Mohamed Amine Khelassi

Execution time budget assignment for mixed criticality systems

Mohamed Amine Khelassi, Yasmina Abdeddaïm Univ Gustave Eiffel, CNRS, LIGM, F-77454 Marne-la-Vallée, France

Abstract-In this paper we propose to quantify execution time variability of programs using statistical dispersion parameters. We show how the execution time variability can be exploited in mixed criticality real-time systems. We propose a heuristic to compute the execution time budget to be allocated to each low criticality real-time task according to its execution time variability. We show using experiments and simulations that the proposed heuristic reduces the probability of exceeding the allocated budget compared to algorithms which do not take into account the execution time variability parameter.

Index Terms-Mixed criticality systems, execution time variability, statistical dispersion parameters.

I. INTRODUCTION

In a mixed criticality system, programs of different criticality are executed on the same processor. The challenge is that low criticality tasks do not disturb the good functioning of the high criticality ones. In real-time scheduling, since the original Vestal's model [START_REF] Vestal | Preemptive scheduling of multi-criticality systems with varying degrees of execution time assurance[END_REF], a classical model has emerged, see [START_REF] Burns | A survey of research into mixed criticality systems[END_REF] for a complete survey. In this model, tasks have several execution times budgets, one budget per possible criticality. If a task does not signal its termination after the execution of its allocated budget at a certain criticality level, the system moves to the next criticality level. In every system criticality level, only tasks of criticality equal or higher to the criticality of the system have to respect their deadlines. Therefore, low criticality tasks can be suspended to allow higher criticality tasks to meet their deadlines. Much research in real-time mixed criticality scheduling assumes that the execution time budgets in every criticality level are provided and focus on the challenge of maximizing the execution of low criticality tasks while guaranteeing that all high criticality tasks always meet their deadlines. The execution time budgets are often defined as estimates of the worst-case execution time at different certification requirements but few studies have focused on how to determine these budgets.

In this paper our aim is to determine the execution time budget of low criticality tasks so that all the tasks always respects their deadlines. If a job does not complete within its assigned budget, the job is suspended. We propose a heuristic to determine the execution time budgets, this heuristic uses an estimation of the variability of execution time of the task to determine the budgets. Many definitions have been proposed for the quantification of execution time variability, in [START_REF] Hughes | Variability in the execution of multimedia applications and implications for architecture[END_REF] and [START_REF] Koch | Neuroflight: Next generation flight control firmware[END_REF] the execution time variability is quantified as the ratio between run-time measurements and either the best, worst or mean execution time. In [START_REF] Nélis | The Variability of Application Execution Times on a Multi-Core Platform[END_REF], [START_REF] Bin | Controlling execution time variability using cots for safetycritical systems[END_REF], the execution time variability is considered to be the factor between the worst-case execution time computed in isolation and along with other workload. These approaches give a quantification that does not give information on how the execution times are distributed as we would like to have. In this work we use statistical dispersion parameters to quantify the execution time variability of a task. The idea is to characterize the variability of execution times more accurately, so that we can use it to determine the budgets to be allocated to tasks. Compared to probabilistic approaches [START_REF] Davis | A survey of probabilistic timing analysis techniques for real-time systems[END_REF] that rely on the computation of the probabilistic response time to determine if the probability of a deadline miss respects a given threshold, our objective is to use an estimate of the variability of execution times without using the complete distribution. Indeed, the methods that use the full distribution to compute the probabilistic response time have a high complexity for exact methods or have to make assumptions on the shape of the distributions for analytical methods. The contributions of the paper are: [START_REF] Vestal | Preemptive scheduling of multi-criticality systems with varying degrees of execution time assurance[END_REF] We propose a definition of execution time variability and a method for its quantification using statistical dispersion parameters, [START_REF] Burns | A survey of research into mixed criticality systems[END_REF] We propose a heuristic that uses the execution time variability to solve the scheduling problem of a mixed criticality system, (3) We evaluate our approach using simulations and benchmarks executed on an ARM-Cortex A53.

II. RELATED WORK

Concerning the use of statistical dispersion parameters in real-time systems, in [START_REF] Alcon | Timing of autonomous driving software: Problem analysis and prospects for future solutions[END_REF], [START_REF] Reghenzani | Timing predictability in high-performance computing with probabilistic real-time[END_REF] the authors use statistical dispersion parameters to measure the variability in the experimentation part to compare different execution time distributions of different modules of a system. The difference with our proposition is that in our work statistical dispersion parameters are part of the task model and not a measure used to estimate the efficiency of an approach. Decreasing the computation times of LO-criticality tasks in order to avoid sacrificing all LO criticality tasks when the system is in high criticality has been considered in [START_REF] Burns | Towards a more practical model for mixed criticality systems[END_REF]- [START_REF] Gu | Dynamic budget management with service guarantees for mixed-criticality systems[END_REF]. In these papers, a new model is used for mixed criticality systems in which the budget of low criticality tasks is smaller when the system is in high criticality than their budget when the system is in low criticality. Our approach differs mainly in the following aspects. Firstly, in our model, all non-critical tasks have a single budget assigned to them. This budget has a practical significance and is linked to the distribution of task execution times. Secondly, our aim is to propose an approach that is agnostic to the scheduling algorithm used. Thirdly, the only monitoring we need to perform during execution is to stop jobs if they do not terminate at their assigned budget.

Our model has some similarities with the approaches using probabilistic tasks models for mixed criticality systems [START_REF] Singh | Mixed criticality scheduling of probabilistic real-time systems[END_REF]- [START_REF] Draskovic | Schedulability of probabilistic mixed-criticality systems[END_REF] as we also consider that we have a distribution of execution times of the tasks. However, we don't use the distribution to compute the probability of deadlines misses, but we use it to guide us in the choice of budgets to assign to tasks so that deadlines are always met.

III. TIME VARIABILITY QUANTIFICATION

In this section we present our method to quantify the execution time variability of a program P . Let S(P ) be a set of execution times of program P . This set can be generated for example using repeated executions of the program in a specific configuration of the processor and/or using estimates of execution times for example in the worst and/or in the best case, however the way this set is computed is out of the scope of this paper. We note C P the random variable taken its values in S(P ) with p(C P = x) being the probability of occurrence of x in S(P ). Note that we do not make any assumption concerning the distribution function of this random variable.

We define the execution time variability as being quantified by a statistical dispersion parameter. In Definition 1 we deliberately do not specify which statistical dispersion parameter to use as we want to remain flexible in the choice of the parameter to be used.

Definition 1 (Execution time variability): The execution time variability of a program P is defined by T V a statistical dispersion parameter of C P the execution time random variable of program P .

Dispersion parameters are used in statistics to measure the tendency of the values of a distribution to be scattered on either side of a value. For example the coefficient of variation measures the dispersion of a set of data around the mean. The higher the value of the coefficient of variation, the greater the dispersion of the values around the mean. The skewness parameter [START_REF] Hippel | Skewness[END_REF] describes which side of the distribution has a longer tail. For unimodal distributions, symmetric distributions should have a skewness value near zero, negative skewness values means that data are more shifted towards the maximum value, positive skewness values means that data are more shifted towards the minimum value of the distribution [START_REF]Standards and Technology, Measures of Skewness and Kurtosis[END_REF].

In this work we propose our own dispersion parameter in Definition 2. Our goal is to provide a dispersion parameter that is not sensitive to the type of distribution the random variable follows. This parameter is inspired from the classical coefficient of variation. In our case, it measures the dispersion of a set of data around the largest value instead of the mean.

Definition 2 (Coefficient of variation to the maximum): The coefficient of variation to the maximum V W CET of X a random variable with n occurrences {X 1 , . . . X n } is:

V W CET = n i=1 (Xi-M ) 2 n M * 100 with M = max 1≤i≤n X i (1)
Given that the VWCET is based on the sum of deviations between the WCET and the other data values, the smaller it is the less likely the data is far away from the WCET.

To illustrate the VWCET and skewness parameters, Figure 1 represents histograms of execution times (in terms of number of cycles) of three sorting programs, merge sort, quick sort and insert sort of the Mälardalen [START_REF] Gustafsson | The Mälardalen WCET benchmarks -past, present and future[END_REF] benchmark. The execution times of the sorting algorithms were measured in a bare-metal environment using the Zynq UltraScale+ MPSoC ZCU104 Evaluation Kit. For every algorithm we use the same input and all the algorithms are executed in the same configuration of the processor i.e. the caches are disabled and one core executes the sorting program and all other cores execute the same program. We can see that the shape of the histograms are are left-biased, centered, or right-biased. For every program we compute the VWCET and skewness statistical parameters.

We can see in Table I that the the sKw value is close to zero for quick sort, negative for insert sort and positive for merge sort confirming the shapes of Figure 1. The V W CET value of merge sort is the larger, then follows the V W CET value of insert sort and the V W CET quick sort is the smallest one.

