M C Arthi

Dr Kavitha Shanmugam

Research Scholar

Financial Inclusion via Mobile Banking -A Comparison Between Kenya and India

Keywords: Financial Inclusion, Digital payments, M-Pesa, Mobile banking, Mobile payments, Payment Banks

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

Introduction

The need for financial inclusion and provision of digital technology to access Government services is increasingly becoming important to address social inclusion and economic uplifting of the poor. The influencers for financial inclusion are; IT & financial literacy (Nanda Kajole, 2018) internet availability (Kieran Mervyn et al, 2014), relatively young population who can learn latest technology fast, smart phones at affordable prices, GDP per capita (Njuguna Ndung'u, 2018) and easy access to the services. In this article, we will address how the banking needs of the rural and economically weaker sections of society are taken up in India & Kenya and how effective are they in the current state of implementation. Innovative financial services enabled with technology, which provides with easy affordable access and hassle free services are adopted by both countries. It is important that customers are well educated on the values of the services offered and satisfied with them. The traditional banking services are inconvenient, more costly and inaccessible for people in the base of the economic pyramid.

In 2012, the World Bank estimated that, one in three mobile money users worldwide were in Kenya although Kenya has less than 1% of the world's population (Jane K. [START_REF] Winn | Mobile Payments and Financial Inclusion: Kenya, Brazil, and India as Case Studies[END_REF] Kenya was able to achieve financial inclusion by providing a simple remittance services that works seamlessly and reliabe on even the most basic mobile handsets. M-Pesa bypassed the complexity of modern banking at a time when traditional bricks and mortar banks in advanced market economies are finding it ever more difficult to compete with alternative financial services (Jane K. [START_REF] Winn | Mobile Payments and Financial Inclusion: Kenya, Brazil, and India as Case Studies[END_REF].

For M-Pesa in Kenya, the value proposition for users was fast, cheap, reliable remittance services from urban and rural areas, while for agents it was fee income. (Jane K. [START_REF] Winn | Mobile Payments and Financial Inclusion: Kenya, Brazil, and India as Case Studies[END_REF]. For Payment banks in India, the value proposition for users are zero balance account, convenience, ease of transfer to any part of India, less transaction charges and security (Bhansali [START_REF] Shrey | Role of Payment Banks in India: Opportunities and Challenges International Journal of Advances in Management and Economics[END_REF] Most M-Pesa customers said that they chose the service because of its low cost (Morawczynski & Pickens, 2009) and M-Pesa's pricing structure encourages users to experiment with the service while extracting value from the transactions that customers value the most (Ignacio Mas & Amolo Ng'weno, 2010).

Both the services use branchless banking systems, taking advantage of increasingly universal real-time mobile communications networks to bring banking services into everyday retail stores, thereby removing the need for banking infrastructure in the communities where poor people live and work [START_REF] Dermish | Branchless and Mobile Banking Solutions for the Poor: A Survey of the Literature[END_REF] The vision of universal branchless banking services is associated with high-volume, low-value transactions by customers using cash with values like quick, convenient, and most importantly, guaranteed service (Ahmed Dermish, et al, 2011)

2

Financial Inclusion in India -An overview Due to the initiative, as of 2017 (data from Business standard), 80% of the Indian people have bank account; about same number of people use mobile phone. The economic revolution of moving away from cash and go to digital transactions have started in India for the past few years and due to COVID-19 situation in India, there is more need to do contactless transactions.

With the limitation of banks not able to open branches in all parts of India, especially rural areas, the need for quick access to bank account for payments, savings and transfer of money is to be fulfilled. With almost all Indian citizens holding mobile phone, the most economic and quicker solution for financial inclusion is to focus on mobile banking solution (Pallab Sikdar, Amresh Kumar, 2017)

Payment Bank History and Current Scenario

To [START_REF] Bengelstorff | A Global Success from Kenya[END_REF]. Rural Kenyans with no banking opportunities leaped from the agricultural age straight into the digital world of tomorrow  Financial Benefit: M-Pesa increased the volume of money flowing in and out of the communities, speeding up money velocity, which boosted local consumption. Therefore, there was an increased volume and variety of food and agricultural inputs in local markets thus bolstering food security.  More jobs: More youth and women found employment (Rob [START_REF] Matheson | Study: Mobile-money services lift Kenyans out of poverty[END_REF] in the rapidly expanding agent network as businessperson became M-Pesa agents in addition to their core business.  Security: In terms of physical security, users reported reduced muggings and thefts, and women reported being able to use M-Pesa to accumulate cash securely (Adrine Muhura, 2019).

Role of Mobile money Agents for M-Pesa

The M-Pesa agents pay a major role in serving the unbanked in Kenya by offering mobile money products and services and helping the country's GDP and increasing the income of households by bringing banks to people's doorstep especially for rural population where the need is higher (refer figure 2). The agents compete with other competitors like commercial banks, Cooperative societies for promoting and sustaining M-Pesa both in rural and urban Kenya (Brian [START_REF] Muthuiora | Enabling Mobile Money Policies in Kenya, Fostering a Digital Financial Revolution[END_REF]. From figure 3, it is very clear the domination of mobile money agents and making it evident that M-Pesa growth in Kenya largely depend on the strength of the mobile money agents. The payment bank transaction is India is comparable with the volume and value of transactions in Kenya although overall mobile transactions volume and value in India including commercial banks is more than 10 times than Kenya's transactions (refer to figure 5 and figure 6). What we can infer is that, the Kenya's transactions trend and value is more or less had become stable over the last one year, whereas in India, there is a variation still showing and we had still not reached maturity on payment banks. In addition, the current Covid-19 situation creates an uncertainty in the volume and availability of money in the hands of customers to do mobile transactions. However, considering population size of the countries, comparing to Kenya, India has to go a long way in financial inclusion as we see only two of the five payment banks are in top 10 banks in volume and value of transactions as per RBI reports.

Conclusion

We can visualize a common trend between the two countries mobile banking services (M-Pesa from Kenya and Payment banks from India), but payment banks popularity as mobile banking solution in India is still a long journey. Payment banks had to compete with schedule commercial banks whereas in Kenya, Safaricom's M-Pesa is almost a monopoly with 64% market share compared other competitors in Kenya (see Annual report, Communications Authority of Kenya). The road ahead for Payment Banks is challenge not only in terms of value or volume but also from profitability perspective. Systematic pricing structure, cross-selling products like insurance, micro finance/lending via tie-ups, are the opportunities ahead of them

Fig. 1 . 3 Financial

 13 Fig. 1. Roadmap of Payment Banking System establishment in India Source: Reserve bank of India and Payment banks sites

Fig. 2 .

 2 Fig. 2. Financial Access Points by type -Rural vs Urban view

Fig. 3 .Fig. 4 Comparing

 34 Fig. 3. Mobile Money Agents Domination in Access points in Kenya *Source: FSD Maps, Kenya, GSMA's Mobile Money for the Unbanked (MMU)

Fig. 5 .Fig. 6 .

 56 Fig. 5. Comparing the trend of mobile transaction between two countries *Source: Central Bank of Kenya, Reserve Bank of India

2.1 Banking for the unbanked Financial

  inclusion in Indian subcontinent was one of the focus areas for the Government and it is taken up with more rigor starting 2014, thanks to the effort of the Government. Pradhan Mantri Jan Dhan Yojana (PMJDY) is an initiative taken by Government of India, which is open to all Indian citizen with the focus of affordable banking

solution for the poor and needy. It aims to provide a basic bank account, remittances, credit and insurance to the citizens. This scheme was launched on 28 August 2014. A record number of 11.50 crore bank accounts opened under PMJDY as on 17th January 2015 against the original target of 7.5 Crore set by 26th January, 2015 (MK.Soumiya Devi et al, 2015) which show the success of the scheme in India.

  overcome the challenges faced due to the restrictions to use Scheduled Commercial banks, in September 2013, a "Committee on Comprehensive Financial Services for Small Businesses and Low Income Households", headed by Nachiket Mor, was formed by the RBI (RBI Press release, 2013). By January 2014, the Nachiket Mor committee submitted its final report with recommendation for a new category of bank called "Payment Banks" with a condition that 25% of its branches must be in the unbanked rural area (Bhansali Shrey et al, 2018). The banks must use the term "Payment Banks" in its books and files in order to differentiate it from other types of bank (RBI report, 2014). Currently there are five active Payments Banks in India namely,

3.1 M-Pesa -Mobile payment solution from Kenya

  Pesa is simple for users with mobile phone number become account number with Safaricom. User credits amount to the account from retail outlets or authorized resellers called agents and top up their phone. A pin is used to protect the account, which can be accessed to send or receive funds (Isaac Mbiti & David N. Weil, 2011). To withdraw cash, the user visits the agents' outlet and makes the transaction (see Safaricom). According to the Central Bank of Kenya, the value of all transactions in 2019 is 44 percent of the country's GDP In cooperation with Safaricom M-Pesa offers savings accounts, paying 3% to 6.65% per annum interest (see Safaricom). Many people started earning interest for the first time in their lives  Mobile payments: Utility bill, cable TV rent, online purchases and supermarket shopping, can be done with M-Pesa. Money transfer to M-Pesa is simple and easy using the cell phone.

	Registration for M-
	over 23
	million users as of the year 2019 (Vodafone, press releases, 2020). The Central Bank
	of Kenya backed M-Pesa development and rollout, as means for financial inclusion and
	kept other commercial banks' opposition at bay, as financial inclusion is a higher im-
	mediate objective over the interests of the banking sector (Elsie Onsongo & Johan
	Schotm, 2017). It had become the widespread person-to-person transaction channel
	such that it accounted for 66.56 percent of the throughput volume in Kenya's National
	Payment System (Brian Muthuiora, 2015). It now provides services such as; Deposit
	and withdrawal of funds, transfer of money to users and non-users, payment of bills,
	gateway to loans, transfer money between services and bank accounts and purchasing
	airtime or what we call as mobile talk time and data (see Safaricom).

M-Pesa is a mobile payments system first developed in Kenya, which effectively offers a banking system based on mobile phones. It was started in 2007, by Vodafone Group plc and Safaricom, the largest mobile network operator in Kenya (Elsie Onsongo & Johan Schotm, 2017). Since then, M-Pesa has seen a rise in popularity with Benefits  . Interest on savings account:  Cell phone becoming virtual banks: In 2007, several million Kenyans owned a cell phone with a Safaricom number. Then this phone number actually became a virtual bank account number (Anja

Abbreviations


Roshni K K1
email: roshni_kk@ethirajcollege.edu.in

Shobana T2
email: shobana_t@ethirajcollege.edu.in

Shruthi R3
email: shruthi_r@ethirajcollege.edu.in

The Impact Of Digital Marketing On Exploratory Buying Behavior Tendencies (EBBT)

A marketing study is not complete without studying the buyer's behavior. With more people moving to the digital platform for day to day purposes, digital marketing is gaining momentum like never before. Hence this study aims to study the effect of digital marketing on Exploratory Buying Behavior Tendencies (EBBT) of the consumers. Since the ambit of digital marketing ambit is huge, only internet marketing, mobile marketing, E-Mail marketing and Social media marketing were taken for the study. Data from 110 respondents were collected and the results were analyzed using weighted average, correlation and regression. The study concludes that out of the four components in digital marketing, internet marketing has the highest influence on consumers as they buy products.

