Theoretical Informatics and Applications

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DEJEAN'S CONJECTURE AND LETTER FREQUENCY

Jérémie Chalopin1 and Pascal Ochem 2 Abstract. We prove two cases of a strong version of Dejean's conjecture involving extremal letter frequencies. The results are that there exist an infinite "

+ " -free word over a 5 letter alphabet with letter frequency 1 6 and an infinite

" 6 5
+ " -free word over a 6 letter alphabet with letter frequency 1 5 .

1991 Mathematics Subject Classification. 68R15.

Introduction

We consider the extremal frequencies of a letter in factorial languages defined by an alphabet size and a set of forbidden repetitions. Given such a language L, we denote by f min (resp. f max ) the minimal (resp. maximal) letter frequency in an infinite word that belongs to L. Extremal letter frequencies have been mainly studied in [START_REF] Kolpakov | On repetition-free binary words of minimal density[END_REF][START_REF] Ochem | Letter frequency in infinite repetition-free words[END_REF][START_REF] Richard | On the entropy and letter frequencies of ternary square-free words[END_REF][START_REF] Tarannikov | The minimal density of a letter in an infinite ternary square-free word is 0.2746[END_REF]. Let Σ i denote the i-letter alphabet {0, 1, . . . , i -1}. We consider here the frequency of the letter 0. Let n(v) denote the number of occurrences of 0 in the finite word v. So the letter frequency in v is n (v) |v| . We say that the letter frequency in the infinite word w is q if for every > 0, there exists an integer n such that for every finite factor v of w of length at least n , we have

n(v)
|v| -q < . The repetition threshold is the least exponent α = α(k) such that there exists an infinite (α + )-free word over Σ k . Dejean proved that α(3) = 7 4 . She also conjectured that α(4) = 7 5 and α(k) = k k-1 for k ≥ 5. This conjecture is now "almost" solved: Pansiot [START_REF] Pansiot | A propos d'une conjecture de F. Dejean sur les répétitions dans les mots[END_REF] proved that α(4) = 7 5 and Moulin-Ollagnier [START_REF] Moulin-Ollagnier | Proof of Dejean's conjecture for alphabets with 5[END_REF] proved that Dejean's conjecture holds for 5 ≤ k ≤ 11. Recently, Currie and Mohammad-Noori [START_REF] Mohammad-Noori | Dejean's conjecture and Sturmian words[END_REF] also proved the cases 12 ≤ k ≤ 14, and Carpi [START_REF] Carpi | On Dejeans conjecture over large alphabets[END_REF] settled the cases k ≥ 33. For more information, see [START_REF] Choffrut | Combinatorics of words[END_REF].

In a previous paper, we proposed the following conjecture which implies Dejean's conjecture.

Conjecture 1.1. [START_REF] Ochem | Letter frequency in infinite repetition-free words[END_REF] (1) For every k ≥ 5, there exists an infinite

f max < 103 440 = 0.23409090 • • • < 1 4 [7]. That is why Conjecture 1.1.2 is stated with k ≥ 6.
In this paper, we prove the first case of each part of Conjecture 1.1:

Theorem 1.2.
(1) There exists an infinite 5 4 + -free word over Σ 5 with letter frequency 1 6 .

(2) There exists an infinite 6 5 + -free word over Σ 6 with letter frequency 1 5 . The C++ sources of the programs and the morphisms used in this paper are available at: http://www.lri.fr/ ~ochem/morphisms/.

Structure and encoding

In the following, a k-word will denote a k k-1 + -free word w over Σ k , for k ≥ 5. We easily check that in a k-word, the distance between two consecutive occurrences of the same letter is either k -1, k, or k + 1. This implies that if there exists an infinite k-word with letter frequency 1 k+1 (resp. 1 k-1 ), then there exists an infinite k-word such that the distance between consecutive occurrences of 0 is always (k + 1) (resp. (k -1)). Notice that 0's cannot be regularly spaced if the letter frequency is 1 k . Such k-words in which 0's are regularly spaced are the catenation of factors of size k + 1 (resp. k -1) of the form 0π Let p = p 0 p 1 p 2 . . . be the code of w and we suppose that p 0 is the identity. We now encode p by the word

1 . . . π k-1 π 1 (resp. 0π 1 . . . π k-2 ),
c = c 0 c 1 c 2 • • • ∈ Π * k such that p i+1 = c i (p i ).
Notice that whereas any permutation in Π k may appear in p, only a small subset S ⊂ Π k of permutation can be used as letters in c. This is because the latter permutations rule the transition between two consecutive factors w i and w i+1 in w, and then w i w i+1 has to be k k-1 + -free.

In the following, we will call coding word a word over S * consisting of the transition permutations for a k-word. Moreover, if the transition corresponding to a coding word c is the identity of Π k , we will say that c is an identity. Remark 2.1. If a coding word c is an identity, then every conjugate (cyclic shift) of c is also an identity.

Proof of main result

Let us consider the possible transitions for 5-words with letter frequency 1 6 . There are exactly two of them:

• 012341024312 corresponds to the transition permutation 2431 (noted 0).

• 012341032143 corresponds to the transition permutation 3214 (noted 1). There are also exactly two possible transitions for 6-words with letter frequency 1 5 : • 0123405132 corresponds to the transition permutation 51324 (noted 0).

• 0123405213 corresponds to the transition permutation 52134 (noted 1). In both cases, we have |S| = 2 and we construct a suitable infinite code c as the fixed point of the following binary endomorphisms:

• For 5-words with letter frequency 1 6 : 0 → 010010010100101001001001010100101001001001010010101001001001010 1 → 100101001001010100101001001010100101001001001010010010010100101

• For 6-words with letter frequency 1 5 : 0 → 0010010100111000110100010 1 → 1000100111000100110100011 These morphisms m satisfy the following properties:

(1) m is q-uniform, that is, for all i ∈ Σ 2 , we have |m(i)| = q.

(2) m is synchronizing, which means that for any a, b, c ∈ Σ 2 and s, r ∈ Σ * 2 , if m(ab) = rm(c)s, then either r = ε and a = c or s = ε and b = c. (3) for all i ∈ Σ 2 , we have m(i) = if i and the factor if is an identity (thus f i is also an identity by Remark 2.1). Let Φ denote the decoding function. In the case of 6-words, we thus have Φ(0) = 0123405132, Φ(1) = 0123405213 and Φ(c = m ω (0)) = w. We have checked that for every factor x of c of size at most 2kq, Φ(x) is

k k-1 + -free.
Let f be a smallest repetition in w of exponent strictly greater than k k-1 . This repetition in w implies that there is a repetition r = is in c whose prefix i is an identity. Since |s| ≥ 2q, s contains a full m-image. So |i| and |s| are multiples of q because m is synchronizing. By property 3 and Remark 2.1, we can assume without loss of generality that |i| starts at the beginning of an m-image. Then our repetition is of the form r = is = m(i )m(s ) = m(i s ) = m(r ). By property 3, r is a repetition whose prefix i is an identity, and thus the factor Φ(r ) is a repetition that appears in w. The exponent of Φ(r ) is

(k±1)(|r |+1) (k±1)|i | , the exponent of f is less than (k±1)(|r|+3) (k±1)|i| = (k±1)(q|r |+3) (k±1)q|i | , and we have (k±1)(|r |+1) (k±1)|i | ≥ (k±1)(q|r |+3) (k±1)q|i | if q ≥ 3
. This is a contradiction because the exponent of Φ(r ) is greater than the exponent of f and Φ(r ) is strictly smaller than f .

Concluding remarks

Theorem 1.2 shows the existence of two types of infinite words, but does not prove that there exist exponentially many such words (which is probably true). On the other hand, the growth rate of these words is significantly smaller than those of k k-1 + -free words. For example, the growth rate of 5-words is about 1.159 [START_REF] Ochem | Upper bound on the number of ternary square-free words[END_REF], whereas 1.048 is a rough upper bound on the growth rate of 5-words with letter frequency 1 6 . Other cases of Conjecture 1.1 might be harder to settle. For 6-words with letter frequency 1 7 , we have |S| = 3, and it is impossible to construct an infinite code using only two of these three transition permutations. We have not been able to find a Σ * 3 → Σ * 3 morphism with suitable properties for them.

k k- 1 + 1 + 4 +

 114 -free word over Σ k with letter frequency 1 k+1 . (2) For every k ≥ 6, there exists an infinite k k--free word over Σ k with letter frequency 1 k-1 . It is easy to see that the values 1 k+1 and 1 k-1 in Conjecture 1.1 are best possible. For 5 -free words over Σ 5 , we obtained

  where π is a permutation of the elements [1, . . . , k -1]. Let Π k denote the set of permutations of [1, . . . , k -1]. The k-word w can thus be encoded by the word p ∈ Π * k consisting in the catenation of the permutations that correspond to the factors of size k + 1 (resp. k -1) in w.

LIF, CNRS Université de Provence, CMI, 39 rue Joliot-Curie, 13453 Marseille, France jeremie.chalopin@lif.univ-mrs.fr

LRI, CNRS Université Paris-Sud 11, Bât 490, 91405 Orsay Cedex, France ochem@lri.fr © EDP Sciences 1999


Esther Boissin

Mathilde Josserand

Wim De Neys

Serge Caparos

Debiasing thinking among non-WEIRD reasoners

Keywords: reasoning, heuristics & biases, debiasing paradigm, Himba of Namibia

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

Introduction

Decades of research on decision-making have shown that we do not always reason correctly.

