Mehran Mohammadi

Santiago Velasco-Forero

François Willot

Mateus Sangalli

Thomas Walter

Jesus Angulo

Choquet Capacity networks for random point process classification

Keywords: Choquet capacity, Max-Plus Algebra, Deep Learning

Introduction

There is a significant demand for analytical techniques that can handle vast amounts of image data in particular in engineering science [START_REF] Chinesta | Virtual, digital and hybrid twins: a new paradigm in data-based engineering and engineered data[END_REF]. For instance, images from satellites, geological maps, microscope images of metals and materials, cellular tissue, and data from computerized tomography [START_REF] Etiemble | Numerical prediction of multiscale electronic conductivity of lithium-ion battery positive electrodes[END_REF] all require automatic and quantitative methods due to the sheer volume of data involved. In some scenarios, the information observed is set of coordinates in 2D or 3D, and the feature of interest depends on the spatial relationship between these points. For example, individual brain activation maps [START_REF] Worsley | Searching scale space for activation in pet images[END_REF][START_REF] Poline | Analysis of individual brain activation maps using hierarchical description and multiscale detection[END_REF], RNA Localization patterns [START_REF] Imbert | Pointfish: Learning point cloud representations for rna localization patterns[END_REF], positions on random sensor networks [START_REF] Ram | On the path coverage properties of random sensor networks[END_REF], realizations of random processes [START_REF] Pawlasová | Supervised nonparametric classification in the context of replicated point patterns[END_REF], among others. On one hand, mathematical morphology [START_REF] Serra | Image Analysis and Mathematical Morphology[END_REF][START_REF] Soille | Morphological Image Analysis: Principles and Applications[END_REF][START_REF] Najman | Mathematical Morphology: From Theory to Applications[END_REF] offers ideas that are applicable in this context, particularly due to its interpretation in terms of the size/shape relationship of the object of interest. On the other hand, data-driven learning methods have changed the way image problems are solved. Especially the deep models allow to solve recognition problems with performance sometimes superior to humans [15]. In this paper, we follow the results of [START_REF] Sangalli | Scale equivariant neural networks with morphological scale-spaces[END_REF][START_REF] Velasco-Forero | Learnable empirical mode decomposition based on mathematical morphology[END_REF][START_REF] Velasco-Forero | Fixed point layers for geodesic morphological operations[END_REF][START_REF] Mondal | Morphological network: How far can we go with morphological neurons? In: 33rd British Machine Vision Conference 2022[END_REF], to present a new application of morphological models learned from data for the characterization of random point processes. The Neyman-Scott point process is presented in section 2. The proposed Choquet Capacity model is described in section 3. Section 4 demonstrate the benefits of the proposed model in comparison with classical convolutional neural networks (CNNs) for classification and regression problems on images generated by Neyman-Scott point process. We conclude in section 5.

Neyman-Scott point processes

In the present work, we consider Neyman-Scott point processes [START_REF] Waagepetersen | An estimating function approach to inference for inhomogeneous neyman-scott processes[END_REF][START_REF] Vere-Jones | Stochastic models for earthquake occurrence[END_REF][START_REF] Westcott | Results in the asymptotic and equilibrium theory of poisson cluster processes[END_REF][START_REF] Ripley | Modelling spatial patterns[END_REF] as a way to generate point patterns that are statistically different from the classical Poisson model of points [START_REF] Daley | An introduction to the theory of point processes: volume I: elementary theory and methods[END_REF][START_REF] Daley | An introduction to the theory of point processes: volume II: general theory and structure[END_REF][START_REF] Moller | Statistical inference and simulation for spatial point processes[END_REF]. The Neyman-Scott model, like Poisson's one, is a particular type of Boolean random set [START_REF] Stoyan | Stochastic geometry and its applications[END_REF]. Poisson point processes on R 2 depend on a single parameter θ, called the intensity. The intensity θ represents the mean number number of points per unit area. In the Poisson point model, the presence of points in two disjoint domains is uncorrelated which lead to the following Poisson-distributed law for the number of points η(Ω) in a domain Ω,

P{η(Ω) = k} = (θ|Ω|) k k! exp -θ|Ω| , (1)
where |Ω| is the surface area of Ω. This leads to the formula:

T X (K ) := P(X ∩ K ̸ = ∅) = 1 -P{K ⊆ X c } = 1 -exp -θ|K| , (2)
for the Choquet capacity T (K ) of a realization X = {x 1 , x 2 , ...} of a Poisson point process and a compact set K [START_REF] Choquet | Theory of capacities[END_REF]. Neyman-Scott point processes are different from Poisson point processes in that they generate clustered sets of points. This is achieved in two steps. In the first step, a standard Poisson point process of intensity θ p is generated, leading to a set of parent points x (p) ∈ R 2 (p ∈ N). In the second step, we associate each parent point with a set of n p daughter points x

(p) 1 , . . . , x (p)
np . The Neyman-Scott point process is made of all daughter points; parent points are not included. Neyman-Scott models are accordingly parametrized by θ p , the probability law for the n p and that of the daughter points, given in term of the multivariate probability law for the x (p) i -x (p) . For simplicity, we consider a constant distribution n p ≡ n. We also assume that the vectors x (p) i -x (p) are i.i.d. variables drawn from a multivariate random distribution in R 2 , with uniform angular distribution in [0; 2π]. Distances from parent to daughter points are drawn from a Pareto law with parameters α, r m such as:

P{|x (p) i -x (p) | > r} = r m r α H(r -r m ), α > 0, r m > 0, (3)
where H(•) is the Heavyside function, equal to one on R + and zero elsewhere.

Here r m controls the typical size of the clusters whereas the parameter α monitors the spatial dispersion. Thus, r has infinite variance when α ≤ 2 and infinite Importantly, the Choquet-Matheron-Kendall theorem established independently by [START_REF] Kendall | Foundation of a theory of random sets[END_REF][START_REF] Matheron | Random Sets and Integral Geometry[END_REF] states that any stationary and ergodic random set, such as Poisson or Neyman-Scott point processes, is completely determined by the functional T (K ). In the next section, we propose a model that use (2) to find optimal K in a discrete setting (denoted as K), in the sense of learning from labeled data for supervised problems. Note that this learning process makes use of convolutional layer where K are optimized in the real set. Accordingly, we propose regulatization terms to encourage the network to learn binary kernels.