Introduction

In an era where information is omnipresent, digital marketing is inevitable. Digital marketing is the use of online platforms reach out to the consumers. Though are many components of digital marketing only a few of them have become the buzzword. To begin with, internet marketing is crucial as India has the second largest number of internet users after China. It concentrates on making the website attractive for consumers, posting advertisements in various sites. Mobile marketing is the next major component as the smart phone penetration is increasing day by day. Mobile marketing is done through applications, SMS and push notifica-tions. Messages containing information regarding products and offers are sent to consumers. E-Mail marketing involves in sending personal and at times exclusive E-Mails to consumers to their mail inbox. E-Mail also contains information about new products in the form of E-Newsletters. Social media marketing is new when compared to the other three components of digital marketing. They include marketing on Facebook, Twitter, Instagram, LinkedIn, etc. This method is more attractive people tend to spend much of their online time on these social media platforms. With such digital media platforms, consumers tend to explore more products as more information is available to them in just a click. This makes them to take the risk of buying and trying unknown, other brand products. Moving on to buying behavior, exploring new products is salient factor influencing consumers. They search for innovative products offered, thereby leading to exploratory buying behavior. The concept of Exploratory Buying Behavior Tendencies (EBBT) was proposed by Hans Baumgartner and Jan-Benedict E.M. Steenkamp in the year 1996. They identified that buying behavior can be motivated when the consumer expects that the end result of using a product will excite him. EBBT is a Two-Factor model. The first factor-Exploratory Acquisition of Products (EAP) deals with sensory stimulation of a consumer. All the five senses of a consumer i.e. sight, taste, smell, hearing and touch must be satisfied when he consumes the product. The second factor -Exploratory Information Seeking (EIS) prods the cognitive stimulation of the consumer. By getting additional information on a variety of products, he gets to choose a product which deliver his needs. As a consumer receives information on internet, mobile phone, E-Mail and social media, the artificial intelligence present in all these media brings in exact information that he needs. This thirst for exploring new products seem to be endless for consumers. For example, if a person is searching for features of a particular brand of mobile phone in internet, details regarding other models of the same brand and different brand's phones immediately pops up in other websites and social media sites. Hence, the study attempts to analyze the impact of digital marketing on EBBT.

2 Review of Literature Baumgartner and Steenkamp (1996) in paper 'Exploratory consumer buying behavior: Conceptualization and measurement' analyzed and proposed a two-factor concept of exploratory consumer buying behavior in which the authors explained that exploratory acquisition of products is different from exploratory information seeking. The method of Analysis used was correlation and two factor analysis. They narrowed down on 41 EAP factors and 28 EIS factors from 89 prospective factors. The results of six studies with students from two different countries, one a major American university and another university in the Netherlands shows that the scale has good psychometric properties and that its relationships with other constructs and actual exploratory behaviors confirm to theoretical expectations.

Waheed and Jianhua ( 2018) in their study 'The linkage between e-marketing and consumers' exploratory buying behavior tendencies have investigated the effect of EBBT toward E-Marketing in addition to the sub-factors of the E-Marketing model in China was inspected with gender as a moderate variable. Data from 1,600 respondents were collected over a period of 7 months. Pearson's correlation was used to analyze the data using SPSS with SEM technique. The results stated the positive relationships of EBBT toward all E-Marketing factors but relatively less positive effect than the direct effect of E-Marketing factors to EBBT.

Claro G Gañac, (2018) in his paper 'Investigating Consumer Optimum Stimulation Level and Exploratory Online Buying Behavior' concentrated on elucidating the shooting increase of Internet shopping not only because of the rapid advances in technology but also due to the deep-rooted susceptibility of consumers to involve in exploratory buying and consumption. The merging of the decisionmaking process with exploratory buying behavior, lead to the formation of a theoretical framework for OSL driven online buying. They have made use of the generic purchase decision model as a framework to measure Internet-domain exploratory shopping behavior. The method of Analysis used was Pearson's correlation. The results exhibit that the weakest connection with exploratory buying behavior across the spread of information acquisition to purchase behavior, whereas consumer innovativeness revealed the highest correlation with impulsive buying behavior.

Research Gap

As digital marketing is the new normal, consumer buying behavior has drastically evolved over the period of time. Even though previous studies concentrate on Internet, mobile and E-Mail marketing, they have failed to touch upon social media marketing. Therefore, this study covers the influence of social media marketing over consumer's buying behavior.

Objectives

-To find out the relationship between components of digital marketing and EBBT.

-To analyze the key factors digital marketing and EBBT.

Methodology

Primary data has been collected through structured questionnaire from 110 respondents using convenience sampling technique. The data has been collected through on online survey having questions measured on five-point Likert scale ranging from strongly agree to strongly disagree. The questionnaire consists of three parts. Part-A has four statements on specific demographic variables. Part -B has 24 statements on digital marketing and 10 statements on EBBT. Part-C has questions on general demographic variables. Correlation and factor analysis have been used for analyzing data.

Results and discussions

To measure the reliability and validity of the questionnaire, Cronbach's alpha, Kaiser-Meyer-Olkin (KMO) and Bartlett test were performed. The value of 0.869 for Cronbach's alpha suggests consistency of the items in the scale. The KMO measures was 0.807, which signifies that the scales of the variables in the questionnaire is understood by all the respondents and they have correctly answered. Additionally, Bartlett's test of for sphericity has a high chi-square value which justifies the sample suitability.

Sample profile:

The table below describes the demographic profile of the respondents with respect to gender, age, educational qualification and monthly income of 110 respondents. It is inferred from the table 2 that most of the respondents are female (57.3%). Majority (84.5%) of the respondents belong to the age group of 22-30 years and 7.3% of them belong to the age group of 30-40. 49.1% of the respondents have completed their UG and 37.3% have completed their PG. 43% of the respondents have their monthly income Rs.20,000 and 20.9 % have their monthly income from Rs 40,000 to Rs.60,000.

Finding of objective 1

In order to identify the relationship between digital marketing and consumers' exploratory buying behavior tendencies (EBBT), Factor analysis of e-marketing with EBBT was examined using five factors: Internet marketing (IM), E-mail marketing (EM), mobile marketing (MM) and social media marketing (SM). The Table 3,4,5 and 6 depicts that there exists a relationship between drivers of digital marketing and dimensions of EBBT. Out of the above four factors, Social Media Marketing is highly correlated with EBBT followed by E-mail marketing, internet marketing and mobile marketing.

Findings of objective 2:

Conclusion

To begin with, internet marketing is crucial as India has the second largest number of internet users after China. Mobile marketing is the next major component as the smart phone penetration is increasing day by day. Social media marketing is more attractive as people tend to spend much of their online time on the social media platforms. Taking all these factors into account, the study shows that social media marketing has a tremendous impact in consumers buying behavior. Thus, companies who are already in the market or new companies entering the market can explore marketing options in various social media platform like Instagram, Facebook, WhatsApp, Twitter, LinkedIn, Pinterest and even YouTube.

As consumers move to these platforms for networking, it makes easy for them to shop as AI in these sites are easily able to identify the consumers' needs through their search results. Even though, internet shopping offers the same, consumers are directly able to interact with the sellers which is not the case in most of the internet sites. Simultaneously, the consumers are opting for detailed information of the products rather than sensory features. Therefore, consumers look out for information of various products before buying them.

Bibliography

Baumgartner, H., Steenkamp, J.E.M .: Exploratory consumer buying behavior: Conceptualization and measurement. Intern. J. of Research in Marketing, 13, 121-137(1996).

  Waheed, A., Jianhua, Y.: Achieving consumers' attention through emerging technologies: The linkage between e-marketing and consumers' exploratory buying behavior tendencies. Baltic Journal of Management, 13(2), 209-235(2018). Gañac, C.G.: Investigating Consumer Optimum Stimulation Level and Exploratory Online Buying Behavior. DLSU Business & Economics Review, 28(1), 67-85(2018).

Table 1 .

 1 Table showing reliability and KMO analysis

	RELIABILITY STATISTICS
	Cronbach's Alpha	N
	0.868	6
	Kaiser-Meyer-Olkin (KMO) and Bartlett test

Table 2 .

 2 Table showing demographic variables

	S.NO	DEMOGRAPHIC VARIABLES	% OF RESPONDENTS
	General demographic
	variables:
		Gender:
	1	Male Female	41.80% 57.30%
		Prefer not to say	0.90%
		Age:
		20 years -30 years	84.50%
	2	30 years -40 years	7.30%
		40 years-50 years	5.50%
		Above 50 years	2.70%
		Educational status:
		UG	49.10%
	3	PG	37.30%
		M.Phil./Research	2.70%
		Professional qualification	10.90%
		Income:
		Less than Rs 20,000	42.70%
	4	Rs 20,000 -Rs 40,000 Rs 40,000 -Rs 60,000	27.30% 20.90%
		Rs 60,000 -Rs 80,000	5.50%
		More than Rs 80,000	3.60%

Table 3 .

 3 Correlation Matrix between Internet Marketing and EBBT

Table 4 .

 4 Correlation Matrix between Mobile Marketing and EBBT

Table 5 .

 5 Correlation Matrix between E-Mail and EBBT

Table 6 .

 6 Correlation Matrix between Social Media Marketing and EBBT

Table 7 .

 7 Total Variance Explained

	Component	Initial Eigenvalues	Extraction Sums of Squared Loadings	Rotation Sums of Squared Loadings
		Total	% of Variance Cumulative %	Total	% of Variance Cumulative %	Total	% of Variance Cumulative %
	1	10.46 30.759 30.759 10.46 30.759 30.759 5.14 15.122 15.122
	2	3.17 9.324	40.083		3.17 9.324	40.083 4.81 14.133 29.256
	3	2.445 7.192	47.275 2.445 7.192	47.275	3.8 11.189 40.445
	4	2.228 6.554	53.829 2.228 6.554	53.829 2.88 8.456	48.901
	5	1.727 5.081	58.909 1.727 5.081	58.909 1.92 5.66	54.561

Table 8 .