IV. MIXED CRITICALITY BUDGET ASSIGNMENT PROBLEM

In this section we show how execution time variability can be exploited in the real-time scheduling problem of mixed criticality systems. The goal is to define the execution time budgets to be assigned to tasks in such a way that: (1) the real-time task set is schedulable, i.e. all the tasks meet their deadlines whatever their criticality (2) critical tasks terminate at the latest at their allocated budget and (3) less critical tasks maximize the instances where they terminate the latest at their allocated budget. We consider that if a job of a task does not terminate after the execution of its assigned budget, the job is stopped. In our model the jobs of the less critical tasks stopped only if during the execution they exceed their budget, otherwise they are executed and are guaranteed to meet their deadlines. We consider a task set Γ of n mixed criticality realtime tasks executed using a scheduling algorithm Sched. We do not specify whether the execution platform is uniprocessor or multiprocessor, however the scheduling algorithm must be sustainable with respect to task execution times, i.e. if the task set is schedulable using algorithm Sched, the task set is still schedulable using algorithm Sched if the task execution times are smaller. Each task τ i ∈ Γ is a tuple

(Budget i , T V i , L i , D i , T i ) with:
• Budget i : set of possible budgets to be assigned to τ i with p i (b) is the probability that the budget b is not exceeded

• T V i : execution time variability of τ i • L i ∈ {LO, HI}: τ i criticality, LO less critical than HI • D i ∈ N * : relative deadline of τ i • T i ≥ D i : task period i.e.
, its minimum inter-arrival time. We consider that T V i , the execution time variability, has been computed using given execution time random variables C i . We suppose that the random variables C i of all the tasks are independent. The independence is necessary for the computation of the scores of Equation 2. The set Budget i is a subset of possible budgets of execution times selected from the possible values of C i . Each set of budgets of a task τ i contains m possible budgets noted {b 1,i , . . . b m,i } ordered in a decreasing order of budgets with b 1,i = W CET i is the worst-case execution time of task τ i with p(b 1,i ) = 1.

Given a task set Γ of n mixed criticality real-time tasks, a budget assignment for Γ is a vector B = (B 1 , B 2 , . . . , B n ) with ∀i ∈ 1 . . . n, B i ∈ Budget i is the execution time budget assigned to task τ i . We define Γ B as the task set where every task τ i is defined by (B i , L i , D i , T i ) with B i , L i , D i and T i are the execution time, the criticality, the deadline and the period of τ i respectively.

Given a task set Γ of n mixed criticality real-time tasks the score of a budget assignment B = (B 1 , B 2 , . . . , B n ) computes the probability that all the tasks do not exceed their assigned budget and is defined as

Score(B) = i∈1...n p i (B i ) (2)
The 1) The task set Γ B is schedulable according to the scheduling algorithm Sched 2) Score(B HI ) = 1

V. BUDGET ASSIGNMENT HEURISTIC

We consider the problem of finding the budget assignment of a task set Γ such that Γ is mixed criticality schedulable and the probability of respecting the budget assignment for low criticality tasks is maximized. An optimal solution to the problem is to assign to all the high criticality tasks τ i their largest value as a budget and find the optimal assignment for low criticality tasks among m nbLO possible assignments where nbLO is the number of low criticality tasks. In this case the schedulability test of the scheduling algorithm is executed in the worst-case O(m nbLO ) times. The more the number of low criticality tasks grows, the longer the computation time is, the latter is also impacted by the complexity of the used scheduling test. For this reason, we propose a non optimal greedy heuristic, Algorithm 1, where the schedulability test is executed in the worst-case O(m × nbLO) times. In this algorithm the low criticality tasks are handled in a decreasing order of their execution time variability parameter. The idea is to start by reducing the budget of the tasks that are the less shifted towards their worst execution time. In Algorithm 1, first in (lines 2-5) if the task set with the minimal assignment budget is not schedulable the problem is not schedulable.

Otherwise a low criticality task with the highest execution time variability is selected. The budget for this task is decreased until the task set is schedulable (lines 8-14). If the task set is still not schedulable, the next task with the highest execution time variability parameter is selected.

Algorithm 1: Time variability heuristic

Input : Γ = {τ i , i ∈ 1..n} Output: system is not schedulable or B = (B 1 , . . . B n )

1 Set = set of all low criticality tasks We illustrate our budget assignment algorithm using the following example.

2 ∀τ i , if τ i ∈ Set, B i = b m,i else B i = b 1,i ; 3 if Γ B not schedulable using algorithm Sched then 4 return system is not schedulable 5 end 6 ∀τ i ∈ Set, B i = b 1,i 7 while Γ B not schedulable using algorithm Sched do 8 i is the index of τ i with ∀τ j ∈ Set, T V i ≥ T V j 9 for r ∈ 2..m do 10 B i = b r,i , 11 if Γ B is
Let τ 1 , τ 2 be two LO criticality tasks and τ 3 a HI criticality task scheduled in a mono-processor according to rate monotonic scheduling algorithm. For every task τ i , S(τ i ) is a set of 100 executions times of τ i with:

• S(τ 1 ) is composed of 10 occurrences of value 1, 20 occurrences of value 2 and 70 occurrences of value 3. • S(τ 2 ) is composed of 40 occurrences of value 1, 50 occurrences of value 2 and 10 occurrences of value 3. • S(τ 3 ) is composed of 10 occurrences of value 1, 10 occurrences of value 2 and 80 occurrences of value 3. The VWCET of τ 1 and τ 2 computed using Formula 1 are respectively V W CET 1 = 0.258, V W CET 2 = 0.48. We don't compute V W CET 3 because the assigned budget of HI criticality task do not depend on the variability of the task.

Let Γ be a mixed criticality task set, as defined in Section IV, composed by τ 1 = ((1, 2, 3), 0.258, LO, 6, 6), τ 2 = ((1, 2, 3), 0.48, LO, 9, 9) and τ 3 = ((1, 2, 3), V W CET 3 , HI, 12, 12).

Using response time analysis [START_REF] Audsley | Applying new scheduling theory to static priority pre-emptive scheduling[END_REF] the worst-case probability deadline miss of τ 3 is 0.204 which is quite high for a high criticality task. We need to reduce LO criticality task execution times to ensure the schedulability. Algorithm 1 applied to Γ first assigns to all the tasks the budget 3. Then the task τ 2 is selected as it is the task with the largest time variability among the set of low criticality tasks. The budget of τ 2 is decreased until the budget 1 as the task set is schedulable with this budget. Algorithm 1 returns the set of budgets B 1 = 3, B 2 = 1 and B 3 = 3 for the tasks τ 1 , τ 2 and τ 3 respectively. The score Score(B HI ) = 1 because the probability of not exceeding the budget B 3 = 3 is equal to 1 and Score(B LO ) = 1 × 0.4 = 0.4 with 1 is the probability of not exceeding the budget B 1 = 3 and 0.4 is the probability of not exceeding the budget B 2 = 1. Note that the budgets computed with Algorithm 1 are the budgets with the optimal score.

VI. SIMULATION-BASED EVALUATION

In this section, we present simulation-based experiments to evaluate the proposed heuristic. We consider uniprocessor earliest deadline first algorithm. We can consider without loss of generality that all the tasks are of low criticality because all the algorithms we are comparing assign to the high criticality tasks WCET as budget. For each task we generate a maximal random utilization that ranges from 1 to 1.45 using [START_REF] Emberson | Techniques for the synthesis of multiprocessor tasksets[END_REF]. We generate periods using a uniform distribution in the range [START_REF] Koch | Neuroflight: Next generation flight control firmware[END_REF]102] and a deadline using a uniform distribution in the range [T i /2, T i ]. We generate the minimal random utilizations by reducing the maximal utilizations by a value that ranges between 1 and 45 percent. We use a truncated normal distribution, where the mean is a random (uniform) value ranging from the Best case execution time BCET to the WCET of the task, and the standard deviation is a random value proportional to the span of execution times (W CET -BCET )/(x) where x follows a uniform distribution in the range [START_REF] Burns | A survey of research into mixed criticality systems[END_REF]40]. We generate 1000 task sets composed of 6 tasks. For each task τ i ,

Budget i = {W CET i , b 1 , b 2 , M edian i }
where b 1 and b 2 are the 80th and 60th percentiles respectively and M edian i is the median value. We discard task sets with i=1...n BCETi Ti > 1, and task sets which do not produce a solution with at least one algorithm in the list of the algorithms that we compare.

We generate three categories of task sets:

• Scenario 1: the first 80% of the task distributions have a skewness value greater than 2, the next 10% have a skewness value between 2 and -2, and the final 10% have a skewness value less than -2. • Scenario 2: the first 10% of the task distributions have a skewness value greater than 2, the next 10% have a skewness value between 2 and -2, and the final 80% have a skewness value less than -2. • Scenario 3: distributions are generated randomly. We test Algorithm 1 in two cases:

• VWCET: Algorithm 1 with ∀τ i , T V i = V W CET (C i ),
• sKw: Algorithm 1 with ∀τ i , T V i is the skewness of C i . We compare Algorithm 1 to the following algorithms that don't consider variability parameter as a criteria to select the task to reduce it's time budget:

• Medians: low tasks are assigned the median as a budget.

• Opt: optimal solution by testing all the configurations and applying the schedulability test O(m nbLO ) times. • Periods: similar to Algorithm 1 but tasks are selected in an ascending order of periods in line 8. • Deadlines: similar to Algorithm 1 but tasks are selected in an ascending order of deadlines in line 8. • Random: similar to Algorithm 1 but tasks are selected randomly in line 8. Figures 3,4 and 5 depict the scores obtained by the different algorithms in the scenarios 1, 2 and 3 respectively suing box plot representation. The mean score is also represented in each figure. We conclude that:

1) Assigning all the budgets of all the tasks to their median value budget result in a score close to 0, 2) Using the execution time variability gives better score than using the periods, or deadlines or a random, 3) VWCET gives better scores than the skewness parameter 4) The difference between the scores of the optimal solution and the ones obtained by the heuristics have the same order of magnitude in all the scenarios. This suggests that our approach is not highly affected by the shape of execution times distribution. In Figure 2, we compare the time needed to compute the set of assigned budgets of the optimal algorithm Opt and the Algorithm 1. We can see that from task sets with 8 tasks, the execution time of Opt increases exponentially.

