Sébastien Lefort
email: sebastien.lefort@lip6.fr

Marie-Jeanne Lesot
email: marie-jeanne.lesot@lip6.fr

Elisabetta Zibetti
email: ezibetti@univ-paris8.fr

Charles Tijus
email: tijus@univ-paris8.fr

Marcin Detyniecki
email: marcin.detyniecki@lip6.fr

Interpretation of approximate numerical expressions: Computational model and empirical study

Keywords: Approximate Numerical Expressions, Imprecision, Median Estimation, Cognitive Number Salience

Approximate Numerical Expressions (ANEs) are linguistic expressions involving numerical values and referring to imprecise ranges of values, illustrated by examples such as "about 100 ". In this paper, a general principle is proposed to interpret uncontextualised ANEs as intervals of denoted values. It is based on an empirically justified combination of characteristics of numerical values, both arithmetical and cognitive, and in particular, taking into account the cognitive salience of numbers. This general principle is instantiated in two computational models that can be extended so as to take into account the applicative context.

An empirical study is conducted to assess the performances of the two models, comparing them to state-of-the-art methods, on real interpretations collected through an on-line questionnaire. Results validate the proposed characteristics used to build the models and show that they offer the best performances in estimating the median interval chosen as representative of the collected intervals.

Introduction

Approximate numerical expressions (ANEs) are vague linguistic expressions of the general form "about x" where x is a number. They are used in daily life to denote an imprecise range of values, for instance to give imprecise pieces of knowledge regarding space (e.g., Berlin is located at about 900km from Paris), time (e.g., the meeting will last around 2 hours) or any numerical evaluation (e.g., the cost will be about 200 euros; the audience gathered around 200 participants). In the field of Human-Computer Interfaces and approximate reasoning, ANEs raise the issue of their interpretation. Indeed, as intelligent systems whose interaction mode rely on natural language become more and more present in daily life, interpreting such vague expressions is a relevant task. Application domains include database querying, such as Geographic Information Systems [? ? ] (e.g., looking for an area whose surface is about 100m 2 ) or more generally flexible queries or expert systems, such as medical expert systems [? ] (e.g., interpreting the information of patient saying he has fever since approximately 48 hours).

In the general domain of vague expressions, beyond the ones involving numerical values, Lasersohn [? ] introduces the notion of pragmatic halo as a formalisation of vagueness. While the denotation of precise expressions corresponds to the entity that is explicitly given, the denotation of vague expressions also includes entities of the same semantic type, that are implicitly conveyed by the expression. The pragmatic halo of an expression is defined as the union of the entities explicitly denoted and the implicit ones. For instance, in the sentence "I left home at 7:56am", the precise expression "7:56am" exactly denotes 7:56 and no other moment (e.g., 7:57 or 7:58). On the contrary, in the sentence "I will be there at around 6pm", the expression "around 6pm" is vague and not only denotes 6:00pm exactly, but also an implicit range of moments around 6pm (e.g., [5:45, 6:15]).

From a logical perspective, it is considered [? ] that a proposition involving a vague expression is satisfied (i.e., its truth value equals true) if the actual entity belongs to the expression pragmatic halo. Therefore, if I actually arrive at 5:49pm, the proposition "I will be there at around 6pm" is true because 5:49pm is included in the pragmatic halo of the expression "around 6pm". In this theoretical framework, interpreting an ANE is equivalent to determining the range of values that makes a proposition including this ANE true, i.e., the range of values explicitly and implicitly denoted by the ANE. For instance, the task of determining the range of distances that should be considered for the query of a hotel located at about 100km.

Three approaches have been proposed to represent the interpretation of ANEs: as fuzzy numbers (see e.g., [? ? ]); as probability distributions of possible intervals, as suggested by Ferson et al. [? ]; as intervals (see e.g., [? ? ? ]). This paper belongs to the latter framework and proposes a general principle instantiated in two computational models to interpret ANEs as intervals.

Approximate Numerical Expressions (ANEs) are a specific kind of vague expressions related to numerical expressions, named imprecise expressions by Solt [? ]. Three parts can be distinguished in an ANE: its semantic and pragmatic contexts, and the reference value. To illustrate them, let us consider the example of a car seller saying "this car is worth about 10,000 euros": the semantic context corresponds to what is evaluated, the car, and to what is counted, money. The pragmatic context is the car seller trying to sell a car. Another pragmatic context might be the buyer trying to negotiate the price. In these two cases, the interpretation of the ANE "about 10,000 euros" may be different. Finally, the reference value, or nominal value, of the ANE is the number itself: 10,000. Although contexts of ANEs affect their interpretation [? ? ? ], i.e., the intervals they denote, the models proposed in this paper are only based on their reference value and do not consider the factors related to their contexts. The aim here is to provide core models based on context-independent factors, i.e., the numerical properties of the expression, that can be extended to be instantiated in specific contexts.

The aim of this paper is to model the intervals of denoted values, corresponding to pragmatic halos, of ANEs. More specifically, the goal is to determine the values of the endpoints of the intervals I(x) = [x -, x + ] denoted by ANEs of the form "about x", where x is a non-zero natural number, i.e., x ∈ N * = N \ {0}, in an uncontextualised framework, i.e., not considering factors related to semantic and pragmatic contexts. This paper proposes two computational models, the Log-Linear Model (LLM) and the Rank Model (RKM), based on an original and empirically justified combination of characteristics of numerical values. Both models rely on a general principle whose first originality is to take into account a cognitive characteristic of ANEs, besides the arithmetical ones, capturing the notion of number salience.

A second one comes from the proposed combination method, based on Pareto frontiers. A third one is the performed experimental validation, comparing the estimation performances of both proposed models to the ones of the models from the literature, obtained on real data collected from an on-line questionnaire.

The paper is structured as follows: Section 2 is dedicated to the description of related works and existing models. The general principle and the two proposed models are presented in Section 3. The experimental setup is described in Section 4. Section 5 presents the results of the experimental study. Finally, conclusions and future works are discussed in Section 6.

Related works

This section first presents the cognitive bases of number representations in human beings as it is exposed in the cognitive psychology literature. Three computational models from the literature are then presented: the ratio model The authors also argue that the visual and verbal forms are not semantic, the meaning of numbers derive from the the analogical magnitude representation, through a mapping process. This observation implies that putting two numbers in relation, for instance in comparison tasks, can only occur at the analogical magnitude representation.

Further, the analogical magnitude representation can be represented as a logarithmically compressed mental line, where quantities are encoded according to the Weber-Fechner law [? ]. Formally, two quantities x 1 and x 2 ∈ N are distinguished if their difference is greater than a fraction c, called Weber fraction, of the larger of both, i.e., if:

|x 1 -x 2 | max(x 1 , x 2 ) > c (1)
The indistinguishability between close quantities results in imprecision when This perspective leads to consider the ANE interpretation issue as a formal problem as in the case of the Scale-Based Model [? ? ] (see Section 2.3).

For instance in the case of "about 8150", the imprecision related to the analogical magnitude representation of 8150 might be high, close to the one of 8000, leading to a wide interval of denoted values. One the contrary, with regards to the verbal or the visual representations and as a numerical fact, 8150 is a more precise description of what is counted or evaluated than 8000.

Therefore, the imprecision corresponding to about 8150 might be lower than the one corresponding to about 8000, leading to a narrower interval of denoted values. Because interpreting an ANE involves both the analogical magnitude and the verbal representation, it is reasonable to think that the interval of denoted depends on a compromise between the outputs of these representations.

The two perspectives of ANE interpretation are supported by the findings of Ferson et al. [? ]. Indeed, they showed that the order of magnitude, as a function of the ANE reference value, and the roundness, related to the number of zeros at the right of the last significant digit, are good predictors of the intervals corresponding to ANEs (see Section 2.4).

Ratio model (RM)

The first approach in modelling the interpretation of ANEs as intervals consists in defining the width of the interval of an ANE as a fixed percentage of its value [? , p. 116]. Denoting s the considered percentage, it is formally defined as:

I RM (x) = [x -x • s, x + x • s] (2)
For instance, using s = 10%, "around 300 " is interpreted as [270,330].

The ratio model (RM) is derived from the Weber-Fechner law involved in the human number cognition (see Section 2.1). Indeed, as a consequence of this law, for a given reference value x, all values whose difference to x is lower than s • x, i.e., v such that |v -x| ≤ s • x, where s is related to the Weber fraction c, are indistinguishable from x.

It can be observed that the ratio model is solely based on the reference value of the ANE. Therefore, it can provide counter-intuitive results for numbers with multiple significant digits. For instance, in the case of x = 8150, with s = 10%, one gets I RM (8150) = [7335, 8965]: the width of this interval (1630) can be considered as too high with respect to the relative magnitude of x (50). This issue, that does not occur for numbers with a single significant digit, is due to the fact that the information conveyed by numbers with multiple significant digits is more precise. The scale-based model, presented in the next subsection, addresses this issue.

Scale-based models (SBM)

The scale-based model [? ? ? ] depends on defined levels of granularity through the use of a scale system S = {s 1 , . . . , s n }, where s i are granularity levels such that s i < s i+1 . S can be complex, when the factor between two granularities is not constant, such as the time scale-system (e.g., S = {1 , 5 , 1/4h, 1/2h, 1h, . . .}) or simple, such as the decimal system (S = {1, 10, 100, . . .}), where s i+1 = 10 • s i .

The scale system represents different granularity levels for the interpretation of a numerical expression. For instance, in the decimal system, the numerical expression "1000 " can be interpreted at the 1, 10, 100 or 1000 granularity levels.

At the 1 granularity level, the expression is precise, "1000 " exactly denotes 1000, whereas at the 1000 granularity level, "1000 " denotes a larger range of values (e.g., [500,1500]).