For instance, if in a sample of 1000 people, with 995 females and 5 males, a randomly selected person is described as liking beer and football, many people would assume that this person is male based on stereotypical associations cued by the descriptive information ("men like beer and football"). If we only consider the description, that may be a reasonable assumption. In general, there might be more males than females who like beer and football. However, some females also like this, and there were far more females (995) than males (only 5) in the sample. The base-rate information should thus push the scale to the "female" side. Unfortunately, people often ignore base-rate information and over rely on their stereotypical prior beliefs (e.g., [START_REF] Kahneman | On the psychology of prediction[END_REF]. This tendency to favour personal beliefs over logical and mathematical principles (e.g., base rates) has been shown to bias reasoning in various situations [START_REF] Evans | Rationality in the selection task: Epistemic utility versus uncertainty reduction[END_REF][START_REF] Kahneman | Thinking, Fast and Slow[END_REF][START_REF] Morewedge | Associative processes in intuitive judgment[END_REF]. However, as in many fields, most previous work has been conducted on samples of WEIRD (Western Educated Industrialized Rich and Democratic societies, [START_REF] Henrich | The weirdest people in the world?[END_REF] participants, representing only about 15% of the world population [START_REF] Arnett | The neglected 95%: Why American psychology needs to become less American[END_REF]. Interestingly, the sparse data regarding non-WEIRDs' reasoning performance often suggest an even stronger propensity to base their reasoning on personal beliefs [START_REF] Luria | Towards the problem of the historical nature of psychological processes 1[END_REF][START_REF] Luria | The neuropsychology of memory[END_REF][START_REF] Scribner | Modes of thinking and ways of speaking: Culture and logic reconsidered[END_REF][START_REF] Trémolière | A 21st century cognitive portrait of the Himba, a remote people of Namibia[END_REF], but see also [START_REF] Maddux | For whom is parting with possessions more painful?: Cultural differences in the endowment effect[END_REF]. An iconic example comes from Luria's work in which illiterate Uzbekistan people showed very poor performance on deductive reasoning problems such as "In the Far North, where there is snow, all bears are white. Novaya Zemlya is in the Far North and there is always snow there. What color are bears there?" (e.g., one participant stated: "I've seen a black bear, I've never seen any others", Luria, 1976, p. 109). Similar poor accuracies were observed with African and central American illiterate people [START_REF] Cole | Cultural differences and inferences about psychological processes[END_REF][START_REF] Scribner | Modes of thinking and ways of speaking: Culture and logic reconsidered[END_REF][START_REF] Sharp | The influence of educational experience on the development of cognitive skills as measured in formal tests and experiments[END_REF].

Although these studies typically indicate that non-WEIRD groups show more biased reasoning, the precise reason is less clear. One possibility is that non-WEIRD people lack the ability to integrate logic into their reasoning. As [START_REF] Luria | Towards the problem of the historical nature of psychological processes 1[END_REF][START_REF] Luria | The neuropsychology of memory[END_REF] famously argued, logical or abstract reasoning could be intrinsically linked with literacy. In the absence of literacy, people's responses would be necessarily driven by their immediate practical experiences; They might be incapable of decontextualizing the information contained in the problem and integrate more abstract, logical principles [START_REF] Luria | Towards the problem of the historical nature of psychological processes 1[END_REF][START_REF] Nisbett | Culture and systems of thought: Holistic versus analytic cognition[END_REF][START_REF] Scribner | Unpackaging literacy. Writing: The nature, development, and teaching of written communication[END_REF].

Alternatively, the more biased responding in non-WEIRD reasoners might be explained by a performance, rather than a competence deficit. That is, while illiterate people from non-WEIRD societies may possess the necessary logical knowledge and may have the ability to integrate it into their reasoning, they would simply not realize that it is useful for solving the problem at hand, particularly for unfamiliar tasks. Interestingly, indirect evidence from studies with WEIRD samples supports the performance hypothesis. Recent debias studies have shown that single-shot interventions, consisting of a brief reminder of the correct solution, can easily boost WEIRD participants' reasoning (e.g., [START_REF] Boissin | From bias to sound intuiting: Boosting correct intuitive reasoning[END_REF][START_REF] Boissin | Debiasing system 1: Training favours logical over stereotypical intuiting[END_REF][START_REF] Claidière | Argumentation and the diffusion of counter-intuitive beliefs[END_REF][START_REF] Hoover | Algebraic reasoning and bat-and-ball problem variants : Solving isomorphic algebra first facilitates problem solving later[END_REF][START_REF] Morewedge | Debiasing decisions: Improved decision making with a single training intervention[END_REF][START_REF] Purcell | Domain-specific experience and dual-process thinking[END_REF][START_REF] Trouche | Arguments, more than confidence, explain the good performance of reasoning groups[END_REF]; see also [START_REF] Isler | How to activate intuitive and reflective thinking in behavior research? A comprehensive examination of experimental techniques[END_REF][START_REF] Isler | Activating reflective thinking with decision justification and debiasing training[END_REF]Yilmaz & Saribay, 2017a). Given that these interventions are extremely short-often lasting no more than 5 minutes-the finding that this nevertheless suffices to remediate participant's reasoning suggests that the bias does not result from a lack of knowledge or so-called "mind" or "storage" gap (e.g., [START_REF] De Neys | The 'whys' and 'whens' of individual differences in thinking biases[END_REF][START_REF] Stanovich | Rationality and the reflective mind[END_REF]. Clearly, it is unlikely that a simple reminder suffices to learn the underlying logical principles ex nihilo. That is, if the short explanation works and allows people to respond correctly, it might be because the critical knowledge was already acquired [START_REF] Boissin | From bias to sound intuiting: Boosting correct intuitive reasoning[END_REF][START_REF] Boissin | Debiasing system 1: Training favours logical over stereotypical intuiting[END_REF][START_REF] Mata | An easy fix for reasoning errors : Attention capturers improve reasoning performance[END_REF].

Its relevancy simply needed to be stressed. These debiasing results might imply that a similar performance account may also be at play in non-WEIRD samples. Some non-WEIRD studies even provide initial support for this interpretation (e.g., [START_REF] Dias | Reasoning from unfamiliar premises a study with unschooled adults[END_REF]see Mercier, 2011). Notably, one study among indigenous Mayan groups showed that non-WEIRD reasoners can accurately perform adapted probabilistic evaluations [START_REF] Fontanari | Probabilistic cognition in two indigenous Mayan groups[END_REF], suggesting that they possess the basic probabilistic knowledge essential for correct responding in various reasoning tasks (e.g., the base-rate task). Moreover, [START_REF] Castelain | Evidence for benefits of argumentation in a Mayan indigenous population[END_REF] demonstrated that non-WEIRD reasoners can perform better when core logical principles are explicitly brought to their attention. Nonetheless, evidence is still limited and further validation of the performance hypothesis within non-WEIRD samples is required.

In the present study, we tested the performance-competence question concerning non-WEIRD samples' reported poorer reasoning performance directly, by adopting a debiasing paradigm similar to [START_REF] Boissin | Debiasing system 1: Training favours logical over stereotypical intuiting[END_REF] study with a non-WEIRD sample: The Himba of Northern Namibia (Africa). The Himba are semi-nomadic herders with limited exposure to western culture, including limited participation in western-style education (schooling) and low levels of industrialization. They predominantly speak the "Otjiherero" dialect, often live in small, collectivist villages (up to 40 adults), and do not use written language [START_REF] Trémolière | A 21st century cognitive portrait of the Himba, a remote people of Namibia[END_REF] or monetary exchanges [START_REF] Cameron | The influence of media on Himba conceptions of dress, ancestral and cattle worship, and the implications for culture change[END_REF].

Figure 1 illustrates a typical Himba village.

We tested the reasoning of Himba people using the base-rate neglect task (e.g., see the introductory example) which has often been used to study biased thinking in WEIRD samples. We used adapted, familiar material in which typical stereotypical Himba beliefs were pitted against logical baserate information. We measured the debiasing intervention effect by contrasting pre-and postintervention performance. The intervention was structured as in previous WEIRD studies in which the role of base-rates on likelihood estimation was briefly explained. In line with previous non-WEIRD studies, we expected accuracy to be very low before training. If a performance deficit explains the reasoning errors, we expect the Himba to benefit from the intervention and show higher post-training accuracy (just as in WEIRD samples). On the contrary, if illiterate Himba individuals do not have the basic competence to use logical considerations to overrule personal beliefs, the short intervention should not help them, and they should show stable (low) performance across the study. Furthermore, by taking advantage of natural ecological variation in the Himba region we could also test factors that may modulate the efficiency of the training. That is, although most adult Himba are illiterate and have received no formal schooling, schooling is not completely absent in the region.

The constitution of Namibia makes primary education mandatory and compels the state to establish and maintain free schools. This led to the implementation of standardised curricula in which basic reading, writing and maths skills are taught (see the 2017 Namibian Ministry-of-Education report).