VII. BENCHMARK-BASED EVALUATION

We performed our experiments on the Zynq UltraScale+ MPSoC ZCU104 platform, featuring a quad-core Arm Cortex A53 processor. We use FreeRTOS [START_REF] Barry | Freertos[END_REF], an open source, small, scalable and well documented real-time OS, along with ESFree [START_REF] Kase | Efficient scheduling library for freertos[END_REF] as the scheduling library in order to incorporate advanced ESFree also provides an extended version of the FreeRTOS task control block (TCB) that includes additional information for managing periodic tasks such as current execution time, WCET and deadline of the task. FreeRTOS measures time using a tick count variable. The RTOS tick interrupt increments the tick count with strict temporal accuracy allowing the kernel to measure time to a resolution of the chosen timer interrupt frequency. In our case we choose a resolution of 1000 Hz, this translates into a time resolution of 1 ms, consequently execution time smaller than 1 ms cannot be measured. The tick interrupt calls an application defined hook function where we check which task is currently under execution and increase it's execution time variable by 1 ms. We generate a set of execution times by executing 6 programs from TACLeBench [START_REF] Falk | Taclebench: A benchmark collection to support worst-case execution time research[END_REF] and Mälardalen [START_REF] Gustafsson | The Mälardalen WCET benchmarks -past, present and future[END_REF] benchmarks under the following configuration: (1) we consider asymmetric multiprocessing with FreeRTOS, meaning each core of a multicore processor runs its own independent instance of FreeRTOS (2) we do not vary the inputs of the programs (3) caches are disabled (4) cores 1, 2 and 3 execute [START_REF] Bechtel | Denial-of-service attacks on shared cache in multicore: Analysis and prevention[END_REF] a program that iterates Table II gives for every program its deadline and its period. All the programs are considered of LO criticality. For each program i, the budgets are Budget i = {W CET i , b 1 , ..., b 7 , M edian i } where b 1 , ... , b 7 are the 99th, 97th, 95th, 90th, 80th, 70th, 60th percentiles respectively and M edian i is the median value.

Using the generated execution times we apply:

• VWCET: Algorithms 1 with ∀τ i , T V i = V W CET (C i ),
• sKw: Algorithms 1 with ∀τ i , T V i is the skewness of C i .

A response time analysis [START_REF] Audsley | Applying new scheduling theory to static priority pre-emptive scheduling[END_REF] is used as schedulability test. We compare Algorithm 1 to the same algorithms used in the simulation-based evaluation i.e. Medians, Opt, Periods, Deadlines and Random. the score computed using Formula 2. We can see that Algorithm 1 using variability parameters sKw and VWCET yields better results than the other algorithms Medians, Random and Periods. When comparing the optimal solution computed using algorithm Opt to the proposed heuristic that uses time variability we can see that the optimal solution in order to maximize the overall scores of the tasks reduces the time budget of 5 of the 6 tasks, in difference to the proposed heuristic where 3 of the 6 tasks will always meet their time budget. Using algorithm Periods only 2 of the 6 tasks will always meet their time budget. The greater the number of tasks whose jobs always meet the assigned budget, the fewer tasks we need to monitor during execution. Using FreeRTOS OS and the library ESFree, we executed the 6 programs during 10 minutes using rate monotonic preemptive scheduling algorithm. If a job do not terminate at its assigned budget computed using Algorithm 1, the job is stopped. We count for every task τ i the number of stopped jobs and computes ratio i the ratio between the number of stopped jobs and the number of activated jobs of task τ i . Table IV presents the comparison of the probability of meeting the execution time budget per task to the ratios obtained. For VWCET the observed task performance aligns well and sometimes slightly exceeds the theoretically computed time budgets, this conclusion holds true for skewness as well. We can deduce that the overhead induced when stopping a job is not high.

VIII. CONCLUSION

This paper introduces an approach for the scheduling problem of mixed criticality systems. We show that the time variability of execution times can be used to compute the execution time budget to be allocated to low-criticality tasks in order to prevent these tasks from disrupting the functioning of high-criticality tasks, while ensuring that low-criticality tasks are not always stopped. We show that the proposed heuristic outperforms the budget selection strategies that are agnostic to the shape of the distribution of execution time budgets. In the future, we plan to apply our approach in an online algorithm where task budgets are adjusted during the execution.

Fig. 1 .

 1 Fig. 1. Execution times (cycles) of programs executed in an ARM Cortex 53

  score Score(B LO ) = i∈1...n Li∈LO p i (B i ) is the score of the budget assignment of LO criticality tasks and Score(B HI ) = i∈1...n Li∈HI p i (B i ) is the score of the budget assignment of HI criticality tasks. Definition 3 (Mixed criticality schedulability): Given a task set Γ of n mixed criticality real-time tasks, the task set Γ is schedulable w.r.t. the budget assignment B = (B 1 , . . . , B n ) and the scheduling algorithm Sched if and only if:

Fig. 2 .Fig. 3 .

 23 Fig. 2. Computation time of Opt compared to Algorithm 1

Fig. 4 .Fig. 5 .

 45 Fig. 4. Algorithms comparison for Scenario 2

VWCET

  AND SKW OF MERGE SORT, QUICK SORT AND INSERT SORT

		Merge sort	Quick sort	Insert sort
	VWCET	7,911	0,459	4,75
	sKw	9,559	-0,399	-4,186
		TABLE I

  Table III depicts the scores obtained for every program using every algorithm. The score Π is

	Prog	Medians	Random	Periods	sKw	VWCET	Opt
	epic	0.51	0.73		0.73	1	1	0.97
	fft	0.50	0.99		0.71	0.80	0.71	0.99
	iSort	1	1		1	1	1	1
	ludcmp	0.51	0.72		0.72	0.72	0.72	0.98
	matrix	0.50	0.72		1	1	0.80	0.90
	sha	0.53	0.72		0.99	0.72	1	0.97
	Π	0.03	0.27		0.37	0.42	0.41	0.83
				TABLE III
			ALGORITHMS SCORES
	Program i		epic	fft	iSort	ludcmp	matrix	sha
	p(B i ) (VWCET)	1	0.71	1	0.72	0.80	1
	ratio i (VWCET)	1	0.69	1	0.75	0.82	0.99
	p(B i ) (sKw)	1	0.80	1	0.72	1	0.72
	ratio i (sKw)	0.99	0.84	1	0.75	1	0.72
				TABLE IV
	THE PROBABILITIES OF MEETING THE EXECUTION TIME BUDGET
								Prog	epic	fft	iSort	ludcmp	matrix	sha
								T(ms)	450	430	70	440	677	460
								D(ms)	152	18	10	52	390	247
								TABLE II
								PROGRAM'S PARAMETERS


domain: sde sdv

text:
IMBE Tania De Almeida
email: tania.dealmeida@univ-lille.fr

Xavier Arnan

| Yvan Capowiez

Mickaël Hedde

François Mesléard

Thierry Dutoit

| Olivier Blight

Ants in restoration ecology: Why, what's and the way forward

Keywords: active restoration, ecological engineering, ecological functions, functional traits, life-history traits, monitoring

Global changes call for more nature-based solutions, especially in nature conservation involving ecological restoration. Current methods essentially based on civil engineering are both expensive and costly in non-renewable energy consumption and pollution terms. The non-sustainability of these techniques is leading to the direct use of certain species to restore degraded ecosystems. Ants, because of their central role in ecosystem functioning and their occurrence on almost all terrestrial ecosystems, are promising candidates for environmental monitoring and such ecological restoration projects. We provide here a narrative review of the ecological functions performed by ants, and we take stock of how ants are currently considered in passive and active restoration. We then propose a trait-based approach to facilitate their use by practitioners in future restoration projects. We list and discuss both life-history traits relevant for environmental monitoring and functional traits known to affect abiotic (physical and chemical soil properties) and biotic (plant and fauna communities) components.

| INTRODUCTION

Ecological restoration plays a vital role in conserving biodiversity worldwide [START_REF] Aronson | Ecosystem restoration is now a global priority: Time to roll up our sleeves[END_REF]. The aim is to increase biodiversity and ecosystem services by recovering the structure and function of human-degraded ecosystems [START_REF] Benayas | Enhancement of biodiversity and ecosystem services by ecological restoration: A meta-analysis[END_REF]. Removing the disturbance that degraded the ecosystem is the first step in a restoration project. Active restoration projects manage the ecosystem through a range of human interventions to accelerate/influence the successional trajectory of recovery (see glossary in Table 1 for definition of terms).

