Aurelien Tellier

Lawrence Uricchio

Inference of genome-wide processes using temporal population genomic data

Recommendation

) provides an additional method in the population genomics toolbox to quantify the extent of linked positive and negative selection using temporal data.

 or Deep Learning (Sanchez et al. 2021). These methods are based on one sample in time and the use of the coalescent theory to reconstruct the past (demographic) history. However, it is

also possible to obtain for many species temporal data sampled over several time points. For species with short generation time (in experimental evolution or monitored populations), one can sample a population every couple of generations as exemplified with Drosophila melanogaster (Bergland et al. 2010). For species with longer generation times that cannot be easily regularly sampled in time, it becomes possible to sequence available specimens from museums (e.g. [START_REF] Cridland | Genome Sequencing of Museum Specimens Reveals Rapid Changes in the Genetic Composition of Honey Bees in California[END_REF] or ancient DNA samples. Methods using temporal data are based on the classical population genomics assumption that demography (migration, population subdivision, population size changes) leaves a genome-wide signal, while selection leaves a localized signal in the close vicinity of the causal mutation. Several methods do assess the demography of a population (change in effective population size, Ne, in time) using temporal data (e.g. [START_REF] Jorde | Unbiased Estimator for Genetic Drift and Effective Population Size[END_REF] which can be used to calibrate the detection of loci under strong positive selection (Foll et al. 2014). Recently [START_REF] Buffalo | Estimating the genome-wide contribution of selection to temporal allele frequency change[END_REF] used genome-wide covariance between allele frequency changes across time samples (and across replicates) to quantify the effects of linked selection over short timescales.

In the present contribution, [START_REF] Pavinato | Joint inference of adaptive and demographic history from temporal population genomic data[END_REF] make use of temporal data to draw the joint estimation of demographic and selective parameters using a simulation-based method (ABC-Random Forests). This study by [START_REF] Pavinato | Joint inference of adaptive and demographic history from temporal population genomic data[END_REF] builds a framework allowing to infer the census size of the population in time (N) separately from the effect of genetic drift, which is determined by change in effective population size (Ne) in time, as well estimates of genome-wide parameters of selection. In a nutshell, the authors use a forward simulator and summarize genome data by genomic windows using classic statistics (nucleotide diversity, Tajima's D, FST, heterozygosity) between time samples and for each sample. They specifically use the distributions (higher moments) of these statistics among all windows. The authors combine as input for the ABC-RF, vectors of summary statistics, model parameters and five latent variables: Ne, the ratio Ne/N, the number of beneficial mutations under strong selection, the average selection coefficient of strongly selected mutations, and the average substitution load. Indeed, the authors are interested in three different types of selection components: 1) the adaptive potential of a population which is estimated as the population mutation rate of beneficial mutations (θb), 2) the number of mutations under strong selection (irrespective of whether they reached fixation or not), and 3) the overall population fitness which is a function of the genetic load. In other words, the novelty of this method is not to focus on the detection of loci under selection, but to infer key parameters/distributions summarizing the genome-wide signal of demography and (positive and negative) selection. As a proof of principle, the authors then apply their method to a dataset of feral populations of honey bees (Apis mellifera) collected in California across many years and recovered from Museum samples [START_REF] Cridland | Genome Sequencing of Museum Specimens Reveals Rapid Changes in the Genetic Composition of Honey Bees in California[END_REF]. The approach yields estimates of Ne which are on the same order of magnitude of previous estimates in hymenopterans, and the authors discuss why the different populations show various values of Ne and N which can be explained by different history of admixture with wild but also domesticated lineages of bees.

This study focuses on quantifying the genome-wide joint footprints of demography, and strong positive and negative selection to determine which proportion of the genome evolves neutrally or not. Further application of this method can be anticipated, for example, to study species with ecological and life-history traits which generate discrepancies between census size and Ne, for example for plants with selfing or seed banking [START_REF] Sellinger | Inference of past demography, dormancy and self-fertilization rates from whole genome sequence data[END_REF], and for which the genomewide effect of linked selection is not fully understood.

Reviewed by Lawrence Uricchio, 07 Oct 2022

I appreciate the author's responses to the first round of reviews. I do think the manuscript is improved, and remains quite interesting.

I have two remaining comments, which correspond to aspects of their reply. They are really quite minor, although they take some space to explain.