Although not always instantiated in practice, through this initiative at least some Himba villagers have received some minimal formal teaching. Relatedly, in the vicinity of the traditional Himba villages, the town of Opuwo (~30,000 inhabitants; [START_REF] Mwinga | Opuwo economic profil[END_REF] is home to some Himba individuals. In contrast with the Himba villages, Himba urban settlements in Opuwo can be equipped with some modern supplies (electricity, running water, mobile phones, and basic furniture). There is also access to public services (e.g., health centre, town hall) and private ones (e.g., food stores, pharmacies, banks). The Himba living in Opuwo regularly engage in the market economy [START_REF] Cameron | The influence of media on Himba conceptions of dress, ancestral and cattle worship, and the implications for culture change[END_REF], by buying food and selling jewellery, which implies some familiarization with numerical processing (e.g., [START_REF] Carraher | Mathematics in the streets and in schools[END_REF][START_REF] Carraher | Written and Oral Mathematics[END_REF][START_REF] Nunes | Street mathematics and school mathematics[END_REF][START_REF] Saxe | Candy selling and math learning[END_REF]. By testing both Himba participants living in traditional villages and in the town of Opuwo-and recording their schooling record-we could a posteriori explore the impact of living environment (i.e., an industrialization proxy) and schooling on the debiasing effect, within the same country, language, and ethnic population.

Methods

Participants

A total of 84 Himba individuals (50 females, mean estimated age = 26.6 years, SEM = 1.0, range from 16 to 55 years, see Supplementary Material Section A for the age-estimation method) were recruited during a 2-month research visit to Namibia in October and November 2021. To our knowledge, none of the participants had ever participated in experimental research.

Among the 84 participants, 49 were living in traditional Himba villages and 35 were living in town (i.e., Opuwo). The villages were located at about 60 km from the town of Opuwo, in the area of Otuazuma. Of the tested villagers, 23 (47%) had gone to school (average number of years of schooling = 5.3 years, SEM = 0.6, range between 1 and 9 years). Among the Himba who lived in town, 17 (49%) had gone to school (average number of years of schooling = 6.9 years, SEM = 0.7, range between 1 and 9 years).

In total, 74 participants were assigned to the critical training group. Practical on-site testing and recruitment complications prevented testing a larger sample. Our primary goal was to test a sample of approximately 50 villagers in the training group-equivalent to the sample size in the WEIRD debias studies of [START_REF] Boissin | Debiasing system 1: Training favours logical over stereotypical intuiting[END_REF]. Near the end of the data collection, we also allocated 10 participants to a control group in which the participants received no training intervention (4 lived in town; 6 lived in the villages). Note that given the non-random group allocation and small numbers of participants in the control group, results should be interpreted with caution. However, our rationale was that the control group would still allow us to explore whether mere habituation and test repetition could lead to debiasing effects.

All participants received compensation for their participation (they could choose 3 gift items among the following: Maize meal, sugar, vaseline or soap-roughly equivalent to a 5-euro value). The study was approved by the CER-Paris Cité Ethics Committee (Project ID: 2021-88-********).

Materials

The study consisted of three blocks presented in the following order: A pre-intervention, an intervention, and a post-intervention block. In total, each participant had to solve 10 problems during the pre-intervention block (four conflict, four no-conflict, and two neutral problems, see below). In the post-intervention block the same number of problems, but with different content, were used along with two extra control heuristic problems (see further). At the end of the main study, we asked participants to rate the material to check that it cued the intended stereotypes and beliefs. All material was translated from English to Otjiherero by two different native translators who discussed and agreed upon each translation. All instructions and task problems were orally presented in Otjiherero by one translator. All the problems are presented in the Supplementary Material Section B.

Base rate problems. In total, 19 base-rate problems were created, based on the Bago and De Neys (2017) presentation format. Participants always received base-rate information about the composition of a new village (e.g., "In this new village, there are 8 men and 2 women") followed by the description of one villager which was designed to cue a stereotypical association (e.g., "This randomly chosen person looks after children"). As shown in Figure 2, the problem was presented with a photo illustrating the composition of the village. For the example above, we presented 8 different male portraits and 2 different female portraits. The base rate was consistently 8/2. The portraits, the base rates, and the description were presented together, on a tablet screen. Participants' task was to indicate to which group the random villager most likely belonged to (e.g., men or women in the example above). Participants gave their answer by touching one of two target photos on the touchscreen, which evoked the possible answers (e.g., to distinguish between men and women, we used pictures of male and female torsos). The position of each response photo was counterbalanced between participants and items. Figure 2 illustrates the full problem format. In the pre-and post-intervention blocks, we presented four critical "conflict" items, and four "no-conflict" items. In the conflict items, the base rate probabilities and the stereotypical information cued conflicting responses (see example above). In the no-conflict items, they both cued the same response (i.e., the description triggered a stereotypical trait of a member of the largest group, see Figure 2). In the above example, the correct response to the conflict item is "a male", while for the noconflict items, the correct response is "a female". These no-conflict problems should be easy to solve.

If participants are paying minimal attention to the task and refrain from random guessing, they should show high accuracy to no-conflict problems [START_REF] Bago | The Smart System 1: Evidence for the intuitive nature of correct responding on the bat-and-ball problem[END_REF]. In addition, the no-conflict items allow us to control for a "reversed heuristic" account [START_REF] Boissin | Debiasing system 1: Training favours logical over stereotypical intuiting[END_REF]. That is, it is possible that our training does not succeed in getting participants to give more weight to the base rates per se but simply have them assume that they are being presented with a counter-intuitive task in which the right answer is always the opposite of the cued heuristic/stereotypical response (e.g., "pick what you don't think it is"). This would lead to selection of the correct response on conflict problems. However, it should also result in floored performance on the no-conflict problems (on which the heuristic response is also correct).

Two sets of 16 unique items (8 pre-intervention and 8 post-intervention block items) were used for counterbalancing purposes. For each block, the conflict problems in one set were the noconflict problems in the other, and vice-versa (i.e., the base-rates were reversed). Participants were randomly assigned to one of the two sets. Consequently, none of the pre-and post-intervention problem contents was repeated within-subjects (i.e., participants saw a total of 16 different items each with a unique stereotypical association).

Neutral problems. We also presented four neutral base-rate problems. These problems were designed such that they did not cue any stereotypical association (i.e., the descriptive information was not diagnostic of a particular group). Here is an example of a neutral base-rate problem:

In this new village, there are 8 women and 2 men. The randomly chosen person has two hands. Is the randomly chosen most likely to be:

o a woman o a man
The neutral base-rate items are traditionally used to track people's knowledge of the underlying logical principles or "mindware" [START_REF] Stanovich | Rationality and the reflective mind[END_REF]. Usually, reasoners have little trouble solving them (e.g., De [START_REF] De Neys | Conflict monitoring in dual process theories of thinking[END_REF][START_REF] Frey | Is conflict detection in reasoning domain general[END_REF]. Here the neutral problems allowed us to explore whether a potential learning effect on conflict base-rate problems in which the reasoner needs to discard a conflicting stereotypical association, leads to a more general performance boost on other untrained base-rate problems (e.g., [START_REF] Boissin | Debiasing system 1: Training favours logical over stereotypical intuiting[END_REF]. [START_REF] Boissin | Debiasing system 1: Training favours logical over stereotypical intuiting[END_REF] did not observe such transfer in their debias studies with WEIRD samples.

Control heuristic problems.

We also presented two control heuristic problems during the postintervention block. The control heuristic problems aimed to check that reasoners did not apply a new heuristic strategy such as "always pick the largest group" and no longer regard the description after the training intervention. This strategy would lead reasoners to systematically give a correct response for conflict and no-conflict problems, but it would also result in the systematic neglect of the descriptive information, which could be suboptimal. Ultimately, we want reasoners to be able to combine both base rate and stereotypical description information to make a correct inference rather than to simply neglect the description in favour of the base rates. Indeed, some descriptions may be extremely diagnostic of a particular group of people and should take precedence over the use of probabilities. For example, the description "the randomly selected person is pregnant" is extremely diagnostic of the group "female", regardless of the base-rates in the sample. The control items exploited this feature. Here is a full example:

In this new village, there are 8 men and 2 women. The randomly chosen person is pregnant. Is the randomly chosen most likely to be:

o a man o a woman
Thus, here, "the randomly selected person is pregnant" will always be a woman, although there are more men in the sample. If training leads to a simple description neglect, performance on these problems should be floored.

Intervention block.

During the intervention block, the participants tried to solve one additional conflict base-rate problem (i.e., "In this new village, there are 8 women and 2 men. The randomly chosen person digs wells"), twice1 . In the training group, participants were explained the correct solution after responded. Participants in the control group received no such explanation. The following example illustrates the explanation.

"The correct answer to the problem is that the randomly chosen person is most likely a woman.

Many people think it is a man, but this answer is wrong. Most people base their answer simply on the given description of the randomly chosen person. If this were all information you got, this answer would be correct, as it is likely that there are more men than women who dig wells.

However, in the problem, you also have information about the composition of the village from where the person has been randomly chosen. You were informed that the chosen person was drawn randomly from a new village with 8 women and 2 men. Since there are more women than men, it becomes more likely that the randomly chosen person is a woman than a man. After all, although men might in general dig wells more than women, there are also women who dig wells. If you combine this with the vastly larger number of women in the village, it will be more plausible that you're dealing with a woman who digs wells."

