Lara Narbona Sabaté

Geoffrey Mesbahi

Guillaume Dezecache

Cristiane Cäsar

Klaus Zuberbühler

Mélissa Berthet

Lara Narbona Sabaté

Animal linguistics in the making: the Urgency Principle and titi monkeys' alarm system

Keywords: Callicebus nigrifrons, vocalisations, sequences, computational linguistics, formal linguistics

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INTRODUCTION

The investigation of meaning (or 'semantics') in nonhuman animal vocalizations has played a central role in the field of animal communication since the foundational work on vervet monkey alarm calls (Seyfarth et al. 1980a(Seyfarth et al. , 1980b)). The question and debate of what animal calls mean is still ongoing as shown by the recent discussions on the notion of functionally referential communication (e.g., [START_REF] Wheeler | Functionally referential signals : A promising paradigm whose time has passed[END_REF][START_REF] Townsend | Functionally referential communication in mammals : The past, present and the future[END_REF] or the debates around the specificity of animal calls [START_REF] Fichtel | Anti-predator behavior of group-living Malagasy primates : Mixed evidence for a referential alarm call system[END_REF]Schlenker et al. 2016b;[START_REF] Dezecache | Working hypotheses on the meaning of general alarm calls[END_REF].

Debates surrounding the concept of meaning in animals come from practical and theoretical difficulties encountered when investigating animal semantics. First, methods commonly used in animal communication to reveal linguistic-like capacities in nonhuman animals sometimes fail to reveal these capacities even in humans [START_REF] Prat | Animals have no language, and humans are animals too[END_REF]. Second, some have discussed the use of linguistic terminology (e.g., 'semantics') in animal communication (e.g., [START_REF] Seyfarth | The central importance of information in studies of animal communication[END_REF][START_REF] Kershenbaum | Acoustic sequences in non-human animals : A tutorial review and prospectus[END_REF][START_REF] Scott-Phillips | Meaning in animal and human communication[END_REF][START_REF] Fitch | Why formal semantics and primate communication make strange bedfellows[END_REF], mainly because of conceptual and theoretical disagreements. Third, since some species combine calls into sequences (see review in Zuberbühler & Lemasson 2014) allocating meaning to an utterance becomes even more complicated, due to the fact that researchers must investigate not only the meaning of the individual component parts but also that of the sequences, and identify potential combination rules (e.g., [START_REF] Engesser | Meaningful call combinations and compositional processing in the southern pied babbler[END_REF][START_REF] Suzuki | The syntax-semantics interface in animal vocal communication[END_REF].

One attempt to overcome these difficulties is the emerging field of animal linguistics, which applies methods from linguistics to animal communication. This is based on the vision that an interdisciplinary collaboration between biologists and linguists can result in a shared terminology and methodology, and that this will foster a more thorough investigation of the communicative capacities of nonhuman animals. Two main linguistic domains are of particular relevance. First, computational linguistics offers mathematical tools that can help to detect underlying structures in complex vocal sequences (see [START_REF] Kershenbaum | Acoustic sequences in non-human animals : A tutorial review and prospectus[END_REF]). Such methods have been successfully applied to several communication systems [START_REF] Kershenbaum | Entropy rate as a measure of animal vocal complexity[END_REF][START_REF] Alger | A novel statistical method for behaviour sequence analysis and its application to birdsong[END_REF][START_REF] Berthet | Titi monkeys combine alarm calls to create probabilistic meaning[END_REF]. Second, formal linguistics provides tools to investigate the combination rules linked to the meaning of individual calls, in order to determine the semantics of the resulting sequences. In a series of analyses, [START_REF] Schlenker | Monkey semantics : Two 'dialects' of Campbell's monkey alarm calls[END_REF]Schlenker et al. ( , 2016aSchlenker et al. ( , 2016bSchlenker et al. ( , 2016c[START_REF] Schlenker | Titi semantics : Context and meaning in titi monkey call sequences[END_REF] proposed to (1) investigate the literal meaning of calls based on the circumstances of their emission (a concept otherwise redefined in [START_REF] Dezecache | Working hypotheses on the meaning of general alarm calls[END_REF], (2) identify the pragmatic inferences that enrich their meanings, and (3) establish the rules that structure the sequences and contribute to their semantics. Following this approach, Schlenker et al. (2016b) proposed the 'Informativity Principle', which states that if one call conveys more information than another call, then the most informative call should be used whenever possible. If the less informative call is emitted, then one can infer that the more informative call is not applicable (otherwise, the emitter would have uttered it). A concrete example comes from the alarm system of male blue monkeys (Cercopithecus mitis). They emit 'pyow' calls in a wide variety of alerting situations, such as the presence of a terrestrial predator or during agonistic interactions. The literal meaning of pyow is thus 'alert'. However, when a raptor is present, pyows are typically not emitted (even though the presence of a raptor is an alerting situation). Rather, male blue monkeys emit 'ka' calls. Thus, these calls are specific to the presence of aerial predators. 'Ka' calls are more informative than 'pyow' calls: while 'pyow' refers to a large set of alerts, the emission of 'ka' is linked to a much-restricted set of circumstances (namely, the presence of a raptor). The Informativity Principle states that, since individuals tend to be as informative as possible when communicating, the least informative call ('pyow') is only emitted when its most informative counterpart ('ka') is not applicable. As such, the enriched meaning of 'pyow' is 'alert but not raptor-related' (otherwise, the 'ka' call would have been emitted).

Another principle proposed by Schlenker et al. (2016b) is the 'Urgency Principle'. The principle postulates that, in a threatening situation, urgent information (e.g., nature or location of the threat) should be communicated as soon as possible. As a consequence, calls conveying urgent information should come before those conveying non-urgent information in the call sequence (Schlenker et al. 2016b(Schlenker et al. , 2016c)). One concrete example is the male putty-nosed monkey alarm system (Cercopithecus nictitans). Male putty-nosed monkeys emit 'hack' calls when confronted with an aerial threat (typically, a raptor), while 'pyow' is emitted to unspecific, general alerts. They also emit 'pyow-hack' sequences to elicit group movements (Arnold & Zuberbühler 2006a, 2006b). The rules of combination of these "pyow-hack" sequences are puzzling. The system does not seem to be idiomatic, since the sequences are slowly emitted and not stereotyped. However, it is not compositional either, since the conjunction of a raptor-related call ('hack') and a general alert call ('pyow') in no obvious way refers to a group movement. Schlenker et al. (2016aSchlenker et al. ( , 2016b) ) argued that this combination might be ruled by the Urgency Principle. According to this principle, if a raptor was present, calls conveying urgent information about the threat (here, 'hack') should come before calls that do not (here, 'pyow'). On the contrary, 'pyow-hack' sequences can only be emitted when group movement is required, but no raptor is present (otherwise 'hack' would have been emitted before 'pyow'). The Urgency Principle is useful here to determine that the meaning of 'hack' is not 'raptor', as could be expected from the context of emission of individual calls, but rather 'non-ground movement related alert': when a 'hack' is uttered first, it refers to the most urgent non-ground movement related alert, i.e., a raptor.

The Informativity Principle and the Urgency Principle have demonstrated their heuristic value for the alarm calling behaviours of several nonhuman primate species. Specifically, they propose principles based on competition among meaningful vocalizations to help understand why one call is emitted rather than another. Nevertheless, it remains unclear whether the framework proposed by Schlenker et al. (2016b) represents a general principle in animal communication, which would require testing it with further data. Second, the understanding of cognitive mechanisms underlying call production and comprehension, which could support the authors' hypotheses, is lacking. Third, the Urgency Principle is currently tailored to species whose sequence structure relies solely on call ordering. However, an increasing amount of evidence suggests that call sequences can convey reliable information through a large set of other encoding mechanisms, such as call intervals, repetition of elements, or probabilities (see review in [START_REF] Engesser | Combinatoriality in the vocal systems of nonhuman animals[END_REF]. For the Urgency Principle to be more largely applicable to animal systems, it needs to be refined to integrate any sort of encoding mechanisms. Finally, these theoretical principles are post-hoc speculations, which provide an interpretation of the sequences of calls once they have been produced. It remains to be tested whether they are relevant to the studied species and whether they account for the mechanisms underlying call production.

Here, we test the hypothesis that the Urgency Principle can be applied to nonhuman animal communication more generally. First, we propose an extended version of the Principle stating that encoding mechanisms (hereafter, 'mechanisms') conveying urgent information should take place before those that do not in the sequence. Second, we propose to investigate whether this extended Urgency Principle reflects the organisation of the alarm sequence of a nonhuman primate, the titi monkey, using methods borrowed from computational linguistics. Titi monkeys (Callicebus nigrifrons) are an ideal candidate species for this exercise. Their alarm vocal system has been well investigated by biologists and linguists (Cäsar et al. 2012a(Cäsar et al. , 2012b[START_REF] Cäsar | Titi monkey call sequences vary with predator location and type[END_REF][START_REF] Schlenker | Titi semantics : Context and meaning in titi monkey call sequences[END_REF][START_REF] Berthet | Contextual encoding in titi monkey alarm call sequences[END_REF][START_REF] Berthet | Titi monkeys combine alarm calls to create probabilistic meaning[END_REF][START_REF] Commier | Commentary: Titi semantics: Context and meaning in titi monkey call sequences[END_REF].