 8 Component Matrix a

					Component
		1	2	3	4	5	6	7	8
	SM Q4	0.757				0.318
	SM Q2	0.729				0.362
	SM Q3	0.705	0.407
	MM Q3	0.701			0.387
	EM Q2	0.689	-0.56
	EM Q1	0.678	-0.568
	EM Q5	0.661	-0.552
	SM Q6	0.66					-0.336
	SM Q5	0.655	0.447
	EM Q4	0.654	-0.307
	EM Q3	0.649	-0.517
	IM Q3	0.641			-0.432
	EM Q6	0.633	-0.511
	MM Q1	0.631
	SM Q1	0.628	0.39
	MM Q4	0.615			0.519
	MM Q5	0.581			0.576
	MM Q6	0.564			0.477
	IM Q6	0.549				-0.352 -0.336
	MM Q2	0.542		0.32	0.41
	IM Q4	0.537					0.379
	IM Q2	0.536	0.333		-0.336
	IM Q1	0.529	0.385			-0.337	0.322
	EBBT Q8	0.515				0.312			0.482
	IM Q5	0.462	0.358 -0.322		-0.322
	EBBT Q3	0.414			0.309
	EBBT Q7	0.37		-0.597
	EBBT Q10			0.576			0.475
	EBBT Q4	0.324		-0.559
	EBBT Q6			-0.483		0.462		0.41
	EBBT Q5			0.438
	EBBT Q9		0.371 0.374			0.395	-0.332
	EBBT Q2			0.412				0.476	-0.304
	EBBT Q1			-0.348		0.322		-0.372	0.367
		Extraction Method: Principal Component Analysis.
			a 7 components extracted.

Table 8 .

 8 Rotated Component Matrix aThe Eigen value table has been sub divided into three sections i.e. initial Eigen value, extracted sum of square loadings and rotated sum of squared loadings. The factor accounts for 69.72% of the variance extractions sum of squared loadings shows the number of rows in the panel of the table corresponding to the number of factors retained. Rotations sum of squared loadings represents the distribution of the variance after the varimax rotation. Factors are rotated for easier interpretation. The idea of rotation is to reduce the number of factors of which the variable under investigation have high loading. Exploratory Factor analysis results show that all the statements are accepted i.e. Social media motivates consumers to interact with others for information sharing has the lowest value of 0.314. The results of exploratory factor analysis reveal that Social media advertising motivates buying tendencies is 0.852 has a significant impact over exploratory buying behavior. Results also shows that digital marketing has an impact Over exploratory buying behavior tendencies.

						Component
		1	2	3	4	5	6	7	8
	EM Q2	0.884
	EM Q1	0.843
	EM Q5	0.839
	EM Q3	0.82
	EM Q6	0.807
	EM Q4	0.651
	SM Q2		0.852
	SM Q3		0.846
	SM Q5		0.788
	SM Q4	0.314	0.757
	SM Q1		0.742
	SM Q6		0.613	0.446
	MM Q5			0.815
	MM Q4			0.789
	MM Q6			0.727
	MM Q2			0.686
	MM Q3	0.314	0.318	0.652
	MM Q1			0.512 0.468
	IM Q1				0.743
	IM Q4	0.389			0.643
	IM Q3	0.398	0.396		0.57
	IM Q5				0.557		-0.405
	IM Q2		0.469		0.475	-0.326
	EBBT Q4					0.649
	EBBT Q3					0.584
	EBBT Q5					-0.496	0.359
	EBBT Q10						0.841
	EBBT Q9						0.695
	IM Q6		0.328		0.394		-0.413
	EBBT Q6							0.796
	EBBT Q8							0.736
	EBBT Q2								-0.736
	EBBT Q1				-0.342				0.601
	EBBT Q7				0.322	0.496			0.498
		Extraction Method: Principal Component Analysis.
		Rotation Method: Varimax with Kaiser Normalization.
			a. Rotation converged in 10 iterations.


Federica Garin
email: federica.garin@inria.fr

Generic Delay-L Left Invertibility of Structured Systems

Keywords: Network systems, Structured systems, Structural systems, Structural observability, Linear time-invariant systems, Smith-McMillan form, Left invertibility, Unknown input, Delayed observers, Cyber-physical security

This paper studies structured systems, namely linear systems where the state-space matrices have zeros in some fixed positions and free parameters in all other entries. This paper focuses on time-invariant systems in discrete time affected by an unknown input, and their delay-L left invertibility, namely the possibility to reconstruct the input sequence from the output sequence, assuming that the initial state is known, and requiring that the inputs can be reconstructed up to L time steps before the current output. Building upon classical results on linear systems theory and on structured systems, a graphical characterization is obtained of the integers L for which a structured system is generically delay-L left invertible.

INTRODUCTION

The study of generic properties of structured systems has been an active research area since the 70's and has received a wide recent attention within the multi-disciplinary community studying complex network systems.

A structured system is a linear system whose state-space matrices have a fixed pattern of zeros (representing known lack of interaction between some states), and the other entries are free parameters. The zero pattern can be equivalently described with a directed graph, where edges represent non-zero entries. The goal is to find graphical conditions ensuring that some system-theoretic property is true generically, where generically (or for almost all parameters) means for all parameters except possibly those lying in a proper subvariety of the parameter space. Since a proper subvariety has zero Lebesgue measure, there is the following probabilistic interpretation: given a property that is generically true, if the parameters are chosen at random (according to any continuous distribution) the probability of the property being true is one.

The early results on structured systems, initiated by the seminal paper [START_REF] Lin | Structural controllability[END_REF], were devoted to controllability and observability. A rich literature was then developed, which is well summarized in the book [START_REF] Murota | Matrices and Matroids for Systems Analysis[END_REF] and in the survey papers [START_REF] Dion | Generic properties and control of linear structured systems: a survey[END_REF] and [START_REF] Ramos | Structural systems theory: an overview of the last 15 years[END_REF].

This paper focuses on structured systems with an unknown input. Such input may represent a fault, an un-modeled part of the system, or a malicious external attack. The latter interpretation has brought significant attention to systems with an unknown input in the research area of cyber-physical security. In particular, papers [START_REF] Pasqualetti | Attack detection and identification in cyber-physical systems[END_REF] and [START_REF] Weerakkody | A graph-theoretic characterization of perfect attackability for secure design of distributed control systems[END_REF] study perfect attacks, namely attacks that can go completely unde-tected, since they produce the same output as a legitimate trajectory without input; extensions to near-perfect attacks are proposed in [START_REF] Mo | False data injection attacks in control systems[END_REF], to include attacks producing small albeit non-zero residuals in the attack detector. In [START_REF] Pasqualetti | Attack detection and identification in cyber-physical systems[END_REF], assuming that the initial state is known, the authors characterize perfect attackability as the lack of left invertibility, and then they study generic left invertibility of structured systems, including an extension to descriptor systems. Instead, [START_REF] Weerakkody | A graph-theoretic characterization of perfect attackability for secure design of distributed control systems[END_REF] focuses on the optimization problem related to sensor placement, where the goal is to ensure left invertibility with the smallest number of dedicated sensors, each measuring one local state, and furthermore minimizing the communication between sensors.

When the initial state is not known or in the presence of noise, the notion of left invertibility should be accompanied by the one of strong observability, namely the possibility to reconstruct the initial state from the output measurements, despite the presence of the unknown input. This is useful to construct Luenberger-like observers or Kalmanlike filters, suitably adapted to cope with the unknown input, as in [START_REF] Sundaram | Delayed observers for linear systems with unknown inputs[END_REF], [START_REF] Yong | A unified filter for simultaneous input and state estimation of linear discrete-time stochastic systems[END_REF] or [START_REF] Garin | Unbiased filtering for state and unknown input with delay[END_REF].

The simultaneous occurrence of the two properties of strong observability and left invertibility has been studied under the names of state-and-input observability or inputand-state observability. For structured systems, generic input-and-state observability has been characterized in [START_REF] Boukhobza | State and input observability for structured linear systems: A graph-theoretic approach[END_REF]; see also Weerakkody et al. (2017a) and [START_REF] Garin | Structural delay-1 input-and-state observability[END_REF] for a simpler rephrasing of their characterization and for further results, in Weerakkody et al. (2017a) about optimal sensor placement, and in [START_REF] Garin | Structural delay-1 input-and-state observability[END_REF] about the additional property of delay-1 left invertibility, discussed below.

Above-mentioned works on generic state-and-input observability (except Garin (2017)) do not take into account the delay of the left invertibility: For which L can we reconstruct inputs u(0), . . . , u(k -L) from the initial state x(0) and measurements y(0), . . . , y(k)?

This notion of delay is crucial when one implements observers for recursively reconstructing the input and the state, as highlighted in [START_REF] Sundaram | Delayed observers for linear systems with unknown inputs[END_REF]. Classical studies by Massey and Sain have found an algebraic condition characterizing delay-L left invertibility as the rank of a matrix involving the matrices of the state-space representation of the system [START_REF] Massey | Inverses of linear sequential circuits[END_REF]. Also see [START_REF] Sain | Invertibility of linear timeinvariant dynamical systems[END_REF] for a thourough discussion of the counterpart of delay-L left invertibility for continuous-time systems.

Results about generic delay-L left invertibility have only focused on delay 1, with a characterization in [START_REF] Garin | Structural delay-1 input-and-state observability[END_REF] for linear time-invariant systems. A series of papers summarized in the thesis [START_REF] Gracy | Input and State Observability of Linear Network Systems with Application to Security of Cyber Physical Systems[END_REF] have studied various aspects of delay-1 left invertibility together with strong observability.

The novelty of this paper is the study of generic delay-L left invertibility for structured systems, for any given delay L. Such problem was first addressed in [START_REF] Garin | Generic delay-L left invertibility of structured systems with scalar unknown input[END_REF], where a graphical characterization was obtained for the particular case where the input is scalar. The current paper solves the problem in the most general case. It builds upon classical results in linear systems theory, together with the literature on the generic structure at infinity of structured systems, to obtain a simple graphical condition which characterizes the values of L for which the structured system is generically left invertible. This result can also be used in combination with the characterization of generic input-and-state observability [START_REF] Boukhobza | State and input observability for structured linear systems: A graph-theoretic approach[END_REF], Weerakkody et al. (2017a), [START_REF] Garin | Structural delay-1 input-and-state observability[END_REF]), to obtain the graphical conditions under which the system can generically have a delay-L observer as in [START_REF] Sundaram | Delayed observers for linear systems with unknown inputs[END_REF]. This paper is organized as follows. In Section 2 we recall the definition of delay-L left invertibility, and its classical characterization in terms of the rank of suitable matrices involving the (A, B, C, D) matrices of the state-space representation of the system. In Section 3 we show an alternative characterization, involving the Smith-McMillan form at infinity of the system's transfer function. Then, in Section 4 we recall results on the generic zero-orders at infinity of structured systems; such results, together with the ones in Section 3, give us the desired graphical condition that characterizes the values of L for which the structured system is generically left invertible (Theorem 7). Section 5 illustrates the results in an example.