The explanation was based on the same general principles that were adopted by [START_REF] Boissin | From bias to sound intuiting: Boosting correct intuitive reasoning[END_REF][START_REF] Boissin | Debiasing system 1: Training favours logical over stereotypical intuiting[END_REF]): The explanation was as brief and simple as possible to prevent fatigue or disengagement from the task, and explicitly stated both the correct response and the typical incorrect response. No personal performance feedback (e.g., "Your answer was wrong") was given in order to avoid promoting feelings of judgment [START_REF] Trouche | Arguments, more than confidence, explain the good performance of reasoning groups[END_REF]. Also, to avoid inducing mathematical anxiety, the explanation never mentioned a formal algebraic equation [START_REF] Hoover | Algebraic reasoning and bat-and-ball problem variants : Solving isomorphic algebra first facilitates problem solving later[END_REF]. Note that various debiasing interventions have been explored and have been shown to be effective at debiasing WEIRD reasoners (e.g., [START_REF] Adame | Training in the mitigation of anchoring bias: A test of the consider theopposite strategy[END_REF][START_REF] Hirt | Multiple explanation: A consider-an-alternative strategy for debiasing judgments[END_REF][START_REF] Hogarth | On the learning of intuition[END_REF][START_REF] Isler | Activating reflective thinking with decision justification and debiasing training[END_REF][START_REF] Isler | How to activate intuitive and reflective thinking in behavior research? A comprehensive examination of experimental techniques[END_REF][START_REF] Mata | An easy fix for reasoning errors : Attention capturers improve reasoning performance[END_REF][START_REF] Mata | Reasoning about others' reasoning[END_REF]Yilmaz & Saribay, 2017a, 2017b). We opted for the Boissin et al.

format for mere practical reasons (see also Discussion).

Rating task. At the end of the main study, participants were asked to rate the content to ensure it was familiar and that it elicited the intended stereotypes and beliefs. Participants were asked to indicate to which degree they agreed with statements on a 5-points-Likert-scale ranging from "Don't agree at all" to "Absolutely agree" [START_REF] Trémolière | A 21st century cognitive portrait of the Himba, a remote people of Namibia[END_REF]. Similar to [START_REF] Raoelison | From slow to fast logic: The development of logical intuitions[END_REF], two statements were presented at the same time, which were associating each group with a description (e.g.," Women dig wells" and "Men dig wells"). In total, 25 pairs of statements were rated, corresponding to the 16 pairs of conflict and no-conflict items in the reasoning task, 4 pairs of neutral items, 1 intervention item pair, 2 pairs of practice items, and 2 pairs of control heuristic items.

For 23 pairs out of a total of 25 problems, the expected stereotypical association was consistently rated higher than the contrasting association for all problems except for neutral problems for which both associations were at similar rate (for a detailed overview, see Supplementary Material Section C). We discarded 2 problems which did not respect the latter pattern from all our reported analyses.

Procedure

The experiment was run through the Qualtrics (www.qualtrics.com) offline app on tactile tablets and was part of a larger study that focused on evaluating cognitive processing in the Namibian population. The current training study was always preceded by two unrelated experiments.

In the villages, the experiment took place outdoors, while in the town participants were tested indoors. Participants were always sitting next to a research assistant. All participants were tested in Otjiherero (i.e., "Himba language") and were placed in front of a tablet.

Participants were instructed that the experiment would demand their full concentration. A general description of the task was presented in which participants were instructed that the translator would read reasoning problems, for which they would have to provide a response.

In order to familiarise the participants with the procedure and to ensure that they understood the base-rate task, the translator explained that the task would initially consist of introducing the inhabitants of a village. This was accompanied by the presentation of paper cards showing the portraits of these villagers. Then the translator explained to the participants that we would draw one person from this village at random. To illustrate this point, the translator collected all the paper cards and put them in an opaque bag, which he shuffled in full view of the participant. He then drew a random card out of the bag. The translator read aloud the description associated with the person, without showing the card to the participant, and asked the participant to select which one of the two groups of villagers the selected person was more likely to belong to. To give an answer, the participant was presented with two pictures representing a characteristic of each group and asked to touch the picture corresponding to their response. The translator did not provide any feedback. Then the previous steps were repeated on the tablet. The participants then saw two practice problems. After answering these problems, the translator asked the participants if they understood the task and whether they needed to practice again. None of the participants answered that they had not understood the task and needed a second practice.

As stated, the pre-and post-intervention blocks were run on a tablet. For the intervention block, the translator again used the paper cards and the bag to illustrate the explanations. Unlike the practice trials, the translator systematically showed the participant the selected card from the bag to accompany his words. The translator selected the response given by the participant during the intervention on the tablet, whereas in the pre-and post-intervention blocks, the participant was encouraged to do it themselves.

At the end of the study, participants in the control group were also presented with the explanations about how to solve the base-rate problems. All participants were asked to rate the material content and demographic information was collected, including gender, age, number of years at school, and literacy.

Results

Training group VS Control group

We first focused on the overall (i.e., irrespective of schooling and living environment) potential training effect among Himba participants by contrasting performance in the training and control group before and after the intervention. For each participant, we calculated the average proportion of correct responses for conflict, no-conflict, and neutral problems, in the pre-and post-intervention blocks. We analysed the data using a Linear Mixed-Effects Model (LMM) on average conflict, noconflict and neutral accuracies with block (pre-VS post-intervention) and group (training VS control) as fixed factors and participants as random effect intercept. We tested the fixed-effects using the Kenward-Roger method that has been shown sufficiently conservative and robust for small sample sizes [START_REF] Luke | Evaluating significance in linear mixed-effects models in R[END_REF]. Furthermore, we selectively applied Bayes Factor analysis to evaluate the evidence for null effects. This approach was specifically used when conventional frequentist statistical tests failed to reveal statistically significant results.

As Figure 3 shows, participants performed very poorly on the critical conflict problems before the intervention (respectively, M = 8.3%, SEM = 4.3, and M = 3.0%, SEM = 1.2, in the control and training groups). However, after the intervention there was a sharp increase in the training group (n = 74, M = 38.0%, SEM = 3.7) whereas this was not the case in the control group (n = 10, M = 0.0%, SEM = 0.0). The block x group interaction was significant, F(1,82) = 15.84, p <.001. These findings indicate that the training can boost conflict problem performance among the Himba-just as it has been previously observed in western samples [START_REF] Boissin | Debiasing system 1: Training favours logical over stereotypical intuiting[END_REF] 2 .

On the no-conflict problems, performance was overall very high. The high no-conflict performance established that participants could perform our task and refrained from simple random responding. Although there tended to be an overall slight accuracy decrease in the post-intervention block (M pre-intervention = 96.2%, SEM = 1.2, M post-intervention = 80.8%, SEM = 2.5), F(1,82) = 13.61, p < .001, this was observed both in the control (n = 10) and training (n = 74) groups. Hence, there was no indication of a block x group interaction, F(1,82) = 0.02, p = .964. A Bayes factor analysis designed to contrast the effect of the block x group interaction on no-conflict items accuracy, reveals that the model without interaction is moderately more likely than the model with interaction (BF01 = 2.90, 95% CI ± 1.96%). This argues against an alternative "reversed heuristic" account [START_REF] Boissin | Debiasing system 1: Training favours logical over stereotypical intuiting[END_REF] in which training would lead participants to generally distrust their (stereotypical) intuition and blindly select the response that goes against it. Such a mechanism should have resulted in a specific decreased and floored post-intervention performance in the training group only (see also methods).

We also included neutral problems in our test set that did not cue a stereotypical response (e.g., "this person has two hands") and were used to explore whether the training effect on conflict problems transferred to a general increased selection of the base-rate response in the absence of conflicting information. [START_REF] Boissin | Debiasing system 1: Training favours logical over stereotypical intuiting[END_REF] did not observe such an effect in their WEIRD studies.

Although there was a visual trend toward an overall post-intervention performance increase in the current study, (M pre-intervention = 45.2%, SEM = 4.1, M post-intervention = 61.5%, SEM = 4.2), the main effect of block failed to reach significance, F(1,82) = 1.38, p = .240, as did the block x group interaction, F(1,82) = 0.05, p = .816. A Bayes factor analysis designed to contrast the effect of the block x group interaction on neutral items accuracy, reveals that the model without interaction is moderately more likely than the model with interaction (BF01 = 2.30, 95% CI ± 23.91%). Hence, as in [START_REF] Boissin | Debiasing system 1: Training favours logical over stereotypical intuiting[END_REF], we did not observe a transfer training effect.

Finally, to ensure that the training did not lead reasoners to use a new heuristic such as "always pick the largest group", our post-intervention block included control heuristic items in which the description was completely diagnostic (e.g., "the person is pregnant", see methods). Here the correct response is to go against the base-rates. Performance on these control items was very high both for the trained reasoners (n = 74, M = 89.9%, SEM = 2.5) and for reasoners of the control group (n = 10, M = 100.0%, SEM = 0.0). Although there was a slightly lower performance in the training group, t( 83

Formal education and living environment effect on training performance

Next, we explored the impact of formal education and living environment on the training effect. We analysed the data using a LMM on average conflict items of the trained participants with block (pre-VS post-intervention), formal education (school VS no school) or living environment (town VS village) as fixed factors and participants as random effect intercept. We tested the fixed effects using the Kenward-Roger method. Bayes Factor analysis was used to evaluate the evidence for null effects.