The relevant granularity level for interpreting an ANE is context-dependent and is linguistically expressed using approximators [? ]. For instance, the approximator "exactly" refers to the lowest granularity level, usually the unit, while "about" refers to the largest granularity level the expression belongs to (e.g., the level of thousands for "about 2000 "). The latter, denoted Gran C (x), is defined as:

Gran C (x) = sup {si∈S|x mod si=0} s i (3)
The values denoted by an ANE about x are the ones that are closer to the reference value x than to any other number on the largest granularity level of the ANE. For instance, the granularity level of 2000 in the decimal system is 1000. The values denoted by "about 2000 " thus range from 1500 to 2500.

Consequently, the width of the interval equals the largest granularity level the ANE belongs to, Gran C (x). Because this model consider intervals centered around the ANE reference value, the corresponding interval is defined as:

I SBM (x) = x - Gran C (x) 2 , x + Gran C (x) 2 (4)
For instance, I SBM (300) = [250, 350] and I SBM (8150) = [8145,8155]. This approach has the advantage of taking into account the granularity of the ANE through a set of scales. However, it does not address the issue of the value of the last significant digit: all ANEs at the same granularity level result in the same interval width. Yet, one may expect that the interval of "about 100 ", for instance, would be narrower than the interval of "about 800 ".

Regression model (REGM)

The two previous models are theoretical and have not been, to the best of our knowledge, experimentally tested for the approximator "about". Unlike this theoretical approach, the empirical one proposed by Ferson et al. [? ] relies on data collected using Amazon Mechanical Turk. Participants of this study have been asked to evaluate the endpoints of intervals corresponding to semantically contextualised ANEs (e.g., "Bats make up about 20% of all classified mammal species globally."). The aim of the authors is to test the relevance of predictors of the interval width. Several approximators, all analysed through the same model, are considered, including "about". The intervals are estimated according to the following equation:

I REGM (x) = x - 10 L(x) 2 , x + 10 L(x) 2 (5)
where L(x) is computed as:

L(x) = ω 1 + ω 2 • O m (x) + ω 3 • R(x) + ω 4 • f (x) + ω 5 • O m (x) • R(x) + ω 6 • O m (x) • f (x) + ω 7 • R(x) • f (x) + ω 8 • O m (x) • R(x) • f (x) (6)
where ω 1 to ω 8 are parameters empirically set by performing a linear regression on the collected dataset. O m (x) is the ANE order of magnitude (O m (x) = log 10 (x)), R(x) its roundness, i.e., the decimal place of the last significant digit (e.g., R(13) = 1 and R(130) = 2), and f (x) ∈ {0, 1} its "fiveness". A number has the fiveness property if its last significant digit is 5.

The three variables O m (x), R(x) and f (x) have been empirically selected as the variables that make the linear regression statistically significant with the collected data [? ].

According to the authors, the regression model REGM can also be applied to test the relevance of the predictors regarding the endpoints of intervals and not only their width, although they do not report the results [? ].

This model presents the advantage of allowing the adaptation to different contexts by learning parameters on a dataset. However, it can be noted that in the conducted experiments, the semantic context is not controlled. Indeed, several semantic contexts are mixed, which may result in interactions between the factors related to the context and the ones related to the ANE reference number.

Proposed model

This section first describes the proposed general principle that provides a pool of candidates for the interval endpoint values. Its originality is twofold.

Firstly, we propose to anchor the choice of the ANE characteristics to take into account, introduced in Section 3.1, on a cognitive ground and to introduce a numerical measure of number complexity to capture the notion of number cognitive salience. Secondly, the aggregation consists in a compromise between the selected characteristics, based on Pareto frontiers, as described in Section 3.2.

Two models are then proposed to select, among the candidate values provided by the general principle, the endpoints of the intervals denoted by ANEs:

a discretised log-linear model, presented in Section 3.3, and a rank model, described in Section 3.4. Both models are designed to be extended so as to adapt to specific semantic and pragmatic contexts.

Characteristics of ANEs

The ANEs considered in this study are of the form "about x", where x ∈ N * .

The aim is to estimate I(x) = [x-∆ -(x), x+∆ + (x)], the interval corresponding to about x, which is the range of values denoted by about x. The width of I(x)

is therefore defined as |I(x)| = ∆ -(x) + ∆ + (x).

Arithmetical characteristics

In the decimal system, x can be written as x = q i=0 a i •10 i , where a i ∈ 0, 9 . Several arithmetical characteristics can be attached to a positive integer, as defined in Table 1. The magnitude is the actual value of x; its number of significant digits is noted N SD(x). The way the human analogical magnitude representation encodes quantities (see Section 2.1) leads to consider the notion of magnitude as a key factor in ANE interpretation.

The granularity Gran(x) is the order of magnitude of x, i.e., the power of ten x belongs to. This definition of granularity can be compared to the one from the scale-based model, Gran C (x) (see Section 2.3). Indeed, Gran C (x) = Gran(x) when the selected scale system is the decimal one. The granularity as

Characteristic

Formal definition Example

Magnitude

x 8150 Granularity Gran(x) = 10 i * where 10

i * = min{i|a i = 0} Last significant digit LSD(x) = a i * 5 Relative magnitude RelM ag(x) = a i * • 10 i * 50 Number of significant digits N SD(x) = q -i * + 1 3 Complexity Cpx(x) = N SD(x) -B(x) 2.5
Table 1: Characteristics of a positive natural number x = q i=0 a i • 10 i , illustrated in the case of x = 8150 in the last column. B(x), used in the definition of complexity, is defined in Eq. ( 7).

we define it is also related to the notion of roundness proposed by [? ] in the regression model (see Section 2.2): Gran(x) = 10 i * and R(x) = i * (see Table 1).

The value of the last significant digit of x is noted LSD(x). The relative magnitude RelM ag(x) is the product of granularity and the value of the last significant digit. It is meant to reflect the relevant part of x for ANE interpretation from the point of view of the verbal representation of numbers (see Section 2.1). Indeed, one may expect that the width of the interval of "about 30.000.050 " is comparable to that of "about 150 ", 50 being the common part.

Cognitive characteristic

Beyond arithmetical characteristics, we propose to introduce a complexity measure, Cpx(x), to capture the cognitive salience of numbers, i.e., the fact that some numbers are more easily evoked than others. Indeed, it has been observed from corpus analyses than some numbers occur more frequently than others [?

? ].
In the Arabic number form, two characteristics of numbers influence their frequency of occurrence. Firstly, the more significant digits a number has, the more complex it is to generate and the less frequently it occurs. Indeed, from a cognitive perspective, frequent numbers would rely on lexically stored infor-mation, while infrequent numbers would require a step-by-step generation [? ].

Therefore, at constant number of digits, the frequency of round natural numbers is observed to be higher than the frequency of non-round natural numbers [? ]. Secondly, numbers whose last significant digit is 5 and, to a lower extent, 2, occur more frequently [? ].

Taking into account these observations, we propose to define a complexity measure, a posteriori justified by the experimental results reported in Section 5, for natural numbers expressed in the decimal scale system, depending on the number of significant digits N SD(x) and the value of the last significant digit LSD(x), to capture the salience of numbers. We propose to formalise the complexity of a natural number as its number of significant digits minus a bonus if its last significant digit is 2 or 5 and its number of significant digits is at least 2. Therefore, the complexity of all natural numbers with a single significant digit is the same and equal to one.

For symmetry reasons around multiples of 10, we propose to consider the case where the last significant digit is 8 the same way as the case where the last significant digit is 2: any number whose last significant digit is 2 is of the form (10 • k + 2) • 10 i * (e.g., 320 = (30 + 2) • 10); we propose to stress its symmetrical with respect to (10 • k) • 10 i * , i.e.,the numbers whose last significant digit is 8 (10 • k -2) • 10 i * (e.g., 280 = (30 -2) • 10). This symmetry argument applied to a number whose last significant digit is 5 does not require special care as

LSD((10 • k -5) • 10 i * ) = LSD((10 • k + 5) • 10 i * ) = 5.
The experimental results described in Section 5 a posteriori justify this proposition.

The bonus function thus distinguishes three categories, depending on the value of the last significant digit LSD(x) and respecting the order of frequency of appearance:

B(x 1 ) > B(x 2 ) > B(x 3 ), for x 1 , x 2 , x 3 ∈ N * such that LSD(x 1 ) = 5, LSD(x 2 ) ∈ {2, 8} and LSD(x 3 ) / ∈ {2, 5, 8}.
Because the proposed exploitation of complexity measure is ordinal (see Section 3.2), the chosen values are only constrained by the induced order. We arbitrarily propose to set these values at 0.5, 0.25 and 0. The bonus function is therefore formalised as:

B(x) =          0.5 if LSD(x) = 5 and N SD(x) > 1 0.25 if LSD(x) = 2 or LSD(x) = 8 and N SD(x) > 1 0 otherwise (7)
Figure 1 illustrates the complexity value Cpx(x) for all integers between 500 and 600 as pluses. We consider a number as more salient than another if its complexity is lower. For instance, 500 is more salient than 501 because Cpx(500) = 1

and Cpx(501) = 3.

The Pareto frontiers based principle

This subsection presents the general principle on which the two proposed models, described in the next subsections, are based. Its aim is to provide a pool of relevant candidate values to be the endpoints of the interval denoted by an ANE.

Principle

The rationale of the proposed model is that a good interval endpoint candidate simultaneously satisfies two constraints that can be contradictory: they must be both salient (i.e., have a low complexity) and close to the ANE reference value. The possible incompatibility of these requirements can be, for instance, illustrated in the case of "about 500": 499 is closer to 500 than 450, but it is less salient (Cpx(499) = 3 and Cpx(450) = 1.5).