Choquet Capacity models

Discrete random set

Usually random sets are defined in the sense of Euclidean geometry, as subsets of R d . However, in many applications, as in image processing, the domain is discrete, as pixels structures in 2D, Ω = Z 2 . The discrete random set (DRS) X on Z 2 is now defined by X := {(i, j) ∈ Z 2 : x i,j = 1}. The discrete-capacity functional of a discrete random set X is defined by [START_REF] Matheron | Random Sets and Integral Geometry[END_REF]:

T X (K) := P(X ∩ K ̸ = ∅) = 1 -P{K ⊆ X c } := 1 -Q X (K) (4)
for every K ∈ S the collection of all bounded subsets of Z 2 . From (4), it is easy to see that: a)

0 ≤ T X (K) ≤ 1 for every K ∈ S, b) T X (∅) = 0, and c) T X (K 1 ) ≤ T X (K 2 ), ∀K 1 , K 2 ∈ S such that K 1 ⊆ K 2 .
Other important properties can be shown for DRS [START_REF] Singh | Discrete random functions: Modeling and analysis using mathematical morphology[END_REF][START_REF] Goutsias | Morphological analysis of discrete random shapes[END_REF]. According to the Choquet-Kendall-Matheron theorem, any discrete random set X is uniquely characterized by its discrete-capacity functional T (K), defined over all bounded subsets K ∈ S of Z 2 . Furthermore, we note that:

P(X ∩ K ̸ = ∅) = P(x ∈ δ K(X)) (5)
where K denotes the K mirrored in the origin. The capacity in ( 4) is accordingly given by the morphological dilation of X by K [START_REF] Serra | Image Analysis and Mathematical Morphology[END_REF][START_REF] Jeulin | Random texture models for material structures[END_REF].

Fig. 2: Schematic of proposed Choquet Capacity model.

Choquet Capacity layer

This section describes a layer that can be used for the computation of ( 5), in the context of neural networks. To do so, we use the extension of convolutional models in the ( , +) algebra, called max-plus convolutions [START_REF] Akian | Max-plus algebra[END_REF], or dilation layer [START_REF] Velasco-Forero | Learnable empirical mode decomposition based on mathematical morphology[END_REF].

The dilation layer replaces discrete classical convolutions and can be compared to fuzzy measures in so-called fuzzy integral neural networks [START_REF] Islam | Enabling explainable fusion in deep learning with fuzzy integral neural networks[END_REF]. For an input image f, the dilation layer is defined as:

δ K (f )(x) := y∈Z 2 [f (x -y) + K(y)] = y∈Z 2 [f (y) + K(x -y)] ( 6
)
where K is a matrix of real values called kernel or structuring function. In the cases where the image f represents a DRS (denoted by f), and the kernel K takes on values in {-1; 0} 1 , we have that f(y) + K(x -y) = 0 if K ⊆ f and 1 elsewhere. By sliding the kernel through the domain and calculating its average, assuming ergodicity, we obtain an estimation for the value of Q f (K) in (4):

Qf (K) := 1 M M j=1 δ K (f)(x j ) ( 7
)
where M is the number of pixels in the image. In the following, equation ( 7) is called a Choquet Capacity layer with convolutional kernels K.

Hereafter, we propose to study the performance of the Choquet Capacity layers to learn n k kernels, K := {K i } n k i=1 , in the framework of neural networks. Accordingly, for each input image f , a vector containing the n k values computed by ( 7) is used as features, transformed linearly by a dense layer (or many for multilayer perceptrons (MLP) models) with parameter W, and finally passed trough an activation function adapted to the regression or classification problem. Fig [START_REF] Bengio | Estimating or propagating gradients through stochastic neurons for conditional computation[END_REF] shows a schematic of the proposed Choquet Capacity model for a classification problem. The classical training mechanism on neural networks seeks to find the optimal model's parameters (W, K), arg min

(K,W) N ℓ=1 loss(Y ℓ , Y ℓ ) + Reg(W, K) (8)
where Y ℓ is the prediction of the network for the ℓ-th sample, Y ℓ is the ground truth, N is the number of samples and Reg(W, K) is a regularization term on the parameters of the model. In its simplest form, the parameters (W, K) are initialized to random values and updated by means of a stochastic gradient descend, the gradient being computed by backpropagation [15]. Note that the shape K in ( 5) is a bounded subset, whereas each of the matrix K in ( 6) is a maxplus convolutional kernel with real values. Therefore, we propose two alternatives that address this difference during the training of the model: i) Regularization: the term Reg in ( 8) is set to penalize kernel values that are far from a binary equivalent, in our case of max-plus algebra on binary images, 0 or -1. Two different regularization functions are considered for R:

Reg 1 (K) = λ 1 n k i=1 (K i + 1) + λ 2 n k i=1 (K i + 1) 2 , (9a) Reg 2 (K) = λ 1 n k i=1 (K i + 1)K i + λ 2 n k i=1 (K i + 1) 2 K 2 i , (9b)
where n k in the number of kernels and λ 1 ≥ 0 and λ 2 ≥ 0 are the hyperparameters to control the effect of regularization. ii) Quantization of the kernels during optimization. We use an integrated value filter to enforce kernel values in [-1, 0]. We use the straight-through estimator proposed in [START_REF] Bengio | Estimating or propagating gradients through stochastic neurons for conditional computation[END_REF][START_REF] Yin | Understanding straightthrough estimator in training activation quantized neural nets[END_REF]. Predictions are computed by rounding the kernel to the closest integer (-1 or 0) whereas the gradient in the backpropagation is left unchanged.

The different approaches above are compared to one another and to classical convolutional networks in the next section. We would like to study the effect of including more layers into the model. This is called the depth of the network in deep learning [START_REF] Goodfellow | Deep learning[END_REF] 4 Numerical experiments

In the present section, we quantify the performances of the various convolutional networks. Two problems are addressed: first, that of classification of realizations of Neyman-Scott point patterns w.r.t. Poisson; second, the prediction of the dispersion parameter α in a regression problem2 .

Classification of point pattern

In this experiment, we set the image size to 500 × 500 pixels and θ = 0.05 in number of points per pixel unit. We choose n = 50 and let θ p = θ/n so that the density of points is the same irrespective of the model chosen. We also fix r m = 0.1 × 500 = 50 and let α varies between 1 and 100. All point patterns are statistically homogeneous. We perform the classification task on two classes of Poisson and Neyman-Scott point patterns with different values of the parameter α and conduct a comparative analysis with the CNNs. The data set contains 500 images for each class, with a train-to-validation ratio of 4:1. For simplicity, we use one layer in each neural network, with four filters of size 7 × 7 pixels. For supervised classification, a common choice of loss function is the binary cross-entropy also known as log loss, defined by

logloss(Y, Y ) := - 1 N N ℓ=1 Y ℓ log q ℓ 0 + (1 -Y ℓ ) log q ℓ 1 ( 10
)
where (q ℓ 0 , q ℓ 1 ) denote the prediction of the network for the ℓ-th samples, Y ℓ the ground truth of the ℓ-th samples and N is the number of samples. The models are trained to predict the class between images produced by Poisson processes and Neyman-Scott models with different values of α parameter. Figure 3 shows the boxplot of validation accuracy for models training from different initialization values. Interestingly, the Choquet Capacity layer demonstrates competitive performance, especially when the parameter α decreases, so that realizations of the Neyman-Scott point pattern are similar to that of a Poisson distribution. When regularization is applied, the model tend to have values {-1, 0}, but they are still values that prevent us from indicating that the solution is binary. For quantization, K contain only values in {-1, 0} in Figure (4)(c), although no distinctive shape is observed. However, performance significantly degraded, when regularization or quantization is used, specially in more difficult cases.