The projects involve three phases: the pre-operational phase of defining the restoration objectives and the ecological functions to restore; the operational phase, which consists in setting up the project; and the post-operational phase, when the success of the restoration is monitored [START_REF] Choi | Theories for ecological restoration in changing environment: Toward "futuristic" restoration[END_REF][START_REF] Clewell | Ecological restoration: Principles, values, and structure of an emerging profession[END_REF]. The operational phase often involves the use of civil engineering principles and techniques based on major technical interventions such as public works machinery (e.g., extraction, loading, and/or compacting machines). These techniques entail extensive consumption of non-renewable resources (e.g., diesel and petrol) and significant emissions of pollutants (e.g., CO 2 , NO x ). To avoid the environmental consequences of such civil engineering, one component of or approach to restoration ecology, called "ecological engineering," consists in directly using organisms to restore degraded ecosystems [START_REF] Aronson | Ecological restoration and ecological engineering: Complementary or indivisible?[END_REF][START_REF] Mitsch | Ecological engineering: A field whose time has come[END_REF]. This nature-based solution inspired by, supported by, or copied from nature uses internal mechanisms to change the ecosystem in the desired successional direction, thereby ensuring the provision of ecosystem services [START_REF] Nesshöver | The science, policy and practice of nature-based solutions: An interdisciplinary perspective[END_REF]. Since ecological engineering generally promotes the use of species that directly or indirectly modify the physics and chemistry of their habitat, thereby influencing other organisms [START_REF] Jones | Positive and negative effects of organisms as physical ecosystem engineers[END_REF], a key challenge is identifying the appropriate species to introduce and/or promote.

To ensure delivery of the ecological functions targeted, the identification of the functions provided by native species is essential [START_REF] Sotka | Not so fast: Promoting invasive species to enhance multifunctionality in a native ecosystem requires strong(er) scrutiny[END_REF]. One way to do this is to study their functional traits. Functional traits that are defined as any measurable morphological, physiological, behavioral, or phenological features of an individual, from cell to whole-organism level [START_REF] Pey | A thesaurus for soil invertebrate trait-based approaches[END_REF][START_REF] Violle | Let the concept of trait be functional![END_REF], make it easier to link a species to the functions it performs in its habitat [START_REF] Cadotte | Beyond species: Functional diversity and the maintenance of ecological processes and services[END_REF][START_REF] Gagic | Functional identity and diversity of animals predict ecosystem functioning better than species-based indices[END_REF]. Traits can reflect environmental tolerance, that is, life-history traits, and/or affect ecosystem processes, that is, functional traits [START_REF] Hedde | A common framework for developing robust soil fauna classifications[END_REF]. Thus, measuring and understanding the diversity of species traits could greatly improve restoration decision making [START_REF] Cadotte | Beyond species: Functional diversity and the maintenance of ecological processes and services[END_REF]. However, despite the recent effort to adopt such a trait-based approach, there is still a gap between their study and their actual use in restoration projects [START_REF] Carlucci | Functional traits and ecosystem services in ecological restoration[END_REF][START_REF] Merchant | Four reasons why functional traits are not being used in restoration practice[END_REF], particularly with fauna.

Globally, the use of animals, particularly invertebrates, to monitor the recovery of an ecosystem or to actively restore it, has been underexplored compared with plants [START_REF] Auclerc | Fostering the use of soil invertebrate traits to restore ecosystem functioning[END_REF][START_REF] Snyder | Current and potential roles of soil macroinvertebrates (earthworms, millipedes, and isopods) in ecological restoration[END_REF]. Among possible explanations for this neglect of invertebrates, the first may be resource limitation to the development of some invertebrates after a disturbance [START_REF] Jouquet | Utilization of earthworms and termites for the restoration of ecosystem functioning[END_REF]. Second, restoration of insect populations is sometimes based on the hypothesis that insects will follow the restoration of plant communities. However, this overlooks the importance of direct or indirect interactions between invertebrates, with some species being key to the establishment and maintenance of others (De Almeida, Mesléard, et al., 2020).

Third, collecting enough individuals may be challenging (e.g., breeding them and finding enough source populations), and such difficulties may hamper their reintroduction in active restorations [START_REF] Jouquet | Utilization of earthworms and termites for the restoration of ecosystem functioning[END_REF]. To date, earthworms, because of the key role they play in soil, are the soil organisms most frequently used in ecological engineering [START_REF] Jouquet | Utilization of earthworms and termites for the restoration of ecosystem functioning[END_REF] and have successfully restored soils (e.g., [START_REF] Butt | Earthworms in soil restoration: Lessons learned from United Kingdom case studies of land reclamation[END_REF], as have termites occasionally [START_REF] Jouquet | Utilization of earthworms and termites for the restoration of ecosystem functioning[END_REF].

By contrary, ants remain underexploited in ecological engineering.

Yet, their social life and related traits, generally forming long-term sedentary colonies, as well as their recognized ecological value make them a good choice for ecological engineering projects. Ants occur throughout all continents except Antarctica and are among the most diverse and abundant organisms on Earth [START_REF] Hölldobler | The ants[END_REF]. In all the different terrestrial ecosystems they occupy, ants play key ecological roles as soil engineers, predators, and/or regulators of plant growth and reproduction (De Almeida, Mesléard, et al., 2020;[START_REF] Del Toro | The little things that run the world revisited: A review of ant-mediated ecosystem services and disservices (hymenoptera: Formicidae)[END_REF]. These effects suggest that ants have enormous untapped potential for applications in active restoration projects.

The aim of this paper is to highlight ants' potential for restoration ecology and to provide keys for their future use. We first provide a narrative review of the ecological functions performed by ants, and we take stock of how ants are currently considered in passive and active restoration. We made a quick review of the ISI Web of Science database by searching for "restoration ants," "ant traits," because our objective was not to conduct a comprehensive review.

We then propose a trait-based approach to facilitate their future use.

We list and discuss both life-history traits relevant for environmental monitoring and functional traits known to affect the soil (physics and chemistry), the plant community, and the soil fauna relevant in ecological engineering.

| ANT ECOLOGICAL FUNCTIONS

Ants play a central role in ecosystem functioning [START_REF] Del Toro | The little things that run the world revisited: A review of ant-mediated ecosystem services and disservices (hymenoptera: Formicidae)[END_REF][START_REF] Elizalde | The ecosystem services provided by social insects: Traits, management tools and knowledge gaps[END_REF]. Their engineering activity affects not only each ecosystem component separately but also their relationships (De Almeida, Mesléard, et al., 2020). By their structural engineering activities, particularly during nest construction and maintenance, ants can move large amounts of underground soil to superficial layers [START_REF] Dostál | Ant-induced soil modification and its effect on plant below-ground biomass[END_REF]. Earthworms are considered to be more important bioturbation agents than ants in several ecosystems, as shown by [START_REF] Taylor | Ant and earthworm bioturbation in cold-temperate ecosystems[END_REF] in cold-temperate ecosystems.

In their study, [START_REF] Wilkinson | Breaking ground: Pedological, geological, and ecological implications of soil bioturbation[END_REF] states that over half of the studies recording bioturbation rates of 10-50 t ha À1 year À1 and exceeding 100 t ha À1 year À1 in some tropical locations. However, the bioturbation role of ants is not negligible. In a recent review, [START_REF] Viles | Ants as geomorphological agents: A global assessment[END_REF] showed that the median amount of soil moved by ants in bioturbation/mounding is about 1.5 t ha À1 year À1 , with a minimum of 0.05 t ha À1 year À1 for Lasius niger [START_REF] Persson | Bioturbation in different ecosystems at Forsmark and Oskarshamn[END_REF]) and a T A B L E 1 Glossary of terms.

Terms Definition

Active restoration

Where damage is intermediate or high, all causes of degradation should be removed or reversed and additional active interventions should be carried out to correct abiotic damage and trigger biotic recovery [START_REF] Mcdonald | International standards for the practice of ecological restoration-including principles and key concepts[END_REF] Disturbance "A natural or anthropogenic event that changes the structure, content, and/or function of an ecosystem, usually in a substantial manner" [START_REF] Clewell | Ecological restoration: Principles, values, and structure of an emerging profession[END_REF] Ecological engineering

"The manipulation and use of living organisms or other materials of biological origin to solve problems that affect people" [START_REF] Clewell | Ecological restoration: Principles, values, and structure of an emerging profession[END_REF] Ecological restoration

"The process of assisting the recovery of an ecosystem that has been degraded, damaged, or destroyed" [START_REF] Gann | International principles and standards for the practice of ecological restoration[END_REF] Ecosystem services "The aspects of ecosystems utilized (actively or passively) to produce human well-being" [START_REF] Fisher | Defining and classifying ecosystem services for decision making[END_REF]. The Millennium Ecosystem Assessment divided ecosystem services into four categories: supporting, regulation, provisioning, and cultural services (Millennium Ecosystem Assessment, 2005) Passive restoration "Where damage is relatively low and topsoil retained, or where sufficient time frames and nearby populations exist to allow recolonization, plants and animals may be able to recover after cessation of certain types of degradation" [START_REF] Gann | International principles and standards for the practice of ecological restoration[END_REF] maximum of 68.38 t ha À1 year À1 for Aphaenogaster longiceps [START_REF] Richards | Aphaenogaster ants as bioturbators: Impacts on soil and slope processes[END_REF]. Ants can also concentrate organic matter and produce large quantities of organic waste that are deposited inside the nest or on the soil surface [START_REF] Frouz | The effect of ants on soil properties and processes (hymenoptera : Formicidae)[END_REF]. Through these activities, ants can alter soil physico-chemical properties and water infiltration, indirectly affecting the surrounding vegetation [START_REF] Dostál | Ant-induced soil modification and its effect on plant below-ground biomass[END_REF].