Therefore, optimising both distance and salience at the same time is not achievable and a trade-off must be considered: the general principle we propose relies on the assumption that when interpreting and ANE "around x", a compromise is made between the salience of the endpoint values and their distance to x. This assumption implies that for a given distance, the salience of the endpoint values is maximised; similarly, at a given salience, the distance of the endpoint value is minimised. For instance, given the ANE "about 500 ", The selected values are the ones that are not dominated by any other value.

intervals
The notion of dominance is a partial order: formally v dominates v (denoted

v ≺ v) iff d x (v ) < d x (v) ∧ Cpx(v ) ≤ Cpx(v).
They constitute the Pareto frontier, formally defined as

P e (x) = {v ∈ V e (x)/∀v ∈ V e (x) v ⊀ v}. The Pareto frontier P e (x) = [y 1 , . . . , y n ]
, with e ∈ {-, +}, is a list ordered by increasing distance to the reference value x,

∀id x (y i ) < d x (y i+1
). Note that P e (x) is a finite set because the complexity of the candidate values cannot be below 1, when they have a single significant digit, and cannot be above the number of significant digits of the ANE reference value. For instance, in the case of "about 560 ", as illustrated on Figure 1, the minimum complexity of candidates is 1 (e.g., 500 or 600) and the maximum is 3 (e.g., 559 or 561). Although there exists an infinite number of candidates with these complexity values, only one of them, the nearest to x, is selected because it dominates all the others. The values in the left Pareto frontier are in decreasing order because they are ordered according to their distance to the reference value.

In the case of x = 560, 540 is not considered as a good left endpoint candidate because it is dominated by 550, a value which is both closer and more salient than 540: 550 represents a better compromise between distance and salience.

One can notice that this model, taking into account the complexity of numbers, naturally captures the asymmetry experimentally observed (see Section 5.1). Indeed, in the case where x have several significant digits, salient numbers are not symmetrically distributed around x. This is illustrated on In contrast to the scale-based models (see Section 2.3), and like REGM, this empirical model is adaptable to different semantic contexts because the coefficients of the regression are learned from collected intervals. The principle of the model is first introduced, before the learning phase is presented.

Principle

The model estimates the interval endpoint values by means of a regression function. As suggested in Section 3.1 and in Section 2, the relative magnitude RelM ag(x) and the magnitude x of an ANE are involved in its interpretation.

We propose to use the logarithms of these dimensions as variables in the regression function. The obtained real values are then discretised to the nearest integer on the Pareto frontiers corresponding to the considered ANE.

The real value is provided by a regression function that represents the estimation of the absolute distance of the interval endpoints from the ANE reference value x, noted ∆ C (x) and computed as:

∆ C (x) = exp(α 1 • log(RelM ag(x)) + α 2 • log(x) + β) (8)
where α 1 , α 2 and β are regression parameters. Their learning phase is described in the next subsection.

At this stage, a symmetrical interval whose endpoints are real values can be constructed as

I = [x - R , x + R ] = [x-∆ C (x), x+∆ C (x)].
Because salient endpoints are more relevant than real ones, we propose to discretise them to the closest values in the Pareto frontiers, according to the following function:

D(v) = arg min yi∈P (x) |v -y i | (9)
where P (x) is the union of the two Pareto frontiers P (x) = P -(x) ∪ P + (x).

The final interval corresponding to the ANE about x is then:

I LLM (x) = [D(x - R ), D(x + R )] (10)
where x - R = x -∆ C (x) and x + R = x + ∆ C (x). Note that the resulting interval I LLM (x) maybe asymmetric because the distribution of the values on the Pareto frontiers may be different when considering the right or the left endpoint (see Figure 1).

Regression parameters

The regression parameters α 1 , α 2 , and β are learned from an experimentally collected database of intervals corresponding to ANEs, e.g., as the one described in Section 4. In a second step, a linear regression according to the least square algorithm is performed to find optimal values of α 1 , α 2 , and β, adjusting for the function deriving from Equation (8): The position of the selected endpoint is computed according to the following equation:

log(∆ e (x)) = α 1 • log(RelM ag(x)) + α 2 • log(x) + β (11)
r P (x) = round   log(RelM ag(x)) -1 + N SD(x) k=1 k   (12)
The estimation of the interval corresponding to the ANE is then

I RKM (x) = [y - r P (x) , y + r P (x) ]. r P (x) is limited in [1, n e ],
i.e., r P (x) = min(max(1, r P (x)), n e ), with e ∈ {-, +}, where n -and n + respectively are the length of P -(x) and P + (x), in the case where r P (x) < 1 or r P (x) > n e .

Experimental settings

An empirical study has been conducted to collect experimental data in the form of intervals corresponding to ANEs. The latter allow to: (i) validate the relevance of the characteristics of ANEs used in both proposed models; (ii) assess, through an experimental study, the performances of the two models proposed in this paper, and to compare them to the models from the literature.

Although the content of the questionnaire is different from the one used by [? ] to collect intervals, the collection methods is similar. Indeed, ANEs are presented to the participants, who explicitly give the endpoints values of the corresponding intervals.

The first subsection presents the material and methods used to collect and process the data. The next two subsections discuss the proposed quality criteria: endpoint prediction accuracy, relative distance, median prediction accuracy and score of median error. The last two subsections are dedicated to the experimental procedure designed to compare the models and their parameterisation.

Data collection and analysis

Material. 24 ANEs, whose reference values are listed in Table 2, are considered in the study. Their reference values are selected in order to cover different combinations of characteristics, to avoid biases towards a specific one: several values of the last significant digit at a granularity level (e.g., 20/30/40/50/80), several granularity levels at a same last significant digit (e.g., 80/800/8000), several numbers of significant digits at the same relative magnitude (e.g., 50/150/8150).

ANEs are not semantically contextualised, no cues are given to participants as to what is measured or counted.

An online questionnaire that includes the 24 selected ANEs has been designed to collect data. Instructions, given in French to the participants, are "Selon vous, entre quelles valeurs (MINIMALE-MAXIMALE) se trouve 'environ x' ? ", which can be translated as "In your opinion, what are the MINIMUM and MAXIMUM values associated with 'about x' ? ". Each participant fills the 24 items corresponding to the 24 ANEs. The order in which they are presented is randomly set for each participant.

Finally, each participant is asked to answer the two following questions, meant to control inter-individual variability:

• Use of mathematics: to examine whether a daily use of mathematics at works or in the studies may affect interpretation of ANEs, participants are asked if their work or study involves mathematical skills.

• Subjective level of mental arithmetic skills: as complementary information to the previous question, participants are asked to auto-evaluate, on a fivepoint Likert scale, whether they think they are good at mental arithmetic. x Gran(x) LSD(x) RelM ag(x) N SD(x) Table 2: Reference values of ANEs used in the questionnaire (see Section 4.1) and their characteristics: magnitude (x), granularity (Gran), value of the last significant digit (LSD), relative magnitude (RelM ag) and number of significant digits (N SD), as defined in Section 3.1.

20
Data cleaning. In order to exclude outlier pairs (minimum and maximum) from the set, data are processed according to the following three steps procedure:

Step 1. 84 intervals are considered as outliers because either:

• they are inadequate ([0, infinity])

• the right endpoint is below the reference value or the left endpoint is above the reference value, formally: Step 2. For both endpoints of each ANE, the mean and standard deviation of the data remaining after step 1 are computed. Any endpoint value beyond three standard deviations of the mean is considered as an outlier. This step leads to exclude 180 intervals.

I + p (x) < x or I - p (x) > x (e.
Step 3. 10 participants are considered as untrustworthy and are excluded because more than 70% of their interval endpoints are missing values or outliers.

From 3504 intervals I p in the original corpus, 3177 (91%) are used as dataset for the experimental validation of the two proposed models, LLM and RKM.

The next subsections describe the quality criteria we propose to perform this validation.

Endpoint prediction criteria

We propose two quality criteria to assess whether the intervals estimated by interpretation models fit the data provided by the participants of the empirical study: endpoint prediction accuracy (P A) and relative distance (RD).

X represents the set of considered ANEs and P(x) the set of participants whose interval is taken into account (i.e., not considered as an outlier) for ANE "about x". We note ∆M ed e (x) the median of the distances ∆P e p (x) over all participants p ∈ P(x) and ∆M e m (x) the absolute distance of the endpoint value (for e ∈ {-, +}) from the ANE reference value x, estimated by a model m.

Endpoint prediction accuracy

The endpoint prediction accuracy is a criterion designed to assess how well a model correctly estimates the participants' answers.

A good prediction occurs when the estimated absolute distance from the ANE reference value is equal to the one from the participant: ∆M e m (x) = ∆P e p (x). A relative error is not penalised. Two rates of relative error were tested: 10% and 5%. We observed that a rate of 5% leads to a heavy penalisation of the models that produce real numbers as estimations: the ratio model and the regression model. Therefore, we propose to set the rate of relative error at 10%. Formally, for model m, and endpoint e of the ANE about x, the prediction accuracy, to be maximised, is defined as: The global score of model m, to be maximised, is obtained by averaging P A over the two endpoints e ∈ {-, +} of all ANEs about x ∈ X :

P A(m) = 1 2 • |X | • x∈X e∈{-,+} P A(m, x, e) (14)

Relative distance

A distance measure is proposed as another way to assess the proximity of the estimated endpoint values with the ones given by the participants.

To take into account the error with regards to the characteristics of the ANE, we propose to define this distance as relative to its relative magnitude RelM ag(x). Indeed, an error of 10 units, for instance, appears as more significant for x = 100 than for x = 1000. On the other hand, the same error is considered to be as significant for 8150 as for 50. This implies that the normalisation factor should not depend on the magnitude but on the relative magnitude of the considered ANE, leading to consider |∆M e m (x) -∆P e p (x)|/RelM ag(x). Averaging over the two endpoints e ∈ {-, +} and all participants p ∈ P(x), as for P A, we propose to define RD for model m, to be minimised, as:

RD(m) = 1 2 • |X | • x∈X e∈{-,+} 1 |P(x)| • p∈P(x) |∆M e m (x) -∆P e p (x)| RelM ag(x) (15)

Median prediction criteria

As detailed in Section 5.1, the empirical study shows that no consensus can be observed regarding the interpretation of ANEs, and that a high variability occurs in the collected intervals. It is thus relevant to discuss the definition of the interval an ANE interpretation model should aim at. The first subsection details the motivation of the proposed approach, that focuses on the median of the observed endpoint values. The next two subsections introduce the quality criteria designed to assess whether this goal is reached.