1. The spirit of my comment about including some additional discussion of Poisson Random Field and MK-based work is because many readers will naturally compare this approach to PRF, which is still very widely used for inferring similar models. I agree that PRF is less likely to be useful for such short time periods (~10 generations), and that it doesn't account for linked selection. But it can infer genome-wide selection and demographic parameters. Many readers looking at your Figure 1 will wonder specifcally how/when your method improves on PRF since it solves a model that is (conceptually) identical to models that were solved with PRF (e.g. Boyko et al 2008). Under which scenarios should we prefer an approch like yours to PRF? Highliting the deficiencies of SFS-based analyses for short timescale changes in population size and stating when specifically PRF-based approaches will fail due to linkage would be helpful. How many generations is too few to infer an expansion (as in your model) with PRF? How much linked selection is needed before the problem is severe with PRF? Even a couple of sentences would be helpful.

2. The authors write in their reply "Therefore, we are more interested in the short term effects of negative selection on allele frequency changes than in long term effects on the site frequency spectrum." I think the issue here is that the allele frequency dynamics themselves are not independent of the population history in burn-in. With low recombination and a high rate of beneficial alleles, the dynamics may look more like clonal interference. If there is a lower rate of positive selection but high backgriound selection, the trajectories of the beneficial alleles may be subject to less perturbation by interference than an equivalent reduction in Ne due to positive selection, but the effect depends on the selection coefficients. I appreciate the section on limitations that discusses background selection, but it is not very clear in the manuscript how this will affect inference under this model. Some discussion of the ways that negative seleciton might affect the parameter inference would be helpful.

Reviewed by anonymous reviewer, 13 Oct 2022

The authors have mostly addressed my concerns. However, I was taken aback by the reluctance to estimate the robustness to deleterious mutations given how easy it is to simulate nowadays. It would have been straightforward to do. In their defense, the authors have added discussion about this potential limitation, in particular suggesting that heterogenous background selection and deleterious mutations could be problematic. The approach has potential, so I do hope that the authors explore these questions in the future to validate the goals behind their approach. -l 126, I think you should also add the sample size here.

Evaluation round

-l 127-128, from the text it seems like you sampled individuals at time t1 and t2, with tau = t2-t1. But from Figure 1 A), it seems like you have been sampling between t1 and t2. In addition, tau is missing on the Figure 1. I would change Figure 1 A) by remplacing "sampling period" by "tau".

-l 133, I would change "c0" to "r0" for the recombination rate as I think it's a more classic notation.

-l 225-249, I do not think all variables have been clearly defined in the previous sections and there might be some inconsistent notations, e.g. is r="c0" ? I think it would help readers if the authors moved Table S1 into the main text (but r is missing from table S1).

-l 285, Once the paper is accepted/recommended please add a link here in the text to the individual VCF files.

-Table 1, what is written in column N (and add it to the table description) ? -l 290-309, I didn't find where the sample size you are using was written. Could you add it in the text or in Table 1 ? Results :

-Figure 4, I think there is a typo in the legend, c) and a) should be exchanged.

-Figure 4, I think having a small table summarizing Figure 4 with the most likely (or whith highest density) ratio of Ne/N would help in understanding the results. Discussion :

-l 440-498, The authors discuss the similarity between Ne and N and they mention the possibility of excluding lower values of theta_b indicating acting selection during the study (by giving explanation on why signature of selection might be harder to find). Could it really not be neutral ? -I don't know if authors have read the recent paper of Barroso et al (https://doi.org/10.1101/2021.09.16.460667) on the inference of the mutation rate map from whole genome data. In their study they interpret the variation of the Ne along the genome as a the variation of the mutation rate. However, if you assume the mutation to be constant along the genome, then they would infer the variation of Ne along the genome. I wonder if the inferred mutation rate map would be an interesting summary statistic for you ABC-rf ? I would recommand the authors to read the study as it could be relevant in the light of their work (and maybe add it to the discussion).

Reviewed by anonymous reviewer, 14 Jan 2022

In their preprint, Pavinato et al. describe an ABC Random Forest approach to jointly infer demography and selection in population genomics, two-time points temporal data. Overall I like the approach a lot, especially the fact that it can be extended to more realistic situations that for example include deleterious mutations. Reading the preprint I had an issue with semantics, that has to do with the fact that the authors do not really infer demography per se, but rather provide an approach that allows to estimate selection while also jointly estimating the influence of "whatever" past demography occurred. This is different from what most people in the field understand by "inferring demography", which usually means explicitly inferring population size changes over time. This needs to be made clearer throughout the manuscript.