DELAY-L LEFT INVERTIBILITY, INHERENT DELAY

Here we recall the definition and the classical algebraic characterization of delay-L left invertibility and of the inherent delay for discrete-time linear time-invariant (LTI) systems, from [START_REF] Massey | Inverses of linear sequential circuits[END_REF]; see [START_REF] Sain | Invertibility of linear timeinvariant dynamical systems[END_REF] for the continuous-time interpretation of the same.

Consider the discrete-time LTI system

x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k) (1)
where x(k) ∈ R n is the state, u(k) ∈ R p is the unknown input, and y(k) ∈ R m is the output. Definition 1. For an integer L ≥ 0, the system (1) is delay-L left invertible if the initial unknown input u( 0) is uniquely determined by the initial state x(0) and the output sequence y(0), . . . , y(L).

The system (1) is left invertible if there exists an integer L ≥ 0 for which it is delay-L left invertible.

The inherent delay of system (1), hereby denoted L in , is the smallest integer L ≥ 0 such that the system is delay-L left invertible.

Notice that delay-L left invertibility implies delay-K left invertibility for all K ≥ L (by definition). Hence, a system is delay-L left invertible if and only if L ≥ L in . For this reason, we will focus our attention on studying the inherent delay.

The characterization of delay-L left invertibility from [START_REF] Massey | Inverses of linear sequential circuits[END_REF] requires to define

M L =     J 0 0 0 . . . 0 J 1 J 0 0 . . . 0 . . . . . . . . . . . . J L J L-1 J L-2 . . . J 0     ,
where J 0 = D and J = CA -1 B for all ≥ 1 are the Markov parameters of the system.

The key remark is that

M L    u(0) . . . u(L)    =    y(0) . . . y(L)    -   C CA . . . CA L-1   x(0) .
For this reason, with the above notation, together with the definition rank M -1 = 0, delay-L left invertibility is characterized as follows.

Proposition 1. [START_REF] Massey | Inverses of linear sequential circuits[END_REF], Thm. 4). For any integer L ≥ 0, the system (1) is delay-L left invertible if and only if rank

M L -rank M L-1 = p.
This immediately implies the following characterization of the inherent delay. Proposition 2. The inherent delay L in of system (1) is the smallest integer

L ≥ 0 such that rank M L -rank M L-1 = p.
The inherent delay L in is infinite if the system is not left invertible, and otherwise it is at most n, as shown in [START_REF] Massey | Inverses of linear sequential circuits[END_REF], Coroll. 2. This implies that the system (1) is left invertible if and only if rank M nrank M n-1 = p.

INHERENT DELAY AND SMITH-MCMILLAN FORM AT INFINITY

The characterization described in Sect. 2 of (delay-L) left invertibility and of the inherent delay is based on the state-space description of the system. However, it is also useful to study characterizations involving the transfer function

G(z) = C(zI -A) -1 B + D.
Left invertibility is equivalent to the fact that G(z) is a leftinvertible rational matrix, which explains the name 'left invertibility' (and also the name 'TFM left invertibility' used in some literature for the same notion, where 'TFM' stands for 'Transfer Function Matrix'). This equivalence, stated below as Prop. 3, can be found for example in the book [START_REF] Trentelman | Control Theory for Linear Systems[END_REF], where it is proved for the continuous-time case, but the same proof argument can be easily applied also to the discrete-time case. Proposition 3. [START_REF] Trentelman | Control Theory for Linear Systems[END_REF] It turns out that also inherent delay can be determined with properties of the transfer function G(z), as we will show in the remainder of this section with the use of the Smith-McMillan form at infinity. We only consider proper transfer functions, which correspond to causal state-space representations as in (1).

We recall here the well-known Smith-McMillan factorization at infinity of a rational function G(z), which corresponds to the Smith-McMillan factorization at zero of H(z) defined as H(z) = G(z -1 ), see e.g. [START_REF] Kailath | Linear Systems[END_REF], Chapter 6. Given a proper rational function G(z), there exist two biproper 1 matrices U (z) and V (z) such that

G(z) = U (z)Λ(z)V (z) with Λ(z) =         z -n1 z -n2 . . . z -nr 0 0 0        
where n 1 , . . . n r are integers, 0

≤ n 1 ≤ • • • ≤ n r .
This factorization is not unique for what concerns U (z) and V (z), but Λ(z) is unique, and is called the Smith-McMillan form at infinity of G(z). The non-negative integers n 1 , . . . , n r are called the zero-orders at infinity of G(z).

We will use the results in [START_REF] Van Dooren | On the determination of the Smith-Macmillan form of a rational matrix from its Laurent expansion[END_REF], where a characterization of the zero-orders at infinity is obtained from the coefficients of the Laurent series at infinity, i.e., the Laurent series of H(z) = G(z -1 ) at zero.

Since (zI -A) -1 = z -1 h≥0 (z -1 A) h , the Laurent series at infinity of G(z) is the following:

G(z) = D + h≥1 CA h-1 B z -h , i.e.,
1 a biproper rational matrix is a square rational matrix which is proper, invertible, and whose inverse is proper

G(z) = h≥0 J h z -h , (2)
where J h 's are the Markov parameters defined in Sect. 2.

The results in [START_REF] Van Dooren | On the determination of the Smith-Macmillan form of a rational matrix from its Laurent expansion[END_REF] involve block-Toeplitz matrices T i 's defined as follows:

T i =     J 0 J 1 J 2 . . . J i 0 J 0 J 1 . . . J i-1 . . . . . . . . . . . . 0 0 0 . . . J 0     ,
where matrices J h 's are the ones involved in the Laurent expansion at infinity (2), and hence they are the Markov parameters of the system.

The property stated below as Prop. 4 reveals the relation between the zero-orders at infinity and the rank indexes, where the ith rank index is defined as rank T i -rank T i-1 . Such property is proved in [START_REF] Van Dooren | On the determination of the Smith-Macmillan form of a rational matrix from its Laurent expansion[END_REF], where it is used as the main tool for an efficient and stable algorithm to compute the Smith-McMillan form. More specifically, [START_REF] Van Dooren | On the determination of the Smith-Macmillan form of a rational matrix from its Laurent expansion[END_REF] gives this property in Sect. III for the Smith-McMillan form at a finite α, and then Sect. IV explains how to extend this result to α = ∞ by studying H(z) = G(z -1 ) at α = 0. Proposition 4. [START_REF] Van Dooren | On the determination of the Smith-Macmillan form of a rational matrix from its Laurent expansion[END_REF]). Given a proper rational matrix G(z), denote by r its normal rank and by n 1 , . . . , n r its zero-orders at infinity and consider the above-defined block-Toeplitz matrices T i 's, letting rank T -1 = 0.

For any integer i ≥ 0, the ith rank index is equal to the number of zero-orders whose value is less or equal to i, i.e., rank T i -rank T i-1 = # j ∈ {1, . . . , r} s.t. n j ≤ i .

Since n 1 ≤ • • • ≤ n r , the expression for the rank index in Prop. 4 can be equivalently re-written as follows:

if i < n 1 , then rank T i -rank T i-1 = 0; if i ≥ n 1 , then rank T i -rank T i-1 = max j ∈ {1, . . . , r} s.t. n j ≤ i .
It is clear that matrices T i involved in Prop. 4 and matrices M i involved in Prop. 1 are similar, since M i can be obtained from T i by permuting blocks of columns and then blocks of rows. Hence, rank T i = rank M i . For this reason, from Prop. 4, together with the characterization of delay-L left invertibility in Prop. 1, we easily obtain the following result.

Proposition 5. The inherent delay of system (1), with transfer function G(z), is infinite if the normal rank of G(z) is smaller than p, and else is equal to n p , its largest zero-order at infinity.

Proof. By Prop. 1, together with the remark rank T j = rank M j , the inherent delay is the smallest integer L such that rank T L -rank T L-1 = p. From Prop. 4, we obtain that rank T L -rank T L-1 < r for all L < n r and rank T Lrank T L-1 = r for all L ≥ r, where r is the normal rank of G(z). This ends the proof of the claim.

GENERIC INHERENT DELAY OF A STRUCTURED SYSTEM

A structured system is a family of LTI systems sharing a same imposed zero-pattern for their A, B, C, D matrices, while all entries that are not fixed to zero are free parameters, that is, distinct real-valued parameters, which can be chosen arbitrarily. A structured system is usually represented by a directed graph (digraph) G, see e.g. [START_REF] Dion | Generic properties and control of linear structured systems: a survey[END_REF]. The vertex set of G is U ∪ X ∪ Y , where U = {u 1 , . . . , u p } is the set of input vertices, X = {x 1 , . . . , x n } is the set of state vertices, and Y = {y 1 , . . . , y m } is the set of output vertices. Edges of G correspond to the entries of matrices A, B, C, D that are free parameters: a parameter a ij in position (i, j) in matrix A corresponds to an edge (x j , x i ), representing the influence of state x j (k) on the state x i (k+1); a parameter b ij in position (i, j) in matrix B corresponds to an edge (u j , x i ); a parameter c ij in position (i, j) in matrix C corresponds to an edge (x j , y i ); and a parameter d ij in position (i, j) in matrix D corresponds to an edge (u j , y i ). For an example, see Sect. 5.

Given a structured system, we say that some property is true generically if it is true for all systems in the family, except possibly for some systems that correspond to a proper subvariety of the parameters space. The goal of this paper is to find graphical conditions on G that characterize whether or not the corresponding structured system is generically delay-L left invertible. The main results will follow from merging the system-theoretic properties described in Sections 2 and 3 with results from the literature on structured systems, as described below. Such results will involve the so-called linkings, defined as follows.

Definition 2. A U -Y linking in G is a collection of vertexdisjoint paths in G, each originated in a vertex in U and ending in a vertex in Y . The size of the linking is the number of paths.