Rationale of the median interval

As vague expressions, the interval denoted by an ANE has no sharp endpoints: the transition between acceptable intervals and unacceptable ones is progressive and might be variable.

One can distinguish between objective and subjective aspects of variability in endpoints. Firstly, from an objective point of view, variability in interpretation across participants can be observed (see Section 5. To overcome these issues, we propose to model the endpoints of the interval corresponding to an ANE as a statistical distribution. However, the quantity of training data to learn relevant models of this type might be huge. Therefore, we propose to estimate parameters of this distribution, such as the mean or the median.

Estimating the median interval appears to be relevant in both aspects of progressiveness and variability: indeed, it corresponds to a value given by one or several participants and is therefore not an arbitrary value. As observed in the empirical study (see Section 5.1), participants tend to give salient numbers as endpoint values: estimating the median leads to naturally produce such values that make sense to the users. Moreover, the median is a robust parameter with regards to extreme values. On the contrary, the mean may result in precise values, less natural to human users, while being less robust with regards to extreme values.

The next subsections present the quality criteria we propose to assess whether the estimations provided by interpretation models fit the observed medians.

Median prediction accuracy

We first propose to use the accuracy score of the median prediction as a quality criterion of media estimation.

We consider a median estimation ∆M e m (x) of model m for endpoint e ∈ {-, +} of ANE "about x" as correct if the error, as for P A, does not exceed 10% of the actual median. This quality criterion, to be maximised, can be formalised as:

M A(m) = 1 2 • |X | x∈X e ∈ {-, +} |∆M e m (x) -∆M ed e (x)| ∆M ed e (x) ≤ 0.1 (16)

Score of median error

In case of incorrect median estimation, one should be able to evaluate the degree of error, i.e., the distance between the prediction and the actual median.

To address this issue, we propose a criterion that assesses the balance between participants' answers that are above and below the estimated endpoint value.

These quantities can be formally defined as The score of the model then depends on the absolute difference between N + and N * + and the absolute difference between N -and N * -. Averaging over the two endpoints and all considered ANEs, the median error, to be minimised, can be defined as: A cross-validation procedure is performed on benchmarks consisting in 1000 runs of learning and test steps, where all models, whether they require a learning step or not, are evaluated against the same test dataset:

N + (x, e) =
M Err(m) = 1 2 • |X | • x∈X e∈{-,+} |N + (x,
1. Participant: the learning step of REGM and LLM is performed on the intervals given by 75% of randomly selected participants, the intervals from the remaining 25% constitute the test dataset. This benchmark is meant to assess whether the models that need a learning step are able to generalise across participants.

2. ANE: the learning step of REGM and LLM is performed on the intervals given by all participants on 17 (66.7%) randomly selected the ANEs.

The intervals of all participants corresponding to the 7 remaining ANEs are used as test dataset. This benchmark is designed to measure the robustness of the log-linear model because the regression parameters are estimated on only 17 • 2 = 34 points. It is also designed to evaluate the generalisation ability of models that need learning across ANEs.

The ratio model takes into account only the magnitude of the ANEs, one can expect that it deals betters with numbers with a single significant digit, for which RelM ag(x) = x (e.g., x = 8000, RelM ag(8000) = 8000), than numbers with multiple significant digits, for which RelM ag(x) = x (e.g., x = 8150, RelM ag(x) = 50). Similarly, scale-based model uses the relative magnitude but not the magnitude. One can therefore expect that this model better deals with multiple significant digits than with a single significant digits. To avoid biases towards models that are advantaged by reference values with a single significant digit only, or reference values with multiple significant digits only, a constraint is included in the random selection: the learning and test sets must include a mix of both types.

In order to determine which model shows the best performances in each benchmark, statistical analyses using ANOVA tests with model as factor, and Tukey's HSD post-hoc tests [? ] are performed. The significance threshold is set at p = .01.

Parameterisation of considered models

Several combinations of parameters are tested for the parameterised models.

The results reported in Section 5 are the ones obtained with the parameters that give the best scores on the four criteria. Indeed, it has been observed that changing the parameters of a model results in a change in the same direction for all four scores. RM gives the best performances with parameter s = 5%. For SBM, the decimal system S = {1, 10, 100, . . .} is the one that better fits the data.

REGM only provides the size of the intervals and no information about their locations or symmetry around the ANEs. We make the assumption that they are centered around the considered ANEs.

In the learning step, LLM requires the choice of a representative of the intervals corresponding to an ANE. In order to account for the variability observed in the collected intervals (see Section 5.1), we propose use the median as aggregation operator of the intervals to constitute the training triples used be the proposed Log-Linear Model (see Section 3.3)

Preliminary validation of the regression model

As a preliminary validation of REGM, the linear regression proposed by [? ] (see Eq.( 6), Section 2.4) was performed on the overall set of the collected data. Table 3 presents the coefficients obtained and the ones reported by the authors [? ]. Four variables of the regression on seven appear to be statistically significant on the considered data: order of magnitude (O m (x), associated to ω 2 ), roundness (R(x), associated to ω 3 ), product of order of magnitude by roundness (O m (x) • R(x), associated to ω 5 ), and product of roundness by fiveness (R(x) • f (x), associated to ω 7 ). The coefficient of determination is much lower on the collected data (R 2 = 0.272) than the one reported by the authors (R 2 = 0.741). These results suggest that the REGM proposed by [? ] does not fit well the data. This may be due to the difference in the contexts or in the language used to collect the data. Indeed, while

Experimental results

This section presents the results of the experimental study. The first subsection briefly introduces the analyses performed on the collected data in order to test the hypotheses concerning the relevant characteristics of ANEs and the distributions of the intervals endpoints. The next subsection deals with the results of the performance assessment of the five interpretation models.

Data description and analyses

Statistical analyses were performed on the data in order to check whether the hypotheses on which the two proposed models are designed are relevant.

The main conclusions are reported here.

Characteristics of ANEs. Results of statistical analyses, not detailed here for space reasons, show that the magnitude, the granularity, the value of the last significant digit and the number of significant digits influence the width of the intervals. More specifically, the greater the magnitude, the granularity, the value of the last significant digit or the number of significant digits of an ANE reference value, the wider its interval.

Variability in the intervals. It is observed that participants tend not to agree on the intervals denoted by ANEs: on average, the 136 respondents give 15.4 different answers per endpoint. The number of different values for each endpoint ranges from 9 ("about 20 " for left endpoint) to 22 ("about 8150 " for right endpoint).

Symmetry of intervals. The symmetry of the collected intervals largely depends on the considered ANE. Indeed, even if on the whole dataset, 74.2% of the intervals are centered around the reference number, intervals of some ANEs, such as 440 or 4730, are significantly less often symmetric (63% and 50% respectively).

This observation can be explained by the fact that these ANEs are less round, i.e., have more significant digits, than others (e.g., 500, 8000) and do not satisfy the fiveness property. Indeed, the collected data show that participants tend to give salient numbers whose complexity is low (e.g., 400 and 450 for 440; 4700 and 4750 for 4730), as endpoint values. Because these salient numbers may be at different distances from the ANE reference value, the resulting intervals may be asymmetric.

However, most endpoint values (84.4%) are included in the Pareto frontiers as defined in Section 3.4, providing an empirical validation of our general principle.

Summary. The results empirically support our choices concerning the characteristics of the ANEs used in the Pareto frontiers model and the two estimators LLM and RKM. Indeed, relative magnitude, as a combination of granularity and the value of the last significant digit, and the number of significant digits appear to be key factors in the interpretation of uncontextualised ANEs. The notion of complexity seems relevant as participants tend to minimise the complexity of the numbers corresponding to the endpoint values they give. Finally, the variability observed in the data suggests that a single interval per ANE cannot satisfy all participants. This variability therefore supports the view of a distribution representation approach of intervals endpoints and the relevance of estimating the median interval as a meaningful parameter of the distribution.

Scores on

Results on the benchmarks

Tables 4 and5 present the performances of the five models, Figures 2 and3 graphically illustrate them. Results are similar in both the Participant and the ANE benchmarks, in which the proposed RKM offers the best performances.

The next subsections discuss the results of each criterion in turn.

Endpoint prediction accuracy. The scores obtained by LLM and RKM are the best ones and do not statistically differ from each other according to the ANOVA tests performed. These scores can be compared to the baseline, which is calculated as the mean relative frequency of the endpoint values mode (see Section 4.2): 28.1% (σ = 6.9). The high scores of LLM and RKM can be due to the fact that the intervals they provide can be asymmetric while the three other models provide intervals centered around the ANE reference value.

RM and REGM perform poorly on endpoint prediction accuracy. This can be explained by the fact that they tend to make real-numbered estimations while participants tend to give integers as interval endpoints. Moreover, RM is well suited for numbers with a single significant digit only, resulting in poor performances concerning numbers with multiple significant digits.

Relative distance. The ANOVA tests reveal a significant effect of the model factor on the RD scores. More precisely, post-hoc analyses show that this difference is due to RM and that scores of other models do not differ significantly.

Like for endpoint prediction accuracy, because RM does not take into account the relative magnitude of the ANE reference value, it shows poor performances for ANEs with multiple significant digits. These performances are all the lower as the difference between the magnitude and the relative magnitude of an ANE reference value is high. Indeed, the highest relative distances occur for "about 4730 " (RD = 6.92) and "about 8150 " (RD = endpoint prediction accuracy, this can be due to the fact that the model provides real-numbered estimations while participants tend to give round numbers.

However, the error score reveals that these real-numbered estimations are closer to the actual medians than the estimations provided by SBM and RM. On the contrary, SBM performs better than REGM on prediction accuracy while the prediction errors are more important.