Parameter regression of the Neyman-Scott point processes

In the present subsection, we compare the performances of different neural network models for predicting the value of the parameter α, in a regression problem. We set the image size to 128 × 128 pixels, the density of points to θ = 100/(128 × 128) and r m = 0.02. We also let α vary between 0 and 9. This last is the value that we would like to predict. For the regression problem, the mean absolute error is the mean absolute difference between true value and prediction:

MAE(Y, Y ) := - 1 N N ℓ=1 |Y ℓ -Y ℓ | (11)
is used as loss function. During training the α values are scaled to the interval [0, 1] as it is standard practice in regression models. As a first example, we compare the result between Choquet Capacity and a classical convolutional layer followed by a multi-layer perceptron that consists of two layers of dimension ten and batch normalization [START_REF] Santurkar | How does batch normalization help optimization?[END_REF], followed by a sigmoid activation, see Figure 5. Note that the convolutional layer and our proposition have exactly the same number of parameters. In this case, our Choquet Capacity model obtains predictions that are clearly quantitatively better. As a second illustration in the regression case we studied the effect of network depth, prior to the use of the Choquet Capacity model. Usually, increasing both depth and width (number of filters per layer) helps to improve the model performance until the number of parameters becomes too high and stronger regularization is needed [START_REF] He | Deep residual learning for image recognition[END_REF]. For that, we consider convolutional layers with kernel size of 3×3 pixels and 48 filters are concatenated before the use of global pooling and the MLP. This architecture is represented in Figure 6. We compare different convolutions in the last layer before the global pooling: proposed Choquet Capacity model, classical CNN, and depthwise convolutional (DWCNN). The latter is included because contains exactly the same number of parameters compared to our Choquet Capacity model. In Figure 7, we compare the validation loss of the three networks with the number of parameters in each model shown in logarithmic scale. The number of degrees of freedom is increased by augmenting the depth from one to seven in the convolutional part in each architecture, leading to models with deeper feature extractions. The Choquet Capacity model improves the regression results in all explored cases, for a fixed number of parameters, compared to DWCNN. Finally, Figure 7 also shows than comparable performances to the Choquet model are achieved by classical CNNs at the cost of a much higher number of parameters.

Conclusion

The results obtained in this study are based on a novel approach that makes use of dilation layers with global pooling, inspired from the definition of the Choquet capacity for random sets in the theory of integral geometry. Numerical experiments performed on synthetic datasets which simulate diverse Neyman-Scott point patterns, demonstrate the efficiency of the method, as well as seemingly superior performances in particular in regression problems involving the dispersion parameter in the Neyman-Scott point process. Compared to classical convolutions, the Choquet Capacity networks are somewhat more difficult to train, due to the presence of a nonlinear convolution filters. Yet they are versatile in that they may be combined with classical convolutional networks. As a negative result, the alternatives studied to motivate the estimation of binary kernels result in very important losses in the performance of the models. This topic, as well as the adequacy of Choquet capacity networks to analyse real-world data, will be addressed in future research works.

mean when α ≤ 1 .

 1 Examples of realizations are shown in Fig. (1). The points spatial dispersion in the Neyman-Scott model is markedly more heterogeneous when α is large (see Fig. (1b). When α approaches one, the Neyman-Scott and Poisson models are difficult to distinguish (see Figs. 1).

Fig. 1 :

 1 Fig. 1: Realizations of a Poisson point process (a) and Neyman-Scott models (b-c) with the same density of points. b) Parameters r m = 0.1, α = 100. c) r m = 0.1, α = 3.

Fig. 3 :

 3 Fig. 3: Validation accuracy for the different neural network models in the twoclasses classification problem (Poisson and Neyman-Scott point processes) vs. the parameter α that monitors spatial dispersion.

Fig. 4 :

 4 Fig. 4: Example of learned kernel K = {K 1 , . . . , K 4 } using different proposed alternatives for regulatization of the Choquet Capacity layer.

  (a) Convolutional layer with average pooling (b) Proposed Choquet Capacity model

Fig. 5 :

 5 Fig. 5: Comparison on the prediction over testing training for one layer convolutional with global average pooling network and our Choquet Capacity proposition in the regression problem of the Neyman-Scott point processes. Both models have the same number of parameters and they have been trained with the same protocol.

Fig. 6 :

 6 Fig. 6: In the architecture used in section 4.2, we vary the number of layers referring to classical convolutional layers of kernel size 3 × 3 pixels follow by ReLU activation.

Fig. 7 :

 7 Fig.7: Performance of different neural networks for varying number of layers in each model (See Figure6). Three architectures are compared. In all cases each layer has 48 kernels of size 3 × 3, ReLU is used as activation function, and a final layer with a convolution of size 13 × 13 followed by GlobalAveragePooling, BatchNormalization and a MLP with two layers of dimension ten with a final sigmoid activation. DWCNN refers to a depthwise convolution, CNN to a classical convolution and Choquet Capacity to the max-plus convolutions in 3. DWCNN and Choquet Capacity models have exactly the same number of parameters.

Hereafter, we consider input images lying in the interval [0, 1], so that in practice, -1 plays the same role as -∞ for kernel.

The source code of this section is available at https://github.com/Jacobiano/ ChoquetLayer

Acknowledgements The authors gratefully acknowledge the financial support of Institut Carnot (grant 220000496).


Parvaneh Adibpour
email: parvaneh.adibpour@inserm.fr

Jean-Rémy Hochmann
email: hochmann@isc.cnrs.fr

Infants' understanding of the causal power of agents and tools

Tool-use understanding is classically thought to involve reasoning about the causal mechanisms that relate tools and their effects (e.g., moving food closer). Through cultural evolution and accumulated innovations, however, humans have developed tools that became causally opaque (e.g., touch screens). We show that cultural transmission remains possible because even young infants are equipped with the fundamental capacity to represent arbitrary causal events where an agent acts on the world with a tool. Infants attributed to a ball the power to cause an arbitrary state change but only if that ball was itself launched by an agent.