The magnitude and direction (positive, negative, or neutral) of their effects on soil physico-chemical properties vary between ant species [START_REF] Frouz | The effect of ants on soil properties and processes (hymenoptera : Formicidae)[END_REF] and environments [START_REF] Farji-Brener | The effects of ant nests on soil fertility and plant performance: A meta-analysis[END_REF]. Most studies show an increase in organic matter within ant nests compared with the surrounding soil [START_REF] Azcárate | Harvester ants (Messor barbarus) as disturbance agents in Mediterranean grasslands[END_REF][START_REF] Cammeraat | The effects of ants' nests on the physical, chemical and hydrological properties of a rangeland soil in semi-arid Spain[END_REF]. Similarly, ant activity generally increases soil nitrogen, phosphorus, and potassium contents. However, no effect [START_REF] Leal | Seed dispersal by ants in the semi-arid caatinga of north-East Brazil[END_REF] or even a decrease in nitrogen or potassium content has also been recorded [START_REF] Dostál | Ant-induced soil modification and its effect on plant below-ground biomass[END_REF].

Accordingly, ants have contrasting effects on plants, positively or negatively modifying plant growth (De Almeida, Mesléard, et al., 2020;[START_REF] Saha | Effect of leaf-cutting ant nests on plant growth in an oligotrophic Amazon rain forest[END_REF] or diversity [START_REF] Saha | Effect of leaf-cutting ant nests on plant growth in an oligotrophic Amazon rain forest[END_REF][START_REF] Schütz | Impact of Formica exsecta Nyl. On seed bank and vegetation patterns in a subalpine grassland ecosystem[END_REF] depending on their nutrient requirements.

In addition to these indirect effects on vegetation, ants can profoundly affect plant population and community structure through seed dispersal. Myrmecochory involves more than ten thousand plant species across many terrestrial ecosystems [START_REF] Lengyel | Ants sow the seeds of global diversification in flowering plants[END_REF]. [START_REF] Christian | Consequences of a biological invasion reveal the importance of mutualism for plant communities[END_REF] showed that the preservation of seed dispersal mutualistic interactions is essential to maintaining natural plant communities. It may shape the initial spatial distribution of plants within populations and define the context for future ecological, demographic, and genetic interactions among emerging seedlings [START_REF] Kalisz | Ant-mediated seed dispersal alters pattern of relatedness in a population of Trillium grandiflorum[END_REF]. Granivorous ants too can disperse seeds, but the positive effects on vegetation are balanced by the predation costs to the plant population [START_REF] Arnan | Uncoupling the effects of seed predation and seed dispersal by granivorous ants on plant population dynamics[END_REF]. Although several studies found more seeds [START_REF] Bulot | Refuse pile turnover by harvester ants (hymenoptera: Formicidae) increases seed density and seedling species richness in dry grasslands[END_REF][START_REF] Schütz | Impact of Formica exsecta Nyl. On seed bank and vegetation patterns in a subalpine grassland ecosystem[END_REF] and seed germination in ant nests than in surrounding areas, the relative effect of seed dispersal via ants depends on ant community composition [START_REF] Prior | Mutualism between co-introduced species facilitates invasion and alters plant community structure[END_REF], as well as on seed availability [START_REF] Arnan | Uncoupling the effects of seed predation and seed dispersal by granivorous ants on plant population dynamics[END_REF]. Ants may also act as biotic defences, protecting plants against herbivores and parasites, thus modifying plant population and community dynamics. In return, plants offer benefits such as shelter and food rewards [START_REF] Rosumek | Ants on plants: A meta-analysis of the role of ants as plant biotic defenses[END_REF].

Ants also impact other invertebrates. They can negatively affect populations of some species through predation and competition, while other species feed on ants themselves. For example, ants and spiders are potential mutual predators [START_REF] Gajski | Ant-eating spider maintains specialist diet throughout its ontogeny[END_REF][START_REF] Sanders | Intraguild interactions between spiders and ants and top-down control in a grassland food web[END_REF]. Ants can also have a mutualistic relationship with trophobionts (e.g., aphids, mealybugs, cicadellids). In exchange for protecting trophobionts, ants collect their honeydew, sometimes as their main food source. This relationship can increase the density and diversity of trophobionts, to the detriment of plants [START_REF] Elizalde | The ecosystem services provided by social insects: Traits, management tools and knowledge gaps[END_REF][START_REF] Styrsky | Ecological consequences of interactions between ants and honeydew-producing insects[END_REF]. Ants can also indirectly affect invertebrates by modifying the abiotic component and plant communities. This impact is mostly positive for macro-, meso-, and microfauna [START_REF] Boulton | How ant nests increase soil biota richness and abundance: A field experiment[END_REF][START_REF] Çakır | The negative effect of wood ants (Formica rufa) on microarthropod density and soil biological quality in a semi-arid pine forest[END_REF][START_REF] Sanders | Ecosystem engineering and predation: The multi-trophic impact of two ant species[END_REF], as well as bacterial and fungal communities. The higher abundance of bacteria and fungi may indirectly increase bacterial and fungal predators such as nematodes and collembolans [START_REF] Boulton | Effects of a common harvester ant (Messor andrei) on richness and abundance of soil biota[END_REF][START_REF] Boulton | How ant nests increase soil biota richness and abundance: A field experiment[END_REF]). An increase in plant heterogeneity and biomass in ant patches can thus be responsible for an increased abundance of detritivores such as Collembola and plantfeeding Coleoptera, and indirectly for that of their predators (e.g., Araneae and predator Coleoptera) (De Almeida, Mesléard, et al., 2020).

While we have stressed the positive effects of ants on the ecosystem, some ants can also cause disservices. For example, leafcutting ants can improve soil conditions [START_REF] Farji-Brener | Leaf-cutting ants and forest groves in a tropical parkland savanna of Venezuela: Facilitated succession[END_REF] and assist in seed dispersal [START_REF] Farji-Brener | Seedling recruitment in a semiarid Patagonian steppe: Facilitative effects of refuse dumps of leafcutting ants[END_REF]) but can also drastically cut new growing vegetation [START_REF] Corrêa | How leaf-cutting ants impact forests: Drastic nest effects on light environment and plant assemblages[END_REF]. Similarly, wood ants (Formica rufa group) can positively affect soil porosity [START_REF] Frouz | The effect of ants on soil properties and processes (hymenoptera : Formicidae)[END_REF] and moisture content [START_REF] Frouz | The effect of nest moisture on daily temperature regime in the nests of Formica polyctena wood ants[END_REF], which may increase microbial activity [START_REF] Jilkova | Respiration of wood ant nest material affected by material and forest stand characteristics[END_REF][START_REF] Stadler | Ant-mediated effects on spruce litter decomposition, solution chemistry, and microbial activity[END_REF]. They are also well known as herbivore predators, which has led to breeding them for biological control of pests such as the pine processionary moth [START_REF] Pavan | Utilitá delle Formiche del gruppo Formica rufa[END_REF]. However, they can also have negative effects on microarthropod density in semi-arid pine forest (Çakır, 2019). Furthermore, positive effects of ants may be not constant along degradation and restoration gradient (e.g., [START_REF] Holec | The effect of two ant species Lasius niger and Lasius flavus on soil properties in two contrasting habitats[END_REF][START_REF] Tuma | The impacts of tropical mound-building social insects on soil properties vary between taxa and with anthropogenic habitat change[END_REF]. The costs and benefits of using a species to restore a site must always be precisely measured and depend strictly on the local context. et al., 2008), Germany (e.g., [START_REF] Dauber | Colonization of temperate grassland by ants[END_REF], Hungary (e.g., [START_REF] Gallé | The effect of forest age and habitat structure on the ground-dwelling ant assemblages of lowland poplar plantations[END_REF], Czech Republic (e.g., [START_REF] Holec | Ant (hymenoptera: Formicidae) communities in reclaimed and unreclaimed brown coal mining spoil dumps in The Czech Republic[END_REF],

Italy (e.g., [START_REF] Ottonetti | Recolonization patterns of ants in a rehabilitated lignite mine in Central Italy: Potential for the use of Mediterranean ants as indicators of restoration processes[END_REF], and Spain (e.g., [START_REF] Arnan | Response of ant functional composition to fire[END_REF], and Australia (19%) (e.g., [START_REF] Majer | Recolonization by ants in rehabilitated open-cut mines in northern Australia[END_REF]. The use of ants as bioindicators dates from Australian mid-1970s assessments of the restoration of abandoned quarries [START_REF] Andersen | Ants show the way down under: Invertebrates as bioindicators in land management[END_REF][START_REF] Majer | Ants: Bio-indicators of minesite rehabilitation, landuse, and land conservation[END_REF][START_REF] Majer | Recolonization by ants in rehabilitated open-cut mines in northern Australia[END_REF].

Since then, it has spread to other countries, mainly Brazil and Europe [START_REF] Casimiro | What can ants tell us about ecological restoration? A global meta-analysis[END_REF]. While abundance and species richness are the most commonly used ecological metrics for assessing success of ant recovery [START_REF] Borges | Terrestrial invertebrates as bioindicators in restoration ecology: A global bibliometric survey[END_REF][START_REF] Underwood | The role of ants in conservation monitoring: If, when, and how[END_REF], ant functional groups seem to recover more rapidly than species richness [START_REF] Casimiro | What can ants tell us about ecological restoration? A global meta-analysis[END_REF].