Differences between ANEs with a single and with multiple significant digits. We performed complementary analyses, not detailed here, to check whether the global performances of the models presented above are also valid when distinguishing between ANEs with a single and multiple signifcant digits: results reveal very similar scores of LLM, RKM and REGM, for ANEs with a single and multiple significant digits in the Participant benchmark.

RM performs well on ANEs with a single significant digit and poorly on ANEs with multiple significant digits while SBM presents the opposite profile.

These results are consistent with the design of the two models: RM takes into account only the magnitude of an ANE, while SBM only considers its relative magnitude.

Concerning the ANE benchmark, one can observe a difference between ANEs with a single and with multiple significant digits, for all models although the difference is less marked for RKM. Indeed, they perform more poorly on ANEs with multiple significant digits than with a single one. Moreover, standard deviations are higher for ANEs with multiple than with a single significant digit.

The difference between the Participant and the ANE benchmarks may be due to the random selection of ANEs in the latter. Indeed, in this benchmark, the models are evaluated on only 7 randomly selected ANEs. A single error thus implies a 14% contribution to the score which quickly leads to poor performances.

Moreover, as noted for the Participant benchmarks, RM and SBM are more or less adapted to reference values with a single or multiple significant digits.

Their scores therefore highly depends on the random draw of ANEs in the test dataset. Concerning REGM and LLM which require a learning step, the training dataset is reduced in the ANE benchmark to 34 triples while they learn on 48 triples in the Participant benchmark. The regression parameters may therefore be less robust in the ANE benchmark.

One can conclude from these results that the generalisation ability of the models is better across participants than across ANEs. The learning datasets should be carefully constituted to cover a large combination between dimensions of ANEs.

Conclusion and future works

To estimate the intervals corresponding to Approximate Numerical Expressions, this paper proposes two computational models, the Log-Linear model and the Rank model. Both are based on a common underlying general principle that takes into account the cognitive salience of numbers and exploit Pareto frontiers.

An experimental study is conducted, using real data collected from an online questionnaire, which supports the proposed general principle: results reveal that the two models exploiting this principle offer the best performances in the four quality criteria we propose, especially in median prediction accuracy.

The performed analyses also show that there is no consensus across participants with regard to the intervals corresponding to an ANE. The approach proposed in this paper therefore consists in estimating the median as the targeted statistical parameter of the endpoints distributions rather than a single interval that should satisfy all participants.

Moreover, the experimental validation of the two proposed models supports the relevance of the cognitive salience, among arithmetical characteristics of ANEs, as a key factor in ANE interpretation.

Future work will include extensions of the proposed model, in particular so as to take into account the two other factors implied by ANE interpretation:

the semantic and pragmatic contexts [? ]. Indeed, the general principle based on Pareto frontiers can be extended so as to include other variables, related to the context in the compromises step. For instance, in a business context, beyond the arithmetical and cognitive dimensions, a third one can be added to represent the role of the user: seller or buyer. Indeed, the right endpoint of the interval corresponding to the ANE "about 10.000 euros" should be closer to the reference value than the left endpoint if the user is a buyer, representing the fact that he prefers lower prices. Conversely, the right endpoint of the interval should be farther from the reference value than the left endpoint is the user is a seller, representing the fact that he prefers higher prices. This can be achieved by proposing a third criterion to capture this context-dependent dimension, in the same way as the complexity dimension that has been proposed to capture the cognitive characteristic. The Pareto frontiers can then be generalised to address three-criteria optimisation problems. A second approach concerns more specifically the Log Linear model: as it relies on a learning step, it can fit to specific contexts by constituting dedicated learning datasets.

Another possible extension of the Pareto frontiers model consists in generalising the complexity of numerical expressions so as to take into account real-numbered ANEs (e.g., "about 3.14 ") and scales of measurement, such as time (seconds, minutes, quarters of hours, etc.) or space (centimeters, meters, kilometers, etc.), to interpret expressions based on more complex scale systems (e.g., "I will be there in approximately half an hour " or "This rope is approximately one meter long"). The second approach consists in representing ANEs as probability distributions.

One can either, as suggested by Ferson et al. [? ], model the residual variation by such distributions, or directly the whole distributions. However, this approach may need a significant number of ANEs and participants to be able to properly characterise the factors and the parameters involved in any ANE interpretation.

Finally, in this study, participants were asked to explicitly give the intervals corresponding to ANEs. It may be that implicit interpretation of ANEs differs from the explicit one. Future work should address this issue, by collecting intervals implicitly, and testing the proposed models, to ensure that there is no bias in collecting data using explicit questionnaires.

(

  RM) [? ], the scale-based model (SBM) [? ? ], and the regression model (REGM) [? ].2.1. Cognitive number representation in human beings Insofar as ANEs involve numbers, it is first relevant to examine the literature dedicated to the way the human cognitive system encodes and represents numbers and quantities. Based on clinical evidences, Dehaene and Cohen [? ] propose a triple code functional model of numerical cognition. They argue that three categories of mental representations are used to manipulate numbers, depending of the task: the visual Arabic form, involving strings of digits (e.g., 36), the verbal word frame, involving sequences of words (e.g., thirty-six), and the analogical magnitude representation, corresponding to the numerical value. In healthy persons, all three categories of representations can be transcoded to any other [? ]. The type of representation used depends on the nature of the executed task. Those involving arithmetic facts (e.g., 242+34) rely more heavily on the verbal representation, while rounding or approximation tasks involve the analogical magnitude representation [? ].

  estimations are performed. The value of the Weber fraction c varies from one individual to another and with age [? ]. The average value of the Weber fraction would be around 0.11 [? ] or 0.12 [? ].From this triple code functional model of numerical cognition, two ANE interpretation perspectives can be postulated. Firstly, interpreting an ANE consists in intuitively estimating the imprecision implied by its analogical mag-nitude representation. This perspective leads to consider the reference value of the ANE as the relevant factor since it defines the analogical magnitude representation. Secondly, as linguistic numerical expressions, ANEs involve the verbal or the visual Arabic representations, depending on the presentation modality.

3. 2 . 2 .

 22 Building the Pareto frontiers To build one Pareto frontier, we consider as candidates all values v ∈ V e (x) ⊂ N, with e ∈ {-, +} and V -(x) = [1, x) and V + (x) = (x, +∞). These values are compared on two criteria: (i) their absolute distance from the ANE reference value: d x (v) = |v -x|; (ii) their complexity Cpx(v), capturing their salience.

Figure 1

 1 Figure 1 illustrates the two Pareto frontiers corresponding to the ANE "about 560": the black plus signs represent the distribution of (v, Cpx(v)) couples for all values v in the interval [500, 600] (indeed, values lower than 500 or greater than 600 are not relevant as they are all dominated, by 500 and 600 respectively). Note that the horizontal axis representing the candidates v also indirectly represents their distance d x (v): the latter increases linearly from the vertical green line representing x = 560. Thus, the part to the right of the vertical line can also be seen as showing the distribution of (d x (v) + x, Cpx(v)); the part to the left shows the the distribution of (x -d x (v), Cpx(v)) P -(560) and P + (560) are represented by the connected points circled in red, respectively below and above 560, denoted by the green vertical line. As a result, P -(560) = [559, 558, 555, 550, 500] and P + (560) = [561, 562, 565, 570, 580, 600].

Figure 1 Figure 1 :

 11 Figure 1 for x = 560, where 500 is farther from 560 than 600. The two models described in the next subsections exploit the values from the Pareto frontiers as candidates to estimate the endpoint values of the denoted interval.

1 .

 1 The data are first transformed to a set of training triples (x, RelM ag(x); ∆ e (x)), where e ∈ {-, +}. So as to distinguish between the left and right endpoints, we define two training triples per ANE, (x, RelM ag(x); ∆ -(x)) and (x, RelM ag(x); ∆ + (x)), where ∆ -(x) is distance between the ANE reference value x and the representative of the left endpoints of its corresponding interval and ∆ + (x) the distance between x and the representative of the right endpoints. These two triples per ANE are obtained by aggregating all the learning intervals corresponding to the considered ANE. The choice of the aggregation operator is specified in Section 4.5.

3. 4 .

 4 Rank model (RKM)The second model, RKM, exploits the Pareto frontiers values in a different way: it consists in estimating directly the position of the endpoint values on the Pareto frontiers. To that aim, it selects y r P (x) ∈ P e (x) = [y 1 , . . . , y n ] endpoint value and the issue is to estimate the relevant rank r P (x). The latter depends on the relative magnitude RelM ag(x) and the number of significant digits N SD(x) of the considered ANE about x. The use of the number of significant digits in Equation (12) relies on the observation that, in the collected data, the interval width increases with the number of significant digits (e.g., |I(8150)| > |I(50)| and N SD(8150) > N SD(50), see Section 5.1).

Population.