During predator encounters, titi monkeys emit alarm sequences: the first individual to spot the threat emits soft calls, and other group members join into a chorus composed of soft and loud calls that can last up to two hours (M. Berthet pers. obs.; Cäsar 2011). Because these long and multi-caller sequences are difficult to investigate with current methods, previous studies [START_REF] Cäsar | Titi monkey call sequences vary with predator location and type[END_REF][START_REF] Berthet | Titi monkeys combine alarm calls to create probabilistic meaning[END_REF]) have focussed on the first 10 and 30 calls of the alarm sequence (or respectively, during the first 18 and 37 sec): these calls are emitted by one caller only, and are likely to convey enough information about the predatory event for kins to adopt a sound reaction. These 10-and 30-alarm sequences are mostly composed of two alarm soft calls, A-and Bcalls. Based on experimental presentations of natural predators, it was shown that these sequences can encode reliable information about the type of predator (aerial vs terrestrial) and its location (ground vs canopy) [START_REF] Cäsar | Titi monkey call sequences vary with predator location and type[END_REF][START_REF] Berthet | Titi monkeys combine alarm calls to create probabilistic meaning[END_REF]. More importantly, this information seems to be conveyed in a gradual, probabilistic fashion by the proportion of consecutive B-calls [START_REF] Berthet | Titi monkeys combine alarm calls to create probabilistic meaning[END_REF]. Specifically, sequences with a high proportion of consecutive B-calls (BB-grams) mostly refer to terrestrial predators on the ground, while sequences with a low proportion of BB-grams mostly refer to aerial predators in the canopy.

To investigate whether the extended Urgency Principle reflects the organization of titi monkey alarm sequences, we propose to investigate how information about the context of emission of the call unfolds with the emission of a sequence. To this end, we investigate how contextual parameters (predatory and social situation) influence specific features of the sequence as it unfolds, using random forests algorithms. Importantly, we extend the analysis window to up to 50 calls (mean ± SD = 54.84 sec ± 52.14). This choice is two-fold. First, it allows us to ensure that, in case it is strictly encoded after urgent information, non-urgent information has the opportunity to be transmitted. Second, these sequences can be reliably analysed, for few callers are involved in the chorus at this stage and calls do not extensively overlap. We hypothesized that, if the extended Urgency Principle reflected the sequential organization of titi monkeys' alarm call sequences, mechanisms encoding urgent information (i.e., information about the predatory situation) should appear before mechanisms that do not.

MATERIAL AND METHODS

Dataset

We extracted data from the datasets published by [START_REF] Cäsar | Titi monkey call sequences vary with predator location and type[END_REF] and [START_REF] Berthet | Titi monkeys combine alarm calls to create probabilistic meaning[END_REF]. Both datasets were collected at the Reserva Particular do Patrimônio Natural Santuário do Caraça, MG, Brazil (20°05′S, 43°29′W) from six wild groups of black-fronted titi monkeys (Callicebus nigrifrons) habituated to human observers (see more details about the field site and the population in [START_REF] Berthet | Dramatic decline in a titi monkey population after the 2016-2018 sylvatic yellow fever outbreak in Brazil[END_REF]. Monkey groups were presented with three stuffed predators: one aerial predator (caracara, Caracara plancus), and two terrestrial predators (tayra Eira barbara and oncilla Leopardus guttulus). Each species of predator was presented twice to each group: once at its usual location (i.e., in the tree canopy for the aerial predators, on the ground for terrestrial predators) and once at its unusual location (i.e., on the ground for the aerial predators, in the tree canopy for terrestrial predators). Focal group's vocal reactions were recorded.

Experiments were conducted from August 2008 to July 2010 on five groups (A, D, M, P and R groups), and replicated from May 2015 to August 2016 on the same five groups plus the S group. Group composition varied between study periods, due to births, dispersals and deaths (see details in [START_REF] Berthet | Dramatic decline in a titi monkey population after the 2016-2018 sylvatic yellow fever outbreak in Brazil[END_REF]. For full experimental setup and trials exclusion criteria, see [START_REF] Cäsar | Titi monkey call sequences vary with predator location and type[END_REF] and [START_REF] Berthet | Titi monkeys combine alarm calls to create probabilistic meaning[END_REF].

For each recording, we collected 18 contextual parameters: (1) group identity, (2) predator type (aerial or terrestrial), (3) predator species (caracara, tayra, or oncilla), (4) predator location (canopy or ground), (5) predator height (i.e., its distance from the ground), (6) identity of the first individual to call at the predator, (7) height of the first individual to call (i.e., its distance from the ground), (8) distance between the first individual to call and the predator, (9) number of individuals composing the group, and (10-18) demographic features of the group, namely the number of infants, of female and male juveniles, of female and male subadults, of female and male adults, of paired adults and of non-paired adults. We considered individuals as adults from the age of 30 months, sub-adults between 18 and 30 months, juveniles between six and 18 months, and infants if less than 6 months old [START_REF] Cäsar | Anti-predator behaviour of black-fronted titi monkeys (Callicebus nigrifrons)[END_REF].

The final dataset comprised 58 sequences: 23 sequences were collected during the first field period (August 2008-July 2010), and 35 sequences collected during the second field period (May 2015-August 2016).

Sequence coding

We used the vocal repertoire described by Cäsar et al. (2012b), which comprises two alarm call types: A-call and B-call. To assure that call types were reliability coded, L. Narbona Sabaté and M. Berthet performed an inter-observer reliability test: they both coded 199 calls from 12 randomly selected sequences, comprising about 7% of the final dataset. According to standards [START_REF] Landis | The measurement of observer agreement for categorical data[END_REF][START_REF] Hallgren | Computing inter-rater reliability for observational data : An overview and tutorial[END_REF], agreement was almost perfect (Cohen's k = 0.81).

L. Narbona Sabaté extracted the first calls up to 50, and labelled each of them as A-, B-call, or 'other' (undetermined call or other call type), based on listening and visual inspection of the spectrograms. L. Narbona Sabaté also measured each inter-call silence interval. Labelling and measurements were conducted using the PRAAT acoustic analysis software (version 6.1, [START_REF] Boersma | Praat: Doing phonetics by computer[END_REF].

The final dataset comprised 58 sequences. Fifty-two sequences were 50-call long. Five sequences were shorter than 50 calls, because monkeys stopped calling before the emission of 50 calls (four sequences were one-call long and one was 43-call long). Finally, one sequence was only 20-call long because it could not be entirely recorded due to logistic issues (Appendix I in Supplemental Data).

Metric extraction

This study aimed to investigate how semantic information unfolds within titi monkey alarm sequences. The first step of this analysis was to unroll the sequence, and describe how its organization and structure changed with the emission of new calls. To do so, each sequence was cut into subsequences, so that a sequence of x calls was decomposed into x subsequences starting with the first call of the sequence and ranging from one to x calls long. Our final dataset was thus composed of 58 sequences, comprising a total of 2,667 subsequences.

We characterized each subsequence by a set of 24 quantitative variables (henceforth, 'metrics'), following the procedure used in [START_REF] Berthet | Titi monkeys combine alarm calls to create probabilistic meaning[END_REF]. Metrics comprised: (1) the mean call interval, defined as the mean of the inter-call silence durations in the subsequence, (2) the coefficient of variation of call interval, defined as CV = standard deviation/mean of inter-call silence durations, (3-4) the proportion of each A-call and B-call in the subsequence, (5-8) the proportion of each possible two consecutive calls, or 2-grams (AA, AB, BA, BB) in the subsequence, (9-16) the proportion of each possible combination of three consecutive calls, or 3-grams (AAA, AAB, ABB, ABA, BBB, BBA, BAA, BAB) in the subsequence, (17,18) the N-gram slopes (here, 2-gram and 3-gram slopes), to test whether a N-gram is more present in the subsequence than the others. To compute this metric, we drew a graphic representation of the probability of each N-gram (either 2-or 3-gram) sorted by decreasing probability, and we extracted the coefficient of regression: if it was different from 0, then one Ngram was more represented in the sequence, (19) the slope of entropy, calculated using Shannon entropy, to measure the organizational complexity of a subsequence. To compute this metric, we plotted zero-, first-and second-order entropies [START_REF] Mccowan | Quantitative tools for comparing animal communication systems : Information theory applied to bottlenose dolphin whistle repertoires[END_REF]) and extracted a coefficient of regression: a negative slope indicated an important sequential organization and high communication capacities, while a null slope indicated a random organization, with low communicative capacities, (20-23) the probability of transitions between each call types (A to A, A to B, B to A, B to B), ( 24) the last call emitted as a proxy of call order. More details can be found in [START_REF] Berthet | Titi monkeys combine alarm calls to create probabilistic meaning[END_REF].

When calculating proportions of 2-and 3-grams, we ignored calls that were coded as 'other': since this label refers to a large set of cases (non-identifiable calls, calls given by naïve individuals in the background, or non-alarm calls), we remained conservative and removed them from the analysis. Concretely, the computation of 2and 3-grams stopped before the 'other' call and re-started from null right after. For each occurrence of one of these calls, two less 2-grams and three less 3-grams were computed in the analysis.

We calculated proportions and probabilities using a Bayesian correction, in order to estimate the occurrence of rare events despite our small sample size [START_REF] Alger | A novel statistical method for behaviour sequence analysis and its application to birdsong[END_REF]. Briefly, we calculated each proportion of event i as proportion i = (yi + alpha)/(total number of events + k * alpha), with yi the number of events i before Bayesian correction, k, the number of possible events, and α, the prior distribution. We chose α = total number of events/(k * 100) as prior distribution, so that we simulated that each subsequence comprised 1% more events i. For example, to calculate the proportion of A-calls in a subsequence of 10 calls, yi is the number of A-calls in the sequence, the 'total number of events' is the number of calls in the sequence (here, 10) and k is the number of possible call types (here, two: A and B-calls). More details about the method can be found in [START_REF] Alger | A novel statistical method for behaviour sequence analysis and its application to birdsong[END_REF], [START_REF] Berthet | Titi monkeys combine alarm calls to create probabilistic meaning[END_REF], and in the online scripts.

These analyses were conducted using R version 3.6.1 (R Core Team 2020) and the cfp package [START_REF] Neumann | cfp: Christof's function package (0.1.11[END_REF].