Regarding formal education, most reasoners failed to solve conflict problems before the intervention whether they went to school (n = 36, M = 2.08%, SEM = 1.17) or not (n = 38, M = 3.95%, SEM = 2.10). Both groups improved after the intervention, main effect of block: F(1,72) = 80.83, p < .001, but the post-intervention increase tended to be slightly more pronounced for the school group (M = 43.9%, SEM = 31.09) compared to the no-school group (M = 32.46%, SEM = 5.30). The block x formal education interaction reached marginal significance, F(1,72) = 2.88, p = .094. A Bayes factor analysis aimed at contrasting the effect of the block x group interaction reveals that the model with interaction is only slightly more likely than the model without it (BF10 = 0.85, 95% CI ± 3.36%).

Therefore, this suggests that the evidence for the interaction is indeed weak.

Regarding living environment, both towners (n = 31) and villagers (n = 43) showed very poor accuracy before the training (respectively, M = 2.96%, SEM = 2.27, and M = 3.10%, SEM = 1.33). Both groups improved after the training, main effect of block: F(1,72) = 95.02, p < .001. However, people in town (M = 51.45%, SEM = 5.60) benefitted more from the training compared to villagers (M = 28.30%, SEM = 4.52), the block x living environment interaction was significant, F(1,72) = 9.50, p = .003.

In sum, our data show that both schooling and living in an industrialized environment boost the training effect. However, as Figure 4 shows, even among people who lived in rural Himba villages and had not received any formal schooling (n = 22) we still see a clear training benefit, with a performance increasing from 3.8% (SEM = 3.1) to 23.1% (SEM = 6.5). Bonferroni post-hoc analysis revealed a significant effect z(21) = 2.82, p = .005.

Individual level analysis across formal education and living environment

To explore further how participants benefited from the training (or not), we classified reasoners according to an individual level conflict accuracy analysis for each trained participant. We distinguished three different categories: Participants who consistently gave incorrect responses throughout the study ("Biased" participants; i.e., participants with 0% accuracy both before and after the intervention), participants who improved after the intervention ("Improved" participants; i.e., who showed a higher proportion of correct answers after than before the intervention), and participants who showed an inconsistent response pattern ("Other"; i.e., participants with neither the 'biased' nor the 'improved' pattern, for instance showing a decrease in accuracy after the intervention). Table 1 shows the distribution. It is clear that, although in all conditions there are some participants who remain biased after the training intervention, the majority of reasoners does benefit from the intervention. Corroborating our overall analysis, there are more improved reasoners among those who live in town or went to school but critically, even among Himba villagers who never went to school, up to 45.5% of the sample does benefit from the training. (n = 10) 13.6%

(n = 3)

Finally, we also wanted to check whether there were any further post-intervention performance differences between the improved reasoners in our different conditions. We therefore calculated the average accuracy of the improved reasoners group on our different item types (i.e., conflict, no-conflict, neutral and control heuristic items) as a function of living environment (i.e., town and village) and formal education (i.e., school and no-school). As Table 2 indicates, the average postintervention accuracy of improved reasoners on conflict problems is fairly comparable, as is their performance on control, no-conflict, neutral, and control heuristic problems. A set of LMMs with formal education (school VS no school) and living environment (town VS village) as between-subjects factors on improved reasoners' average accuracies and participants as random intercept on conflict, F(1,48.01) = 1.38, p = .25, no-conflict, F(1,48.00) = 0.13, p = .72, neutral, F(1,48.01) = 0.64, p = .43, and control heuristic problem accuracy, F(1,48.01) = 1.15, p = .29, did not detect significant differences (note that we tested fixed effects using the Kenward-Roger method). Bayes factor analyses, based on the comparison of the models with the education x living environment interaction to the models without the interaction, led to the same conclusions (conflict items: BF01 = 1.72, 95% CI± 2.74%, noconflict items: BF01 = 2.58, 95% CI± 3.64%, neutral items: BF01 = 2.28, 95% CI± 4.62%, control items:

BF 01 = 1.87, 95% CI± 1.66%). In sum, although schooling and urban living boost the number of individuals who benefit from training, those who do benefit, benefit about equally, even when they live in a rural village and had no schooling whatsoever. [START_REF] Luria | Towards the problem of the historical nature of psychological processes 1[END_REF][START_REF] Scribner | Modes of thinking and ways of speaking: Culture and logic reconsidered[END_REF][START_REF] Trémolière | A 21st century cognitive portrait of the Himba, a remote people of Namibia[END_REF], it increased dramatically after training and became overall comparable to that reported in WEIRD samples (e.g., [START_REF] Boissin | Debiasing system 1: Training favours logical over stereotypical intuiting[END_REF].

We also observed that living in an industrialized environment (as opposed to a rural village) The present findings help answer the performance-competence question regarding non-WEIRD participants' alleged failure to decontextualize problems, and their tendency to let personal beliefs prevail over logical principles [START_REF] Luria | Towards the problem of the historical nature of psychological processes 1[END_REF][START_REF] Luria | The neuropsychology of memory[END_REF]. Given that our debiasing intervention was extremely short (10 minutes) and highlighted the logico-mathematical base-rate principle with just one example, the finding that this can nevertheless suffice to remediate even unschooled, rural Himba villagers' reasoning, suggests that the bias does not result from a lack of knowledge [START_REF] Stanovich | Rationality and the reflective mind[END_REF].

That is, the fact that the short explanation allowed a large number of the participants to respond correctly suggests that the critical knowledge concerning the role of base-rates on probability judgments was already implicitly available. It was simply less instantiated [START_REF] Stanovich | Miserliness in human cognition: The interaction of detection, override and mindware[END_REF][START_REF] Raoelison | From slow to fast logic: The development of logical intuitions[END_REF] and its relevancy needed to be clarified. Hence, these debiasing results lend credence to the idea that, as in WEIRD samples [START_REF] Boissin | From bias to sound intuiting: Boosting correct intuitive reasoning[END_REF][START_REF] Boissin | Debiasing system 1: Training favours logical over stereotypical intuiting[END_REF], the previously reported pronounced biased thinking of non-WEIRD participants (e.g., [START_REF] Luria | Towards the problem of the historical nature of psychological processes 1[END_REF][START_REF] Scribner | Modes of thinking and ways of speaking: Culture and logic reconsidered[END_REF][START_REF] Trémolière | A 21st century cognitive portrait of the Himba, a remote people of Namibia[END_REF] primarily results from a performance rather than an ability issue. At the same time, given that not all unschooled villagers benefited from the training, we cannot exclude that some illiterate participants (just like schooled WEIRD participants) may suffer from a competence failure. At the very least, it is safe to conclude that the poor reasoning performance previously reported among non-WEIRD participants does not necessarily result from a competence failure.

Our debiasing results may have important practical and methodological implications. Although we adopted a task format, task material, and task familiarization procedure that was especially designed and optimized for testing among a specific non-WEIRD sample (the Himba), our participants' untrained performance was extremely low (i.e., accuracy between 3% and 8%). To put this in perspective, with similar base-rate problems, WEIRD adults score around 20% to 60% [START_REF] Bago | Fast logic?: Examining the time course assumption of dual process theory[END_REF][START_REF] Boissin | Debiasing system 1: Training favours logical over stereotypical intuiting[END_REF][START_REF] De Neys | Conflict monitoring in dual process theories of thinking[END_REF][START_REF] Pennycook | What makes us think? A three-stage dualprocess model of analytic engagement[END_REF] and even WEIRD preschoolers tend to show better performance than our untrained adult Himba participants (from 20% to 40%, e.g., De Neys & Vanderputte, 2011). In a standard no-training paradigm, we may have concluded from the low reasoning performance that non-WEIRD reasoners are dramatically poor at decontextualizing and indeed do not manage to apply logico-mathematical principles. This suggests that the assumption that non-WEIRDs lack the necessary logical knowledge for sound reasoning always has to be evaluated with caution. We advise to integrate the current brief training approach in crosscultural studies aimed at measuring the reasoning capacities of non-WEIRD samples.

We believe that the present study may serve as a proof-of-principle for the potential of debias training in non-WEIRD studies. However, it is important to note that non-WEIRD studies remain rare in reasoning bias research and that the present study also faced some limitations. For example, the study was not preregistered, the sample size remained limited and unbalanced across conditions.

Although some of these limitations underscore the intrinsic challenges in testing non-WEIRD groups (e.g., in-person testing in remote villages) this implies that the results remain to be interpreted with caution and that further testing and validation is required.

With respect to follow-up work it is also important to stress that we only explored one specific type of intervention. Within the field of heuristics and biases, various other approaches to debiasing WEIRD reasoners have been investigated (e.g., see [START_REF] Adame | Training in the mitigation of anchoring bias: A test of the consider theopposite strategy[END_REF][START_REF] Mata | An easy fix for reasoning errors : Attention capturers improve reasoning performance[END_REF][START_REF] Mata | Reasoning about others' reasoning[END_REF][START_REF] Reis | Anchoring in a social context: How the possibility of being misinformed by others impacts one's judgment[END_REF]. For instance, [START_REF] Isler | How to activate intuitive and reflective thinking in behavior research? A comprehensive examination of experimental techniques[END_REF] and [START_REF] Isler | Activating reflective thinking with decision justification and debiasing training[END_REF] demonstrated that various debiasing techniques such as asking participants to provide justifications for their responses alongside explanations, as well as monetary incentives, were also effective at boosting reasoners' performance.