  Participants were recruited through an announcement diffused on mailing-lists. 146 adults volunteered to freely take part in the study, 102 women and 44 men, aged 20 to 70 (mean= 38.6; standard deviation= 14.2). All were native French speakers. Data preprocessing. The answer given by participant p for the ANE "about x" is noted I p (x) = [I - p (x), I + p (x)]. The answers are used to compute two absolute distances ∆P e p (x) between the ANE reference value x and the endpoints e ∈ {-, +} of the interval: ∆P e p (x) = |I e p (x) -x|. Consequently, I p (x) = [x -∆P - p (x), x + ∆P + p (x)] and |I p (x)| = ∆P - p (x) + ∆P + p (x). The analyses reported in the following sections are based on the absolute distances. This variable is more relevant than the interval width, because it allows to compare intervals together, without losing the symmetry information. Indeed, two widths may be equal although the values of the endpoints are different (e.g., |[95, 110]| = |[90, 105]|).

  g., I(800) = [700, 750] or I(800) = [810, 850]) • an endpoint is greater than ten times or lesser than one tenth the reference value, formally I + p (x) > 10 • x or I - p (x) < x 10 (e.g., I(100) = [9, 1101])

  considered in this paper produce a single estimation per endpoint, the maximum score a model can reach corresponds to the relative frequency of the endpoint values mode over the participants.

  |{p ∈ P(x)|∆P e p (x) > ∆M e m (x)}| and N -(x, e) = |{p ∈ P(x)|∆P e p (x) < ∆M e m (x)}|. The model m should be such that N + (x, e) = N -(x, e) for all x, e. However, because participants' intervals endpoints are not uniformly distributed, but rather distributed on few values, a perfect balance may not be possible. For instance let's consider the case of the ANE "about 100 ", the distribution, over 60 intervals, of the left endpoint values might be 20 times 40, 30 times 45 and 10 times 48. The median of this distribution is ∆M ed -(100) = 45. Even if the model correctly predicts this median value, a perfect balance is not achieved because 20 endpoint values are below and 10 are above. The score would therefore not reflect the fact that the median is actually correctly estimated. To overcome this issue, we propose that the balance score of estimation takes into account the balance of the actual median, i.e., the two quantities N * + (x, e) = |{p ∈ P(x)|∆P e p (x) > ∆M ed e (x)}| and N * -(x, e) = |{p ∈ P(x)|∆P e p (x) < ∆M ed e (x)}|.

  e) -N * + (x, e)| + |N -(x, e) -N * -(x, e)| |P(x)| (17) 4.4. Experimental procedure Using the four quality criteria P A, RD, M A and M Err described above and the empirically collected intervals as data, we compare the performances of the two models using the Pareto frontier principle we propose (see Section 3): the log-linear model (LLM) and the rank model (RKM), to the ones of the three models from the literature discussed in Section 2: the ratio model (RM) [? ], the scale-based model (SBM) [? ? ] and the regression model (REGM) [? ].

Figure 2 :

 2 Figure 2: Mean scores of models in Participant benchmark: endpoint prediction accuracy P A (top, left), relative distance RD (top, right), median prediction accuracy M A (bottom, left), and median error M Err (bottom, right). It can be observed that the two proposed models, RKM and LLM, show the best performances while the Ratio model (RM) offers the worst, in each criterion.

Figure 3 :

 3 Figure 3: Mean scores of models in ANE benchmark: endpoint prediction accuracy P A (top, left), relative distance RD (top, right), median prediction accuracy M A (bottom, left), and median error M Err (bottom, right). The two proposed models, RKM and LLM, show the best performances in each criterion.

  Two other representations of ANEs should also be investigated to model the variability in the collected intervals. The first one is proposed by Lakoff [? ] and consists in using fuzzy numbers as representation of ANEs. From this point of view, the median of the endpoints distribution is of particular interest because it corresponds to the 0.5-cut of the fuzzy numbers in the random set view of elicited membership functions proposed by Bilgiç and Türkşen [? ]. Moreover, values located on the Pareto frontiers can be used to characterise other α-cuts [? ].

Table 3 :

 3 . Coefficients of the REGM linear regression (see Eq.(6), Section 2.4), reported by Ferson et al. [? ], and obtained on the collected data. Bold values are the significant ones in the regression performed on the collected data.

	Parameter Reported by [? ] Obtained on collected data
	ω 1	-0.208	0.375
	ω 2	0.428	0.220
	ω 3	0.281	-0.00485
	ω 4	0.0940	0.977
	ω 5	0.0147	0.0807
	ω 6	-0.0640	-0.244
	ω 7	-0.0102	-0.496
	ω 8	0.0404	0.138

  Ferson et al. [? ] embed ANEs in several semantic contexts, the ANEs of this study are not contextualised.

	linear regression coefficients for REGM. This can be explained by the fiveness
	variable f (x), a property which only occurs five times in the collected corpus:
	some learning sets do not offer enough ANEs with this property to correctly
	estimate coefficients related to this variable. Runs in which such aberrant coef-
	ficients are observed, resulting in estimated interval endpoints located at at least
	10.000 units from the ANE reference value, are not included in the analyses. On
	1000 runs, 88 lead to aberrant fiveness coefficients. Therefore, the means and
	standard deviations of REGM performances in the ANE benchmark are based
	on 912 runs.
	Moreover, the participants of this study are native French speakers while the
	workers of the Ferson et al. [? ] study are English speakers.
	Further, some random selections in the ANE benchmark result in aberrant

Table 4 :

 4 Score means and standard deviations in parentheses of the four criteria for each model on the Participant benchmark. Bold scores are statistically the best ones according to the ANOVA tests. Several scores are bold if no significant difference is revealed by post-hoc analyses.

Table 5 :

 5 Score means and standard deviations in parentheses of the four criteria for each model on the ANE benchmark. Bold scores are statistically the best ones according to the ANOVA tests. Several scores are bold if no significant difference is revealed by post-hoc analyses. = 8150), because it does not consider the ANE magnitude. The non significant difference between the score of this model and the others can thus be explained by the low number of such ANEs in the corpus.

	Median prediction accuracy and median error. Statistical analyses reveal that
	RKM shows the best performances, both in median prediction accuracy, and in
	median error, providing an empirical validation of our proposed model.

with the highest difference between relative magnitude and magnitude, while the scores for other ANEs range from 0.06 to 0.73. A twofold explanation can account for the statistically non-significant difference in the relative distance scores of the four other models. Firstly, REGM and LLM are regression-based models. Although not based on the same ANE characteristics, they both aim at minimising the distance between observed data and the linear regression, leading to estimations that are close to the collected intervals. Secondly, for SBM and RKM, results reveal differences for specific ANEs, depending on the characteristics taken into account by the model: SBM shows poor performances in high relative magnitude and high magnitude ANEs (e.g., x REGM offers a poor prediction score but an average error score. As for


Samuel Beaulieu

Antoine Comby

Dominique Descamps

Baptiste Fabre

Gustavo A Garcia

Romain Géneaux

Alex G Harvey

Francois Légaré

Z Mašín

Laurent Nahon

A F Ordonez

S Petit

B Pons

Y Mairesse

O Smirnova

V Blanchet

Photoexcitation Circular Dichroism in Chiral Molecules

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

many

Chirality is ubiquitous in nature and fundamental in science, from particle physics to metamaterials. The most established technique of chiral discrimination -photoabsorption circular dichroism -relies on the magnetic properties of a chiral medium and yields an extremely weak chiral response. We propose and demonstrate a new, orders of magnitude more sensitive type of circular dichroism in neutral molecules: photoexitation circular dichroism. It does not rely on weak magnetic effects, but takes advantage of the coherent helical motion of bound electrons excited by ultrashort circularly polarized light. It results in an ultrafast chiral response and the efficient excitation of a macroscopic chiral density in an initially isotropic ensemble of randomly oriented chiral molecules. We probe this excitation without the aid of further chiral interactions using linearly polarized laser pulses. Our time-resolved study of vibronic chiral dynamics opens a way to the efficient initiation, control and monitoring of chiral chemical change in neutral molecules at the level of electrons.

The macro-world gives us many examples of chiral dynamics created by helical structures which convert rotations in a plane into translational motion orthogonal to it, from the Archimedes screw to plane propellers and household fans. In the micro-world, the electrons bound inside chiral molecules should develop a similar helical motion when excited by planar rotation of the electric field of circularly polarized light. Electronic excitation by circularly polarized light has been used to distinguish right-handed from left-handed molecules since 1896 [START_REF] Cotton | Recherches sur l'absorption et la dispersion de la lumire par les milieux doués du pouvoir rotatoire[END_REF] . The technique, called the photoabsorption circular dichroism (CD) [START_REF] Barron | Molecular light scattering and optical activity[END_REF] , is based on the difference in the absorption of leftand right-circularly polarized light in chiral molecules and remains the go-to tool [START_REF] Berova | Circular dichroism: principles and applications[END_REF] for analysing properties of biological molecules, providing indispensable information on their structure, kinetics and thermodynamics, interaction with the environment and with other molecules. However, it does not rely on the helical nature of bound electron currents 4 , but uses the helical pitch of the light wave instead. This pitch, given by the wavelength of the absorbed light, λ 2500 Å (1 Å = 10 -8 cm), is barely noticeable on the molecular scale of ∼ 1 Å, leading to very weak signals, three to four orders of magnitude less than the light absorption itself. Formally, the chiral-sensitive part of the light-induced excitation requires the excited electrons to respond to both the electric and the magnetic field of the light wave 5 , see Fig. (1a).

Remarkably, in spite of extraordinary recent advances in developing new methods for chiral discrimination that do not rely on the magnetic properties of the medium [7][8][9][START_REF] Lux | Circular dichroism in the photoelectron angular distributions of camphor and fenchone from multiphoton ionization with femtosecond laser pulses[END_REF][START_REF] Lehmann | Imaging photoelectron circular dichroism of chiral molecules by femtosecond multiphoton coincidence detection[END_REF][START_REF] Pitzer | Direct determination of absolute molecular stereochemistry in gas phase by coulomb explosion imaging[END_REF][START_REF] Herwig | Imaging the absolute configuration of a chiral epoxide in the gas phase[END_REF][START_REF] Patterson | Enantiomer-specific detection of chiral molecules via microwave spectroscopy[END_REF][START_REF] Yachmenev | Detecting Chirality in Molecules by Linearly Polarized Laser Fields[END_REF][START_REF] Comby | Relaxation Dynamics in Photoexcited Chiral Molecules Studied by Time-Resolved Photoelectron Circular Dichroism: Toward Chiral Femtochemistry[END_REF][START_REF] Nahon | Valence shell one-photon photoelectron circular dichroism in chiral systems[END_REF][START_REF] Nahon | Determination of accurate electron chiral asymmetries in fenchone and camphor in the vuv range: Sensitivity to isomerism and enantiomeric purity[END_REF] , none has relied on detecting the helical motion of bound electrons. Is it possible to excite and probe such motion? We demonstrate both theoretically and experimentally that one can (i) induce chiral stereodynamics of bound electrons without the help of magnetic field effects, a new phenomenon we call PhotoeXcitation Circular Dichroism (PXCD); (ii) probe it with linearly polarized light without the help of the further chiral interactions that are usually presumed to be a prerequisite for chiral discrimination. Coherent excitation substitutes further chiral interactions at the probe step by coupling to a different quantum state of the same chiral molecule.