Statistical analysis

In order to investigate semantic information as the sequence unrolls, we analysed how the structure of the sequence (its metrics) varied with the number of calls already emitted and the contextual information. To this end, we investigated what contextual parameters (if any) influenced the metrics' values of each subsequence using random forests algorithms. Random forests are machine learning algorithms that predict the value of a variable to explain (here, each metric) based on explanatory variables (here, the contextual information), using a set of multiple decision trees.

We split the dataset into 50 subsets, one for each subsequence length. For each subset, we conducted the following analysis: first, for each metric, the explanatory variables that accounted for most of the metric's variance, if any, were extracted. To this end, we used a Variable Selection Using Random Forest (VSURF) method: for each metric, we ran 50 random forests, each composed of 2,000 decision trees built using six randomly selected contextual parameters as explanatory variables. We then used a three-step variable selection procedure to eliminate contextual variables that were redundant and/or irrelevant to the metric [START_REF] Genuer | VSURF : An R package for Variable Selection Using Random Forests[END_REF]. Remaining contextual variables were the most likely to impact the metric for this subsequence length. In other terms, these contextual parameters were likely to be encoded by the metric at this time of the sequence.

Second, for each metric, we ran a random forest containing 500 decision trees built using the contextual variables selected during the previous analysis. We extracted the percentage of variance explained (here: pseudo R-squared) for each random forest: this value, ranging from zero to 100, indicates the robustness of the prediction. The complete statistical analysis (i.e., the extraction of relevant contextual variables and the computation of random forests) was repeated for each subset, so that we computed 1,197 random forests.

After computing all 1,197 random forests, we selected those whose predictions were the most robust. We plotted all the values of variance explained (sorted by increasing order) and extracted the inflexion point. Random forests with a value of variance explained strictly greater than the inflexion point were considered the most robust and included in the results: they indicated, for each subset, what metric(s) encoded information, and what type(s) of contextual information they encoded, or in other words, how encoded information varied as the sequence unrolled.

The analyses were conducted using R version 3.6.1 (R Core Team 2020), the tidyverse package version 1.3.0 [START_REF] Wickham | Welcome to the Tidyverse[END_REF], the VSURF package version 1.1.0. [START_REF] Genuer | VSURF : An R package for Variable Selection Using Random Forests[END_REF], the randomForest package version 4.6-14 [START_REF] Liaw | Classification and regression by randomForest[END_REF], and the inflection package version 1.3.5 (Christopoulos 2019). The full dataset and the associated statistical scripts are available on a Figshare depository (https://figshare.com/projects/Animal_linguistics_inthe_making_The_Urgency_Principl e_and_titi_monkeys_alarm_system/121914).

RESULTS

After the variable selection step and the robustness step, the inflexion point occurred at 45.76% of variance explained (Appendix II in Supplemental Data). Random forests with a greater percentage of variance explained were considered robust, which was the case for 398 out of 1,197 random forests. While random forests were built with a total of 24 metrics as variables to explain, only 11 metrics were present in the retained random forests (Appendix III in Supplemental Data). Similarly, 12 contextual variables composed the retained random forest, out of the 18 contextual variables proposed. Each retained random forest included 1.11 ± 0.36 (mean ± SD) contextual variables (a schematic summary can be found in Appendix IV in Supplemental Data).

What information is conveyed, and when

Twelve main contextual variables were retained. Two contextual variables (namely the type of predator and the predator species) were predominantly encoded throughout the sequence: they were relevant for subsequences of all lengths and accounted for most of the variance of 148 and 246 of the retained random forests, respectively (Fig. 1, Table 1). Three other contextual variables were of lesser importance in the sequences: they accounted for most of the variance of eight to 11 retained random forests (Table 1). These included the height of the predator (relevant only for subsequences of one to three calls long), the number of juvenile females (relevant for subsequences of eight to 18 calls long), and the number of individuals (relevant for some subsequences from eight calls long) (Fig. 1). Finally, other contextual information, like the location of the predator, the distance between the caller and the predator, the height of the first caller, the group identity, the number of unmated adults, the number of juveniles and the number of subadult males, were sporadically present along the later sequence (from eight to 50 calls long, Fig. 1, Table 1).

How information is conveyed

Eleven metrics were selected by the statistical analysis (Fig. 2). Five of these metrics were relevant to all subsequences' length: the proportion of A-and B-calls, the transition probability from B-to B-calls, the proposition of 2-grams BB, and the proportion of 3-grams BBB. Three other metrics were relevant for more than 70% of subsequences lengths: the transition probabilities from A-to A-calls (relevant for 39 subsequences), the proportion of 2-grams AA (relevant for 37 subsequences) and the proportion of 3-grams AAA (relevant for 45 subsequences). Finally, the last call was relevant for 27 subsequences mostly comprised within the first 20 calls, and the proportions of 3-gram BAB and BBA were only relevant for two subsequences each (nine to 10 and eight to nine, respectively). Interestingly, predator's type, species and height were encoded by all metrics except the proportions of 3-grams BAB and BBA. These two metrics exclusively coded for social information and were relevant for subsequences of eight to 10 calls long (Table 1, Fig. 1). The number of juvenile females was encoded by a metric that also coded for predator type and species: the proportion of 3-grams AAA (Table 1). Schlenker et al.'s (2016b) Urgency Principle states that calls conveying information about the nature or location of a threat should come first. This principle helped draw satisfactory conclusions about the formal properties of a specific primate call system, but it has not been further tested with other species. This study investigated whether the Urgency Principle reflected the organization of another animal communication system, the titi monkey alarm calls. Since some species (including titi monkeys) do not only rely on call order to convey information [START_REF] Engesser | Combinatoriality in the vocal systems of nonhuman animals[END_REF], we proposed to extend the Urgency Principle to other known encoding mechanisms that are relevant to these species. This extended Urgency Principle states that encoding mechanisms that convey urgent information should take place before those that do not, regardless of the mechanism. Our hypothesis was that, if the extended Urgency Principle reflected the sequential organization of the titi monkeys' alarm call sequences, mechanisms encoding information about predator type and location should appear early in the sequences.

DISCUSSION

Two main information types were encoded in the sequences. First, information about the predatory situation (mainly, the nature and location of the predator) was encoded since the very beginning of the sequence (Fig. 1), using eight mechanisms that occurred throughout the whole sequence, and the last call which mostly took place at the beginning of the sequence (Fig. 2). Second, social information (mainly the number of individuals and the number of juvenile females) was sporadically encoded later (Fig. 1), using one mechanism that was salient throughout the whole sequence (proportion of 3-gram AAA, which also encoded information about the predatory situation), one mechanism that mostly occurred at the beginning of the sequence (last call, which also encoded for predator type, species and height), and two mechanisms (the proportion of 3-grams BAB and BBA) that appeared later in the sequence (Fig. 2). Overall, mechanisms encoding for urgent information took place since the beginning of the sequence, while mechanisms exclusively encoding for less-urgent information arose later.

Our results confirm that the Urgency Principle may be applicable to animal communication systems more generally. Alarm calls or sequences are designed to convey rapid and reliable information to conspecifics about the presence of a nearby threat. In some species, individuals react more strongly and faster to the alarm calls of conspecifics than to the calls of a predator [START_REF] Barrera | Reliability of public information : Predators provide more information about risk than conspecifics[END_REF] or to the presence of the predator itself [START_REF] Mclachlan | Speedy revelations : How alarm calls can convey rapid, reliable information about urgent danger[END_REF]. It is not surprising that alarm sequences are structured so that urgent information is conveyed at the beginning: this provides an effective warning to conspecifics, especially those that are naive to the presence of the threat [START_REF] Griesser | Do warning calls boost survival of signal recipients? Evidence from a field experiment in a group-living bird species[END_REF]. Similarly to titi monkeys, New Holland honeyeaters encode urgent information (there, the urgency of the threat) in the very first call of their alarm sequences [START_REF] Mclachlan | Speedy revelations : How alarm calls can convey rapid, reliable information about urgent danger[END_REF].

While information about the nature of the predator was encoded throughout the sequence, information about its location was mostly present at the start of the sequence. Two hypotheses can explain this difference. First, it is possible that information about predator location is of lesser importance than information about predator species or type. While information about the predator's nature is crucial to adopt an adaptive reaction (e.g., mobbing, specific escape strategy) and may need to be redundantly sent to ensure reception by conspecifics (McLachlan & Magrath 2020) and effective coordination of the group, information about a predator location may only be needed at the beginning to make sure that all members have correctly spotted the predator. The second hypothesis is that mechanisms encoding predator location are related to identification mistakes. Even if tayra and oncillas are good climbers [START_REF] Brosset | Observations sur l'éthologie du tayra Eira barbara (carnivore)[END_REF][START_REF] Sunquist | Oncilla[END_REF], predators in the canopy mostly are raptors (M. Berthet pers. obs.). Raptors typically elicit sequences beginning with A-calls (Appendix I in Supplemental Data). Individuals that spot something in the canopy may quickly emit Acalls, even if they are not sure about the nature of the threat, in case this is a raptor («better be safe than sorry» strategy, [START_REF] Ferrari | Predation risk and antipredator strategie[END_REF]. They may later switch to B-calls if determining that the threat is a terrestrial one. If so, location would be encoded as a byproduct of the predator identification process, while being nonetheless relevant to conspecifics [START_REF] Berthet | Titi monkeys combine alarm calls to create probabilistic meaning[END_REF]).