I have no concern with the overall approach that does sound very promising. However, I have two main concerns regarding the practical usability when faced with actual datasets that will likely present 1) recombination rate heterogeneity and 2) varying levels of deleterious mutations across loci.

1) Recombination. The authors use uniform recombination in their approach. We know however that many genomes have highly heterogeneous distributions of recombination events, and those even often co-vary with with functional elements in genomes. For example in human genomes, recombination hotspots tend to co-coccur together with regulatory elements. This is an example where not accounting for this heterogeneity is a big issue given that any approach with uniform recombination will then likely underestimate adaptation that occurs in these functional elements. The authors need to test with simulations with heterogeneous recombination how much their approach is affected, especially when the recombination hostpots tend to occur near where the adaptation is expected in the first place.

2) Deleterious mutations. In the same spirit, instead of just discussing very briefly about this limitation, the authors could actually run a few simulation to see how their approach performs when the test, simulated genomes actually include deleterious mutations. What happens when the deleterious mutations are distributed evenly across loci, and what happens when the deleterious mutations are heterogeneously distributed across loci. The deviations from the expected values especially for adaptation should then provide more solid ground to discuss the limitations in much more detail than it is done at the moment. I am suggesting thes two things (recombination and deleterious mutations) since it would not take too many additional simulations with SLIM to get a more precise sense of how not accounting for them biases the estimates.

In addition I just have some minor comments about a few things that need to be better defined in the Introduction. First, the authors need to better describe how Random Forest works and why this works better in the context of ABC. Second, the authors need to better define early in the introduction what latent variables are, how they work, and what their usefulness is in the approach.

Reviewed by Lawrence Uricchio, 12 Jan 2022

"Joint inference of adaptive and demographic history from temporal population genomic data" by Pavinato et al explores the inference of demographic parameters and selection parameters using random forests and Approximate Bayesian Computation. They present an analysis of their simulation-based inference procedure as well as an application of their method to genomic timeseries data from honey bees. The manuscript addresses an interesting and important problem, and is overall clear and well-written. I very much appreciated the thorough investigation of the performance of their method. I have some questions and suggestions for the authors. I want to emphasize that although I have several comments, I do very much appreciate the authors' work and found the manuscript quite interesting.

Main comments:

1. Overall I found the descriptions of the analyses to be clear, but how they fit with the existing literature on joint inference of selection and demography was sometimes less clear. Some important areas of the literature on joint selection/demography inference are not mentioned in the manuscript (described more in the next point). It would be very helpful if the authors could further clarify how the approach that they take (random forests with ABC) provides an advance over the range of previous approaches. Their method aims to directly account for the effects of linked selection (which I agree is an important goal), but whether their method actually performs better in a practical sense than methods that do not explicitly model linked selection does not seem to be assessed. The effect of the random forests on the performance relative to ABC without the inclusion of the random forests is also not mentioned, although this may be a less feasible comparison given computational efficiency limitations.

2. The authors state in the introduction that "Methods for demographic inference assume that most of the genome evolves without the influence of selection and that any deviation from the mutationdrift equilibrium observed in the data was caused by demographic events". I agree in the sense that the effects of linked selection are often ignored in such analyses, but there are many studies that attempt to disentangle selection/demography by dividing the genome into putatively selected sites and putatively neutral sites.

Such methods are derived from the PRF (Poisson Random Field) and/or MK (McDonald-Kreitman) framework and assume that demographic events shape patterns of variation at putatively neutral sites (often taken as synonymous alleles within genes) while both selection and demography shape putatively selected sites (sometimes taken as nonsynonymous alleles within genes). Patterns of variation at the "neutral" sites are used to fit the demographic model while the "selected" sites are used to fit the selection model. There are many such papers (e. The realization that putatively neutral sites may not be sufficiently 'neutral' for such analyses (due to direct or indirect selection) has also been a long-standing topic of discussion, and some studies have sought to address this issue. For example, Gazave et al 2014 (PNAS) sequenced regions far from genes in humans to try to fit demographic models that were less affected by selection. Torres, Szpiech, & Hernandez 2018 (Plos genetics) found that background selection has a substantial influence on demographic inference in humans. Messer & Petrov 2013 (PNAS) developed a method to infer adaptation rate and tested its robustness to linked selection and non-equilibrium demography. There are is also at least one paper that seeks to infer recurrent selection and demography using a combination of ABC and PRF methods (N. Singh et al 2013 Genetics, "Inferences of demography and selection...")