A global model of ant community dynamics based on nine functional groups was developed for Australian communities according to environmental stressors (factors limiting ant productivity) and disturbance (factors eliminating ant biomass) [START_REF] Andersen | A classification of Australian ant communities, based on functional groups which parallel plant life-forms in relation to stress and disturbance[END_REF][START_REF] Andersen | Ants show the way down under: Invertebrates as bioindicators in land management[END_REF][START_REF] Hoffmann | Responses of ants to disturbance in Australia, with particular reference to functional groups[END_REF]. Ant community functional composition provides information on the state of ecosystem degradation. Disturbances also have strong indirect effects on competitive hierarchies through changes in dominant species abundance.

However, the Australian functional groups developed by Andersen (1995) combine life-history traits, taxonomy, and competitive abilities, and this inclusion of taxonomy limits their application to other ant communities [START_REF] Hedde | A common framework for developing robust soil fauna classifications[END_REF]). An approach based on quantitative traits that avoid taxonomy would allow more accurate evaluation of ant community responses to environmental change, regardless of biogeographic region.

| Ants in ecological engineering projects

Because of their multi-component impacts, ants have strong potential in ecological engineering projects (De Almeida, Mesléard, et al., 2020;[START_REF] Lu | Modifying effect of ant colonization on soil heterogeneity along a chronosequence of tropical forest restoration on slashburn lands[END_REF][START_REF] Nicolai | Ecological engineers' nests benefit plant recovery following fire in a semiarid grassland, New Mexico, USA[END_REF]. However, to date, only one study has Through their bioturbation and organic matter concentration activities, ants can help restore degraded soils early in the succession. As seed dispersers they may be key to the recolonization and dispersion of plants. In passive restoration, ants have been shown to accelerate restoration [START_REF] Nicolai | Ecological engineers' nests benefit plant recovery following fire in a semiarid grassland, New Mexico, USA[END_REF][START_REF] Nicolai | Harvester ant nests improve recovery performance of drought impacted vegetation in grazing regimes of semiarid savanna, Texas[END_REF]. Nicolai (2019) demonstrated that 2 years after a wildfire, Pogonomyrmex rugosus (Emery, 1895) ant nest edges improved resources leading to faster recovery of grass inflorescences than on burned controls. As areas of high grass reproduction during recovery, ant patches may act as a source of seed dispersal into the surrounding habitat [START_REF] Nicolai | Ecological engineers' nests benefit plant recovery following fire in a semiarid grassland, New Mexico, USA[END_REF]. In a tropical forest, ant nest soil enrichment in microbial carbon was also found to increase with restoration age (up to 53 years) [START_REF] Lu | Modifying effect of ant colonization on soil heterogeneity along a chronosequence of tropical forest restoration on slashburn lands[END_REF]. Such high soil enrichment in organic matter, available nitrogen, ammonium, and nitrate at an early stage might promote spontaneous forest restorations. At a later restoration stage, ants improve plant development by strongly stimulating microbial growth and opening up space for their establishment [START_REF] Lu | Modifying effect of ant colonization on soil heterogeneity along a chronosequence of tropical forest restoration on slashburn lands[END_REF].

Although ant impacts on soil physico-chemical properties are mostly small-scale and limited to the nest, they can play a significant role in maintaining spatial heterogeneity and plant species richness at site scale [START_REF] Platt | The colonization and formation of equilibrium plant species associations on badger disturbances in a tall-grass prairie[END_REF]. For example, they impact resource heterogeneity within a landscape by creating diversity in nutrient concentrations [START_REF] Dostál | Population dynamics of annuals in perennial grassland controlled by ants and environmental stochasticity[END_REF] and thus potentially enhancing the restoration of degraded grasslands [START_REF] Lane | Diminishing effects of ant mounds on soil heterogeneity across a chronosequence of prairie restoration sites[END_REF]. Their nests may foster recolonization by other organisms, mainly plants but also fauna [START_REF] Kachamakova | The role of ant nests in European ground squirrel's (Spermophilus citellus) postreintroduction adaptation in two Bulgarian mountains[END_REF][START_REF] Nicolai | Ecological engineers' nests benefit plant recovery following fire in a semiarid grassland, New Mexico, USA[END_REF], throughout the degraded ecosystem. Moreover, their impacts are detected not only during nest occupation but also up to more than 20 years after colony disappearance [START_REF] Kristiansen | Abandoned anthills of Formica polyctena and soil heterogeneity in a temperate deciduous forest: Morphology and organic matter composition[END_REF][START_REF] Lane | Diminishing effects of ant mounds on soil heterogeneity across a chronosequence of prairie restoration sites[END_REF], thus increasing their spatial and temporal environmental impact.

BOX 1 Study case of using ants in active restoration.

In 2009, an oil leak destroyed 5.5 ha of a Mediterranean grassland [START_REF] Bulot | A new transplantation protocol for harvester ant queens Messor barbarus (hymenoptera: Formicidae) to improve the restoration of species-rich plant communities[END_REF]. After soil restoration (see [START_REF] Bulot | Ecological restoration by soil transfer: Impacts on restored soil profiles and topsoil functions[END_REF] for details), a fine crust formed at the soil surface, limiting plant establishment. To restore soil physico-chemical properties and the vegetation component, a trait-based approach was applied to identify a candidate ant species combining all the traits required to restore the functions of interest. We determined that the selected species should exhibit a range of traits related to soil decompaction and increased soil nutrients, as well as seed dispersal, and be abundant in the original ecosystem. Of the most abundant native species, those that disperse seeds and nest in the soil were identified from the regional literature [START_REF] Cerdan | Etude de la biologie, de l'écologie et du comportement des fourmis moissonneuses du genre Messor (Hymenoptera, Formicidae) en Crau[END_REF], and two species were selected: M. barbarus and Pheidole pallidula (Nylander, 1849). Granivorous ants are known to increase soil nutrient content, the workers' large mandible gap allows the species to displace large soil particles and seeds, their permanent nests enhance the effect on soil, and their dispersion through nuptial flight accelerates colonization. Although both species disperse through nuptial flight and have high densities of workers, M. barbarus was finally chosen because of its larger mandible gap and larger body size [START_REF] Bulot | A new transplantation protocol for harvester ant queens Messor barbarus (hymenoptera: Formicidae) to improve the restoration of species-rich plant communities[END_REF]De Almeida, Blight, et al., 2020). In this case study, the degraded area was surrounded by the original grassland. Given that M. barbarus makes permanent and temporary trails up to 40 m long [START_REF] Detrain | Recolte de graines et dynamique du réseau de pistes chez la fourmi moissonneuse Messor barbarus[END_REF], the founding queens were transplanted to a maximum of 25 m from the Mediterranean grassland to ensure that the ants were able to forage in the original ecosystem (see [START_REF] Bulot | A new transplantation protocol for harvester ant queens Messor barbarus (hymenoptera: Formicidae) to improve the restoration of species-rich plant communities[END_REF]

| THE WAY FORWARD: A TRAIT-BASED APPROACH FOR FUTURE PROJECTS

Traits can reflect environmental tolerance, that is, life-history traits, and/or affect ecosystem processes, that is, functional traits [START_REF] Hedde | A common framework for developing robust soil fauna classifications[END_REF] that make easier to link a species to the functions it performs in its habitat [START_REF] Cadotte | Beyond species: Functional diversity and the maintenance of ecological processes and services[END_REF][START_REF] Gagic | Functional identity and diversity of animals predict ecosystem functioning better than species-based indices[END_REF]. In ecological restoration, a trait-based approach diverges from classical restoration methods that focus on restoring species composition of pre-disturbance communities to focus on restoring ecosystem functioning. The application of such methods is receiving growing attention worldwide [START_REF] Carlucci | Functional traits and ecosystem services in ecological restoration[END_REF]. However, one of the reasons for their current limited use is the lack of knowledge about the link between functional traits and ecological functions performed within the ecosystem [START_REF] Carlucci | Functional traits and ecosystem services in ecological restoration[END_REF].

Because of their social life form, ant traits should be quantified at both the level of the individual and that of the colony [START_REF] Elizalde | The ecosystem services provided by social insects: Traits, management tools and knowledge gaps[END_REF], and selected according to the restoration needs. The type of restoration and its objectives will depend on the state of the degraded ecosystem (Table 1; Figure 1). When disturbance of the target ecosystem is slight enough to be dealt with by removing the disturbances (i.e., passive restoration), community and population changes need to be monitored to ensure that all target functions been restored [START_REF] Holl | When and where to actively restore ecosystems[END_REF]. When only the biotic component is deeply degraded, biotic interactions (e.g., seed dispersal, predation, competition, facilitation, symbiosis) will require restoration. If disturbance has degraded both abiotic and biotic components, the priority is to restore soil physical and chemical properties [START_REF] Heneghan | Integrating soil ecological knowledge into restoration management[END_REF] by identifying the ecological functions of interest (e.g., soil structure, water infiltration, soil fertility) and then the biotic component. After the abiotic and/or biotic components are restored, community and population changes need to be monitored, in particular ant traits characterizing the stages of the restoration (Figure 1).

| Ant traits relevant in environmental monitoring

Certain characteristics define a species as a "bioindicator species," the most notable being ease of measurement, sensitivity to environmental disturbances, and predictable responses to environmental disturbances [START_REF] Niemi | Application of ecological indicators[END_REF]. Ants generally fulfil these three criteria.