The two results given below in Prop. 6 are stated in the survey paper [START_REF] Dion | Generic properties and control of linear structured systems: a survey[END_REF], see Theorem 2. The first result, about left invertibility, has been proved in van der Woude (1991a) and the second one, about the zero-orders at infinity, has been independently proved in [START_REF] Commault | Disturbance rejection for structured systems[END_REF] and [START_REF] Van Der Woude | On the structure at infinity of a structured system[END_REF]. All such proofs were given for the case without direct feedthrough of the input to the output, i.e., D = 0. However, Theorem 2 in [START_REF] Dion | Generic properties and control of linear structured systems: a survey[END_REF] is stated for a system as in (1), and a careful examination of the proofs in van der Woude (1991a) and [START_REF] Commault | Disturbance rejection for structured systems[END_REF] confirms this result, since all proof arguments remain valid also in the presence of D. Also, above-cited results are given for continuoustime systems, but they are based on the transfer function, which is the same in the discrete-time case. Proposition 6. [START_REF] Dion | Generic properties and control of linear structured systems: a survey[END_REF], Thm. 2). Given a structured system with digraph G, let G(z) = C(zI -A) -1 B+D be the corresponding transfer function. Denote by s the largest size of a U -Y linking in G. For h = 1, . . . , s, denote by α h the minimum number of X-vertices in a U -Y linking of size h in G, and set α 0 = 0. Then

(1) Generically, G(z) has normal rank equal to s.

(2) Generically, the zero-orders at infinity of G(z) are n 1 , . . . , n s given by n h = α h -α h-1 .

From Prop. 6, together with Prop. 5 from Sect. 3, we immediately obtain the following characterization of generic inherent delay, which is the main result of this paper. Theorem 7. Consider a structured system with digraph G.

Denote by s the largest size of a U -Y linking in G. For h ∈ {1, . . . , s}, denote by α h the minimum number of Xvertices in a U -Y linking of size h in G, and set α 0 = 0. Generically, the inherent delay of the structured system is

L in = ∞ if s < p and L in = α p -α p-1 if s = p.
This means that, if s = p, then the structured system is generically delay-L left invertible if and only if L ≥ α pα p-1 . Remark 8. The statements of Proposition 6 and Theorem 7 do not discuss what happens for those particular parameters for which the properties are different from the generic case.

Concerning item (1) in Proposition 6, the following result is true: for any choice of parameters, the normal rank of G(z) is smaller than or equal to s. Indeed, Theorem 1 in van der Woude (1991a) shows that the normal rank can be smaller than the maximum normal rank only over a proper subvariety; equivalenty, the generic normal rank is equal to the maximum normal rank, no choice of parameters allows to exceed the generic normal rank.

The above remark on the normal rank implies the following one on left invertibility: if s < p, then the system is not left invertible (equivalently, the inherent delay is infinite), for any choice of the parameters.

Concerning the inherent delay when s = p, instead, no conclusion can be driven about its value for some particular parameters: the inherent delay can be smaller or larger than the generic one, as we will show in an example in Section 5.

A final remark is about the computational complexity. As discussed in [START_REF] Dion | Generic properties and control of linear structured systems: a survey[END_REF], Section 10, the computations of s and α i 's involved in the statements of Prop. 6 and Thm. 7 can be cast as optimization problems for which there is a rich literature of polynomial-time algorithms (maximum flow, and minimum-cost flow, respectively).

EXAMPLE

In this section, we give an example of structured system, to illustrate the definitions and results given in Sect. 4, in particular Theorem 7 and Remark 8.

Consider the structured system with the following matrices:

A =     
0 a 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 43 0 0 0 0 a 53 a 54 0

     , B =      0 b 12 b 21 0 0 b 32 0 0 0 0      , C = c 11 0 c 13 0 0 0 0 0 c 24 c 25 , D = 0 0 0 0 .
The corresponding digraph G is depicted in Figure 1.

We want to apply Theorem 7 to find the generic inherent delay of this structured system. In this example, the size of the input is p = 2. minimum number of X-vertices, formed by the two paths (u 1 , x 2 ), (x 2 , x 1 ), (x 1 , y 1 ), and (u 2 , x 3 ), (x 3 , x 5 ), (x 5 , y 2 ).

u 1 u 2

x 1

x 2

x 3

x 4 y 1 y 2 x 5 Fig. 3. Red edges represent an U -Y linking of size 1 with minimum number of X-vertices: the path (u 2 , x 3 ), (x 3 , y 1 ).

We can see in Figure 2 that in G there exists an U -Y linking of size two, i.e., there are two vertex-disjoints paths, each originated in a vertex in U and ending in a vertex in Y . Clearly, no U -Y linking can have a size larger than the size of U , and hence the maximum size of a U -Y linking in this example is s = p = 2. The linking in Figure 2 has four X-vertices (x 1 , x 2 , x 3 and x 5 ) and this is the smallest number of X-vertices among all U -Y linkings of size two. Indeed, the only other U -Y linkings of size two have the same first path, and the second path which is either (u 2 , x 3 ), (x 3 , x 4 ), (x 4 , y 2 ), which has the same number of X-vertices, or (u 2 , x 3 ), (x 3 , x 4 ), (x 4 , x 5 ), (x 5 , y 2 ), which has more X-vertices. Hence, in this case α 2 = 4.

We then compute α 1 = 1 by looking at Figure 3, which shows the U -Y linking of size 1 (i.e., the path from some u in U to some y in Y ) with the smallest number of Xvertices.

With s = p = 2, α p = α 2 = 4 and α p-1 = α 1 = 1, by Theorem 7 this structured system has generic delay α p -α p-1 = 3.

We can also find the inherent delay by using the results from Sect. 2, involving matrices M 0 , . . . , M 5 , in order to explore some cases with non-generic parameters (param-eters in some proper subvariety, for which the results are different from the generic ones). Simple computations give

M 5 =                  
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 β 0 0 α 0 0 0 0 0 0 0 0 0 γ 0 0 0 0 0 0 0 0 0 0 0 0 β 0 0 α 0 0 0 0 0 0 0 δ 0 γ 0 0 0 0 0 0 0 0 0 0 0 0 β 0 0 α 0 0 0 0 0 0 0 δ 0 γ 0 0 0 0 0 0 0 0 0 0 0 0 β 0 0 α 0 0 0 0 0 0 0 δ 0 γ 0 0 0 0 Matrices M i for i = 0, 1, 2, 3, 4 are the square submatrices of M 4 obtained by restricting the attention to first 2(i + 1) rows and 2(i + 1) columns.

                  , with α
It is then easy to compute the rank of such matrices, from which the inherent delay can be obtained by finding the smallest L such that rank M L -rank M L-1 = 2. Different results will be obtained, for different parameters.

Generically, α, β, γ and δ are non-zero, and hence rank M 0 = 0, rank M 1 = 1, rank M 2 = 2, rank M 3 = 4, rank M 4 = 6 and rank M 5 = 8. This confirms that the inherent delay is generically equal to 3, as obtained above by applying Theorem 7.

We can now look at some particular parameters, for which the inherent delay is different from the generic one.

First, for some parameters, we have α = 0 and non-zero β, γ, δ, so that rank M 0 = 0, rank M 1 = 0, rank M 2 = 2, rank M 3 = 4, rank M 4 = 6 and rank M 5 = 8. Hence, for such parameters the inherent delay is equal to 2, which is smaller than the generic one.

For some other parameters, we have γ = 0 and non-zero α, β, δ, so that rank M 0 = 0, rank M 1 = 1, rank M 2 = 2, rank M 3 = 3, rank M 4 = 5 and rank M 5 = 7, which gives an inherent delay of 4, larger than the generic one.

There are also some parameters for which the system is not left invertible (the inherent delay is infinite), for example when γ = δ = 0 and α, β are non-zero, which gives rank M 0 = 0, rank M 1 = 1, rank M 2 = 2, rank M 3 = 3, rank M 4 = 4 and rank M 5 = 5. Recall that n = 5 and that a system is left-invertible if and only if rank M nrank M n-1 = p, which fails in this case.

CONCLUSION

In this paper, we have studied discrete-time linear timeinvariant structured systems with an unknown input, and their delay-L invertibility. Exploiting results from linear systems theory and from structured systems theory, we have given a graphical condition that characterizes the generic inherent delay, namely the non-negative integer L in such that the structured system is generically delay-L left invertible if and only if L ≥ L in (or infinite if no such finite integer exists). We have also presented an example that illustrates how some particular parameters can exist such that the inherent delay is smaller than the generic one, and some others such that the inherent delay is larger than the generic one.

Fig. 2 .

 2 Fig. 1. Digraph G of the structured system in Section 5.

  = c 11 b 12 + c 13 b 32 , β = c 11 a 12 b 21 , γ = c 24 a 43 b 32 + c 25 a 53 b 32 and δ = c 25 a 54 a 43 b 32 .

ACKNOWLEDGEMENTS

The author wishes to thank Alain Kibangou and Sebin Gracy for earlier joint work that motivated this study. This research has been partially supported by the Agence Nationale de la Recherche (ANR) via grant "Hybrid And Networked Dynamical sYstems" (HANDY), number ANR-18-CE40-0010.


François Osiurak
email: francois.osiurak@univ-lyon2.fr

Nicolas Claidière

Alexandre Bluet

Joël Brogniart

Salomé Lasserre

Timothé Bonhoure

Laura Di Rollo

Néo Gorry

Yohann Polette

Alix Saude

Giovanni Federico

Natalie Uomini

Emanuelle Reynaud

Technical reasoning bolsters cumulative technological culture through convergent transformations

come   L'archive ouverte pluridisciplinaire

Introduction

Today technology pervades human life, and it is often taken for granted that technology and science progressively become more advanced and more refined through time. Yet, the origin of this capacity remains a fascinating mystery. Other primates sometimes use tools [e.g., chimpanzees [START_REF] Mcgrew | Chimpanzee material culture: Implications for human evolution[END_REF], capuchins [START_REF] Visalberghi | Tool use in Cebus[END_REF], orangutans (3)] and have sometimes been shown to be doing so for very long periods of time [START_REF] Mercader | Excavation of a chimpanzee stone tool site in the African rainforest[END_REF]. There is also evidence that, like humans, nonhuman primates learn to use tools through social learning, by the observation of tool-using conspecifics [START_REF] Whiten | Cultures in chimpanzees[END_REF]. However, human technology is strikingly different because non-human primate tool use does not evolve and does not gradually become more efficient and more complex through time [START_REF] Tomasello | Cultural learning[END_REF]. The term Cumulative Technological Culture (CTC) has been coined to refer to the progressive increase in the complexity and/or efficiency of tools and techniques that are too complex to be invented by a single individual [START_REF] Boyd | The cultural niche: Why social learning is essential for human adaptation[END_REF][START_REF] Dean | Identification of the social and cognitive processes underlying human cumulative culture[END_REF][START_REF] Whiten | Cultural evolution in animals[END_REF]. Interestingly, there is increasing evidence that some non-technological aspects of animal culture can cumulatively evolve [START_REF] Sasaki | Cumulative culture can emerge from collective intelligence in animal groups[END_REF][START_REF] Garland | When does cultural evolution become cumulative culture? A case study of humpback whale song[END_REF][START_REF] Williams | Evidence for cumulative cultural evolution in bird song[END_REF], raising the question of how and why technology became cumulative in humans and not in other primates.