The effects of non-equilibrium demography on selection inferences using genetic time series data

  #1 DOI or URL of the preprint: https://doi.org/10.1101/2021.03.12.435133 Version of the preprint: 2 Author's Reply, 07 Sep 2022 Download author's replyDownload tracked changes file Introduction -l 50, I would define "type I error rate" in the text for clarity -l 80-95, you mention your methods to be accurate and that random forest has solved the computation issue, yet you didn't presented your input data. It would be usefull to have a sentence/paragraph describing the input data. Methods -l 107, if you assume a constant population size of N, please mention it more clearly here.

  g. one could start with Williamson et al 2005 PNAS and Keightley & Eyre-Walker 2007 Genetics, or a more recent paper such as Racimo & Schraiber 2014 Plos Genetics).

Conflict of interest:

The recommender in charge of the evaluation of the article and the reviewers declared that they have no conflict of interest (as defined in the code of conduct of PCI) with the authors or with the content of the article. The three reviewers and myseld acknowledge the efforts you have made in preparing this revision which addresses satisfactorily most critical points and comments. The reviewers have few last minor comments for you to incorporate before I can proceed with the recommendation. I anticipate that this can be done relatively quickly. I anticipate to evaluate your reply to these comments myself.

Reviews

Best regards and sorry for the delay in obtaining all reviews.

Aurelien Tellier

Reviewed by anonymous reviewer, 14 Sep 2022

Summary of the manuscript:

The authors developp an ABC-random forest approach to simultaneously infer census population size (N), effective population size (Ne) and the mutation rate of benefical mutations (theta_b) from temporal population genomic data. To do so they simulate genomic data under selection using Slim to train their ABC-rf and then demonstrate the accuracy of their approach on simulated data in different cases. They then apply their approach on genomic data of different bee population which were collected at different time points. The authors find effective population size very similar to census size and further discuss this result.

General comments :

In this version of the manuscript, I found the material and method section clearer. Some pointed limits/issues were not addressed but I align with the editor's decision. I understand the authors could not addressed all comments as it would be an unreasonable amount of work and go beyond the scope of this study. Hence, I'm happy the authors mentioned that they will try to address some of the comments in a future project/manuscript, which I am looking forward to read.

Minor comments :

-l 308-309, Please add a link here to the VCF files used in this study (e.g. https://github.com/mnavascues/ABC_TimeAdapt/tree/master/data). I am sorry if I overlooked how to access to the data, and authors can ignore my comment. Yet, I looked in the manuscript authors have referred to, but could also not find the mentionned VCF files (I could only find how to access the raw reads). If authors downloaded the data, I would appreciate if the link was written in the manuscript. If the authors were given the data, I would appreciate if they would make the VCF files accessible for everyone. By sharing the data, other people can try their method on a dataset, participating the diffusion of their work and of science. I believe sharing the VCF files would fit with the spirit and the recommandation guidelines of PCI. I am also okay if the files are shared after publication of the paper and/or acceptance in PCi (but it would be nice to state it clearly).

Decision by Aurelien Tellier, posted 17 Jan 2022

Dear authors, I am sorry for the delay in the evaluation of your manuscript. I had trouble finding reviewers at first, but then found three very enthusiastic reviewers, but the end of the year holidays delayed their reviews. Your manuscript is very interesting and I hope you will consider submitting a revised version. You will find that all reviewers are very enthusiastic about the manuscript and very supportive (see attached reviews). The reviewers evaluated thoroughly your manuscript and suggest a number of minor changes to improve clarity of the methods and results. These should be easily doable. In addition, two reviewers suggest potential additional simulations to test for the effect of heterogeneous recombination, heterogeneity of deleterious mutations in genes and demography on the accuracy of your estimations. While I am aware that it would require a substantial amount of work to do all these tests, I would recommend to test by simulations the effect of heterogeneous recombination on your method accuracy. The other potential biases can be dealt with in the discussion (you will find also some useful suggestions to improve the discussion and add some additional references).