Infants thus consider that the power to cause arbitrary events ultimately belongs to intentional agents but agents can transfer this power to tools.

Introduction

Tools are objects that are manipulated by agents with the intention to cause an effect in the world.

Early tools in primate and human evolution, as in nowadays wild animals, were used to acquire food, groom conspecifics, or for self-defense [1]. In these situations, the causal mechanisms that relate tool use and their effects (e.g., moving food closer, removing parasites) are largely transparent and can be accounted for by intuitive theories [2][3][4][START_REF] Spelke | Principles of object perception[END_REF][START_REF] Spelke | Core knowledge[END_REF]. But modern humans show a unique flexibility and versality in their use of tools.

Through cultural evolution, each generation improved the previous generation's tools, so much that underlying causal mechanisms have become opaque. The spectrum of possible effects of modern tools is virtually infinite, from switching on the light, to playing music or having Granma appear on a computer screen. This opacity may constitute a learning problem for children, especially if one considers that tool use requires reasoning about the causal mechanisms relating the tool and its effects [START_REF] Osiurak | The elephant in the room: What matters cognitively in cumulative technological culture[END_REF]. In the present study, we propose that cultural transmission of tools is made possible by infants' disregard of causal opacity and an early understanding of the large causal power of agents and tools.

We asked whether infants understand that an intentional agent can cause an arbitrary state change on a third object through the use of an intermediary object, the tool. This understanding involves the integration, in a complex structure, of a chain of causal relations and three thematic roles: the agent acts on the tool (or instrument), which in turn affects the patient's state. of occluded events, 20% of contact events and 20% of gap events. Experiment 1 (A) tested causal inference for direct launching events. In Experiment 2 (not depicted), infants only saw the contact (50%) and gap (50%) events from Experiment 1. Experiments 3 (B) and 4 (C) tested causal inference for indirect state change events. The yellow object had face features and was introduced as a self-propelled intentional agent in Experiment 3 but not in Experiment 4.

Before addressing the issue of tool use, Experiment 1 introduced a method relying on pupillometry to replicate classic findings showing that infants infer a causal interaction, the contact between two objects, to account for the visible motion of a ball [START_REF] Ball | The perception of causality in the infant[END_REF][START_REF] Kotovsky | Reasoning about collisions involving inert objects in 7.5-monthold infants[END_REF][START_REF] Wang | Should all stationary objects move when hit? Developments in infants' causal and statistical expectations about collision events[END_REF][START_REF] Muentener | Infants' causal representations of state change events[END_REF]. The inference of contact and temporal contiguity is taken as a signature of understanding the causal relationship between two events [START_REF] Leslie | The perception of causality in infants[END_REF][START_REF] Leslie | Do six-month-old infants perceive causality?[END_REF][START_REF] Michotte | The perception of causality (TR Miles, Trans.)[END_REF]. In contrast to classic looking time studies, pupillometry provides timing information, which can improve the interpretation of the results [START_REF] Hochmann | Negative mental representations in infancy[END_REF]. Eight-month-old infants (N = 20) watched repeated movies where, shortly after a yellow ball went behind an occluder, from the left, a second ball initially positioned at the right boundary of the occluder began to move (Fig. 1A). Rare interleaved unoccluded events served to probe infants' representation of what happened behind the occluder. Previous research showed that unexpected events trigger pupil dilation in infants, indicating surprise [START_REF] Hochmann | Negative mental representations in infancy[END_REF][START_REF] Hochmann | Representations of Abstract Relations in Infancy[END_REF]. If infants inferred a contact between the two balls behind the occluder, they should be more surprised -exhibiting larger pupil dilation -by rare unoccluded Gap movies, in which the first ball did not contact the second ball, than by rare unoccluded Contact movies, in which the first ball contacted the second. To confirm that responses to uncoccluded movies related to the interpretation of occluded movies, and not to the mere visual properties of Gap and Contact movies, a second group of infants (Experiment 2, N = 21) only watched the Contact and Gap movies. If Gap movies elicit more pupil dilation than Contact movies, independently from the processing of occluded movies, the same results should be obtained in Experiment 2 as in Experiment 1. Instead, we predicted larger pupil dilation for Gap compared to Contact events in Experiment 1 than in Experiment 2.

Next, in Experiment 3, we studied infants' interpretation of a tool-use event, where an agent used a ball to trigger the flickering of a cube. A prior study concluded that infant cognition does not encompass the notion that a mere contact between two inanimate objects could instigate a state change such as flickering [START_REF] Muentener | Infants' causal representations of state change events[END_REF]. This finding suggests that state-changes lie beyond the explanatory power of intuitive physics, which seems primarily focused on elucidating object motions. In contrast, infants consider that an intentional agent -e.g. a human hand -can elicit a state-change upon direct contact [START_REF] Muentener | Infants' causal representations of state change events[END_REF][START_REF] Liu | Origins of the concepts cause, cost, and goal in prereaching infants[END_REF]. Here, we tested whether infants consider that agents can elicit state-change with a tool. Eight-month-old infants (N = 40) were first introduced with a self-propelled intentional agent [START_REF] Saxe | Secret agents: Inferences about hidden causes by 10and 12-month-old infants[END_REF]. Next, they watched occluded movies in which the agent launched a ball, which went behind an occluder, from the left (Fig. 1B). On the other end of the occluder, a partly occluded cube began flickering after a short delay. Again, interleaved unoccluded events served to probe infants' representation of what happened behind the occluder. In rare unoccluded Contact movies, infants watched the agent launching the ball that then contacted the cube, which immediately began flickering. In rare unoccluded Gap movies, infants watched the agent launching the ball, which stopped before contacting the cube. The cube began flickering with a delay. If infants conceived that the ball is the proximate cause of the cube's flickering, they should have inferred a contact between the ball and the cube behind the occluder, and temporal contiguity between that contact and flickering. In consequence, Gap movies should be more surprising and elicit larger pupil dilation than Contact movies.

Finally, tools are defined as objects that are manipulated by agents with the intention to cause an effect.

Following this definition, we predict that the representation of a tool-use event necessitates the involvement of an intentional agent. In Experiment 4, infants (N = 40) watched the same sequence of movies as in Experiment 3, except that a yellow sphere, with no indication of agency, replaced the intentional agent (Fig. ,1C). We predicted weaker or null effects in this group.

Results

Pupil dilation results are displayed in Fig. 2. Experiments 1-2 -In Experiment 1, 8-month-old infants (N=20) watched repeated launching events involving two balls. Contact events (20%), when the first ball contacted the second, and Gap events (20%), when the first ball stopped before touching the second, were intermixed with a majority of Occluded events (60%) that were compatible with both the Contact and Gap events, as the fate of the first ball was hidden by an occluder (Fig. 1). Infants exhibited larger pupil dilation for Gap events than for Contact events between 3050 and 5233 ms (cluster mass permutation test, P = .01 one-tail). In Experiment 2, 8-month-old infants (N=21) only saw unoccluded Gap (50%) and Contact (50%) events. There was no significant difference of pupil dilation in response to the two types of events.