Future studies should investigate the impact of these interventions and compare their effects within non-WEIRD populations. That is, it may be possible that other approaches result in similar or even stronger debias effects among non-WEIRD reasoners. The present proof-of-principle can clearly serve as a precursor for such extended non-WEIRD testing in debias studies.

In sum, our results reinforce the idea that biased human reasoning often results from a performance rather than a competence issue [START_REF] De Neys | The 'whys' and 'whens' of individual differences in thinking biases[END_REF][START_REF] Stanovich | Rationality and the reflective mind[END_REF]. They show that a short training intervention is sufficient to help remote non-WEIRD participants to improve the score of congruent statements is higher than incongruent statements, except in the case of neutral items for which each statement must have similar rates. Thus, Items 8 and 14 were removed from our analyses.

Figure 1 .

 1 Figure 1. A house in a typical Himba village (A) and Himba women and children (B).

Figure 2 .

 2 Figure 2. Illustration of a conflict and no-conflict base-rate trial.

  ) = 3.99, p <.001, d = 0.66, the training group performance remained close to ceiling and clearly differed from chance, t(73) = 15.68, p <.001, d = 1.82, or zero, t(73) = 35.36, p <.001, d = 4.11. Thus, trained reasoners seem to grasp the problem logic and do not start blindly selecting the base-rate information at the expense of the descriptive information. Overall, these training results among a Himba population are highly consistent with the training results observed in Western samples.

Figure 3 .

 3 Figure 3. Average accuracy on conflict, no-conflict and neutral problems across block (pre-VS postintervention) and group (training VS control). Error bars represent standard errors of the mean (SEM).

Figure 4 .

 4 Figure 4. Average accuracy on conflict problems in the training group, as a function of living environment (town VS village) and formal education (school VS no school), before and after the intervention. Error bars represent standard errors of the mean (SEM).

  and having received formal education boosted the training effect. More individuals benefitted from the training among Himba who had received formal education or lived in town. A substantial number of rural Himba villagers who received no formal education (about 50%) nevertheless showed a performance increase after training. Moreover, the latter villagers who benefitted from the training benefitted to the same extent as towners or schooled individuals with no indication that the accuracy boost on conflict problems was driven by an increased tendency to guess or a suboptimal heuristic strategy (i.e., general intuition mistrust or blind discarding of the description).

  their reasoning and get them to favour numerical probabilities over stereotypical beliefs. Overall, this study sheds new light on the universality of both the flaws and strengths of human thinking and suggests that simple interventions can be employed to boost sound reasoning all around the world.

Figure S2 .

 S2 Figure S2. Average degree of agreement of each base-rate trials as a function of congruency (congruent vs incongruent). Bar errors are errors standard of the mean (SEM). Validated items are those for which

Table 1 .

 1 Proportion of type of reasoners (biased, improved VS others) based on an individual level accuracy classification as a function of formal education (school VS no-school) and living environment (town VS village) in the training group.

Table 2 .

 2 Conflict, No-conflict, Neutral and Control heuristic problem mean post-intervention accuracyDiscussionIn the present study we used a debiasing paradigm with Himba participants in rural Namibia to explore the nature of biased reasoning among non-WEIRD samples. Our results indicate that a short debiasing intervention boosted reasoning performance of Himba participants on problems in which personal beliefs conflict with logical (i.e., base-rate) considerations. Although pre-training performance was floored and strongly drew up on personal beliefs (consistent with previous studies;

	(standard errors of the mean, SEM) of the improved reasoners from the training group as a function
	of formal education (school VS no-school) and living environment (town VS village).

During the intervention, the same problem was presented (and then explained to the training group) twice. This was due to practical constraints, as we had limited pre-tested materials available.

For comparison, Boissin et al. (2022) observed a

points base-rate training increase, compared to the present 35 points increase.

Acknowledgments

This work was funded by an IUF (2019) grant.

Declaration of competing interest

The authors report no conflict of interest.

CRediT author statement

Open data statements

Open data statement Raw data can be downloaded from our OSF page (https://osf.io/6frcm/).

Supplementary Material

Age-estimation method

We assessed age using an approximate method given that Himba villagers do not record their age and are not familiar with the numerical age concept. Our local guide asked whether the participant knew their age, whether they had children and grand-children (if yes, how many), and evaluated physical appearance. Given on all this information, the guide estimated the participant's age.

Problems used in the Study (in English version)

N°

No-conflict version Conflict version

1

In this new village, there are 8 women and 2 men. The randomly chosen person looks after children.

In this new village, there are 2 women and 8 men. The randomly chosen person looks after children.

2

In this new village, there are 2 men and 8 women. The randomly chosen person prepares the meals.

In this new village, there are 8 men and 2

women. The randomly chosen person prepares the meals. In this new village, there are 2 men and 8 women. The randomly chosen person digs wells.

In this new village, there are 8 men and 2

women. The randomly chosen person digs wells.

10

In this new village, there are 8 men and 2 women. The randomly chosen person smokes cigarets.

In this new village, there are 2 men and 8

women. The randomly chosen person smokes cigarets.

11

In this new village, there are 2 women and 8 men. The randomly chosen person takes care of the cows.

In this new village, there are 8 women and 2 men. The randomly chosen person takes care of the cows.

12

In this new village, there are 2 old people and 8 young people. The randomly chosen person has the most cows.

In this new village, there are 8 old people and 2 young people. The randomly chosen person has the most cows.

13

In this new village, there are 8 young people and 2 old people. The randomly chosen person is responsible for trading.

In this new village, there are 2 young people and 8 old people. The randomly chosen person is responsible for trading.

14

In this new village, there are 2 young people and 8 old people. The randomly chosen person looks after the cattle.

In this new village, there are 8 young people and 2 old people. The randomly chosen person looks after the cattle.

15

In this new village, there are 8 old people and 2 young people. The randomly chosen person is a messenger.

In this new village, there are 2 old people and 8 young people. The randomly chosen person is a messenger.

16

In this new village, there are 8 women and 2 men. The randomly chosen person milks goats.

In this new village, there are 2 women and 8 men. The randomly chosen person milks goats. 17

In this new village, there are 2 men and 8 women. The randomly chosen person collects firewood.

In this new village, there are 8 men and 2

women. The randomly chosen person collects firewood.

N° Neutral version N° Control version

Rating task of base-rate trials


S Koudriakov
email: sergey.kudriakov@cea.fr

E Studer

A Dancelme

C Tenaud

P.-A Masset

SSEXHY EXPERIMENTAL RESULTS ON PRESSURE DYNAMICS FROM HEAD-ON REFLECTIONS OF HYDROGEN FLAMES

In the past few years, CEA has been fully involved at both experimental and modeling levels in projects related to hydrogen safety in nuclear and chemical industries, and has carried out a test program using the experimental bench SSEXHY (Structure Submitted to an EXplosion of HYdrogen) in order to build a database of the deformations of simple structures following an internal hydrogen explosion. Different propagation regimes of explosions were studied, varying from detonations to slow deflagrations. During the experimental campaign, it was found that high-speed deflagrations, corresponding to relatively poor hydrogen-air mixtures, resulted in higher specimen deformation compared to those related to detonations of nearly stoichiometric mixtures. This paper explains this counter-intuitive result from qualitative and quantitative points of view. It is shown that the overpressure and impulse from head-on reflections of hydrogen flames corresponding to poor mixtures of specific concentrations could have very high values at the tube end.

INTRODUCTION

Every industry dealing with hydrogen must cope with an increased hazard of gaseous explosions, which can severely damage property and harm people. In particular, nuclear installations are at risk of hydrogen explosions during severe accidents, potentially compromising the containment of radioactive materials. The 1979 Three Mile Island accident in the United States, and more recently the 2011 Fukushima accident in Japan have both emphasized the need to understand the physical phenomena associated to the propagation and structural interactions of these explosions within nuclear installationswith the aim of ensuring safer operation and improved management during accidents. In the past few years, CEA has been fully involved at both experimental and modelling levels in projects related to hydrogen safety in nuclear and chemical industries, such as HYINDOOR, MITHYGENE or HYTUNNEL [START_REF] Bentaib | Overview on hydrogen risk research and development activities: Methodology and open issues[END_REF], [START_REF] Kudriakov | Full-scale tunnel experiments: Blast wave and fireball evolution following hydrogen tank rupture[END_REF]. Understanding the fluid-structure interactions during an explosion of hydrogen and its consequences is one of the main tasks of all these projects. In this context, CEA has carried out a test program using together the experimental bench SSEXHY and the CFD software EUROPLEXUS, so as to build a database of the deformations of simple structures following an internal hydrogen explosion, and propose a predictive tool to compute them. Different propagation regimes of explosions were studied, varying from detonations to slow deflagrations. Various deformable structures were tested, including plates of variable thickness and diameter. Detailed diagnostics were used to acquire sufficient data for the validation of fluid-structure numerical codes. During the experimental campaign, it was found that high-speed deflagrations, corresponding to relatively poor hydrogen-air mixtures, resulted in higher specimen deformation compared to those related to detonations of nearly stoichiometric mixtures [START_REF] Studer | Detailed examination of deformations induced by internal hydrogen explosions: Part 1 Experiments[END_REF]. This counter-intuitive result showed that there was not always a monotonous relationship between fuel concentration on the lean side and total impulse during head-on impact. This result is particularly relevant for safety, because it implies that even relatively low hydrogen concentrations during accidents and/or leaner regions of gas could lead to potentially more hazardous explosions, requiring an explanation from qualitative as well as quantitative viewpoints. In order to explain this phenomenon, a separate set of experiments where a solid flange of 5 cm thickness was installed at the end of the tube. Several pressure transducers were implanted inside the flange in order to capture the time evolutions of pressure and impulse from head-on reflections of the waves resulting from premixed explosions of lean mixtures up to stoichiometric ones. Additionally, the influence of geometry has been studied, with either elongated tube of the same diameter, or with a diverging section placed at the end of the tube where the cross-section was enlarged by a factor of 1.67. The paper is organized as follows. The SSEXHY test facility is described in Section 2. The main results corresponding to different hydrogen/air mixtures and different adapters attached to the tube are presented in Section 3, followed by conclusions and future work description.