Schlenker et al.'s (2016b) Urgency Principle was initially articulated to shed

light on the mechanisms underlying information transfer in putty-nosed monkeys, which combined meaningful calls into meaningful sequences using an ordering rule. But nonhuman animals can encode information using a large variety of encoding strategies [START_REF] Engesser | Combinatoriality in the vocal systems of nonhuman animals[END_REF]). Here, we showed that the Urgency Principle can be extended beyond single calls: in titi monkey alarm sequences, mechanisms encoding for urgent information appeared earlier than those that do not. This suggests that the Urgency Principle can be applied to nonhuman sequences that do not rely solely on call ordination to convey information. Interestingly, information related to the predatory event is predominant in the sequences, while social information (mainly, the number of individuals and the number of juvenile females) occurs sporadically. Information about group composition is not crucial in a predatory context, and encoding mechanisms specifically allocated to this information pertain to disruptions of the sequence pattern (proportion of 3-grams BAB and BBA), which could suggest that social encoding is a by-product of the group's call emission. Titi monkeys live in groups composed of one breeding pair and their offspring (Bicca-Marques & Heymann 2013). The group often remains in close physical proximity, which makes it impossible to isolate each individual's vocal contribution from the chorus. As a result, our study investigated the vocal reactions of the groups, comprising between two and seven individuals. The first hypothesis is that each individual's vocal utterance may disrupt others' utterances. [START_REF] Cäsar | Titi monkey call sequences vary with predator location and type[END_REF] showed that individual sequences encoded predator type and location through disruption patterns (e.g., a sequence composed of B-calls indicated a terrestrial predator on the ground while a single A-call at the beginning of the B-sequence indicated a terrestrial predator in the canopy). Several individuals spotting the predator at different times may produce asynchronous alarm sequences that disrupt others' sequence. The second hypothesis pertains to communication development. While call production (i.e., the emission of well-formed acoustic structures) seems to be largely innate in nonhuman primates [START_REF] Gultekin | High plasticity in marmoset monkey vocal development from infancy to adulthood[END_REF]e.g., Snowdon 2009; but see [START_REF] Watson | Vocal learning in the functionally referential food grunts of chimpanzees[END_REF], call usage (i.e., the production of calls in the right situation) is socially learnt [START_REF] Seyfarth | Vocal development in vervet monkeys[END_REF][START_REF] Snowdon | Chapter 7. Plasticity of communication in nonhuman primates[END_REF]: young individuals gradually learn to produce calls in the right context. The larger the titi monkey group, the more immature individuals (including juvenile females), which may increase the chances of emission of inappropriate calls and therefore, create disruption patterns. Further work is needed to test the effect of additional factors such as the variation of distance between individuals or the latency of each individual's call emission, and conclude on the underlying mechanisms.

In this article, we were able to confirm that methods from computational linguistics can be successfully applied to animal communication to process large datasets, highlight underlying structure and unroll information transfer processes. We have also shown that one of the key principles from the animal formal semantics framework (namely, the Urgency Principle) is an encoding strategy that is found in at least one species of nonhuman primates. Further verifications are needed to confirm that these methodologies are entirely reliable, such as testing whether the Informativity Principle is also relevant to nonhuman animals, and extending these investigations to other taxa. Overall, our work provides further evidence that the emerging field of animal linguistics offers promising methods and theories that can help unveil the linguistic properties of nonhuman communication systems.

Fig. 1 .

 1 Fig. 1. -Information encoded from the first to the 50th call of the alarm sequence. Coloured cells represent the information type encoded at each point of the sequence: green cells represent information about the predatory event and yellow cells, social information. N is the number of sequences that were used in the analysis, at each point of the sequence.

Fig. 2 .

 2 Fig. 2. -Encoding mechanisms from the first to the 50th call of the alarm sequence. Blue cells represent encoding mechanisms that are salient at a given point of the sequence. Black cells represent encoding mechanisms that are not possible at specific points of the sequence, because not enough calls are emitted yet. N is the number of sequences that were used in the analysis, at each point of the sequence.

Table 1 .

 1 Information type encoded by each metric. Numbers: subsequences for which a metric (in line) encoded for a type of information (in column), based on the outcome of 398 robust random forests.

				Information about the predatory event				Social information
			Type of predator	Predator species	Height of the predator	Location of the predator	Distance to the predator	Height of the first caller	Group	Number of individuals	Number of unmated adults	Number of juveniles	Number of subadult males	Number of juvenile females
		Last call	12	15	1	3	1	1	3	4	1	6
		Proportion of A-calls	18	32	3
		Proportion of B-calls	18	32	2
	M e	Transition probability from A to A Transition probability from B to B	17 16	22 33	2
	t
	r	Proportion of 2-grams AA	18	19
	i c s	Proportion of 2-grams BB Proportion of 3-grams AAA	15 16	34 29	2									11
		Proportion of 3-grams BAB								2			2
		Proportion of 3-grams BBA								2			2
		Proportion of 3-grams BBB	18	30	1
		Total	148	246	11	3	1	1	3	8	1	6	4	11

ACKNOWLEDGEMENTS

The authors thank Philippe Schlenker and Emmanuel Chemla for helpful discussions, the many field assistants and master students who took part into data collection, and the Santuário do Caraça for their logistic support.

Guillaume Dezecache: https://orcid.org/0000-0002-9366-6287 Klaus Zuberbühler: https://orcid.org/0000-0001-8378-088X Mélissa Berthet: https://orcid.org/0000-0002-3170-1315

DATA ACCESSIBILITY

The full dataset and the associated statistical scripts are available on a Figshare depository (https://figshare.com/projects/Animal_linguistics_inthe_making_The_Urgency_Principl e_and_titi_monkeys_alarm_system/121914).

Summary of the random forest's analysis. Random forests were retained in the analysis if their variance explained was greater than the threshold (see Appendix II).


Minjuan Cai

Sridhar Gowrisankaran

Maros Gregor

Hryhorii Makarov

Tomas Roch

Jinjun Li

Feng Wu

Gilles Mailhot

Marcello Brigante

Olivier Monfort
email: monfort1@uniba.sk

Unraveling the activation mechanism of oxidants using copper ferrite nanopowder and its application in the treatment of real waters contaminated by phenolic compounds

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

Introduction

Phenolic compounds are well known as persistent and harmful pollutants that are widely present in wastewater from various industries, thus presenting a serious threat to ecosystems and human health [START_REF] Zhu | Photocatalytic degradation of 4-chlorophenol over Ag/MFe 2 O 4 (M = Co, Zn, Cu, and Ni) prepared by a modified chemical co-precipitation method: a comparative study[END_REF]. Among these pollutants, bisphenol A (BPA) is an endocrine disrupting compound that can be found in different concentrations in many consumer products such as personal care products, tableware and medical supplies [START_REF] Chen | Efficient heterogeneous activation of peroxymonosulfate by modified CuFe 2 O 4 for degradation of tetrabromobisphenol A[END_REF]. Due to its widespread industrial use, BPA has been detected in environmental matrices (such as soil, sediments, groundwater, surface water, air) and food [START_REF] Mustieles | Bisphenol A and its analogues: A comprehensive review to identify and prioritize effect biomarkers for human biomonitoring[END_REF].

Other phenolic compounds including phenol and p-nitrophenol which is used in wide variety of organic syntheses and as a pesticide, respectively, can be also bioaccumulated, thus being considered highly toxic [START_REF] Panigrahy | A comprehensive review on eco-toxicity and biodegradation of phenolics: Recent progress and future outlook[END_REF][START_REF] Balakrishnan | Heterojunction photocatalysts for the removal of nitrophenol: A systematic review[END_REF]. Recently, advanced oxidation processes (AOPs) have attracted a lot of interest for its high efficiency in the treatment of water pollution [START_REF] Othman | Preparation and catalytic performance of CuFe 2 O 4 nanoparticles supported on reduced graphene oxide (CuFe 2 O 4 /rGO) for phenol degradation[END_REF]. Furthermore, compared to the other existing technologies such as adsorption [START_REF] Abbaszadeh | Treatment of lead-contaminated water using activated carbon adsorbent from locally available papaya peel biowaste[END_REF], flocculation [START_REF] Gadekar | Coagulation/flocculation process for dye removal using water treatment residuals: modelling through artificial neural networks[END_REF], biodegradation [START_REF] Pal | Review of technologies for biotreatment of refinery wastewaters: progress, challenges and future opportunities[END_REF] and membrane filtration [START_REF] Erkanlı | Brackish water recovery from reactive dyeing wastewater via ultrafiltration[END_REF], AOPs are a very promising because they induce rapid degradation of organic pollutants into less harmful substances or completely converting them to carbon dioxide (CO 2 ), water, and inorganic ions [START_REF] Hu | Cobalt-catalyzed sulfate radical-based advanced oxidation: A review on heterogeneous catalysts and applications[END_REF]. Usually, hydroxyl radicals (HO • , E 0 = 1.9-2.7 V vs. NHE) are the most common reactive oxygen species (ROS) in AOPs and are non-selective for the degradation of almost all organic pollutants. In addition, sulfate radicals (SO 4

•-) based AOPs (SR-AOPs) have become a more promising research direction since SO 4

•-has a longer half-life, greater oxidation capacity (E 0 = 2.5-3.1 V) and a wider pH range stability compared to HO • [START_REF] Ling | A Fe(II)/citrate/UV/PMS process for carbamazepine degradation at a very low Fe(II)/PMS ratio and neutral pH: The mechanisms[END_REF][START_REF] Rao | Degradation of carbamazepine by Fe(II)-activated persulfate process[END_REF]. Moreover, SO 4

•are more selective for the degradation of organic compounds containing aromatic group constituents than the hydroxyl radicals [START_REF] Neta | Rate constants and mechanism of reaction of sulfate radical anion with aromatic compounds[END_REF][START_REF] Antoniou | Degradation of microcystin-LR using sulfate radicals generated through photolysis, thermolysis and etransfer mechanisms[END_REF].

