was supported by the SEE-ERA Net Plus program, contract n° 115, from the European Union.