Analyzing Experiments 1 and 2 together, infants overall exhibited larger pupil dilation for Gap than for Contact events between 3700 and 5200 ms (cluster mass permutation test, P = .02 one-tail). In addition, the interaction between Experimental Group (Experiment 1, Experiment 2) and Trial Type (Gap, Contact) was significant between 4333 and 5233 ms (cluster mass permutation test, P = .04 one-tail); i.e., the difference of pupil dilation between Gap and Contact events was larger in Experiment 1 than in Experiment 2. In fact, in that time window, pupil dilation for Gap relative to Contact events was significant in Experiment 1 (t(19) = 2.46; P = .012 one-tail) but not in Experiment 2 (t(20) = .06, P = .47 one-tail).

Experiments 3-4 -In Experiment 3, 8-month-old infants (N=40) watched repeated state change events, involving an intentional agent, a ball and a flickering cube. All events began with the agent launching the ball. Unoccluded Contact events (20%), when the ball contacted the cube, and Gap events (20%), when the ball stopped before contacting the cube, were intermixed with a majority of Occluded events (60%) that were compatible with both the Contact and Gap events, as the fate of the ball was hidden by an occluder (Fig. 1). The analysis was restricted to the time window of interest defined by the effects observed in Experiments 1-2 between 3700 after the gap onset and the end of the trial. Infants exhibited larger pupil dilation for Gap events than for Contact events between 3767 and 4766 ms (cluster mass permutation test, P = .01 one-tail). In Experiment 4 (N=40), instead of an intentional agent, an inanimate yellow sphere launched the ball. There was no significant difference of pupil dilation in response to Contact and Gap events.

Analyzing Experiments 3 and 4 together, infants overall exhibited larger pupil dilation for Gap than for Contact events between 3983 and 4600 ms (cluster mass permutation test, P = .03 one-tail). In addition, the interaction between Experimental Group (Experiment 3, Experiment 4) and Trial Type (Gap, Contact) was significant between 4083 and 4583 (cluster mass permutation test, P = .04 one-tail); i.e. the difference of pupil dilation between Gap and Contact events was larger in Experiment 3 than in Experiment 4. In fact, in that time window, pupil dilation for Gap relative to Contact events was significant in Experiment 3 (t(39) = 2.8; P = .004 one-tail) but not in Experiment 4 (t(39) = .04; P = .48 one-tail).

equivalent to longer looking times in traditional violation of expectation paradigms. In the current paradigm, trials had a fixed duration, making cumulative looking time analyses uninformative.

Nevertheless, we explored the equivalence between pupil dilation and increased looking time by analyzing differences in the number of Gap and Contact trials included in the final analysis of the pupil data. Indeed, each trial was only included if the infant looked long enough at the screen (i.e., more than 65% of the time window of interest, see Analysis section). A higher number of included Gap trials would thus suggest that infants tended to look longer in Gap trials than in Contact trials.

Discussion

The present studies explored how infants conceive the causal power of agents, objects and tools. Experiments 1-2 showed that 8-month-old infants spontaneously attribute to a ball the power to cause another object's motion through contact, respecting the laws of intuitive physics. In addition, these experiments validated our pupillometry methods. Next, Experiments 3-4 showed that 8-month-old infants attribute to a ball the power of causing an arbitrary (not explained by any obvious laws of physics) state change to a cube, but only if the ball was itself launched by an intentional agent. In other words, the inference of the final causal interaction between the ball and the cube depended on the causal launching between the agent and the ball. This fact suggests that the two causal interactions were not represented as two independent sequential events, but were rather integrated into one complex structure: the agent caused the final state change with the ball. The ultimate cause of the state change is thus the agent, who transmitted their causal power to the ball.

The distinction between inanimate objects and intentional agents is a core structure of the mind [3,[START_REF] Spriet | Visual object categorization in infancy[END_REF]. Infants expect inanimate objects to be moved by causal external forces [START_REF] Kotovsky | Reasoning about collisions involving inert objects in 7.5-monthold infants[END_REF][START_REF] Wang | Should all stationary objects move when hit? Developments in infants' causal and statistical expectations about collision events[END_REF][START_REF] Mascalzoni | Innate sensitivity for self-propelled causal agency in newly hatched chicks[END_REF], while intentional agents exert self-propelled motion and rational actions [START_REF] Gergely | Teleological reasoning in infancy: The naıve theory of rational action[END_REF] guided by internal goals [START_REF] Woodward | Infants selectively encode the goal object of an actor's reach[END_REF], create order [START_REF] Kim | From music to animacy: Causal reasoning links animate agents with musical sounds[END_REF][START_REF] Ma | Preverbal infants infer intentional agents from the perception of regularity[END_REF][START_REF] Newman | Early understandings of the link between agents and order[END_REF][START_REF] Schachner | From music to movement: The role of causal reasoning in infants' and adults' representations of music[END_REF] and are prone to social affiliation [27-31] and moral evaluation [START_REF] Sloane | Do infants have a sense of fairness?[END_REF][START_REF] Hamlin | Social evaluation by preverbal infants[END_REF][START_REF] Ting | Principles and concepts in early moral cognition. The social brain: A developmental perspective[END_REF]. Our results further support an additional fundamental conceptual distinction between intentional agents and objects: their respective causal powers (see also [START_REF] Muentener | Infants' causal representations of state change events[END_REF][START_REF] Liu | Origins of the concepts cause, cost, and goal in prereaching infants[END_REF][START_REF] Saxe | Secret agents: Inferences about hidden causes by 10and 12-month-old infants[END_REF]). While the causal power of objects is limited to intuitive physics and possibly to direct actions (Experiments 1 and 4), the power of intentional agents extends to arbitrary state changes and indirect instrumental actions (Experiment 3). In addition, we show that an inanimate object can inherit the agent's causal power, when it acquires the role of a tool, mediating between an agent and a final effect. This suggests that, conceptually, tools occupy a zone of contact between the core domains of inanimate objects and that of intentional agents. However, our results should not be interpreted as meaning that the causal power of agents and tools is fully unconstrained. In particular, the results reported here and elsewhere [START_REF] Muentener | Infants' causal representations of state change events[END_REF] suggest that young infants still expect agents to act by contact, whether directly or through a tool, and the effect to be immediate (temporal contiguity). The primacy of contact and temporal contiguity suggests that the schematic representation of causal interactions involving agents may be derived from those of intuitive physics, maintaining some of their constraints, while opening up the spectrum of possible effects (not just motion).