I look forward to receive your revised version and will make sure the review process get faster in the next round, Many thanks for submitting to PCI, Aurelien Tellier

Reviewed by anonymous reviewer, 08 Dec 2021

Summary of the manuscript:

The authors developp an ABC-random forest approach to simultaneously infer census population size (N), effective population size (Ne) and the mutation rate of benefical mutations (theta_b) from temporal population genomic data. To do so they simulate genomic data under selection using Slim to train their ABC-rf and then demonstrate the accuracy of their approach on simulated data. They then apply their approach on genomic data of different bee population which were collected at different time points. The authors find effective population size very similar to census size and further discuss this result.

General comments :

I found the manuscript interesting and I enjoyed reading it. The authors try to answer a question which is central in the field of population genetics and of conservation biology. However I believe the manuscript can be inmproved by addressing two points. First, I think the Method section is a little bit confusing (see below for further details) and the Material is not well described. Secondly, depending on the recommender's decision, I think few additional analyses could complete their study.

Major comments :

-I find the ABC-rf to be well defined but not the data on which it was applied. In the main text (or in the appendix) I could not find basic information on the input data such as sample size of simulations or of the data (or if sample size at t1 was equal to sample size at t2, which does not seem to be the case according to what is written l 442). I believe the authors should add a table (in appendix or main text) describing what they simulated and the data they have used. Currently, the input data is very cryptic and I'm not sure results could be reproduced from the main text only.

With the recommender's approaval I would also address those two points :

-I think one or two supplementary analyses are missing. The Authors assume census population size to be constant in presence of selection (if I understood correctly). I think it's necessary to run have also been studied (E.g., Schraiber et al 2016 (Genetics) uses an MCMC-based procedure and compares equilibrium/non-equilibrium demography). This is just a subset of the papers in this area, and it's not my intent to be prescriptive about citing particular papers. Overall, it would be very helpful for the reader if the authors could put their method into context with this body of work.

3. The authors focus on a model of beneficial alleles and neutral alleles. I found this a bit surprising, given that negatively selected alleles are likely to represent a much larger fraction of mutations and have a substantial effect on the frequency spectrum (and other summary statistics).

The authors discuss the impact of background selection briefly in the discussion, but I think further justification for the relevance of this beneficial allele model (or simulation of a model that includes deleterious alleles, or an argument as to why negative selection will have limited impact on the authors' summary stats) would be helpful here. It is true that some recent papers (e.g. Schrider, Shanku & Kern 2016 Genetics) have focused on positive selection models, but their purpose was to describe the effects of positive selection on demographic inference directly, rather than develop inference procedures. It seems unlikely to me that a model of beneficial alleles alone can provide useful inferences on real data that are certainly affected by a mixture of beneficial alleles and deleterious alleles. Alternatively, if there are specific insights that a beneficial allele model can provide then it would help to further highlight these. I do appreciate that the authors were clear about this limitation in their discussion section.

Minor comments:

-Line 71: Is the '10,000' here years? Or generations? -Line 80: Another paper that may be of interest here is A. Stern et al Plos Genetics (2019) -Line 90: AABC is also a way to reduce the number of simulations, see Buzbas & Rosenberg 2015 Theoretical Population Biology.

-Line 120: I'm a bit confused by this description of burn-in. Generally the purpose is to reach the dynamic equilibrium state that is determined by the parameter settings.

-Figure 1: Why are there two different colors for the neutral mutations? Do they represent different things?

-Selection is determined by a Gamma distribution here, which has two parameters. Only the mean of the distribution seems to be mentioned here, which seems to leave one free parameter. How are these parameters selected (perhaps this is mentioned and I missed it)? -Line 190: I think 'oscillated' may not be the appropriate word here, perhaps fluctuated? -Line 231: 10 generations seems like a very short time? What is the rationale for this number? -Line 293: The transition from discussing real to simulated data seems very abrupt here? I thank the authors and editors for the opportunity to review this very interesting manuscript.

Sincerely, Lawrence Uricchio


Kimon Keramidas
email: kimon.keramidas@univ-grenoble-alpes.fr

Silvana Mima

Adrien Bidaud

Blue and turquoise hydrogen: bridge options or mirage?