CTC has been considered to be driven by two engines: High-fidelity transmission, the similarity between tools and techniques across episodes of transmission [also called episodic fidelity, see [START_REF] Charbonneau | Understanding cultural fidelity[END_REF]], and innovation [START_REF] Legare | Imitation and innovation: The dual engines of cultural learning[END_REF]. The crucial role of high-fidelity transmission in CTC has been repeatedly stressed with the rationale that, when an innovation appears, it will quickly be lost if it cannot be faithfully transmitted to others [START_REF] Boyd | The cultural niche: Why social learning is essential for human adaptation[END_REF][START_REF] Henrich | The secret of our success: How culture is driving human evolution, domesticating our species, and making us smarter[END_REF][START_REF] Lewis | Transmission fidelity is the key to the build-up of cumulative culture[END_REF][START_REF] Muthukrishna | Innovation in the collective brain[END_REF][START_REF] Tennie | Ratcheting up the ratchet: On the evolution of cumulative culture[END_REF][START_REF] Tomasello | The human adaptation for culture[END_REF]. High-fidelitytransmission has often been assumed to result from unique social-cognitive skills that allow humans to imitate or "infocopy" [ [START_REF] Tomasello | Cultural learning[END_REF][START_REF] Tennie | Ratcheting up the ratchet: On the evolution of cumulative culture[END_REF][START_REF] Henrich | The evolution of prestige: Freely conferred deference as a mechanism for enhancing the benefits of cultural transmission[END_REF][START_REF] Herrmann | Humans have evolved specialized skills of social cognition: The cultural intelligence hypothesis[END_REF]; also called high-fidelity copying or propensity fidelity, see [START_REF] Charbonneau | Understanding cultural fidelity[END_REF]]. Following this view, the role of causal reasoning in CTC has often been minimized by assuming that complex technologies result from the accumulation of many often poorly understood improvements made over generations [START_REF] Boyd | The cultural niche: Why social learning is essential for human adaptation[END_REF][START_REF] Derex | Causal understanding is not necessary for the improvement of culturally evolving technology[END_REF][START_REF] Harris | The role of causal knowledge in the evolution of traditional technology[END_REF] combined with rare intentional improvements achieved through causal reasoning [START_REF] Henrich | Demography and Cultural Evolution: How adaptive cultural processes can produce maladaptive losses: The Tasmanian case[END_REF]. Although the term causal reasoning is widely employed in the literature, we have stressed [START_REF] Osiurak | The elephant in the room: What matters cognitively in cumulative technological culture[END_REF] that the term technical reasoning may be more appropriate in the field of CTC because it refers to a specific form of causaland analogicalreasoning directed toward the physical world. Thus, the term technical reasoning will be hereafter used to refer to this specific kind of reasoning.

Importantly, the similarity observed between tools or techniques in CTC needs not come from the existence of copying mechanisms [START_REF] Sperber | Explaining culture: A naturalistic approach[END_REF][START_REF] Sperber | An objection to the memetic approach to culture in Darwinizing culture: The status of memetics as a science[END_REF]. An alternative explanation, the cultural attraction theory [START_REF] Sperber | Explaining culture: A naturalistic approach[END_REF][START_REF] Scott-Phillips | Four misunderstandings about cultural attraction[END_REF][START_REF] Claidière | How Darwinian is cultural evolution? Phil[END_REF], is that humans are endowed with cognitive mechanisms that are not specifically designed for copying but that transform and adapt what is socially learned to the ends and characteristics of the learning individual. Under such a view, highfidelity transmission is achieved through convergent transformations, that is, individuals with the same goal will give rise to similar cultural products through shared cognitive skills, goals, and environments [START_REF] Charbonneau | All innovations are equal, but some more than others: (Re)integrating modification processes to the origins of cumulative culture[END_REF][START_REF] Claidière | Convergent transformation and selection in cultural evolution[END_REF]. This view has recently received support from theoretical, experimental, and field studies on cultural evolution in humans [START_REF] Morin | How portraits turned their eyes upon us: Visual preferences and demographic change in cultural evolution[END_REF][START_REF] Morgan | Biased transformation erases traditions sustained by conformist transmission[END_REF][START_REF] Miton | Universal cognitive mechanisms explain the cultural success of bloodletting[END_REF][START_REF] Miton | Motor constraints influence cultural evolution of rhythm[END_REF][START_REF] Gandon | Copying errors of potters from three cultures: Predictable directions for a so-called random phenomenon[END_REF][START_REF] Nyhof | Spreading non-natural concepts: The role of intuitive conceptual structures in memory and transmission of cultural materials[END_REF] and non-human primates [START_REF] Claidière | Convergent transformation and selection in cultural evolution[END_REF][START_REF] Claidière | Cultural evolution of systematically structured behaviour in a non-human primate[END_REF] and is also in line with studies on language evolution [START_REF] Kirby | Innateness and culture in the evolution of language[END_REF]. For instance, arguments transmitted along transmission chains could become degraded and then fully reconstructed through deductive reasoning [START_REF] Claidière | Argumentation and the diffusion of counter-intuitive beliefs[END_REF]. With respect to CTC, the cultural attraction theory translates into the fact that individuals use technical-reasoning skills to solve complex technological problems and that high-fidelity transmission is the result of convergent transformations.

To experimentally examine the role of convergent transformations versus copying mechanisms in CTC, we used a recently developed task that was aimed at providing the sort of complexity encountered in CTC. Derex et al. [START_REF] Derex | Causal understanding is not necessary for the improvement of culturally evolving technology[END_REF] reported a micro-society study designed to investigate the role of causal understanding in CTC. The task consisted of optimizing a wheel system (Fig. 1A). This task was multidimensional because the speed of the wheel depended on its moment of inertia (i.e., a wheel with its four weights close to its center is faster than a wheel with its four weights farther from the center) and the position of its center of mass (i.e., the initial acceleration is better when the wheel is unbalanced with the center of the mass located ahead and above the axis of the wheel; fig. S1). The participants performed the task as members of chains of five participants (Fig. 1B). The experimenter transmitted to each participant (except those of the first generation) the weight configurations and their associated speeds from the last two trials of the previous participant (hereafter called Configurations+Speed condition). Participants' understanding of the wheel system was assessed with an understanding test consisting of selecting the fastest wheel configuration out of several presented. The results indicated that the wheel system became progressively optimized, and although Derex et al. [START_REF] Derex | Causal understanding is not necessary for the improvement of culturally evolving technology[END_REF] found no improvement in understanding, we performed a partial replication (41) motivated by some methodological issues and demonstrated that the improvement was linked to the participants' understanding of the technology (i.e., their understanding increased through time, they were able to transfer this understanding to other problems, and they were better than controls).

The findings reported in both Derex et al. [START_REF] Derex | Causal understanding is not necessary for the improvement of culturally evolving technology[END_REF] and our prior study [START_REF] Osiurak | Technical reasoning is important for cumulative technological culture[END_REF] gave limited insight into the process through which participants arrived at better configurations because the participants had access to all the information they needed (wheel configurations and associated speeds) to reproduce with high fidelity the wheel systems produced by their predecessors. Nevertheless, our findings [START_REF] Osiurak | Technical reasoning is important for cumulative technological culture[END_REF] indicated that even when individuals had access to all the information needed to blindly copy, they formed a causal representation of the physical system that they used to produce improvements. The role of technical reasoning may appear redundant here but could reveal its full potential if the information transmitted between individuals is incomplete or incorrect [START_REF] Osiurak | The elephant in the room: What matters cognitively in cumulative technological culture[END_REF]. Partial information combined with technical-reasoning skills might allow an individual to converge toward the same technical solution or another technical solution that maintainsor even improvesthe technology [START_REF] Osiurak | The elephant in the room: What matters cognitively in cumulative technological culture[END_REF].

The present study aimed to test this possibility through two micro-society conditions, in which participants had five trials to improve a wheel system, as described above (14 chains of five participants in each condition; Fig. 1, A andB). After their five trials, the participants' understanding of the physical system was assessed with an understanding test (twelve items for center of mass and twelve for inertia). In the Speed-Only condition, the only information that was transmitted to the next participant was the last two wheel speeds of the previous participant in the chain (i.e., no information at all about the configurations associated; Fig. 1C). In the Configurations+Speed+Noise condition, the participants were given two weight configurations and their associated speeds. The configurations came from the previous participant in the chain but were modified by randomly changing the position of the four weights closer to or further from the center of the wheel (Fig. 1C). Therefore, the participants had access to partial or degraded information in both conditions. If copying is crucial for CTC and technical reasoning is non-necessary, no improvement of the physical system should be observed in both conditions, nor should an increase in understanding be found. By contrast, if technical reasoning is important for completing partial or biased information, then an improvement of the physical system should be observed in both conditions, and it should be accompanied by an increase in understanding.

Results and Discussion

The results supported the second prediction. The wheel speed increased over

2B).

A link was also found in both conditions between the wheel speed (the best speed of the last two trials) and the total understanding score (Speed-Only: Wheel-speed estimate, 0.55; 95% CI: 0.39 to 0.73; Configurations+Speed+Noise: Wheel-speed estimate, 0.65; 95% CI: 0.46 to 0.84; Fig. 3B). To examine convergence, we computed an intra-generation similarity score, which reflected the similarity between the wheel configurations of the participants of the same generation. We found that this intra-generation similarity score increased over generations within both conditions but not between conditions (Speed-Only: Generation estimate, 1.41; 95% CI: 0.72 to 2.10; Configurations+Speed+Noise: Generation estimate, 1.75; 95% CI: 1.25 to 2.18; Between: Generation estimate, 0.67; 95% CI: -0.14 to 1.36; Fig. 3C). Taken together, these results indicate that high-fidelity transmission can arise through an increased understanding of the task that allow individuals to converge to optimal solutions [for similar results in a Configurations+Speed condition, see [START_REF] Osiurak | Technical reasoning is important for cumulative technological culture[END_REF]; see also fig. S3].

A careful examination of the evolution of wheel speed over the 25 trials (Fig. 2A) provides additional information about the dynamics of the cognitive processes involved in each condition. In the Configurations+Speed+Noise condition, the participants in the secondto-fifth generations maintained the wheel speed from their first trials at the same level as that of the last trials of their predecessors. This pattern, which is similar to that of the Configurations+Speed condition of Osiurak et al. [(41); fig. S4; see also [START_REF] Derex | Causal understanding is not necessary for the improvement of culturally evolving technology[END_REF]], suggests that the participants tended to usebut also to be canalized by [ [START_REF] Derex | Causal understanding is not necessary for the improvement of culturally evolving technology[END_REF][START_REF] Osiurak | Technical reasoning is important for cumulative technological culture[END_REF] see also [START_REF] Bonawitz | The double-edged sword of pedagogy: Instruction limits spontaneous exploration and discovery[END_REF][START_REF] Wood | Copy me or copy you? The effect of prior experience on social learning[END_REF][START_REF] Wood | Does a peer model's task proficiency influence children's solution choice and innovation?[END_REF]]social information to begin to form a causal representation of what could be an effective physical system. The early impact of social information on the formation of this causal representation is confirmed by the presence of a link between the total understanding score and the wheel speed (the best speed of the first two trials; Wheel-speed estimate, 0.25; 95% CI: 0.01 to 0.45;

Fig. 3A).