How (and when) infants come to understand contactless tools, such as the TV remote control, is a question for future studies. Note however, that in our experiments, contact and temporal contiguity covaried. It remains possible that infants would accept contactless causal actions in presence of temporal contiguity, and if a goal is clearly apparent [START_REF] Southgate | Infants attribute goals even to biomechanically impossible actions[END_REF]. More generally, future work is needed to describe the spectrum of agents' possible actions and circumscribe their causal power.

A long tradition in developmental psychology has argued that understanding instrumental actions, including those involving a tool, develops towards the end of the first year of life and consolidates in the second year of life, in parallel with children's first-person motor behavior (e.g., [START_REF] Piaget | The origins of intelligence in the child[END_REF][START_REF] Sommerville | Experience matters: The impact of doing versus watching on infants' subsequent perception of tool use events[END_REF][START_REF] Sommerville | Pulling out the intentional structure of action: the relation between action processing and action production in infancy[END_REF]). Those results have been used to support views of action understanding as relying on motor simulation [START_REF] Hunnius | What are you doing? How active and observational experience shape infants' action understanding[END_REF][START_REF] Woodward | Mirroring and the development of action understanding[END_REF]. In contrast, converging with observations that object function supports early object individuation [START_REF] Futó | Communicative function demonstration induces kind-based artifact representation in preverbal infants[END_REF][START_REF] Stavans | Four-month-old infants individuate and track simple tools following functional demonstrations[END_REF], our results

show that 8-month-old infants already possess the cognitive capacities to understand instrumental actions.

Far from constituting a learning problem, the opacity of the causal mechanism that relates contact with the state change may have allowed infants to by-pass simulation mechanisms and demonstrate their early understanding of tools. In sum, the notion of a tool appears to be a basic element of the mind, present early in infancy, prior to children's production of instrumental actions. Incidentally, our results further suggest that the representation of a thematic role instrument precedes the acquisition of explicit morphosyntactic markers of that role [START_REF] Lidz | The role of incremental parsing in syntactically conditioned word learning[END_REF][START_REF] Rissman | Thematic roles: Core knowledge or linguistic construct?[END_REF].

Humans are in effect the only species to have developed complex causally-opaque tools. Even so, we cannot exclude that the capacities observed here in young infants are shared with other species, especially non-human primates. Wild chimps, for instance, have demonstrated an understanding of cooking devices [START_REF] Warneken | Cognitive capacities for cooking in chimpanzees[END_REF], and laboratory monkeys are capable to acquire modern tools such as joysticks and touchscreens, albeit after extended trainings periods [START_REF] Baraduc | Schema cells in the macaque hippocampus[END_REF]. Yet, subtle differences may exist between species regarding their resilience to causal opacity, and their need for a detailed understanding of the causal mechanisms involved in tool use. The phenomenon known as over-imitation -i.e., the assumption that all elements in an observed action sequence play a causal (even if opaque) role in fulfilling the artifacts' function -was proposed as a mechanism that allows children to learn the use of novel tools, even when causal mechanisms are not readily apparent [START_REF] Hernik | Functional understanding facilitates learning about tools in human children[END_REF][START_REF] Lyons | The hidden structure of overimitation[END_REF][START_REF] Lyons | The scope and limits of overimitation in the transmission of artefact culture[END_REF][START_REF] Clay | Is overimitation a uniquely human phenomenon? Insights from human children as compared to bonobos[END_REF]. Empirical studies found that over-imitation is weaker in non-human primates than in 3-and 5-year-old human children, suggesting reduced resilience to causal opacity [START_REF] Horner | Causal knowledge and imitation/emulation switching in chimpanzees (Pan troglodytes) and children (Homo sapiens)[END_REF][START_REF] Mcguigan | Imitation of causally-opaque versus causallytransparent tool use by 3-and 5-year-old children[END_REF], hence possibly higher difficulty in representing arbitrary causal events. Further comparative studies are needed to progress on these issues.

Finally, our results highlight the potential interest of the timing information provided by pupillometry in investigating infants' reasoning. It holds the promise to lift (in part) the ambiguity held in the interpretation of classical looking-time measures in the violation of expectation paradigm. The name itself of that paradigm implies that infants expect a precise event and are surprised when something else happens; here, they would predict a contact between the ball and the cube, and be surprised when seeing a gap between the two objects. Increased looking time would thus signal the detection of a prediction error.

Nevertheless, the same effects are also generally compatible with a posteriori evaluations of observed events, in which infants do not predict what will happen but only detect incongruencies between events a posteriori. In Experiment 3, the incongruency could be the observation of flickering without an apparent cause in gap events. When pupil dilation is triggered by perceptual changes and/or the violations of welldefined predictions, the effects are registered around 1000-1500 ms in infants [START_REF] Hochmann | Negative mental representations in infancy[END_REF][START_REF] Hochmann | Representations of Abstract Relations in Infancy[END_REF][START_REF] Hochmann | The invariance problem in infancy: a pupillometry study[END_REF][START_REF] Wetzel | Infant and adult pupil dilation in response to unexpected sounds[END_REF]. In comparison, the pupil responses observed here occurred late -i.e., starting more than 3000 or 4000 ms after the ball stopped in Gap events, that is about 2000 ms later than what simple perceptual predictions would elicit.

Such late effects make it unlikely that infants reacted to the gap because they formed a detailed prediction of a contact between the ball and the cube. A possible scenario is that the state-change in occluded events triggered curiosity, asking "why is the cube flickering?", and unoccluded events provided alternative answers that infants were capable of evaluating. In this scenario, pupil dilation shows that infants considered a contact between the ball and the cube a better explanation for the flickering than a spatiotemporal gap, provided the ball was launched by an agent. The inference of a contact as the cause of flickering may result from these a posteriori evaluations.

In conclusion, along cultural evolution, humans have produced tools, whose power transcends the limits of intuitive physics, to the point of causing apparently arbitrary effects. Our results suggest that the abilities to create and understand these tools are rooted in an early capacity to understand and represent even arbitrary and opaque causal effects, as long as an intentional agent can be conceived as the ultimate cause.

Material and Methods

Participants: Sample size in Experiment 1 followed a power analysis of the results of [START_REF] Muentener | Infants' causal representations of state change events[END_REF] (Experiment 1) conducted on G*Power. That experiment replicated a previous study [START_REF] Ball | The perception of causality in the infant[END_REF], with a sample size of 20 infants.

An effect size of d = 1.11 was computed from a t-value of 2.42 and degree of freedom of 19. To obtain a similar effect a sample size of N > 11 is necessary. We thus kept the sample size of N = 20 participants for Experiments 1 and 2.