Some ant traits can provide information on resource availability. appropriate for assessing such responses [START_REF] Arnan | Response of ant functional composition to fire[END_REF][START_REF] Wiescher | Assembling an ant community: Species functional traits reflect environmental filtering[END_REF]. For example, body size gives information on the soil compaction with a higher proportion of small ants in compacted soil [START_REF] Schmidt | Ant assemblage and morphological traits differ in response to soil compaction[END_REF]. Dark colors, large bodies, and physiological tolerance of low temperatures are traits associated with lowtemperature climatic zones. Ants tend to be smaller in warm and wet habitats [START_REF] Gibb | Habitat disturbance selects against both small and large species across varying climates[END_REF]. Similarly, hair can increase tolerance of dehydration [START_REF] Willot | Total internal reflection accounts for the bright color of the Saharan silver ant[END_REF][START_REF] Wittlinger | Hair plate mechanoreceptors associated with body segments are not necessary for threedimensional path integration in desert ants, Cataglyphis fortis[END_REF]. Monitoring these traits in a degraded ecosystem will provide information on the evolution of microclimate conditions in degraded and restored habitats. Monitoring requires traits to be measured directly on individuals present in the study areas, but it is important to consider that some traits are subject to seasonal variability, such as diet [START_REF] Lázaro-González | Short-term ecological and behavioural responses of Mediterranean ant species Aphaenogaster gibbosa (Latr. 1798) to wildfire[END_REF] and thermal resistance [START_REF] Bujan | Seasonal plasticity of thermal tolerance in ants[END_REF]. As measuring traits can be very time-consuming, body size could be the first choice for monitoring environmental changes. This trait is relatively easy to measure. It could be the first trait studied, and one or two other more tedious traits, such as hair or color, could be combined with it.

Ant traits are closely related to environmental complexity [START_REF] Nooten | Habitat complexity affects functional traits and diversity of ant assemblages in urban green spaces (hymenoptera: Formicidae)[END_REF]. Ant body size and leg length tend to be negatively correlated with habitat complexity, with longer legs and larger heads in open spaces than in complex habitats [START_REF] Weiser | Ecological morphospace of New World ants[END_REF][START_REF] Yates | Morphological traits: Predictable responses to macrohabitats across a 300 km scale[END_REF]. Mandibular and eye traits also vary with habitat complexity: longer apical teeth and smaller mandibles in relation to their overall length were recorded in open habitats at the ant community level [START_REF] Gibb | Does morphology predict trophic position and habitat use of ant species and assemblages?[END_REF][START_REF] Yates | Morphological traits: Predictable responses to macrohabitats across a 300 km scale[END_REF]. At this same scale, [START_REF] Salas-Lopez | Land-use change effects on the taxonomic and morphological trait composition of ant communities in French Guiana[END_REF] found that mandible length and width, apical tooth length, and eye size increased with land-use intensification. Moreover, in areas disturbed by fires, average colony size, worker size, worker polymorphism, and ratio of queen to worker size were greater than in unburned areas [START_REF] Arnan | Response of ant functional composition to fire[END_REF]. Ants with longer antenna scapes, which are important for perceiving the environment and locating resources or prey items [START_REF] Silva | Morphological patterns and community organization in leaf-litter ant assemblages[END_REF][START_REF] Weiser | Ecological morphospace of New World ants[END_REF], prevailed in habitats with dense herb/grass layer [START_REF] Nooten | Habitat complexity affects functional traits and diversity of ant assemblages in urban green spaces (hymenoptera: Formicidae)[END_REF].

4.2 | Ant traits relevant to restore ecological functions

| Abiotic component

The first trait to consider when restoring the abiotic component is the ant nesting habit, selecting ground-nesting species. Ants' capacity to restructure soil depends on their physical capacity to transport material [START_REF] Dostál | Ant-induced soil modification and its effect on plant below-ground biomass[END_REF]. During nest construction and maintenance, workers use their mandibles to transport mineral particles and soil pellets outside the nest. The size of their mandible gap, which is positively correlated with body size, influences the formation of texturecontrast soils [START_REF] Oliveras | Reduction of the ant mandible gap range after biotic homogenization caused by an ant invasion (hymenoptera: Formicidae)[END_REF] (Figure 1; Table 2), determining the size of the soil particles excavated. The larger the mandible gap, the more able ants are to transport large particles [START_REF] Dostál | Ant-induced soil modification and its effect on plant below-ground biomass[END_REF][START_REF] Martín-Perea | Messor barbarus ants as soil bioturbators: Implications for granulometry, mineralogical composition and fossil remains extraction in Somosaguas site (Madrid basin, Spain)[END_REF]. Nest depth can also influence the size of soil particles excavated. Such excavation increases soil (macro) porosity, which in turn alters the ratio between soil solid and air phases and leads to lower bulk densities [START_REF] Cammeraat | The impact of ants on mineral soil properties and processes at different spatial scales[END_REF].

Moreover, the size of ant nest biopores strongly affects water infiltration and soil erosion from water flowing overland (Lobry de [START_REF] Lobry De Bruyn | The effect of ant biopores on water infiltration in soils in undisturbed bushland and in farmland in a semi-arid environment[END_REF]. Higher soil pore volume in the nest mound promotes preferential flow and thus increases water content at greater depths [START_REF] Macmahon | Harvester ants (Pogonomyrmex spp.): Their community and ecosystem influences[END_REF]. Water infiltration is also related to nest architecture, particularly the sizes of nests and of nest entrances [START_REF] Macmahon | Harvester ants (Pogonomyrmex spp.): Their community and ecosystem influences[END_REF] (Figure 1; Table 2), which vary among ant species [START_REF] Cammeraat | The impact of ants on mineral soil properties and processes at different spatial scales[END_REF].

Ants also help to modify soil chemical properties. Their diet determines the magnitude of soil enrichment. Leaf-cutting and granivorous ants contribute more to soil fertility than omnivorous species, which suggests that the accumulation of food and waste generated from plant sources, for example, in ant refuse piles, may improve soil fertility more than that from animal sources [START_REF] Farji-Brener | The effects of ant nests on soil fertility and plant performance: A meta-analysis[END_REF].

This may be explained by the considerably higher amount of plant biomass than animal biomass in most terrestrial ecosystems and the differences between plants and animals (e.g., decomposition rate, nutrient composition). The enhancement of soil fertility in leaf-cutting and granivorous ant nests may also be due to their large colonies [START_REF] Farji-Brener | The effects of ant nests on soil fertility and plant performance: A meta-analysis[END_REF]Jonkman, 1980aJonkman, , 1980b)).

| Biotic component

Ants involved in seed dispersal directly influence plant community composition and structure. The spatial effects of seed dispersal may depend on the colony foraging strategy [START_REF] Lanan | Spatiotemporal resource distribution and foraging strategies of ants (hymenoptera: Formicidae)[END_REF][START_REF] Macmahon | Harvester ants (Pogonomyrmex spp.): Their community and ecosystem influences[END_REF]. Ant body size is a good predictor of seed dispersal distance [START_REF] Christian | Consequences of a biological invasion reveal the importance of mutualism for plant communities[END_REF][START_REF] Ness | Ant body size predicts dispersal distance of ant-adapted seeds: Implications of small-ant invasions[END_REF], which influences plant establishment, seed germination, and seedling growth [START_REF] Christian | Consequences of a biological invasion reveal the importance of mutualism for plant communities[END_REF][START_REF] Fernandes | Ant removal distance, but not seed manipulation and deposition site increases the establishment of a myrmecochorous plant[END_REF]. Larger species disperse seeds farther [START_REF] Espadaler | Seed dispersal curve of a Mediterranean myrmecochore: Influence of ant size and the distance to nests[END_REF][START_REF] Ness | Ant body size predicts dispersal distance of ant-adapted seeds: Implications of small-ant invasions[END_REF]. Beyond the attractiveness of seeds, their successful transport may depend on ants' mandible gap and thus on their physical ability to carry seeds of a specific morphology or size [START_REF] Oliveras | Reduction of the ant mandible gap range after biotic homogenization caused by an ant invasion (hymenoptera: Formicidae)[END_REF] (Figure 1; Table 2). Leaf-cutting ants, although they can disperse seeds [START_REF] Wirth | Seed dispersal by leaf-cutting ants[END_REF], have attracted more attention for their ability to carry leaves. Their body size and leg size determine leaf-transport rate, relatively long legs being advantageous [START_REF] Burd | Body size effects on locomotion and load carriage in the highly polymorphic leaf-cutting ants Atta colombica and Atta cephalotes[END_REF].

Ants' diet clearly determines their impact on the biotic component. Granivorous species produce more noticeable effects on vegetation, whereas predatory species have a stronger impact on fauna [START_REF] Offenberg | Ants as tools in sustainable agriculture[END_REF][START_REF] Vandermeer | Ecological complexity and pest control in organic coffee production: Uncovering an autonomous ecosystem service[END_REF]. Thus, granivorous ants are expected to have a greater effect at earlier restoration stages (colonization), while predators will have greater impacts at more advanced stages (population control). For predatory or generalist species, the colony foraging strategy and worker body size and mandible gap size determine prey size [START_REF] Cerdá | Predation by ants on arthropods and other animals[END_REF][START_REF] Oliveras | Reduction of the ant mandible gap range after biotic homogenization caused by an ant invasion (hymenoptera: Formicidae)[END_REF].