Sulfate radicals can be produced by homolytic cleavage of peroxydisulfate (PDS) and unsymmetric cleavage of peroxymonosulfate (PMS), which can be activated by either light, heating, or alkaline earth and transition metals [START_REF] Ding | Sulfate radicals induced degradation of tetrabromobisphenol A with nanoscaled magnetic CuFe2O4 as a heterogeneous catalyst of peroxymonosulfate[END_REF][START_REF] Lei | CuFe 2 O 4 @GO nanocomposite as an effective and recoverable catalyst of peroxymonosulfate activation for degradation of aqueous dye pollutants[END_REF][START_REF] Cai | Toward a better understanding of peroxymonosulfate and peroxydisulfate activation using a nano zero-valent iron catalyst supported on graphitized carbon: Mechanisms and application to the degradation of estrogenic compounds in different water matrix[END_REF][START_REF] Jia | Caffeine degradation using peroxydisulfate and peroxymonosulfate in the presence of Mn 2 O 3 . Efficiency, reactive species formation and application in sewage treatment plant water[END_REF]. However, these activation processes limit their application at larger scales than laboratory conditions. For example, some transition metal ions are harmful to organisms, thus heterogeneous catalysts being a potential response to these problems.

Therefore, the development of transition metal-containing heterogeneous catalysts for the activation of PDS and PMS for the degradation of organic pollutants has been a popular research topic [START_REF] Feng | Sulfate Radical-Mediated Degradation of Sulfadiazine by CuFeO 2 Rhombohedral Crystal-Catalyzed Peroxymonosulfate: Synergistic Effects and Mechanisms[END_REF][START_REF] Ding | Nonradicals induced degradation of organic pollutants by peroxydisulfate (PDS) and peroxymonosulfate (PMS): Recent advances and perspective[END_REF][START_REF] Wang | Further understanding the involvement of Fe(IV) in peroxydisulfate and peroxymonosulfate activation by Fe(II) for oxidative water treatment[END_REF][START_REF] Xu | The mechanism of degradation of bisphenol A using the magnetically separable CuFe 2 O 4 /peroxymonosulfate heterogeneous oxidation process[END_REF].

Many ferrites, including spinel ferrites, have been already reported as efficient catalysts for the degradation of organic pollutants [START_REF] Xu | The mechanism of degradation of bisphenol A using the magnetically separable CuFe 2 O 4 /peroxymonosulfate heterogeneous oxidation process[END_REF][START_REF] Guan | Efficient degradation of atrazine by magnetic porous copper ferrite catalyzed peroxymonosulfate oxidation via the formation of hydroxyl and sulfate radicals[END_REF]. MFe 2 O 4 is the typical formula of spinel ferrites, where M = Mn, Ni, Cu, Zn [START_REF] Fröhlich | Preparation and characterization of NiFe 2 O 4 /activated carbon composite as potential magnetic adsorbent for removal of ibuprofen and ketoprofen pharmaceuticals from aqueous solutions[END_REF][START_REF] Ai | Activated carbon/CoFe 2 O 4 composites: Facile synthesis, magnetic performance and their potential application for the removal of malachite green from water[END_REF][START_REF] Shao | Facile synthesis, characterization of a MnFe 2 O 4 /activated carbon magnetic composite and its effectiveness in tetracycline removal[END_REF][START_REF] Ren | Sulfate radicals induced from peroxymonosulfate by magnetic ferrospinel MFe 2 O 4 (M=Co, Cu, Mn, and Zn) as heterogeneous catalysts in the water[END_REF]. Copper ferrite (CuFe 2 O 4 ), has received considerable attention in water treatment due to its high magnetic permeability, excellent chemical and mechanical stability and has been extensively used in PDS and PMS activation [START_REF] Feng | Copper-promoted circumneutral activation of H 2 O 2 by magnetic CuFe 2 O 4 spinel nanoparticles: Mechanism, stoichiometric efficiency, and pathway of degrading sulfanilamide[END_REF][START_REF] Wang | Magnetic ordered mesoporous copper ferrite as a heterogeneous Fenton catalyst for the degradation of imidacloprid[END_REF][START_REF] Zhang | Degradation of tetracycline by activated peroxodisulfate using CuFe 2 O 4 -loaded biochar[END_REF]. Although the excellent efficiency of

Materials and methods

Chemicals

The details of the chemicals used in this study are listed in Text S1. All chemicals were used as received without purification and unless otherwise specified, Milli-Q water (18 MΩ cm) was used in all experiments.

Preparation and characterization of CuFe 2 O 4

CuFe 2 O 4 nanopowder was synthesized using a single-pot hydrothermal method with polyethylene glycol (PEG) as a structure directing agent. Briefly, a 40 mL solution of 50 mM of Cu(NO 3 ) 3 •3H 2 O and 100 mM of Fe(NO 3 ) 3 •9H 2 O was mixed with 1 g of PEG under magnetic stirring. Afterward, the pH was adjusted to 12 using NaOH solution. The solution was then transferred in an autoclave and heated for 24 h at 160 °C. After cooling down, the slurry was centrifuged and the powder was washed with distilled water and ethanol. After drying at 80.0 °C overnight and annealing at 500 °C for 3 h (2.5 °C min -1 ), CuFe 2 O 4 nanopowder was obtained.

The crystalline phase composition of the nanopowder was investigated by X-ray diffraction (XRD) using PANalytical X-ray diffractometer (Cu Kα, λ = 1.5418 Å). The surface morphology was analyzed by scanning electronic microscope coupled to energy dispersive X-ray spectroscopy (SEM/EDX) using Tescan Lyra III instrument. The chemical state of the elements at the surface of the nanopowder was investigated by X-ray photoelectron spectroscopy (XPS) equipped with monochromatic Al Kα X-rays (1486.6 eV) and spectra was measured at ambient temperature with photoemission of 45° from the surface. A low energy electron gun (below 1 eV) had been used for charge neutralization to reduce the effects of charging. The XPS analyses was performed before and after the use of CuFe 2 O 4 to characterize its surface composition but also to clarify the mechanism of PDS and PMS activation. Also, the valence band maximum (VBM) was also determined using ultraviolet photoelectron spectroscopy (UPS) by a He discharge lamp with a photon energy of 21.22 eV. To investigate the stability of the nanopowder, the concentration of leaching metals including Cu and Fe was determined during the pollutant degradation by atomic absorption spectrophotometry (AAS, PerkinElmer). The samples were first filtered using a PTFE 0.22 µm filter prior for analyses while 10 µM of Fe(SO 4 ) and Cu(SO 4 ) solution were prepared and used for the calibration curve. The detection wavelength was set at 248.3 nm and 324.8 nm for iron and copper, respectively.

Degradation setup

High-performance liquid chromatography (HPLC) equipped with a 250 mm x 4.6 mm Nucleodur 100-5 C18 column was used to quantify the concentrations of BPA, phenol, and p-nitrophenol (Table S1) with the UV detector set at 277 nm, 221 nm, and 315 nm, respectively. The mobile phase was composed of water and acetonitrile 40/60 (v/v) in isocratic mode with a flow rate of 1 mL min -1 and column temperature of 40°C. Under these conditions, the retention times of BPA, phenol, and p-nitrophenol were 5.6, 2.7 and 3.1 min. The degradation kinetics were fitted using the pseudo-first-order kinetic model (Eq. 1).

(Eq. 1)

where the C represents the residual concentration of the pollutant at t min, and C 0 was for the initial concentration of the pollutant, k' (min -1 ) is the degradation constant.

During the pollutant degradation, the initial pH was fixed at 6.2, and different experimental parameters including PDS and PMS concentrations (from 0 to 5 mM), catalyst dosage (from 0 to 0.5 g L -1 ) and light vs. dark conditions were investigated. The activation mechanism of PDS and PMS by CuFe 2 O 4 , quenching experiments were performed during the BPA degradation.

Isopropanol (Isop) could simultaneously quench SO 4

•-and HO  with a second-order rate constant of 8.2 × 10 7 M -1 s -1 and 1.9 × 10 8 M -1 s -1 , respectively, while TBA could selectively scavenge HO • with a constant of 6.0 × 10 8 M -1 s -1 [START_REF] Buxton | Critical Review of rate constants for reactions of hydrated electrons, hydrogen atoms and hydroxyl radicals (⋅OH/⋅Oin Aqueous Solution[END_REF][START_REF] Huie | Electron transfer reaction rates and equilibria of the carbonate and sulfate radical anions[END_REF]. It was well known that furfuryl alcohol (FFA) can quench singlet oxygen ( 1 O 2 ) with a second-order rate constants of 1.2 × 10 8 M -1 s -1 ) [START_REF] Mi | Almost 100 % Peroxymonosulfate Conversion to Singlet Oxygen on Single-Atom CoN 2+2 Sites[END_REF], but also can react with HO • and SO 4

•- [START_REF] Cai | Toward a better understanding of peroxymonosulfate and peroxydisulfate activation using a nano zero-valent iron catalyst supported on graphitized carbon: Mechanisms and application to the degradation of estrogenic compounds in different water matrix[END_REF]; therefore, 1 O 2 contribution was evaluated by adding FFA and Isop. Also, PMSO was employed to investigate the potential role of high valent iron that might be present at the surface of the material during degradation reactions. The role of dissolved oxygen was also assessed by comparison with degradation experiments done under N 2 saturated conditions.

To highlight the potential application of CuFe 2 O 4 , real water matrices including tap water (TW) but also sewage treatment plant water (STPW) were used for the degradation of BPA, phenol and p-nitrophenol. The physico-chemical characteristics of TW and STPW matrices were presented in Table S2. Finally, the reusability of CuFe 2 O 4 in the activation of PDS and PMS was evaluated.