Previous studies reported no significant effect when an inanimate object caused a state change [START_REF] Muentener | Infants' causal representations of state change events[END_REF].

Moreover, an instrumental action is intuitively more complex than a direct action, as it involves three objects instead of two, and two relations instead of one. Therefore, we considered that the task in Experiment 3 could be more difficult than in Experiment 1, possibly yielding a smaller effect. We thus doubled the sample size to N = 40 participants for Experiments 3 and 4.

Experiment 1 -Twenty infants participated in Experiment 1 (mean age = 8 months 11 days, range: 7 months 29 days to 8 months 22 days). Four additional infants were tested but excluded from the final analysis for not providing good data for at least one Contact movie and one Gap movie. One additional infant was excluded because of parental interference with the procedure.

Experiment 2 -Twenty-one infants participated in Experiment 2 (mean age = 8 months 11 days, range: 8 months 1 day to 8 months 28 days). One additional infant was tested but excluded from the final analysis for not providing good data for at least one Contact movie and one Gap movie.

Experiment 3 -Forty infants participated in Experiment 3 (mean age = 8 month 12 days, range: 8 months 0 day to 9 months 0 day). Two additional infants were tested but excluded from the final analysis for not providing good data for at least one Contact movie and one Gap movie. 1 day to 9 months 0 day). Two additional infants were tested but excluded from the final analysis for not providing good data for at least one Contact movie and one Gap movie.

All participants were recruited through the consultation of birth records at the local city halls. This study was approved by the local ethics committee "Comité de Protection des Personnes Sud Est II". Parents gave an informed consent before participating in the study and were informed that they could withdraw their consent at any time without any consequence.

Stimuli: Short movies for Experiments 1-4 were created using Keynote software. Stimuli in Experiments 1-2 are adapted from [START_REF] Muentener | Infants' causal representations of state change events[END_REF]. Three movies were created. At the beginning of the Occluded movie, a red ball was visible on the screen, positioned at the right end of and partly occluded by an occluder. A yellow ball, initially placed on the left of the screen began moving at 1600 ms and went behind the occluder at 2030 ms. At 2400 ms, the red ball began moving rightward and left the screen at 3060 ms. The movie ended at 4400 ms. The Contact and Gap movies showed the same sequence of events, except that the occluder was removed, and each movie showed a different scenario of what happened behind the occluder. In the Contact movie, the yellow ball stopped when contacting the red ball at 2400 ms. The red ball began moving immediately. In the Gap movie, the yellow ball stopped short of touching the red ball at 2100 ms. The red ball began moving at 2400 ms. Contact and Gaps movies ended at 7333 ms Four movies were created for Experiment 3. In the Introduction movie, a yellow agent taking the shape of a sphere with two eyes and a mouth entered the screen from the left. It paused at 1700 ms, turned to face the participant and jumped. Then it moved right ward. It repeated the same actions again at 4330 and 6930 ms. The Introduction movie lasted 11860 ms. At the beginning of the Occluded movie, a red ball was placed at the left of an occluder, which partly occluded a blue cube on its right end. The yellow agent entered the screen from the left. It stopped at 1760 ms, when contacting the red ball, which began moving immediately.

The red ball went behind the occluder at 1800 ms. The cube began alternating between blue and yellow at 2530 ms until 3330 ms. This color alternation was accompanied by a repeated glass sound. The Contact and Gap movies showed the same sequence of events, except that the occluder was removed, and each movie showed a different scenario of what happened behind the occluder. In the Contact movie, the red ball stopped when contacting the cube at 2530 ms. The cube began changing color immediately until 3330.

In the Gap movie, the red ball stopped short of touching the cube at 2200 ms. The cube began changing color at 2530 ms until 3330. Contact and Gap movies ended at 7383 ms. Three movies were created for Experiment 4. They had the exact same structure as respectively the Occluded, Contact and Gap movies of Experiment 3, except that the yellow agent was replaced by a yellow sphere that had no face feature.

Fig. 1 -

 1 Fig. 1 -Schematic representation of the stimuli used in Experiments 1-4. In Experiments 1, 3 and 4, infants saw 60%

Fig. 2 -

 2 Fig. 2 -Pupil dilation for the Contact (blue) and Gap (red) events in Experiments 1-4. The baseline is taken in the last 500 ms before the two types of events differ (i.e., when the ball stops in Gap events). Shaded gray areas indicate the time window of interest in which cluster mass permutation tests were performed. For Experiments 1 and 2, this time window extended from the end of the baseline (0 ms) to the end of the trial. For Experiments 3 and 4, the time window of interest was restricted to the period where significant effects were observed in Experiments 1-2. Horizontal black bars indicate periods of significant larger pupil dilation for Gap vs Contact events in each Experimental Group, analyzed individually. Horizontal magenta bars indicate periods of significant interaction between Experimental Group and Trial Type. Light color areas indicate standard errors from the mean.

Table 1 -

 1 Table 1 summarizes the number of Gap and Contact trials infants contributed to the final analyses. For each participant, we computed a difference Average number of Contact and Gap trials that infants saw, and average number of good trials (see Analysis), included in the final analyses.

	Experiment	#

score (#good Gap trials -#good Contact trials) / (#good Gap trials + #good Contact trials); a positive score indicated more Gap trials; a negative score indicated more Contact trials. We then compared the distribution of this score to chance (0). This analysis showed that infants contributed significantly more Gap than Contact trials in Experiment 1 (t(19) = 2.6; P = .009) and in Experiment 3 (t(39) = 2.7; P = .006), while there was no difference in Experiment 2 (t(20) = .1; P = .46) and in Experiment 4 (t(39) = .3; P = .36). The effects in Experiments 1 and 3 cannot result from a trial order effect, as Gap and Contact trials were intermixed. Rather, we propose that these effects reflect a tendency to look longer at Gap trials. In keeping with the interpretation of traditional violation of expectation paradigms, longer looking times indicate surprise. The trial number analysis thus converges with the pupillometry analysis, suggesting that infants are surprised by Gap trials in Experiments 1 and 3, but not in Experiments 2 and 4.

Contact trials #good Contact trials #Gap trials #good Gap trials

	Experiment 1	8.4 (2.6)	2.9 (2.1)	8.5 (2.4)	3.8 (2.5)
	(Direct launching)
	Experiment 2	9.5 (1.3)	5.2 (2.8)	9.4 (1.5)	5.3 (2.8)
	(Control)
	Experiment 3	8.5 (1.9)	5.4 (2.6)	8.8 (1.8)	5.8 (2.1)
	(Indirect state change with Agent)
	Experiment 4	7.6 (2.3)	5.0 (2.3)	7.7 (2.2)	5.0 (2.6)
	(Indirect state change without Agent)
	Standard deviations from the mean are indicated in parentheses.