Length of mandibles being related to diet; longer mandibles lead to more predation events [START_REF] Parr | GlobalAnts: A new database on the geography of ant traits (hymenoptera: Formicidae)[END_REF] (Figure 1; Table 2).

Changes in soil physico-chemical properties and vegetation positively and indirectly influence above-and below-ground fauna T A B L E 2 Ants effect traits and their associated ecological functions on soil, vegetation and fauna components, and unit of measurement. Note: In the literature, some authors refer to the ant body size and others to their biomass. To discuss these two traits, as they are correlated, we have chosen to use only the more commonly used and more easily measurable trait "ant body size."

occurrence, abundance, and community structure [START_REF] Sanders | Habitat structure mediates top-down effects of spiders and ants on herbivores[END_REF][START_REF] Sanders | Ecosystem engineering and predation: The multi-trophic impact of two ant species[END_REF][START_REF] Schuch | Potential positive effect of the ant species Lasius niger on linyphiid spiders[END_REF]. In ant nests, the high rates of organic material accumulation increase food resources for detritivores and indirectly increase microhabitats for soil fauna through vegetation modification. Nests are also a refuge for myrmecophilous species such as lycaenid butterflies (e.g., Maculinea teleius (Bergsträsser, 1779) and M. nausithosus (Bergsträsser, 1779)) [START_REF] Wynhoff | Looking for the ants: Selection of oviposition sites by two myrmecophilous butterfly species[END_REF] or some beetles (e.g., Amidobia talpa (Heer, 1841) and Monotoma conicicollis (Aubé, 1838)) [START_REF] Päivinen | Species richness, abundance and distribution of myrmecophilous beetles in nests of Formica aquilonia ants[END_REF].

Other species are predators of the brood or adults, such as spiders from the genus Zodariinae (e.g., Zodarion elegans (Simon, 1873)) [START_REF] Traxler | The impact of predation by the myrmecophagous spider Zodarion elegans (Araneae: Zodariidae) on the activity pattern of the Mediterranean harvester ant Messor wasmanni (hymenoptera: Formicidae)[END_REF] or some beetles (e.g., Pella humeralis (Gravenhorst, 1806)) [START_REF] Päivinen | Species richness, abundance and distribution of myrmecophilous beetles in nests of Formica aquilonia ants[END_REF].

| Ant traits to consider to facilitate population introduction, establishment, and dispersion

Although in some cases, providing shelters (e.g., stones, wood logs) may be sufficient to reinforce populations by decreasing nesting site competition, when introduction is required, the operational phase will be facilitated by choosing a species easy to collect and introduce. The species should be common near the introduction site, generalist, and pioneer in the ecosystem dynamics. For example, in their study on brown coal mining spoil dumps in the Czech Republic, [START_REF] Holec | Ant (hymenoptera: Formicidae) communities in reclaimed and unreclaimed brown coal mining spoil dumps in The Czech Republic[END_REF] showed that two species, L. niger and Manica rubida, have a marked ability to colonize new habitats and are considered to be pioneer species in this context. Given the negative impact that some exotic ant species can have on their environment (e.g., on vegetation [START_REF] Christian | Consequences of a biological invasion reveal the importance of mutualism for plant communities[END_REF], native ant diversity [START_REF] Wittman | Impacts of invasive ants on native ant communities (hymenoptera: Formicidae)[END_REF], arthropods, and vertebrates [START_REF] Lach | Consequences of ant invasions[END_REF])), we highly recommend choosing native species.

The number of queens per colony and the dispersal method are also important. Collecting queens from colonies that disperse by budding may require digging out the pre-existing colony, which may degrade the donor site. With species that reproduce during nuptial flight, collecting mated queens should be easier and have less impact on the donor site.

As in any introduction process, particular attention must be paid to species involved in hybridogenesis. In this reproductive system, known as "social hybridization" and varying among species [START_REF] Lavanchy | Hybridogenesis[END_REF][START_REF] Romiguier | Convergent evolution of social hybridogenesis in Messor harvester ants[END_REF], workers result from the hybridization of parents of different genetic origin. It is thus important that both lineages are still present on the site (e.g., collection and introduction of both lineages) to ensure successful population growth.

To avoid a population collapse in the years following introduction, nesting habit (hypogaeic, under stones, dead wood, litter, arboreal) is a key factor to ensure species establishment, particularly if shelters have to be created (e.g., dead wood and stone cover). Overall, the environmental conditions of the degraded site need to fulfil the species' requirements, such as thermal and moisture habitat preferences. Territoriality must also be considered to ensure that the candidate species will persist in coexistence with the species already present. Species reproducing by mating flight, although they take more time to build a stable and mature colony than species reproducing by budding, can spread over greater distances. Finally, choosing polymorphic specieswith intraspecific variation in worker body size-may be a solution.

Worker polymorphism has been related to the breadth of functional roles performed by colonies [START_REF] Arnan | Worker size-related task partitioning in the foraging strategy of a seedharvesting ant species[END_REF][START_REF] Mertl | Behavioral evolution in the major worker subcaste of twig-nesting Pheidole (hymenoptera: Formicidae): Does morphological specialization influence task plasticity?[END_REF] and implies greater variability of responses to temperature, which increase the overall period of external activity, and enhances colony success [START_REF] Cerda | Links between worker polymorphism and thermal biology in a thermophilic ant species[END_REF]. As most incipient colonies from mated queens have only nanitic workers, polymorphism is only considered an advantage after the first phase of colony growth.

| CONCLUSION

To develop trait-based approaches in ecological restoration, it is essential to acquire general knowledge of the organisms to use, the functions they performed, the traits related to the functions, and ecosystem services. While the current knowledge on the ecological functions of ants has already been extensively studied as shown here, we have pointed out a list of key traits in ants that are valuable for environmental monitoring and/or that are related to the different functions targeted in restoration projects.

However, some limitations remain and may constrain a wide use of a trait-based approach. The access to ant trait data is one of them.

Indeed, to date, only a few ant databases have been developed including GABI project, which focuses on distribution records [START_REF] Guénard | The global ant biodiversity informatics (GABI) database: Synthesizing data on the geographic distribution of ant species (hymenoptera: Formicidae)[END_REF], and "GlobalAnts," which focuses on ant traits with associated georeferenced assemblage-level data [START_REF] Parr | GlobalAnts: A new database on the geography of ant traits (hymenoptera: Formicidae)[END_REF]. While these databases appear to be interesting and powerful tools in ecological restoration, the large amount of missing data is a limit to their current use [START_REF] Auclerc | Fostering the use of soil invertebrate traits to restore ecosystem functioning[END_REF]. A first step should be to complete the data on traits by focusing on the most important for ecological restoration. Future studies need to address this gap across a wide range of functional traits both at the individual and colony level.

  actively used ants to restore a degraded ecosystem. In 2011, Bulot et al. (2014) reintroduced mated queens of the granivorous ant Messor barbarus (Linnaeus, 1767) in the Plain of la Crau (South of France) to accelerate the restoration of soil physico-chemical properties and vegetation (Box 1).

  . Mated queens were reintroduced at 10-m intervals to reduce the risk of competition once colonies were established. It took about 1 h for four people to collect 600 founding queens of M. barbarus and 2 days to reintroduce them on the Crau plain (Bulot et al., 2014). Seven years after introduction of mated queens, M. barbarus improved soil fertility, drove the soil seed bank towards the reference ecosystem, and significantly increased plant biomass, species richness, and micro-local heterogeneity. By directly and indirectly facilitating plant establishment, M. barbarus accelerated grassland restoration compared to controls without M. barbarus nests, trails, or refuse piles(De Almeida, Blight, et al., 2020).

  Monitoring ant body size, by measuring the traits of individuals from local populations, as well as diet might provide information on the quantity and quality of resources available in a degraded or restored habitat. Although traits linked to feeding behaviors can reveal a species' response to environmental changes, traits directly related to a species' tolerance of environmental changes might be more F I G U R E 1 Main ant individual and colony traits relevant in restoration projects. (a) A slightly degraded ecosystem requires first removal of disturbances, then monitoring of community and population changes to ensure that restoration of all target functions is complete or ongoing. (b) A more intensely degraded ecosystem requires only the biotic component functions to be restored after disturbance removal. (c) A deeply degraded ecosystem (abiotic and biotic components) requires, after disturbance removal, first restoration of soil functions, followed by restoration of biotic component functions. In cases b and c, ant introduction will be effective only (i) when conditions for their establishment and persistence are met or (ii) if they are able to forage at an undisturbed site nearby. For example, granivorous ants should be introduced after the site has been colonized by vegetation or at a distance from an undisturbed site that does not exceed their foraging capacity. After the restoration of each component, community, and population changes also require monitoring. [Colour figure can be viewed at wileyonlinelibrary.com]

ACKNOWLEDGMENTS

This work was supported by Région Sud-PACA (Ph.D. grant) and the Tour Du Valat Research Institute. Thanks to Marjorie Sweetko for reviewing the English of the paper.

DATA AVAILABILITY STATEMENT

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

CONFLICT OF INTEREST STATEMENT

There is no conflict of interest.


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