Results and discussion

Material characterization

The crystalline phase composition of the nanopowder had been examined by XRD (Fig. S1). The nanopowder was predominantly composed of CuFe 2 O 4 (PDF 00-034-0425; space group I41/amd) with some impurities identified as CuO and Fe 2 O 3 (Fig. S1). These separated impurities were the results of the method of synthesis which cannot avoid their formation from the Cu and Fe precursors, but their presence was assumed to not affect the overall efficiency of CuFe 2 O 4 nanopowder. Indeed, the CuFe 2 O 4 diffractions were relatively broads compared to that of the impurities, thus indicating nanosized crystallites with coherent domain size 26 ±13 nm. The SEM pictures of CuFe 2 O 4 (Fig. 1) showed particle size of about 160 nm. In addition, the particles have a cuboidal morphology which was in accordance with the tetragonal structure (Fig. 1).

Fig 1. SEM pictures of CuFe 2 O 4 nanopowder.

The chemical composition and oxidation state in CuFe 2 O 4 nanopowder had been further examined by XPS (Fig. S2). Four main peaks with binding energies of 294, 529, 710 and 933 eV corresponded to the C 1s, O 1s, Fe 2p, and Cu 2p elements, respectively (Fig. S2a). On examining the high resolution XPS spectra, the Cu 2p signal (Fig. S2b) exhibited two symmetric peaks at 933.3 eV and 953.6 eV (along with a satellite peak at 941.4 eV) that corresponded to the binding energies of 2p 3/2 and 2p 1/2 Cu(II), respectively [START_REF] Dong | Monodispersed CuFe 2 O 4 nanoparticles anchored on natural kaolinite as highly efficient peroxymonosulfate catalyst for bisphenol A degradation[END_REF]. For the Fe 2p spectrum (Fig. S2c), the asymmetric peak of Fe 2p 3/2 at 710.9 eV can be deconvoluted into two signal that correspond to binding energies of Fe(II) and Fe(III) [START_REF] Huang | Degradation of atrazine by ZnxCu 1 -xFe 2 O 4 nanomaterial-catalyzed sulfite under UV-vis light irradiation: Green strategy to generate SO 4[END_REF]. Furthermore, the O 1s signal (Fig. S2d) can be deconvoluted into two peaks at 529.4 and 531.5 eV that corresponded to lattice oxygen and surface hydroxyl groups [START_REF] Amini | Synthesis of magnetic Cu/CuFe 2 O 4 @MIL-88A(Fe) nanocomposite and application to dispersive solid-phase extraction of chlorpyrifos and phosalone in water and food samples[END_REF][START_REF] Hu | Adsorption and heterogeneous Fenton degradation of 17α-methyltestosterone on nano Fe 3 O 4 /MWCNTs in aqueous solution[END_REF], while the C1s signal centered at 294 eV (Fig. S2e) was attributed to residual carbon from PEG and/or adsorbed organic species from air [START_REF] Zhang | Degradation of bisphenol A by hydrogen peroxide activated with CuFeO 2 microparticles as a heterogeneous Fenton-like catalyst: Efficiency, stability and mechanism[END_REF]. The XPS results confirmed the high purity of CuFe 2 O 4 nanopowder. In addition, the UPS (Fig. S2f) exhibited that the VBM of CuFe 2 O 4 was located at 4.0 eV, thus CuFe 2 O 4 was a relatively strong oxidant that might activate PMS and PDS with the photogenerated charge carriers i.e., e -and h + .

Effect of reaction conditions on catalytic degradation of BPA

A first set of experiments using 0. where k (UVA) , k (oxidant) , k (CuFe2O4) and k (UVA + CuFe2O4 + oxidant) are the BPA degradation constants (min -1 ) of the mentioned system.

As reported in Fig. 2b, the degradation constants of the multi-factor system

(
different CuFe 2 O 4 dosages (from 0 to 0.5 g L -1 ) were tested, a similar trend was observed for both PDS and PMS. Under dark and UVA light, the increase of catalyst dosage improved the degradation efficiency of BPA in solution within 60 min (Fig. S2). This result could be ascribed to the increase of exposed active sites that would facilitate the activation of PMS and PDS oxidants. The corresponding BPA degradation constants were reported in Fig. 3a andb. Briefly, when the catalyst dosage was 0.5 g L -1 , the degradation constant of BPA in the CuFe 2 O 4 /PMS/UVA system was 0.211 min -1 (Fig. 2a), which was about 5.1, 10.0, and 15.1 times that of 0.2 g L -1 (0.041 min -1 ), 0.1 g L -1

(0.021 min -1 ) and 0.05 g L -1 (0.014 min -1 ) of CuFe 2 O 4 respectively. Similarly, the k' was 0.016 min -1 using 0.5 g L -1 of catalyst in CuFe 2 O 4 /PDS/UVA system (Fig. 3b), which was about 1.2, 2.7, and 3.2 times the value determined with 0.2 g L -1 (0.013 min -1 ), 0.1 g L -1 (0.006 min -1 ) and 0.05 g L -1 (0.005 min -1 ) respectively.

Experiments using different oxidant concentrations (from 0 to 5 mM) show that in the PMS containing systems (Fig. S4a andb), the degradation efficiency of BPA was significantly enhanced when the PMS concentration increased from 0 to 5 mM, while in the case of PDS containing systems (Fig. S4c andd), the degradation efficiency of BPA increased and then decreased as the PDS concentration increases to 5 mM. The highest degradation constant of BPA reached 0.28 min -1 for 5 mM PMS and 0.018 min -1 for 2 mM PDS (Fig. 3c andd). The observed decline in the PDS system suggested that the concentration of oxidant precursors was the controlling factor for the CuFe 2 O 4 /PDS systems and that might be due to scavenging effect of PDS toward generated reactive species such as HO  and SO 4 - [START_REF] Ushani | Sulfate radicals-based advanced oxidation technology in various environmental remediation: A state-of-the-art review[END_REF].

Fig. 3. Reaction rate constant of the BPA degradation with (a,b) different catalyst dosages in 2mM

PMS and PDS and (c,d) different oxidant concentration using 0.5 g L -1 CuFe 2 O 4 . Initial conditions:

[BPA] = 25 µM, pH 0 = 6.2.

Based on the above results, 0.5 g L -1 of CuFe 2 O 4 and 2 mM of PMS/PDS were selected as the optimal conditions for further experiments in order to understand the activation mechanism of oxidant using CuFe 2 O 4 . Under adopted experimental conditions, the decrease of the degradation constant of BPA followed the order: CuFe )

Dark UVA (b)

PDS

UVA light significantly increaseed the degradation of BPA in the both systems. Further experiments were carried out to clarify the activation mechanism.

Oxidant activation mechanism and involvement of reactive species

To gain in-depth mechanistic insights into the oxidant activation in CuFe 2 O 4 /PMS and CuFe 2 O 4 /PDS systems, scavenging experiments had been conducted to identify the generated reactive oxygen species (ROS) both in the dark and under UVA (Fig. 4 andS5).

In the presence of TBA, Isop, or a combination of Isop + FFA, in the CuFe 2 O 4 /PMS system, noticeable reductions in the degradation constants of BPA were observed, with values decreasing from 0.112 to 0.059, 0.045, and 0.018 min -1 , respectively (Fig. 4a). In the presence of UVA light, the BPA degradation constants also decreased from 0.211 to 0.159, 0.047, and 0.027 min -1 following the introduction of TBA, Isop, Isop + FFA (Fig. 4a). Regarding the contribution of these ROS in the degradation of BPA (Text S2), HO • had a predominant role in the dark (47%) while SO 4

•-was identified as the main reactive species under UVA (53%). In addition, the PMS containing systems had been also purged using N 2 to eliminate dissolved O 2 . At such conditions, a decrease of BPA degradation rate of 73% in the dark and 66% under UVA was observed, thus indicating that a part of ROS was derived from dissolved O 2 but also from the action of PMS. On the other hand, in the CuFe 2 O 4 /PDS systems (Fig. 4b), the addition of the different scavengers led to a small decrease of the BPA degradation constant highlighting a limited contribution of HO • , SO 4

•-and 1 O 2 . In addition, the N 2 purging experiments demonstrated a limited effect of oxygen concentration on the degradation of BPA, suggesting the also the presence of non-radical degradation pathway.

The common characteristic of these systems was the BPA degradation enhancement when UVA light was used. Indeed, UVA can further promote the generation of ROS by the cleavage of the peroxo bond of PMS and PDS (Eqs. 4 and 5) [START_REF] Rodríguez-Chueca | Magnetic CoFe 2 O 4 ferrite for peroxymonosulfate activation for disinfection of wastewater[END_REF], but also can lead to the photolysis of surface Fe(III), thus generating HO • (Eq. 6) [START_REF] Kattel | Oxidative degradation of emerging micropollutant acesulfame in aqueous matrices by UVA-induced H 2 O 2 /Fe 2+ and S 2 O 8 2-/Fe 2+ processes[END_REF]. In addition, CuFe 2 O 4 is a known semiconductor photocatalyst with a highly positive VBM (Figure S2f), thus the photogenerated e -and h + can activate the oxidants (Eqs. 7 and 8) [START_REF] Xin | Three-dimensional porous CuFe 2 O 4 for visible-light-driven peroxymonosulfate activation with superior performance for the degradation of tetracycline hydrochloride[END_REF].