Exciting chiral dynamics in bound states

A hallmark of helical motion of bound electrons is the appearance of an induced dipole orthogonal to the polarization plane of the exciting circular light. We first show that an ultrashort pulse creates such a dipole in a randomly oriented molecular ensemble. Let the electric field of the pulse, rotating in the x-y plane, coherently excites two states (Fig. 1b) of a chiral molecule. As shown in the Supplementary Information (SI), the orientation-averaged induced dipole acquires the desired component along the light propagation direction z:

d P XCD z ∝ σ[ d 01 × d 02 ] d 12 sin(∆E 21 t), (1)
Here d 01 , d 02 and d 12 are the dipole transition vectors connecting the ground |0 and the two excited states |1 , |2 (Fig. 1b), ∆E 21 is the energy spacing between the excited states. For more than two states, Eq.(1) will contain the sum over all pairs of excited states n, m, leading to oscillations at all relevant frequencies ∆E nm . As a function of time the induced dipole vector maps out a helix (Fig. 1b) and the z-component of the helical current is

j P XCD z ∝ σ[ d 01 × d 02 ] d 12 ∆E 21 cos(∆E 21 t). (2)
Both d P XCD z and j P XCD z are quintessential chiral observables (see e.g. [START_REF] Barron | True and false chirality and absolute asymmetric synthesis[END_REF][START_REF] Tang | Optical chirality and its interaction with matter[END_REF] ). Indeed, both are proportional to the light helicity σ = ±1 and to the triple product of three vectors

[ d 01 × d 02 ] d 12 .
This product presents a fundamental measure of chirality: it changes sign upon reflection and thus has an opposite sign for left and right enantiomers. For randomly oriented non-chiral molecules

d P XCD z = j P XCD z = 0.
Eqs.(1,2) lead to the following conclusions. First, the coherent excitation of electronic states leads to a charge displacement in the light propagation direction. Hence, a macroscopic dipole The oscillations of the PXCD signal Eqs. (1,2) appear to suggest that probing it requires the combination of ultrafast time resolution and chiral sensitivity. We now show that time-resolving PXCD does not, in fact, require a chiral probe. The coherence underlying PXCD allows a chiral object to 'interact with itself', albeit in a different quantum state, thus mimicking interaction with "another chiral object" and removing any need for other chiral interactions during the probe step.

d P XCD
One such non-chiral probe, termed PhotoeXcitation-induced photo-Electron Circular Dichroism (PXECD), is introduced below.

Probing chiral dynamics in bound states

One way to probe the excited chiral density is to promote the chiral wave-packet to the electron continuum using a linearly polarized pulse (Fig 1c). As shown in the SI, the standard photoionization observable, the photoelectron current averaged over molecular orientations, is: First, the electron current Eq. ( 3) is proportional to the helicity σ of the pump pulse. Second, as transitions to the continuum are described by complex dipole vectors, it contains two triple products. Just like the triple product

J P XECD z (k) = σ[ d 01 × d 02 ] D r 12 (k) sin(∆E 21 τ ) -σ[ d 01 × d 02 ] D i 12 (k) cos(∆E 21 τ ), (3)
[ d 01 × d 02 ] d 12 earlier, both [ d 01 × d 02 ] D r 12 and [ d 01 × d 02 ] D i
of the pump pulse or of the enantiomer is swapped, showing that PXECD is a genuine chiral effect.

The chiral nature of the response arises only if the participating bound states are coherently excited.

Once the coherence is lost, the chiral signal will also disappear.

Importantly, the state of the continuum (Fig 1c ) does not need to be chiral, as it only provides a link between the two chiral bound states. J P XECD z (k) remains chiral even for a plane wave continuum (see the SI), in this case D 12 (k) only has an imaginary component:

J P XECD z,P W (k) = -σ[ d 01 × d 02 ] D i,P W 12 (k) cos(∆E 21 τ ). (4)
The total photoelectron current J P XCD tot = J P XECD z,P W (k)dk measures the helical current excited in bound states j P XCD z (Eq. 2) distorted by the partial alignment of the molecular ensemble induced by the pump (see the SI). One might think that partial alignment of the excited molecular ensemble could already be fully responsible for enabling non-chiral probes of chiral dynamics. It is not true in our case. Indeed, the effect of alignment persists for a single excited electronic state and for the two excited electronic states with collinear dipoles, but in both cases it leads to zero PXECD current. Finally, removing the effect of partial alignment from Eq.(4) shows that the PXECD current remains chiral for every k, while J P XCD tot becomes directly proportional to the chiral component of the helical current in bound states:

J P XCD tot ∝ j P XCD z (see the SI).
Probing the created chiral excitation using photo-electron imaging with linearly polarized light constitutes yet another new phenomenon, PhotoeXcitation-induced photoElectron Circular Dichroism (PXECD). PXECD is reminiscent of the Photoelectron Circular Dichroism (PECD)

7-9, 16-18, 21 , which arises when a circularly polarized light is used to photoionize a chiral molecule.

However, there is a fundamental difference. PECD can only exist if the molecular potential felt by the emitted electron is chiral 7 (the effect becoming negligible for photoelectron energies above 10 eV), while the initial orbital may or may not be chiral at all [START_REF] Ulrich | Giant chiral asymmetry in the c 1 s core level photoemission from randomly oriented fenchone enantiomers[END_REF] . It is also clear from the diagram of PECD in Fig 1 (d). The continuum state cannot merely serve as a non-chiral link, as in this case it will only mediate the coupling of the chiral object, the molecule in the ground state, "to itself" rather than to another chiral object.

In contrast to PECD, the PXECD requires neither chiral light, nor chiral scattering of the photo-electron. Since PXECD does not require the technically challenging use of ultrashort circularly polarized XUV pulses [START_REF] Wang | Femtosecond single-shot imaging of nanoscale ferromagnetic order in Co/Pd multilayers using resonant x-ray holography[END_REF][START_REF] Spezzani | Coherent light with tunable polarization from singlepass free-electron lasers[END_REF][START_REF] Allaria | Highly coherent and stable pulses from the FERMI seeded freeelectron laser in the extreme ultraviolet[END_REF][START_REF] Fleischer | Spin angular momentum and tunable polarization in high-harmonic generation[END_REF][START_REF] Ferré | A table-top ultrashort light source in the extreme ultraviolet for circular dichroism experiments[END_REF] , it opens unique perspectives for ultrafast chiral-sensitive measurements using readily available linearly polarized UV and XUV light from table-top high harmonic generation sources, with no restrictions on photoelectron energies.

We shall now confirm both numerically and experimentally that our scheme provides a sensitive time-resolved probe of chiral molecular dynamics in bound states.

Theoretical analysis in fenchone

To quantify the PXECD effect we performed quantum mechanical calculations on fenchone molecules ) and oscillates at frequencies determined by the energy differences between the states forming the electronic wave-packet. The calculated pump-probe PXECD signal reveals these oscillations (Fig. 2c). While few-femtosecond pulses would be needed to resolve them, the PXECD signal can also be detected with much longer pulses. Fig. 2(d) shows that both PXCD and PXECD survive temporal averaging over 100 fs duration of a probe pulse.

Observation of PXECD in fenchone

In our experiment, a circularly polarized femtosecond pump pulse at 201 nm (6.17 eV photon energy, 80 meV at 1/e bandwidth) photoexcites enantiopure fenchone molecules from a supersonic gas jet in the interaction zone of a velocity map imaging spectrometer. The molecules are excited to their first (s-and p-) Rydberg bands through single-photon absorption (Fig. 3 (a), see the SI). The molecules will tend to relax towards the equilibrium geometry of the Rydberg states, and oscillate around it. Figure 5 illustrates the influence of this change of molecular geometry on the calculated PXECD signal. Even small bond length changes (≤ 7%) lead to significant modification of the PXECD signal. This demonstrates the remarkable sensitivity of PXECD to molecular vibrations, which follow the electronic excitation. At 4 ps (not shown), the PXECD completely vanishes while the Rydberg population is still significant. This result unambiguously reflects the loss of wave-packet coherence which halts chiral dynamics in our experiment.

Vibrational PXCD: experiments in camphor

Is it possible to create PXCD from purely vibrational excitation of a chiral molecule? Theoretically, the two excited states in Eqs.(1,2) needed for PXCD do not have to be different electronic states.

Vibrational states within the same electronic state can also fulfil the PXCD condition as long as their dipoles are not collinear, see Eqs. (1,2). As shown in the SI, this requires the breakdown of the Franck-Condon approximation, which is caused by a strong dependence of the electronic wave-function on the position of the nuclei. In turn, such dependence leads to the appearance of electronic currents stimulated by the nuclear motion, which is triggered by the pump pulse. Thus, vibrational PXCD is intertwined with the underlying chiral motion of electrons. Note that this strong dependence of the electronic wave-functions on the nuclear positions naturally arises in the vicinity of conical intersections between electronic potential surfaces. Thus, we expect that PXECD could be used to excite and reveal coherent chiral dynamics at conical intersections.

To gain further insight into the role of electronic versus vibrational dynamics in PXECD, we performed measurements in (1R)-(+)-camphor, a very similar structural isomer of fenchone. The sand p-Rydberg bands of camphor are upshifted by additional several tens of meV compared to fenchone, preventing direct excitation of the pstates and thus of an electronic chiral wavepacket. Nevertheless, the experiment still reveals a strong PXECD signal, indicating that a chiral vibronic wave-packet has been created in the s-Rydberg band of camphor. The α coefficients in camphor and fenchone are of opposite sign as seen in multiphoton [START_REF] Lux | Photoelectron circular dichroism of bicyclic ketones from multiphoton ionization with femtosecond laser pulses[END_REF] and one-photon PECD [START_REF] Nahon | Valence shell one-photon photoelectron circular dichroism in chiral systems[END_REF] .