Acknowledgements

This work was funded by the collaborative McDonnell Foundation Grant 220020449 and the LABEX CORTEX (ANR-11-LABX-0042). We thank Liuba Papeo, Susan Carey and two anonymous reviewers for comments on previous versions of this manuscript. We thank Céline Spriet, Marion Gasselin, and Emilie Serraille for their help in recruiting and testing infants.

Procedure: Presentation of the stimuli as well as data recordings were controlled by PsyScope X (http://psy.cns.sissa.it). Each trial in the experiment consisted in the presentation of one movie. Two trials were separated by an animation of colorful rotating arrows to attract the attention of participants toward the center of the screen. Each trial began when participants gazed toward the center of the screen for at least 100 ms.

Experiment 1 consisted in a maximum of 52 trials. Movies were shown in a pseudo-random order, with 60% of Occluded movies, 20% of Contact movies and 20% of Gap movies. The first 5 trials always showed Occluded movies; two Gap or two Contact movies never directly followed each other.

Experiment 2 consisted in 10 contact and 10 Gap movies showed in a pseudo-random order, avoiding more than two repetitions of the same movie.

Experiment 3 began with the Introductory movie, followed by up to 52 trials. Movies were shown in a pseudo-random order, with 60% of Occluded movies, 20% of Contact movies and 20% of Gap movies. The first 5 trials always showed Occluded movies; two Gap or two Contact movies never directly followed each other.

Experiment 4 consisted in a maximum of 52 trials. Movies were shown in a pseudo-random order, with 60% of Occluded movies, 20% of Contact movies and 20% of Gap movies. The first 5 trials always showed Occluded movies; two Gap or two Contact movies never directly followed each other.

Analysis:

All data and codes for analyses are shared publicly at osf.io/4caht/.

The pupil diameter for both eyes was recorded for gazes on the screen. For each trial, we considered a baseline time window lasting 500 ms and ending at the moment the Contact and Gap movies began to differ, that is when the ball stopped in the Gap movies (2100 ms after the trial onset in Experiments 1-2; 2200 ms after the trial onset in Experiments 3-4). The end of the baseline was then considered time 0. We excluded trials with less than 65% # of pupil diameter information over the period that begins with the gap onset in Gap events and ends with the trial -i.e., 0-5233 ms in Experiments 1-2, 0-5183 ms in Experiments 3-4 -and/or less than 100 ms of pupil diameter information in the baseline time window. Only the Contact and Gap trials were analyzed. Infants with at least one included trial of each type (Contact and Gap) were included in the final analyses. The average number of good trials per infant is reported in Table 1 for each experiment. Missing data points for good trials were linearly interpolated. Non-parametrical cluster mass permutation tests [START_REF] Hochmann | The invariance problem in infancy: a pupillometry study[END_REF][START_REF] Maris | Nonparametric statistical testing of EEG-and MEG-data[END_REF] were implemented to probe the variation of pupil dilation in response to Contact and Gap movies and the effect of Experimental Group (Experiment 1 vs.

# The choice of this parameter was set for the analysis of Experiment 1 and maintained for the other experiments. In Supplementary Materials, we report a multiverse analysis, demonstrating the solidity of our results when varying this criterion between 50% and 80%.

12 Experiment 2; Experiments 3 vs. Experiment 4). To test the effect of Trial Type (Gap, Contact), we compared the pupil dilation for Gap vs Contact trials at each timepoint with a paired-sample t-test.

Neighboring timepoints showing an effect with a t-value larger than a threshold corresponding to (p<0.05) were clustered and tested for significance using a permutation test, with 1000 random permutations of the trial-type labels on the original data. The significance probability of the original clusters was computed as the number of times the shuffled data produced clusters with higher summed t values than the real data. To test the interaction of Experimental Group (Experiment 1 vs. Experiment 2; Experiment 3 vs. Experiment 4) and Trial Type (Gap vs. Contact), we computed the time-course of the difference of pupil dilation between Gap and Contact trials for each infant. We then compared these values at each timepoint using 2sample t-tests. Neighboring timepoints showing an effect with a t-value larger than a threshold corresponding to (p<0.05) were clustered and tested for significance using a permutation test, with 1000 random permutations of the Experimental Group labels on the original data. The significance probability of the original clusters was computed as the number of times the shuffled data produced clusters with higher summed t values than the real data. The time window where effects were observed in Experiments 1 and 2 served to highlight a window of interest to constrain the analyses in Experiments 3 and 4.

All t-tests reported in the paper are one-tail, as we had strong predictions for each of the tests that we ran.

We predicted that Gap trials should elicit larger pupil dilation than Contact trials in Experiments 1 and 3; we predicted larger differences between Gap and Contact trials in Experiment 1 than in Experiment 2, because we hypothesized that the effect observed in Experiment 1 is due to the interpretation of occluded events that are absent in Experiment 2; we predicted larger differences between Gap and Contact trials in Experiment 3 than in Experiment 4, because we hypothesized that infants consider that the ultimate cause of the state change in Experiment 3 is the Agent, which is absent in Experiment 4.

We also compared the number of Gap vs Contact trials included in each Experiment. This analysis is detailed in Trial Number analysis in the result section.

Supplementary materials

Multiverse Analysis

In order to assess the robustness of our results, we conducted a multiverse analysis, varying the arbitrary criterion for selecting good trials between 50% and 80%. While 50% is probably too lenient, including trials for which infants are not much attentive, 80% resulted in too high an attrition rate in Experiment 1 (about 50%). The chosen value of 65%, as reported in the main text, constitutes a trade-off between these two values for Experiment 1 (a conceptual replication of several looking-time studies [1][2][3][4]). It was set for the analysis Experiment 1 and maintained for the other experiments. For each value of the exclusion criterion, for Experiments 1-4, Table S1 reports the results of a cluster mass permutation test assessing larger pupil dilation for Gap vs. Contact events, and Table S2 reports the results of the analysis of the number of contributed Gap and Contact trials. These analyses show that the resultsespecially those of Experiments 3-4 -are largely independent from the choice for the value of the exclusion criterion. This multiverse analysis thus supports the main conclusions presented in the main text.

Table S1 -Summary results of a multiverse analysis of the pupil dilation. The criterion for trial exclusion is a minimal percentage of pupil data required to include a trial in the final analysis. For each experiment and each value of the criterion for trial exclusion, we report the result of a cluster mass permutation test assessing larger pupil dilation in Gap vs. Contact events. The time window where the effect is observed is indicated when the Pvalue is significant (P <.05) or when it shows a trend (P <.10). n.s.: not significant; i.e., P >.