MOTIVATION -OBJECTIVE -Hydrogen (H2): promising vector in the race to reach to net-zero -Many announced national strategies & investments: RePowerEU, Japanese H2 Strategy… -Current production based on gas and coal (70 MtH2, 750 MtCO2) -How to supply large enough quantities of H2 while making it a lower-GHG vector than the fuels it replaces?

TECHNOLOGIES for H2 production:

-Green H2 (wind/solar + electrolysis): decreasing costs in renewable electricity and electrolysis technologies, but needs H2 buffer storage -Pink H2 (nuclear + electrolysis): relevant only for countries with large nuclear power deployment plans -Blue H2 (steam methane reforming with CCS): challenges of CO2 capture, transport and longterm sequestration -Turquoise H2 (gas/biomass pyrolysis): no emissions, no impact of carbon price; possibility of using existing supply infrastructure; black carbon reuse

RESULTS

METHODS

-POLES: global energy system model (multi-regional partial equilibrium) -results here for USA -Endogenous modelling of H2 demand for energy uses and industry feedstock + 13 H2 production technologies -Scenarios:

  1.5C (global carbon price) + 1.5C with 20-year delay in CCS deployment -Turquoise: nearly as competitive as blue (needs both fossil gas and electricity) -Green: becomes cost-competitive only long term with climate policy 1. Direct emissions: fuel combustion, CH4 reforming  mostly captured in blue (max 95% capture rate) 2. Indirect emissions: upstream fossil fuels supply chain  partly reduced with climate policy 3. Indirect emissions: electricity production  reduced to nearly zero by 2050 with climate policy Total emissions content vs fossil gas, USA: 2020: blue 3x less, turquoise same / 2050: blue 9x less, turquoise 20x less 2050 delayed CCS > CONCLUSIONS & POLICY RECOMMENDATIONS -Turquoise H2 potential bridge, subject to gas market and geopolitical conditions; could get credits from selling carbon black -Blue H2 presents additional risk issues related to CCS infrastructure -Green H2 only true low-GHG option, long-term -Investment in blue/turquoise H2 risks resulting in gas supply infrastructure lock-in -Early investment in green H2 will allow learning and cost reductions


Pierre Hyvernat
email: hyvernat@iml.univ-mrs.fr

Synchronous Games, Simulations and λ-calculus

We refine a model for linear logic based on two well-known ingredients: games and simulations. We have already shown that usual simulation relations form a sound notion of morphism between games; and that we can interpret all linear logic in this way. One particularly interesting point is that we interpret multiplicative connectives by synchronous operations on games. We refine this work by giving computational contents to our simulation relations. To achieve that, we need to restrict to intuitionistic linear logic. This allows to work in a constructive setting, thus keeping a computational content to the proofs. We then extend it by showing how to interpret some of the additional structure of the exponentials. To be more precise, we first give a denotational model for the typed λ-calculus; and then give a denotational model for the differential λcalculus of Ehrhard and Regnier. Both this models are proved correct constructively.

Introduction

Transition systems and simulation relations are well known tools in computer science. More recent is the use of games to give models for different programming languages [1,[START_REF] Hyland | On full abstraction for PCF: I, II and III[END_REF]2], or as an interesting tool for the study of other programming notions [START_REF] Sansom Abramsky | Applying game semantics to compositional software modelling and verification[END_REF]. We have devised in [START_REF] Hyvernat | Synchronous games, deadlocks and linear logic[END_REF] a denotational model of linear logic based on those two ideas. Basically, a formula was interpreted by an alternating transition system (called an interaction system) and a proof was interpreted by a safety property for this interaction system. Those concepts which were primarily developed to model imperative programming and interfaces turned out to be a rather interesting games model: a formula is interpreted by a game (the interaction systems), and a proof by a "non-loosing strategy" (the safety property).

Part of the interest is that the notion of safety property is very simple: it is only a subset of the set of states. However, in terms of games, the associated strategy (whose existence is guaranteed by the condition satisfied by the subset of states) is usually not computable. We will show that it is possible to overcome this problem by restricting to intuitionistic linear logic. More precisely, we will model typed λ-calculus (seen as a subsystem of intuitionistic linear logic) within a constructive setting. The model for full intuitionistic linear logic (ILL) can easily be derived the present work and the additive connectives defined in [START_REF] Hyvernat | Synchronous games, deadlocks and linear logic[END_REF].