By contrast, as shown in Fig. 2A, the participants in the Speed-Only condition seemed to reinvent the wheel at each generation, with the participants in the second-to-fifth generations obtaining a very low performance in their first trial, close to that of the first trial of participants in the first generation. Nevertheless, over their five trials, the participants in the second-to-fifth generations tended to improve the physical system systematically and progressively, leading them to outperform their predecessors. This outcome suggests that the participants in this condition benefited from the social information provided by the speed of previous wheels by comparing it with the performance based on their initial causal representation of the wheel system. Thus, the greater the gap between the performance of their predecessor and their initial performance, the more the participants attempted to modify and thus enhancetheir causal representation of the physical system, thereby allowing them to improve dramatically their wheel system over their own trials. Support for this interpretation comes from the absence of a link in this condition between the total understanding score and the wheel speed, when the best speed of the first two trials is considered (Wheel-speed estimate, 0.16; 95% CI: -0.05 to 0.36; Fig. 3A).

The inertia dimension had a greater impact on the wheel speed than the center of mass dimension. Therefore, the considerable increase in wheel speed over the five trials implied that the participants in the Speed-Only condition explored further the inertia dimension and preferentially enhanced their understanding of this dimension. To investigate this aspect, we computed two exploration scores, one for the inertia dimension and the other for the centerof-mass dimension, which reflected the exploration of each dimension over the five trials. We found that the inertia exploration score increased over generations in the Speed-Only condition (Generation estimate, 3.63; 95% CI: 1.69 to 5.49; Fig. 3E) in parallel with the participants' inertia understanding score (Generation estimate, 2.28; 95% CI: 0.42 to 4.13; Fig. 2D). In broad terms, the mere availability of information about the predecessor's performance was enough to orient the participants toward a technical-reasoning-based exploration [reasoned trial and error [START_REF] Osiurak | The elephant in the room: What matters cognitively in cumulative technological culture[END_REF][START_REF] Vaesen | The cognitive bases of human tool use[END_REF]].

The pattern in the Configurations+Speed+Noise condition was different, with no significant effect of generations on the inertia exploration (Generation estimate, -0.81; 95% CI: -2.49 to 0.81; Fig. 3E) and understanding scores [Generation estimate, 0.26; 95% CI: -0.98 to 1.56; Fig. 2D; for similar results in a Configurations+Speed condition, see [START_REF] Osiurak | Technical reasoning is important for cumulative technological culture[END_REF]]. By contrast, there was an increase in the participants' center-of-mass understanding score (Generation estimate, 1.59; 95% CI: 0.79 to 2.50; Fig. 2C). This increase was not found in the Speed-Only condition (Generation estimate, -0.24; 95% CI: -1.18 to 0.75; Fig. 2C).

Interestingly, the increase in the center-of-mass score reported in the Configurations+Speed+Noise condition was accompanied by a decrease in the center-of-mass exploration score over generations (Generation estimate, -4.88; 95% CI: -8.05 to -1.53; Fig. 3D). The random noise added before transmission fortuitously and almost systematically led to the generation of unbalanced wheel configurations (table S1), which was critical to observe the effect of the position of the wheel's center of mass on its speed. Random noise, along with the canalizing effect associated with the transmission of wheel configurations, seems to have directed the participants' attention to the center-of-mass dimension, leading them to improve their understanding of this dimension. These findings highlight how the introduction of random modifications can favor the understanding of often poorly understood dimensions, which can in turn lead to specific innovations. The presence of a decrease in the center-ofmass exploration score suggests that these innovations did not result from lucky errors or occasional experiments but from the direct contribution of random modifications introduced along with the canalizing effect. This is telling with respect to our hypotheses because if participants were using a strategy in which they introduced random modifications and selected the best outcomes, (1) participants in both conditions would mostly explore the center-of-mass dimension (favored by random modifications) and (2) transmission chains in the two conditions would converge towards the same outcome (but see Fig. 3C).

The present study extends previous findings, which have questioned the crucial role of copying mechanisms [START_REF] Acerbi | Culture without copying or selection[END_REF][START_REF] Caldwell | Social learning mechanisms and cumulative cultural evolution: Is imitation necessary?[END_REF] and/or emphasized the importance of causal understanding/inductive biases [START_REF] Griffiths | Theoretical and empirical evidence for the impact of inductive biases on cultural evolution[END_REF][START_REF] Thompson | Human biases limit cumulative innovation[END_REF][START_REF] Pinker | The cognitive niche: Coevolution of intelligence, sociality, and language[END_REF] and innovative skills in CTC [START_REF] Whiten | Cultural panthropology[END_REF][START_REF] Vale | Why do chimpanzees have diverse behavioral repertoires yet lack more complex cultures? Invention and social information use in a cumulative task[END_REF]. In this experiment, participants with very little or randomly transformed social information managed to improve and converge towards similar outcomes than participants with complete information [ [START_REF] Charbonneau | Understanding cultural fidelity[END_REF][START_REF] Charbonneau | Fidelity and the grain problem in cultural evolution[END_REF] see also [START_REF] Acerbi | If we are all cultural Darwinians what's the fuss about? Clarifying recent disagreements in the field of cultural evolution[END_REF][START_REF] Strachan | Evaluating the relative contributions of copying and reconstruction processes in cultural transmission episodes[END_REF]]. This is remarkable and demonstrate the importance of technical reasoning in producing CTC: Technical reasoning is a reconstruction mechanism that allows us to recover from partial or degraded information obtained through social learning and therefore guarantees the high-fidelity transmission of advantageous technologies (i.e., the high similarity between the wheel configurations). Said differently, technical reasoning can be viewed as a potential cognitive mediator of CTC that allows individuals to filter information acquired either through their own experience (asocial learning) or through social learning, by extracting relevant information and rejecting irrelevant information, irrespective of the origin of this information [START_REF] Osiurak | The elephant in the room: What matters cognitively in cumulative technological culture[END_REF]. Technical reasoning might participate in both the innovative component and the high-fidelity component of CTC, thus implying that the distinction between these two components might be of convenience rather than of cognitive distinctness.

The use of micro-society paradigms has provided significant insights into the origin of CTC. As stressed by Miton and Charbonneau [START_REF] Miton | Cumulative culture in the laboratory: Methodological and theoretical challenges[END_REF], participants in micro-society paradigms "are adept inventors, capable of innovating in a matter of minutes" (p. 4). Consistent with this fact, our participants were able to improve a multidimensional technology, which is not obviously intuitive as shown by the results, as well as the understanding of it in only about 20 min and five trials. Showing that technical reasoning can play a reconstruction role in such paradigms can also renew the question of how technologies have evolved in early hominins.

For instance, the Oldowan industrial complex emerged at around 2.6 million years ago. This industry comprises mainly sharp-edged flakes and the cores from which they were removed [START_REF] Toth | The oldowan reassessed: A close look at early stone artifacts[END_REF]. There is no clear tradition within the Oldowan prior to 2 million years ago [START_REF] Stout | Technological variation in the earliest Oldowan from Gona, Afar, Ethiopia[END_REF]. The Acheulean industry, which appeared around 1.75 million years ago, corresponds to bifacially shape stone tools used as handaxes and cleavers. This industry is characterized by a striking homogeneity, which persisted for around 1.5 million years [START_REF] Beyene | The characteristics and chronology of the earliest Acheulean at Konso, Ethiopia[END_REF][START_REF] Shipton | Generativity, hierarchical action and recursion in the technology of the Acheulean to Middle Palaeolithic transition: A perspective from Patpara, the Son Valley, India[END_REF]. This shift may reflect the emergence of high-fidelity transmission in our lineage. Several proposals have been made for interpreting this shift at a cognitive level, with a particular focus on social cognitive skills [START_REF] Shipton | Before cumulative culture: The evolutionary origins of overimitation and shared intentionality[END_REF][START_REF] Tennie | Early stone tools and cultural transmission: Resetting the null hypothesis[END_REF]. The findings reported here provide an alternative interpretation in suggesting that technical reasoning could have also contributed to the emergence of high-fidelity transmission. Of course, this interpretation must be taken with caution because other aspects can also be fundamental to maintain the stability of technologies over long periods of time, such as theperhaps presupposedlesser cost of copying compared to understanding the technical behavior of conspecifics and the interaction of this learning cost with environmental conditions [START_REF] Boyd | The cultural niche: Why social learning is essential for human adaptation[END_REF][START_REF] Giuliano | Understanding cultural persistence and change[END_REF], the different social-learning strategies [e.g., prestige bias, conformity bias [START_REF] Henrich | The evolution of prestige: Freely conferred deference as a mechanism for enhancing the benefits of cultural transmission[END_REF][START_REF] Henrich | The evolution of cultural evolution[END_REF]], or the superiority of some social-learning conditions over others in the transmission process [e.g., superiority of communication over reverse engineering [START_REF] Morgan | Experimental evidence for the coevolution of hominin tool-making teaching and language[END_REF][START_REF] Osiurak | Physical intelligence does matter to cumulative technological culture[END_REF]]. Regardless, our results show that CTC and the high-fidelity transmission of technology between individuals should not be systematically interpreted as evidence of cognitive mechanisms capable of copying coupled to the random retention of useful modifications.

Materials and Methods

Ethics

The Ethics Committee of the University of Lyon Department of Psychology approved the study, and the procedure was carried out in accordance with the ethical standards of the 1964 Declaration of Helsinki. Informed consent was obtained from all participants after the nature and possible consequences of the study were explained [see [START_REF] Osiurak | Technical reasoning is important for cumulative technological culture[END_REF]].

Participants

One hundred and forty-six students at the University of Lyon took part in the study (M age = 20.84, SD age = 2.91; 102 women). The participants were non-selectively recruited through advertisements posted on social media websites [see [START_REF] Osiurak | Technical reasoning is important for cumulative technological culture[END_REF]].

Experimental apparatus

The wheel system used in the present study was the same as the one used by Osiurak et al.

[ [START_REF] Osiurak | Technical reasoning is important for cumulative technological culture[END_REF]; for an illustration, see https://osf.io/m3d7q/]. A description is provided in fig. S5.