In our experiment, this could be a consequence of PXECD sensitivity to isomerism (see Figure 5 to gauge the sensitivity to nuclear configuration), but it could also be a signature of the different nature of the excited chiral electronic currents in fenchone and camphor. Changing the excitation wavelength from 202 nm to 200 nm does not affect the monoexponential decay of the PES. In contrast, a strong change is observed in the PXECD: the α magnitude is almost twice as large and it is shifted in energy towards the red wing of the photoelectron spectrum. The drastic change observed in the PXECD signal in camphor once the pump photon energy is increased by only 60 meV illustrates the extreme sensitivity of this measurement to the excited vibrational dynamics.

Conclusions and outlook

We have demonstrated two new phenomena. First, we have shown the efficient excitation of a macroscopic bound chiral electron density in the excited states of randomly oriented chiral molecules without the help of magnetic interactions (the PXCD phenomenon). In the dipole approximation the chiral pitch of circularly polarized light vanishes. This means that the creation of the macroscopic chiral density in the isotropic ensemble of chiral molecules is based not on the helical structure of light, but on its planar rotation.

Second, we have shown that the resulting chiral dynamics can be probed without the help of further chiral interactions and thus in an efficient and versatile way. The detection relies on photoelectron circular dichroism arising from the ionization of excited molecules by linearly polarized light pulses (the PXECD phenomenon), but is not limited to this scheme. The application of a linearly polarized XUV probe in PXECD would enable genuine probing of ultrafast chiral bound dynamics, since PXECD does not require chiral interaction in the continuum, which becomes negligible for sufficiently high-energy electrons.

The ensemble-averaged chiral charge density arising in PXCD implies asymmetry in charge distribution along the light propagation direction. Depending on the medium density, this could lead to a very large coherently oscillating macroscopic dipole. The phase of this oscillation is opposite for two enantiomers, leading to macroscopic enantio-sensitive effects. The existence of the enantio-sensitive macroscopic dipole opens the way to the separation of enantiomers in isotropic racemic mixtures in the gas phase.

The PXCD phenomenon opens the way to direct visualization of chiral electronic density using time-resolved X-ray diffraction imaging, both in the gas and condensed phase. Intense ultrafast sources of X-ray radiation, such as Free Electron Lasers, combined with measurements, sensitive to valence-shell dynamics in the gas phase [START_REF] Bredtmann | X-ray imaging of chemically active valence electrons during a pericyclic reaction[END_REF] should lead to few-fs time resolution of chiral charge dynamics.

Finally, PXCD could be used to drive molecular reactions in chiral systems in a stereospecific way, by imprinting a chiral torque via the helicity of the exciting circularly polarized pulse.

The ultrafast charge dynamics triggered by coherent electronic excitation is reminiscent of ultrafast charge migration triggered by photo-ionization [START_REF] Lunnemann | Ultrafast charge migration in 2-phenylethyl-N, N-dimethylamine[END_REF][START_REF] Breidbach | Universal attosecond response to the removal of an electron[END_REF][START_REF] Remacle | An electronic time scale in chemistry[END_REF][START_REF] Kuleff | Charge migration in different conformers of glycine: The role of nuclear geometry[END_REF][START_REF] Lepine | Attosecond molecular dynamics: Fact or fiction?[END_REF][START_REF] Leone | What will it take to observe processes in 'real time'?[END_REF][START_REF] Kuleff | Ultrafast correlation-driven electron dynamics[END_REF] recently observed in ref. [START_REF] Calegari | Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses[END_REF] and speculated to underlie charge-directed reactivity in cations [START_REF] Weinkauf | Nonstationary electronic states and site-selective reactivity[END_REF] . Chiral electron stereo-dynamics in neutral molecules may open similar opportunities for controlling charge and energy flow in molecules at the level of electrons, offering new perspectives for such intriguing problems as asymmetric synthesis, a major challenge in stereochemistry.

Methods

An Even-Lavie valve is used as a pulsed enantiopure fenchone source with helium as carrier gas

z

  and the corresponding chiral density are created in the excited states, with a chiral current oscillating out of phase for the two enantiomers. Second, PXCD requires no magnetic or quadrupole effects. Hence, it is orders of magnitude stronger than standard photoabsorption CD.While photoabsorption CD exploits the helical pitch of the laser field in space, PXCD takes advantage of the sub-cycle rotation of the light field in time and is inherently ultrafast. Indeed, PXCD arises only if the excitation dipoles d 01 , d 02 are non-collinear: for the angle φ between the two transition dipoles, the PXCD (Eqs.(1,2)) is proportional to σ sin(φ). Since σ = ±1, σ sin(φ) = sin(σφ) = sin(σωτ ), where ω is light frequency and τ = φ/ω is the required time for the light field to rotate by the angle φ. PXCD vanishes if the coherence between excited states |1 and |2 is lost and reflects dynamical symmetry breaking in an isotropic medium.

.

  Here τ is the pump-probe delay, D 12 (k) = D r 12 (k) + i D i 12 (k) is the Raman-type photoionization vector (see the SI) which connects the excited bound states via the common continuum and plays the role of d 12 of Eq.(1,2) and k is the photoelectron momentum.

(

  see the SI). First, we simulated the PXCD phenomenon and calculated the excitation of the s-and p-manifold of Rydberg states in fenchone by a circular pump pulse. The resulting electron density of the Rydberg wave-packet is asymmetric in the z-direction in the momentum space. The asymmetry reverses if the helicity of the pump pulse or the handedness of the molecule is reversed. The strength of the PXCD can be quantified by the magnitude of the chiral component of the excited electron density. It is obtained by subtracting the momentum space density D obtained with right (R) and left (L) polarized light: P XCD = 2(D(L)-D(R))/(D(L)+D(R)). Even after averaging over molecular orientations, the calculated PXCD reaches very high values (35%, Fig. 2(a)). The asymmetry of the charge distribution corresponds to a macroscopic dipole moment d PXCD z which reaches 3 Debye (Fig. 2(b)

A

  time-delayed, linearly polarized probe pulse at 405 nm (3.1 eV photon energy, 35 meV FWHM bandwidth) induces one-photon ionization of the excited molecules. The cross-correlation of the pump and the probe pulses is 170 fs. The photoelectrons are accelerated and projected by an electrostatic lens onto a set of dual microchannel plates and imaged by a phosphor screen and a CCD camera. The photoelectron images are recorded alternatively using left (LCP) and right (RCP) circularly polarized pump pulses. The difference (LCP-RCP) and sum (LCP+RCP) of these two images are reconstructed using a least-square fitting algorithm (see the SI). We define the PXECD signal as P XECD = 2(LCP -RCP ) (LCP +RCP ) and the photoelectron spectrum (PES) as P ES = (LCP + RCP )/2. Both are shown in Fig. 3(b) for a 200 fs pump-probe delay. As expected, a significant PXECD signal is observed, reaching 1 % 28 in good agreement with our calculations (Fig. 2(d)). The photoelectron spectrum contains a single broad component, corresponding to ionization from the outermost orbital (vertical ionization potential 8.72 eV). The position of this component does not shift with the pump-probe delay (Fig. 4 (b)) and decays in 3.3 ps, reflecting simple vibronic relaxation of the Rydberg population onto lower states which cannot be photoionized by the 3.1 eV probe photons. The temporal evolution of the PXECD image shows much richer spectroscopic features, which can be analyzed by decomposing it in odd Legendre polynomials (Fig. 4(a)). We note that a sum of first-and third-order Legendre polynomials, with coefficients α and α , is enough to get the PXECD images. Both coefficients maximize around ∼ 50 meV below the maximum of the PES. The PXECD signal (Fig. 4(b)) can be decomposed into two components: below and above the maximum of the PES. The low-energy component of α undergoes a rather smooth decay. On the contrary, its high-energy component decays very quickly and even changes sign around 1 ps. For α the behaviour is opposite, i.e. the high-energy component shows much slower dynamics than the low-energy part. Such time-and electron energy-dependent behaviour is characteristic of internal vibrational torsional motion and may indicate the change of the chiral structure of the molecule induced by such motion. Indeed, the electronic excitation of the molecules is expected to be accompanied by a significant vibrational excitation, since the equilibrium geometries of the 3s and 3p Rydberg states are quite different from that of the ground state.

  to avoid cluster formation. (1R)-(-) and (1S)-(+)-fenchone correspond to (1R,4S) and (1S,4R) fenchone respectively. The 170 fs cross-correlation time as well as the 0 fs delay are determined on the lightest fragment C 4 H 5 O + produced by dissociative ionization with both linearly polarized pump and probe. The high voltage of the repeller electrode was -3kV for the experiment done in fenchone and only -2kV for the experiment done in Camphor, which increases the energy resolution.Note that the latter, along with the energy calibration, that has been determined by photoionizing krypton. Typically the energy resolution is 80 meV at 0.7 eV kinetic energy. The presented results are obtained by scanning the pump-probe delays typically 30 times. At each delay, helicity is

FIGURE 1 :

 1 FIGURE 1: Chiral discrimination schemes. C.C. denotes complex conjugated, i.e. time-

FIGURE 2 :

 2 FIGURE 2: Theoretical analysis of electron chiral dynamics in (1S)-(+)-fenchone. (a) Mo-

FIGURE 3 :

 3 FIGURE 3 : PXECD in fenchone molecules. (a) Absorption of circularly polarized pulse at

FIGURE 4 :

 4 FIGURE 4: Time-Resolved PXECD in fenchone and camphor. (a) Legendre polynomi-

FIGURE 5 :

 5 FIGURE 5: Sensitivity of PXECD in (1S)-(+)-fenchone to the evolution of the chiral molec-

will change sign upon reflection. Thus, the electron current reverses its direction if the handedness