PMS

the generation of high valent iron was first checked since surface Fe(III) might be oxidized into Fe(IV) [START_REF] Cai | Toward a better understanding of peroxymonosulfate and peroxydisulfate activation using a nano zero-valent iron catalyst supported on graphitized carbon: Mechanisms and application to the degradation of estrogenic compounds in different water matrix[END_REF]. Fe(IV) can react with PMSO to produce PMSO 2 (Fe(IV): PMSO 2 =1:1), but in all the investigated systems, no PMSO 2 was detected (Fig. S6) which excludes the role of Fe(IV) in the activation of PMS and PDS. It might be plausible that the leached metal from CuFe 2 O 4 i.e., Cu 2+ and Fe 3+ can be involved in the activation of PMS and PDS via homogeneous reactions. Therefore, the concentrations of Cu 2+ and Fe 3+ in the solutions were determined during the BPA degradation using 0.5 g L -1 CuFe 2 O 4 (Fig. S7). No Fe ion leaching from CuFe 2 O 4 was detected, as the concentrations solution are below the detection limit while the concentration of leached Cu ion in PMS system was about 26.1 mg L -1 after 15 min, which was 15.5 times higher than that of the PDS system in 60 min (1.7 mg L -1 ).

However, the contribution of Cu ions to the degradation of BPA was minor, since about 4% and 15%

HO • SO •- 4 1 O 2 others
degradation extent was observed Cu(II)/PMS and Cu(II)/PMS/UVA, respectively (Fig. S8). These results suggested that the degradation of BPA follows a heterogeneous reaction.

The oxidation state modification of CuFe 2 O 4 in the presence of PMS and PDS was explored using XPS analysis. In fact, the transition metals with higher oxidation state, here Fe 3+ , can generate peroxymonosulfate or persulfate radicals (SO 5

•-and S 2 O 8 •-, Eqs. 9 and 11) and Fe 2+ . This latter can further decomposed PMS/PDS to generate more powerful reactive radicals (SO 4

•-, Eqs. 10 and 12).

In this way, the redox cycle of Fe(III)/Fe(II) being the limit step [START_REF] Das | Reactivity and role of SO 5 •radical in aqueous medium chain oxidation of sulfite to sulfate and atmospheric sulfuric acid generation[END_REF][START_REF] Das | Reduction potentials of SO 3 •-, SO 5 •-, and S 4 O 6 •3radicals in aqueous solution[END_REF][START_REF] Gholami | Facile hydrothermal synthesis of novel Fe-Cu layered double hydroxide/biochar nanocomposite with enhanced sonocatalytic activity for degradation of cefazolin sodium[END_REF]. The high resolution XPS spectra of Fe (Fig. S9) showed some variation in the ratio Fe 3+ /Fe 2+ upon reaction with BPA and/or oxidant.

--(Eq. 9)

--(Eq. 10)

--(Eq. 11)

---(Eq. 12)

On the other hand, the transition metals with lower oxidation state i.e., Cu 2+ , can also be involved during the PMS and PDS activation. Some authors have already suggested the involvement of Cu(III)/Cu(II) redox cycle for the activation of these oxidants [START_REF] Zhang | Production of Sulfate Radical from Peroxymonosulfate Induced by a Magnetically Separable CuFe2O4 Spinel in Water: Efficiency, Stability, and Mechanism[END_REF][START_REF] Li | New insights into the CuFe 2 O 4 /PMS process: The roles of surface Cu(III) species and free radicals[END_REF], but this pathway was excluded in our work since the XPS did not exhibit Cu 3+ formation during the PMS/PDS activation (Fig. 6). Indeed, Cu(I) was detected by XPS after oxidant activation. After the reaction with PMS, the asymmetric Cu 2p 3/2 signal can be deconvoluted into two peaks with binding energy of 932.7 and 933.8 eV which were associated with the Cu + and Cu 2+ [START_REF] Lei | Heterogeneous Degradation of Organic Pollutants by Persulfate Activated by CuO-Fe 3 O 4 : Mechanism, Stability, and Effects of pH and Bicarbonate Ions[END_REF], while before reaction, no Cu + signal was present (Fig. 6). Based on these results, the suggested activation mechanism of PMS and PDS by Cu 2+ followed a Cu(II)/Cu(I) redox cycle similar to that of Fe(III)/Fe(II) (Eqs. 13-16) [START_REF] Zhang | Degradation of tetracycline by activated peroxodisulfate using CuFe 2 O 4 -loaded biochar[END_REF][START_REF] Wang | Insights into the mechanism of peroxydisulfate activated by magnetic spinel CuFe 2 O 4 /SBC as a heterogeneous catalyst for bisphenol S degradation[END_REF]. Furthermore, because of the standard reduction potential of Cu(II)/Cu(I) was lower (0.17 V) than that of Fe(III)/Fe(II) (0.77 V), the reduction of Fe 3+ by Cu + was thermodynamically feasible (Eq. ( 17)), which was an advantage in the regeneration of surface Fe(II) [START_REF] Zhang | Degradation of bisphenol A by hydrogen peroxide activated with CuFeO 2 microparticles as a heterogeneous Fenton-like catalyst: Efficiency, stability and mechanism[END_REF].

--(Eq. 13)

--

(Eq. 14) - - - (Eq. 15) - - - (Eq. 16)
(Eq. 17)

However, this mechanism did not play a major role in the case of the PDS activation by CuFe 2 O 4 either in the dark or under UVA (Fig. 5). Therefore, a non-radical mechanism was assumed and the XPS data confirm this pathway (Fig. 6). In fact, after reaction with PDS, the Cu 2p 3/2 signal remained symmetric, thus no changes were observed (Fig. 6). With the addition of BPA, the signal became asymmetric due to the presence of Cu + (Fig. 6), thus suggesting an electron transfer from BPA to PDS via CuFe 2 O 4 as electron mediator [START_REF] Jawad | Tuning of Persulfate Activation from a Free Radical to a Nonradical Pathway through the Incorporation of Non-Redox Magnesium Oxide[END_REF][START_REF] Tian | A Review of Persulfate Activation by Magnetic Catalysts to Degrade Organic Contaminants: Mechanisms and Applications[END_REF]. In addition, the O 1s signal of CuFe 2 O 4 can be deconvoluted into lattice (bulk O 2-) and surface oxygen i.e., surface hydroxyl group (Fig. S10). In both PMS and PDS containing systems, the number of surface hydroxyl groups increaseed during activation of the oxidants, thus suggesting an important role of this functional groups in both radical and non-radical mechanism.

Reusability of CuFe 2 O 4 and its application in different water matrices

In order to assess the reusability of CuFe 2 O 4 , the degradation of BPA was tested during recycling experiments. As shown in Fig. 7, the CuFe 2 O 4 /PMS/UVA system demonstrated a higher degradation efficiency than the PDS containing system at the first use.

However, the BPA degradation in the PMS containing system significantly decreaseed from 100% to 32% in the second cycle with the reuse of CuFe 2 O 4 (Fig. 7a). On the contrary, CuFe 2 O 4 was much more stable for activating PDS (Fig. 7b), since a slower decrease was observed with BPA degradation efficiency degrading from 72%, to 29% after three cycles. The observed decrease can be due to the leaching of copper which was more significant in the PMS containing system, thus explaining the differences between PMS and PDS containing systems. Also, the XPS analyses showed significant changes at the surface of the CuFe 2 O 4 after its use for PMS and PDS activation, and that might also a reason to explain its low reusability. The degradation efficiency of CuFe 2 O 4 in pure water, TW and STPW was investigated for three phenolic organic pollutants (BPA, phenol and p-nitrophenol) (Fig. 8). In pure water system (Fig. 8a), all target pollutants can be completely degraded within 20 min in CuFe 2 O 4 /PMS/UVA. As expected, the degradation rate of all three pollutants decreased significantly in both the tap water (Fig. 8c) and STPW (Fig. 8e). This effect was due to presence of inorganic anions and natural organic matter, which can partially quench the generated ROS. However, in STPW (Fig. 8e), phenol and p-nitrophenol were degraded up to 90% and 61%, which were good degradation extents after only PMS UVA (a)

system followed the order BPA > phenol > p-nitrophenol, which was the same observed trend as in the case of the PDS system. In this latter system, although the degradation of BPA remained relatively efficient in pure water (100%), TW (75%) and STPW (64%), the degradation rate of phenol and p-nitrophenol significantly decreases in TW and STPW (Fig. 8b, d andf).

The degradation efficiencies observed for the three phenolic compounds can be correlated with their O-H bond dissociation energies (BDE) where the values have been already reported by H. Su et al.

[53] (Table S1). In addition, for phenolic organic contaminants, the BDE of C-H bond is higher than that of O-H bond, thus the compounds with electron donating groups (such as OH) are more reactive. Therefore, BPA which had a smaller BDE and more electron-donating groups compared to phenol and p-nitrophenol was easier to degrade.

STPW -PDS

In this study, the mechanism of PMS and PDS activation using CuFe 2 O 4 both in the dark and under UVA was clarified while the effect of different experimental parameters was also investigated for the BPA degradation to determine optimal conditions. In brief, the selected conditions were 0. Radical quenching experiments suggested that the BPA degradation followed a radical pathway in PMS-containing systems with SO 4

•-as the main reactive species while for PDS-containing systems, a non-radical pathway was predominant. Indeed, XPS analyses exhibited that surface copper played an important role and electron transfer from the pollutant to the catalyst and then to PDS was suggested. The stability of CuFe 2 O 4 exhibited leaching of Cu 2+ (26.1 mg L -1 under PMS system and 1.7 mg L -1 under PDS system), thus highlighting that this catalyst cannot be reused, especially for PMS activation. However, regarding the complexity of tap and sewage treatment plant waters, the activation of PMS and PDS by CuFe 2 O 4 catalyst exhibited excellent performance, thus being a promising strategy to develop for wastewaters treatment and their potential reuse in the circular economy of water.