Procedure

The procedure was basically the same as in [START_REF] Osiurak | Technical reasoning is important for cumulative technological culture[END_REF]. The experiment took place in an experimental room at the University of Lyon (around 30 min in duration). The participants sat at a table placed 2 m from the experimental apparatus. Before the experiment, the participants completed a consent form. After the experiment, they indicated whether they had an academic background in engineering or physics.

Experimental design

Building phase. This phase was similar in both conditions. Instructions were similar to those of Osiurak et al. [ [START_REF] Osiurak | Technical reasoning is important for cumulative technological culture[END_REF]; see https://osf.io/m3d7q/]. The participants had five trials to optimize the speed of a wheel that descended a 1-m-long inclined track. They could move four weights to any of 12 discrete positions along each spoke and were free to choose their own configuration (from 1 to 12, with 1 being the closest position to the center of the wheel and 12 the furthest position from the center of the wheel). After the participants used a marker pen to indicate the positions of the four weights on the wheel (i.e., a paper version of the configuration), the experimenter positioned the weights on the physical wheel accordingly.

The participants were not allowed to move the weights on the physical wheel themselves in order to prevent damage due to potential repeated awkward manipulations. The wheel also needed to be placed correctly in its initial position and the release had to be accomplished without any abrupt movements in order to avoid a modification of the trajectory of the wheel.

Nevertheless, the participants could scrutinize the experimenter moving the weights and releasing the wheel as well as the wheel descending the track. The time it took the wheel to travel down the track was automatically recorded by a computer program (see https://osf.io/m3d7q/). The wheel speed and the associated configuration were then displayed to the participants, who had as much time as they needed to consult their last two configurations and choose the next one. As explained above, we used a paper-and-pencil method to display the wheel speeds and the associated configurations (see https://osf.io/m3d7q/). After three trials, the experimenter reminded the participants in the Speed-Only condition that the wheel speeds of their last two trials would be transmitted to the next participant in the chain. In the Configurations+Speed+Noise condition, the experimenter reminded the participants that their last two configurations and the associated speeds would be transmitted to the next participant in the chain. In this condition, a random noise was introduced by the experimenter into the wheel configurations before transmission, by moving the four weights six positions closer to or farther from the center of the wheel (i.e., the absolute sum of the modifications equaled 6). Thus, if the configuration of the wheel of a participant on their fourth trial was, for instance, [Position Top Weight : 9; Position Front Weight : 9; Position Bottom Weight : 5; Position Back Weight : 5], a random modification of six positions was applied (e.g., +1;-3;+2;0), which modified the configuration of the wheel (i.e., Position Top Weight : 9+1=10; Position Front Weight : 9-3=6; Position Bottom Weight : 5+2=7; Position Back Weight : 5+0=5]. The configurations thus modified and their associated speeds were then transmitted by the experimenter to the next participant. The computer program used to generate these random positions is available at https://osf.io/m3d7q/. The introduction of this random noise frequently generated wheels that did not descend (i.e., speed of 0 m h -1 ). Sometimes, the program could generate two configurations with null speed for the same participant. To ensure that the participants in the second-to-fifth generations received at least one configuration with a wheel that descended, we reran the program until we obtained a wheel with a non-null speed for the second configuration when the speed associated with the first configuration was already null. There were 28 chains of five participants each (i.e., 14 chains in the Speed-Only condition and 14 chains in the Configurations+Speed+Noise).

Testing phase. In both conditions, the participants completed this phase after the building phase [see [START_REF] Osiurak | Technical reasoning is important for cumulative technological culture[END_REF]]. They were instructed that they would be presented with items consisting of four wheels and that they would have to choose which of the four wheels would roll down the rails faster in their opinion. They could take as much time as they needed to complete the test.

They received no feedback. All the participants saw the same items in the same order. The understanding test consisted of 24 items (i.e., 12 inertia items and 12 centre of mass items).

The test is available at https://osf.io/m3d7q/. Finally, the participants had to write a brief theory (i.e., less than 340 characters long) about the functioning of the wheel system, which always started with "The wheel covers the distance faster when…" [for a similar procedure, see [START_REF] Derex | Causal understanding is not necessary for the improvement of culturally evolving technology[END_REF]]. The data collected about these theories are not discussed in the present report.

Statistical analyses

One participant in the Speed-Only condition received an incorrect speed from the previous participant because of an experimental error. The data of this participant were removed, and the participant was replaced by a new participant. We checked for the presence of outliers for each condition separately. As our key predictions concerned the increase in wheel speed over generations, we explored whether some chains behaved differently from the others on this aspect. To do so, we computed the slope associated with each chain with x being the position of the participant in the chain and y the best speed of their last two trials. For each condition, we obtained 14 slopes. We considered as outliers the chains with a slope that did not fall within two standard deviations from the mean. This procedure led us to remove one chain of five participants in the Speed-Only condition, which we replaced with a new chain of five participants. No chain was removed for the Configurations+Speed+Noise condition. In total, the data of six participants were excluded, giving us a final sample of 140 participants (14 chains of five participants for each condition).

Wheel speed corresponded to the best speed of the first or last two trials. Wheels that did not travel down were assigned a speed of 0 m h -1 . To compute the intra-generation similarity score, we first selected the configuration of the wheel that was fastest among the last two trials of each participant. Then, for the within-condition similarity score, we compared this best configuration with the 13 best configurations produced by the other participants of the same generation, and we did so for each participant of each generation. The comparison was based on the sum of absolute differences of the positions of the four weights between two configurations. The positions varied from 1 to 12. Therefore, the maximum absolute differences could be 44 [i.e., (Position 12 -Position 1 )  4 weights]. The sum of absolute differences reflected the dissimilarity between the two configurations. Therefore, we subtracted this sum from 44 to obtain the similarity score. The procedure was the same for the between-condition similarity score except that we compared the best configuration of each participant with the 14 best configurations produced by the participants of the same generation in the other condition. More detail about this intra-generation similarity score is given at https://osf.io/m3d7q/ [for a similar procedure, see [START_REF] Osiurak | The pedagogue, the engineer, and the friend: From whom do we learn?[END_REF][START_REF] Reindl | Young children fail to generate an additive ratchet effect in an open-ended construction task[END_REF]]. The inertia exploration score corresponded to the difference between the smallest and the greatest sum of positions of the four weights on the five trials. The greater this difference, the greater the exploration. The center-of-mass exploration score corresponded to the surface of the convex envelope that contained the centers-of-mass coordinates of the five wheels. For each wheel, the coordinates of its center of mass were computed by subtracting the position of the bottom weight from that of the top weight (x-coordinate) and the position of the back weight from that of the front weight (y-coordinate). The greater this surface, the greater the exploration. More detail about these two exploration scores is provided at https://osf.io/m3d7q/.

In both conditions, we first explored the wheel speed over generations. Wheel speed corresponded here to the best speed of the last two trials. Wheels that did not travel down were assigned a speed of 0 m h -1 . We used regression modelling in R [(69); lmerTest package [START_REF] Kuznetsova | lmerTest Package: Tests in linear mixed effects models[END_REF]] to fit a linear model with 'Wheel speed' as outcome variable, 'Generation' as fixed effect, and 'Chain's identity' as random effect. The same analyses were conducted for the understanding scores (Total, Center of Mass, and Inertia), the intra-generation similarity score (Within-condition and Between-condition), the exploration scores (Center of Mass and Inertia) and the best speed of the first two trials (Two participants in the Speed-Only condition and four participants in the Configurations+Speed+Noise condition were excluded from this analysis because their first two wheels did not descend). We also used regression modelling in R [(69); lmerTest package [START_REF] Kuznetsova | lmerTest Package: Tests in linear mixed effects models[END_REF]] to explore the links between the understanding scores (Total) and the best speed of the first or last two trials. We fitted a linear mixed model with 'Understand Score (Total)' as outcome variable, 'Wheel speed (the best speed of the first or last two trials)' as fixed effect, and 'Generation' and 'Chain's identity' as random effects.

Statistical significance was set at p < .05 and bootstrapping method was used to estimate 95% confidence intervals.

  generations in both the Speed-Only condition [Generation estimate, 3.03 m h -1 ; 95% Confidence Interval (CI): 1.28 to 4.73] and the Configurations+Speed+Noise condition (Generation estimate, 1.77 m h -1 ; 95% CI: 0.53 to 2.96; Fig. 2A, fig. S2) in parallel to the participants' total understanding score (Speed-Only: Generation estimate, 2.04; 95% CI: 0.35 to 3.88; Configurations+Speed+Noise: Generation estimate, 1.85; 95% CI: 0.40 to 3.05; Fig.

Fig. 1 .

 1 Fig. 1. Experimental task and design. A. The task consisted of minimizing the time it took a wheel to travel down an inclined track. The wheel had four radial spokes. On each spoke, a weight could be moved closer to or further from the center of the wheel on 12 positions. B. The participants performed the task as members of chains of five participants. Each of the participants had five trials to improve the wheel system by modifying the wheel configuration. The experimenter transmitted to each participant (except those of the first generation) the information about the last two trials (grey) of the previous participant. After the five trials, the participants'understanding of the wheel system was assessed with an understanding test (12 center-of-mass items and 12 inertia items). C. In the Speed-Only condition, only the wheel speeds of the last two trials (grey) were transmitted to the next participant. In the Configurations+Speed+Noise condition, the participants were given

Fig. 2 .

 2 Fig. 2. Parallel improvement of the wheel system and of its understanding. A. Wheel speed over generations

Fig. 3 .

 3 Fig. 3. Links between the wheel speed and the understanding scores and increase of intra-generation

Center-of-mass items only; D: Inertia items only). The performance obtained by a control group is also shown [see[START_REF] Osiurak | Technical reasoning is important for cumulative technological culture[END_REF]; Mean: Yellow; Standard Error: Blue]. Error bars indicate standard errors.

Funding

This work was supported by grants from the French National Research Agency (ANR) project TECHNITION (ANR-21-CE28-0023-01; FO, NC, NU and ER) and was performed within the framework of the LA E C RTE of Universit de Lyon (ANR-11-LABX-0042; FO and ER), within the program "Investissements d'Avenir" (ANR-11-IDEX-0007; FO and ER) operated by ANR. NU was supported by the Max Planck Society and a grant from the Templeton World Charity Foundation (https://www.templetonworldcharity.org/) #0271. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of Templeton World Charity Foundation, Inc.

Data and materials availability

The data that support the findings of this study and the codes used in this study are available at https://osf.io/m3d7q/. All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials.

Author contributions

Conceptualization

Competing interests

Authors declare that they have no competing interests.

Supplementary Materials

Figs. S1 to S5