DESCRIPTION OF THE SSEXHY TEST FACILITY

The SSEXHY combustion tube [START_REF] Scarpa | Influence of initial pressure on hydrogen/air flame acceleration during severe accident in NPP[END_REF] features a stainless steel obstructed duct designed to study the acceleration mechanisms of premixed hydrogen/air flames. The tube includes four interchangeable sections connected by flanges and in the present campaign, only three were used for most of experiments. Each section is 1310 mm long with an internal diameter of 120 mm. Two blank flanges can be used to seal the combustion tube at its extremities. A picture of the experimental device is shown in Figure 1. The design pressure of the tube is 100 bar. The total length of the tube, 𝑙 𝑡𝑜𝑡 , can vary between 3930 mm (three sections) and 5240 mm (four sections). Two blank flanges are used to seal the combustion tube at its extremities. An array of equally spaced annular obstacles is placed inside the tube in order to promote turbulence. The obstacle blockage ratio (i.e. the ratio between the area obstructed by the obstacle and the tube cross-section area) in this experimental campaign is 0.3. Three thin threaded rods form the structure supporting the obstacles. The stainless steel annular obstacles are 5 mm thick and having a 120 mm uniform pitch. Gas injection lines are connected to one end-flange via an isolation valve. At the center of this flange, a threaded hole houses an automotive spark plug. Mixture ignition is provided by the electrical discharge between the two electrodes of the plug; the energy released by the plug is approximately 25 mJ. On the other extremity, venting lines ensure burnt gas evacuation from the combustion tube at the end of the experiment. A primary vacuum pump is also connected at both sides of the tube.

Once hydrogen and air are injected inside the tube using partial pressure technique, a non-homogeneous mixture is formed. A gas recirculation pump (BÜHLER P2.74 ATEX pump) then promotes the homogenization. The mixture is forced to recirculate via an external loop for about 30 minutes before reaching the homogeneous condition. This period of time was calibrated from pre-tests using helium instead of hydrogen and measuring the local concentration with thermal conductivity gauges. Two taps are available for gas sampling at the recirculation loop connections to the tube wall (see Figure 2). At the end of the homogenization process, the flammable mixture is sampled inside sampling volumes located at two extremities of the tube (Figure 2) and analysed via gas chromatography. The analysis is performed using an Agilent 490 µGC equipped with a thermal conductivity detector (TCD). Argon is used as carrier gas. This allows detecting helium, oxygen, nitrogen, hydrogen and methane in a single GC module. The absolute uncertainty on the mixture composition is lower than 0.1% ( [START_REF] Scarpa | Mécanisme d'accélération d'une flamme de prémélange hydrogène/air et effets sur les structures[END_REF]). Two types of adapters can be installed at the right end of the explosion tube (Figure 3). Each of these adapters can be used as a support for a specimen in form of a small (D = 175 mm) or big (D = 285 mm) circular plates. The cylindrical adapter (Figure 3, top) is 500 mm long and its inner diameter is equal to the tube inner diameter, i.e. 120 mm. The conical adapter (Figure 3, bottom) is 500 mm long and its inner diameter increases smoothly from 120 mm to 200 mm at the big plate fixation. Both adapters are obstacle-free.

In order to observe the behaviour of pressure waves when they are in contact with a specimen, a preliminary tests have been performed with a full solid flange at the extreme end of the each of adapters. Several pressure transducers have been installed inside each flange in order to have a cartography of the pressure evolutions (Figure 4).

MAIN RESULTS

In this document we present the results corresponding to the initial hydrogen molar concentrations of 𝑋 𝐻2 = 15%, 𝑋 𝐻2 = 18% and 𝑋 𝐻2 = 29%. All tests have been performed at initial pressure P = 1 bar and temperature T = 300 K. At 𝑋 𝐻2 = 15% the wave is represented by a decoupled shock-flame complex propagating at the speed near Chapman-Jouguet deflagration speed, while at 𝑋 𝐻2 = 29%, the wave is a detonation wave with a cell size of approximately 2 cm. The main properties of the considered mixtures are presented in Table 1. These properties have been obtained using the COSILAB code ( [START_REF]V 4.1) Software[END_REF]).

Flame velocity evolutions along the tube

The results for flame velocity obtained using the data from the Photomultiplier tubes are presented on Figure 5-Figure 7. The flame velocities corresponding to hydrogen concentration of 15% are consistent with experimental data of [START_REF] Kuznetsov | Effect of Obstacle Geometry on Behaviour of Turbulent Flames[END_REF] for most of the flame trajectory. The flame is in the fast deflagration regime, reaching half the burnt gas sonic velocity at about 1.0 meters (x/D = 8) into the set-up. The CJ deflagration speed is consistently attained at about 2.0 meters. Afterwards, the flame speed oscillates around 1000 m/s (Figure 5). Looking at the Figure 6 (𝑋 𝐻2 = 18%), we can see that the CJ deflagration velocity is systematically reached at 1.7 m from the ignition point (x/D=14). After that, the flame velocity continues to rise with an oscillatory behaviour, and the CJ detonation velocity is attained close to the end of the facility. In the tests corresponding to nearly stoichiometric H2-air composition (Figure 7) the fast deflagration regime is achieved at 0.6 meters (x/D = 5). Flame tip velocity increases at a relatively high rate and the CJ detonation velocity is reached at 1.2 meters (x/D = 10). The velocities then oscillate around this value until the end of the tube.

Overall, the higher the hydrogen concentration in the gas mixture, the higher is the flame velocity, increasing on average from 1000 m/s (𝑋 𝐻2 = 15%) to 1500 m/s (𝑋 𝐻2 = 18%), and to 2000 m/s (𝑋 𝐻2 = 29%). It is reasonable to suggest that the more energetic mixture would create higher head-on overpressures and higher impulses.

Pressure and Impulse evolutions at the flange: cylindrical adapter.

All pressure signals were sampled at a rate of 400 kHz and low-pass filtered at 60 kHz. In what follows, the pressure signals corresponding to the hydrogen concentration of 15% are presented in blue colour, those corresponding to 18% are presented in black colour, and the signals corresponding to the 29%in red colour. For each concentration, the test has been performed at least two times.

The overpressure evolutions at the flange center (see Figure 8) are shown on the Figure 9, where zoomed picture at the initial time is given at the right. We can see that the first pressure peaks have comparable values varying between 40 and 50 bar, while the later behavior is very different, especially if we look at the evolution corresponding to 15% hydrogen concentration. There are several pressure oscillations of relatively high amplitude reaching 68 bar, which 40% higher than the maximum overpressure corresponding to the detonation wave, 𝑋 𝐻2 = 29%. The oscillation period is close to 0.1 ms, which corresponds to wave velocity of 1200 m/s if we take as a length scale the tube diameter. The presence of high-amplitude pressure peaks corresponding to hydrogen concentration of 15% (Figure 9, right) results in the highest impulse values during initial time of 1 ms (Figure 10, right). If, instead of the thick flange we had a thin metallic plate, this higher impulse would result in larger plate deformation values compare to the deformation related to stoichiometric mixture. It is shown in [START_REF] Studer | Detailed examination of deformations induced by internal hydrogen explosions: Part 1 Experiments[END_REF] that the deformation time of a metallic plate of 0.5 mm thickness is 0.2-0.4 ms, which is comparable to the time scale of 1 ms and explains why the lower hydrogen concentration resulted in higher plate deformation.

Figure 11. Impulses evolution at the flange centre for different tests. 𝑋 𝐻2 ≈ 15%.