CuFe 2 O

 2 4 has been explained by its ability activating PDS and PMS into HO • , SO 4 •-, singlet oxygen ( 1 O 2 ), there are still some discrepancies in the literature and the current published reports lack a detailed explanation of the differences in the activation mechanisms of the CuFe 2 O 4 /PMS and CuFe 2 O 4 /PDS systems. Furthermore, the differences in the reaction mechanisms in the dark and under UVA light are not been yet thoroughly compared. In this study, the ability of CuFe 2 O 4 synthesized in the form of nanopowder was evaluated through the degradation of BPA as model pollutant. The effects of UVA radiation, catalyst and oxidant precursors dosage were investigated on the BPA degradation efficiency. The different activation mechanisms and involvement of reactive species generated from PMS and PDS containing systems were elucidated. Finally, the simultaneous degradation of different phenolic compounds (BPA, phenol and p-nitrophenol) using CuFe 2 O 4 were investigated in real water matrices to highlight potential applications. The results of this work aimed to demonstrate that SR-AOP containing CuFe 2 O 4 was a prominent approach for purifying any contaminated water, thus highlighting the potential reuse of the treated waters in the circular economy.

Fig. 2 .

 2 Fig. 2. a) BPA degradation in different systems, b) BPA degradation constant in PDS and PMS

Fig. 4 . 2 .

 42 Fig. 4. The BPA degradation constant (k') of quenching experiments in the various systems. Initial

Fig. 5 .

 5 Fig. 5. Contribution of the reactive oxygen species to the BPA degradation over CuFe 2 O 4 in different

Fig. 6 .

 6 Fig. 6. High-resolution XPS spectra of Cu 2p 3/2 signal for freshly prepared CuFe 2 O 4 , CuFe 2 O 4 in

Fig. 7 .

 7 Fig. 7. The reusability of CuFe 2 O 4 for BPA degradation in PMS system (a) and PDS system (b).

Fig. 8 .

 8 Fig. 8. The degradation efficiency of mixed solution by CuFe 2 O 4 under UVA in virous water matrix:

5 gL - 1 of

 51 CuFe 2 O 4 and 2 mM of PMS/PDS under pH 0 = 6.2 ± 0.1. For the degradation of BPA, CuFe 2 O 4 coupling with PMS demonstrated better degradation than PDS-containing systems. The reaction rate constants of BPA degradation followed the order: CuFe 2 O 4 /PMS/UVA (0.211 min -1 ) > CuFe 2 O 4 /PMS (0.112 min -1 ) > CuFe 2 O 4 /PDS/UVA (0.018 min -1 ) > CuFe 2 O 4 /PDS (0.011 min -1 ).

  5 g L -1 of CuFe 2 O 4 were provided in the presence of 2 mM of PDS and PMS in the dark and under UVA light to assess the ability of the nanopowder to activate the CuFe 2 O 4 and PMS, while 72% of degradation was reached within 1 hour in the presence of PDS. To compare the degradation efficiencies of each system, the BPA degradation constants for each system had been reported in Fig.2b. It was clear that a strong synergistic effect (SE) observed when both catalyst and oxidants occurred either in the dark and UVA light. The SE of the CuFe 2 O 4 /oxidant systems both in the dark and under UVA process can be determined with the following formula:

	(Eq. 2)
	(Eq. 3)

oxidants. As depicted in Fig.

2a

, less than 5% of BPA was degraded after 1h either under UVA light (direct photolysis) or in the presence of CuFe 2 O 4 (photocatalysis) after 1h. In the presence of 2 mM of PDS and PMS under UVA light (and without CuFe 2 O 4 ), the degradation efficiency of BPA slightly increaseed to 13% and 11% after 1hour of reaction. However, the combination of CuFe 2 O 4 with PDS and PMS resulted in a considerable BPA removal both in the dark and under UVA light. In the dark, the CuFe 2 O 4 /PDS system degraded 45% of BPA within 1 hour, while CuFe 2 O 4 /PMS system exhibited a complete BPA degradation within 20 min. Moreover, higher efficiencies were observed in the presence of UVA where complete BPA degradation was obtained after only 15 min using

  CuFe 2 O 4 /oxidant/dark and UVA) were significantly enhanced compared to the addition of single-factor systems. Indeed, the values of SE calculated for PDS/CuFe 2 O 4 , UVA/PDS/CuFe 2 O 4 , PMS/CuFe 2 O 4 and UVA/PMS/CuFe 2 O 4 systems were 3.2, 3.8, 22.4, and 34.1 respectively.

Acknowledgement

This work was partially supported by the Slovak Research and Development Agency (SRDA) under the contract No. APVV-21-0039 and the Scientific Grant Agency of the Ministry of Education, Science, Research and Sport of the Slovak republic (VEGA project No. 1/0319/23), and the Grant of Comenius University Bratislava for Young Scientists No. UK/226/2023. The presented results are obtained in the frame of the French-Slovak bilateral project "SOLAREM" co-financed by PHC Stefanik program (Campus France) and SRDA under the contract No. SK-FR-22-0002. Finally, authors thank the financial support under the Operational Program Integrated Infrastructure for the projects "Advancing University Capacity and Competence in Research, Development and Innovation" (ACCORD, ITMS2014+:313021X329) and "UpScale of Comenius University Capacities and Competence in Research, Development and Innovation" (USCCCORD, ITMS 2014+: 313021BUZ3), co-financed by the European Regional Development Fund. Authors also acknowledge the Ministry of Education of China for providing the financial support for Minjuan CAI stay in France, the IRP CNRS "Processes for Environmental Remediation" and from the LIA by the project I-site CAP 20-25 and PRC program CNRS/NSFC n°270437


Jose Luis

Avila Betancourt

Ana María

Ojeda Salazar

Paul Drijvers

Csaba Csapodi

Hanna Palmér

Katalin Gosztonyi

; Eszter

A case study about the understanding of addition and product of probabilities of a technological high school student

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

A case study about the understanding of addition and product of probabilities of a technological high school student

José Luis Ávila Betancourt 1 and Ana María Ojeda Salazar 1

1 Centro de Investigación y de Estudios Avanzados del IPN, México; joseluis.avila@cinvestav.mx Keywords: Stochastic ideas, probability, understanding, qualitative, high school.

As prior research has pointed out, students' understanding of probability ideas faces up to biases and a deterministic way of thinking that their teaching hardly seems to overcome. We examined a Mexican technological high school syllabus proposal of the rules of addition and product of probabilities, as well as their understanding by a student (referred to as MC) who attended the current probability course and voluntarily received an additional online teaching of probability fundamentals, which included those rules. We posed the question: What are MC's understanding features of addition and product of probabilities rules, and how might that understanding be influenced by the institutional proposal? We aimed at enhancing the teaching approach for these topics.

Theoretical Framework

This study has three main axes: epistemological, social and cognitive. The epistemological axis considered [START_REF] Heitele | An epistemological view on fundamental stochastic ideas[END_REF] proposal of ten stochastic fundamental ideas that should be taught in a continuous spiral curriculum, two of which are the addition and product of probabilities rules. It also considered Steinbring's (2005) view of the epistemological triangle, which connects and interrelates reference, symbol, and concept to constitute the latter. Also, concerning the social axis, Steinbring (2005) clarifies how mathematical ideas are constructed through individuals' social interaction when assigning and exchanging signifiers-signifieds to argue and justify concepts.

The cognitive axis includes Tall's (2013) three worlds of mathematics, which evolve from conceptual embodiment to symbolic operations to axiomatic formalism. More particularly, for [START_REF] Duval | Understanding the Mathematical Way of Thinking -The Registers of Semiotic Representations[END_REF] there are three main cognitive operations ruling the mathematical activity: the semiotic registers' production, treatment, and conversion. Specifically, regarding probability thinking, [START_REF] Tversky | Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment[END_REF] have proposed cognitive biases, such as representativeness, availability and the conjunction fallacy, which can lead to errors when calculating probabilities.

Methodology

This qualitative research was carried out in three phases: syllabus (DEMS-IPN, 2008) documental research, online extra-class teaching sessions, a questionnaire and a semi-structured interview. The data were analyzed and interpreted using criteria categories from the theory: referent, stochastic fundamental ideas, semiotic registers, wording terms, and other mathematical concepts [START_REF] Ojeda | Estrategia para un perfil nuevo de docencia: un ensayo en la enseñanza de estocásticos [Strategy for a new teaching profile: an essay on teaching stochastics[END_REF].

The questionnaire consisted of six problems presented as open questions about the fundamental ideas in focus. It was provided, handwritten answered, digitalized, and sent it back, everything online. The interview was based on MC's answers to the problems, which were shown on screen with her produced registers. It consisted of 42 questions, was carried out online and videorecorded.

Findings

The program emphasizes procedural algorithms for the addition and product of probability ideas, mainly by set operations between events. However, it adopts a deterministic approach to probability, with no recommendations on the use of semiotic registers, such as tree diagrams and contingency tables. Additionally, the program's directions for reference situations for problem-solving is unclear and very general. Consequently, students may develop a piecemeal and incomplete understanding of these ideas, limited to the conceptual embodiment world, which could lead to availability biases.

MC showed an understanding of probability ideas such as probability measure and sample space, still, some misconceptions regarding combinations of events emerged. She struggled to differentiate mutually exclusive from independent events when applying the addition rule. She also had difficulty using the product rule and confused conjunction and conditionality of events, hence interchangeable for her. MC did not recognize the usual words (or/and) related to the union/intersection of events and used conditional probability instead, focusing on the order of events' appearance in the problem text.

Although she correctly used tree diagrams, her execution revealed a reference-sign and sign-concept disruption, which could be due to the conjunction fallacy and the availability heuristic prompted by a colloquial use of terms, placing her in the conceptual embodiment world for both ideas.

Conclusion

These findings suggest that student's understanding of the addition and product of probabilities is characterized by heuristics, lack of representation, and deterministic thinking. To resort to different semiotic registers when teaching the addition and product of probabilities at the technological high school is crucial for the students to understand these rules. A variety of problems to overcome the conjunction fallacy is needed, so to differentiate conjuncted from conditioned and conditioning events, and mutually exclusive from independent events. The wording to refer to combined events needs further research.