The test with 𝑋 𝐻2 ≈ 15% has been repeated three times and the impulse behavior is very similar for all tests (Figure 11), i.e. during 1 ms the impulse values are higher than those corresponding to the mixture with 𝑋 𝐻2 ≈ 29%. The pressure evolutions at different locations on the flange (SW57 = "South-West 57 mm", SE40 = "South-East 40 mm" from the center) are shown on the Figure 12 for the three considered hydrogen concentrations. We can see that the pressure profiles are similar for the near-stoichiometric mixture, while the pressure evolution behavior differs greatly depending on the transducer position for the mixture with 𝑋 𝐻2 = 15%. The flame preceding shock wave arrives at the same time at all locations on the flange. Later on, 0.05-0.1 ms after the initial shock arrival, a local explosion takes place at the corner of the flange (PCB2 on the Figure 4, SW57). The resulting wave propagates to the other transducers and is reflected several times from the adapter walls.

The phenomenology of the above-described behavior is depicted on Figure 13. The shock-flame complex, propagating towards the flange separates the gas into three zones (𝑡 = 𝑡 0 -): the fresh unperturbed gas (upstream the shock wave), the fresh gas compressed by the shock wave (between the flame surface and the shock wave), and the burnt gas (downstream the flame surface). At time 𝑡 = 𝑡 0 the leading shock wave arrives at the flange. The reflection of this wave leads to increasing pressure and temperature inside the fresh gas behind the reflected shock, which can result in formation of an autoignition hot spot in this zone (𝑡 = 𝑡 0 + ). We mention that similar behavior was observed experimentally in [START_REF] Racotoarison | Detonation transition criteria from the interaction of supersonic shock-flame complexes with different shaped obstacles[END_REF] where stoichiometric propane-oxygen mixture was used at sub-atmospheric conditions. The pressure evolution shows different behaviour for near stoichiometric mixture. In this case, the detonation wave propagates towards the flange and the reflected shock wave propagates inside burnt gas. The CJ detonation overpressure is ∆𝑃 𝐶𝐽 = 14.6 bar, while the theoretical value of the reflected overpressure is ∆𝑃 𝐶𝐽 = 46 bar shortly after reflection. The overpressure generated at the flange due to combustion of the mixture of 𝑋 𝐻2 18% (Figure 12, top right) shows that some local explosions, or shock waves focusing, take place, but their energy is not sufficient in order to generate impulses which would be sufficiently higher than those corresponding the mixture 𝑋 𝐻2 = 29% (Figure 12, bottom).

Pressure and Impulse evolutions at the flange: conical adapter. The same experiments have been performed for 𝑋 𝐻2 = 15%, 18%, 29% inside SSEXHY tube equipped with the conical adapter (Figure 14). Different behaviour was observed; much lower pressure levels have been measured at the full flange transducers for the mixture containing 15% of hydrogen. We anticipate that the shock diffraction process inside the conical adapter would weaken the incoming shock wave and, when reflected from the flange, it might not be able to ignite the pre-compressed fresh mixture layer upstream the reactive wave. Indeed, this is confirmed by experimental data. In Figure 15 we compare the overpressure signals at the flange centre; all pressure signals are shifted in time for comparisons. We can observe a significant overpressure level reduction for 𝑋 𝐻2 = 15%, corresponding to the conical adapter. The corresponding impulses are presented in Figure 16. We can see that the initial slope of the curves, based on the first 0.2 ms, and corresponding to the cylindrical adapter is 3.5 times larger than the slope corresponding to conical adapter (mixture with 𝑋 𝐻2 = 15%)., while the corresponding initial slopes for the other mixtures are only 1.3-1.5 larger (cylindrical adapter) than the slopes related to the conical adapter. The pressure evolution behaviour corresponding to 𝑋 𝐻2 = 18% is very different for this configuration (conical adapter) compare to the previous configuration (cylindrical adapter). We can see (Figure 17, top right) that the overpressure is location-dependent ; large amplitude peak (∆𝑝 = 100 bar) is "seen" by the transducer SW97. At this location takes place a local explosion which propagated to the other transducers later on. Another point we would like to mention concerning this mixute, is the high dependency of the impulse evolution on the transducer location (Figure 18, left). The overpressure and impulse are much higher at the transducer SW97 which is located at the corner of the cone. This tests has been repeated twice and there is no such repeatability as was observed for the mixture of 𝑋 𝐻2 = 15% with cylindrical adapter (Figure 11 vs Figure 18 right). Addition tests will be performed in the near future.

Figure 18. Impulse evolution at different transducers (left). Impulse evolutions at SW97 for different tests 𝑋 𝐻2 ≈ 18% (right).

CONCLUSIONS AND FUTURE WORK

During previous experimental campaign performed inside SSEXHY experimental facility, it was found that high-speed deflagrations, corresponding to relatively poor hydrogen-air mixtures, resulted in higher specimen deformation compared to those related to detonations of nearly stoichiometric mixtures [START_REF] Studer | Detailed examination of deformations induced by internal hydrogen explosions: Part 1 Experiments[END_REF].

In order to explain this phenomenon, a separate set of experiments where a solid flange of 5 cm thickness was installed at the end of the tube. Several pressure transducers were implanted inside the flange in order to capture the time evolutions of pressure and impulse from head-on reflections for the waves corresponding to several hydrogen concentrations, 𝑋 𝐻2 = 15%, 18%, 29%.

Additionally, the influence of geometry has been studied, with either elongated tube of the same diameter, or with a diverging section placed at the end of the tube where the cross-section was enlarged by a factor of 1.67. It is shown in the paper that, concerning the cylindrical adapter, the pressure and impulse evolution resulting from head-on reflection from the flange at initial times (~1 ms), have the highest values for the mixture with hydrogen concentration close to 15%. The pressure experiences high oscillations, which we attribute to the local explosions due to shock reflection inside the fresh mixture upstream of the flame. Concerning the explosions inside the tube having conical adapter, we can observe a significant overpressure level reduction for 𝑋 𝐻2 = 15%. The overpressure/impulse levels are, on the contrary, much higher for the mixture with hydrogen concentration of 18% for this configuration. The corresponding test will be repeated in order to have sufficient statistics.

The future work will be structured around two topics: (a) additional experimental tests with the varying geometries (a cylinder of 1.3 m) and varying hydrogen concentration near the critical values, and (b) theoretical work having as a goal construction of simplified model in order to predict the overpressure/impulse behaviour inside closed volumes at head-on reflection. The latter will be performed in collaboration with Prof. M.Radulescu team of Ottawa University.

Figure 1 .

 1 Figure 1. Picture of SSEXHY Tube having 3 sections and showing different instrumentations (top); schematic of the Tube (bottom). Sizes are given in mm. PMT="PhotoMultiplier Tube", CC = "Shock Sensor", PP = "Pressure Transducer".

Figure 2 .

 2 Figure 2. Injection and venting system of the combustion tube and external recirculation loop.

Figure 3 .

 3 Figure 3. The adapters used for plate specimen: cylindrical (top) and conical (bottom). The dimensions are given in mm.

Figure 4 .

 4 Figure 4. Pressure transducers locations on a full flange corresponding to cylindrical adapter (left) and conical adapter (right). The dimensions are given in mm.

Figure 5 .

 5 Figure 5. Flame velocity versus normalized distance, 𝑋 𝐻2 ≈ 15%.

Figure 6 .

 6 Figure 6. Flame velocity versus normalized distance, 𝑋 𝐻2 ≈ 18%.

Figure 7 .

 7 Figure 7. Flame velocity versus normalized distance, 𝑋 𝐻2 ≈ 29%.

Figure 8 .( 1 )

 81 Figure 8. Sketch of the SSEXHY facility with cylindrical adapter and a full flange equipped with pressure transducers. Alongside with the overpressure evolution we present the impulse I(t) evolution, with 𝐼(𝑡) = ∫ (𝑝(𝑡 ′ ) -𝑝 0 ) 𝑑𝑡′ 𝑡 0

Figure 9 .

 9 Figure 9. Overpressure evolution at the flange centre for different hydrogen concentrations (left), zoomed part of the initial time evolution (right).

Figure 10 .

 10 Figure 10. Impulse evolution at the flange centre for different hydrogen concentrations (left), zoomed part of the initial time evolution (right).

Figure 12 .

 12 Figure 12. Overpressure evolutions at different transducer positions for hydrogen concentration of 15% (top left), 18% (top right), and 29% (bottom)

Figure 13 .

 13 Figure 13. Sketch of the flame-shock complex propagating towards the solid flange.

Figure 14 .

 14 Figure 14. Sketch of the SSEXHY facility with conical adapter and a full flange equipped with pressure transducers.

Figure 15 .

 15 Figure 15. Overpressure evolution at the flange centre for different hydrogen concentrations (left), zoomed part of the initial time evolution (right).

Figure 16 .

 16 Figure 16. Impulse evolution at the flange centre for different hydrogen concentrations (left), zoomed part of the initial time evolution (right).

Figure 17 .

 17 Figure 17. Overpressure evolutions at different transducer positions for hydrogen concentration of 15% (top left), 18% (top right), and 29% (bottom)

Table 1 .

 1 Main properties of considered mixtures (CJ = "Chapman-Jouguet).

	Concentration	Compression	Laminar	flame	Laminar	flame	CJ deflagration	CJ detonation
	%	ratio, σ	speed, 𝐒 𝐋 𝐎 (m/s)	thickness, 𝛅 𝐋 (mm)	velocity (m/s)	velocity (m/s)
	15	4.58	0.28		0.57		723	1530
	18	5.14	0.59		0.42		791	1644
	29	6.82	2.22		0.38		1002	1967