The structure of safety properties is in fact richer than the structure of λterms. In particular, safety properties are closed under unions. Since there is no sound notion of "logical sum" of proofs, this doesn't reflect a logical property. However, it is important in programming since it can be used to interpret non-determinism. The differential λ-calculus of Ehrhard and Regnier ( [START_REF] Ehrhard | The differential lambda calculus[END_REF]) is an extension to the λ-calculus, which has a notion of non deterministic sum. We show how to interpret this additional structure.

1 Interaction Systems

The Category of Interaction Systems

We briefly recall the important definitions. For more motivations, we refer to [START_REF] Hancock | Programming as applied basic topology[END_REF] and [START_REF] Hyvernat | Synchronous games, deadlocks and linear logic[END_REF]. Definition 1. Let S be a set (of states); an interaction system on S is given by the following data:

for each s ∈ S, a set A(s) of possible actions; -for each a ∈ A(s), a set D(s, a) of possible reactions to a; -for each d ∈ D(s, a), a new state n(s, a, d) ∈ S.

We usually write s[a/d] instead of n(s, a, d).

Following standard practise within computer science, we distinguish the two "characters" by calling them the Angel (choosing actions, hence the A) and the Demon (choosing reactions, hence the D). Depending on the authors' background, other names could be Player and Opponent, Eloise and Abelard, Alice and Bob, Master and Slave, Client and Server, System and Environment, etc.

One of the original goals for interaction systems (Hancock) was to represent real-life programming interfaces. Here is for example the interface of a stack of booleans:

-S = List(B); -A( ) = {Push(b) | b ∈ B} ∪ {Pop}; -    D( , Push(b)) = { * } D([], Pop) = {error} D(b : s, Pop) = { * } -    n(s, Push(b)) = b : s n([], Pop, error) = [] n(b : s, Pop) = s
This gives in full details the specification of the stack interface. This is more precise than classical interfaces which are usually given by a collection of types: compare with this poor description of stacks:

-Pop : B -Push : B → () which doesn't specify what the command actually do; but only tells how they can be used.

The notion of morphism between such interaction systems is an extension of the usual notion of simulation relation: Definition 2. If w 1 and w 2 are two interaction systems on S 1 and S 2 respectively; a relation r ⊆ S 1 × S 2 is called a simulation if:

(s 1 , s 2 ) ∈ r ⇒ ∀a 1 ∈ A 1 (s 1 ) ∃a 2 ∈ A 2 (s 2 ) ∀d 2 ∈ D 2 (s 2 , a 2 ) ∃d 1 ∈ D 1 (s 1 , a 1 ) s 1 [a 1 /d 1 ], s 2 [a 2 /d 2 ] ∈ r .
This definition is very similar to the usual definition of simulation relation between labelled transition systems, but adds one layer of quantifiers to deal with reactions. That (s 1 , s 2 ) ∈ r means that "s 2 simulates s 1 ". By extension, if a 2 is a witness to the first existential quantifier, we say that "a 2 simulates a 1 ". Note that the empty relation is always a simulation. In practise, to prevent this degenerate case, we would add a notion of initial state(s) and require that initial states are related through the simulation.

To continue on the previous example, programming a stack interface amounts to implementing the stack commands using a lower level interface (arrays and pointer for examples). If we interpret the quantifiers constructively, this amounts to providing a (constructive) proof that a non-empty relation is a simulation from this lower level interaction system to stacks. (See [START_REF] Hancock | Programming as applied basic topology[END_REF] for a more detailed description of programming in terms of interaction systems.)

Recall that the composition of two relations is given by:

(s 1 , s 3 ) ∈ r 2 • r 1 ⇔ (∃s 2 ) (s 1 , s 2 ) ∈ r 1 and (s 2 , s 3 ) ∈ r 2
It should be obvious that the composition of two simulations is a simulation and that the equality relation is a simulation from any w to itself. Thus, we can put: Definition 3. We call Int the category of interaction systems with simulations.

Note that everything has a computational content: the composition of two simulations is just given by the composition of the two "algorithms" simulating w 3 by w 2 and w 2 by w 1 ; and that the algorithm for the identity from w to w is simply the "copycat" strategy.

Notation

Before diving in the structure of interaction systems, let's detail some of the notation.

-An element of the indexed cartesian product a∈A D(a) is given by a function f taking any a ∈ A to an f (a) in D(a). When the set D(a) doesn't depend on a, it amounts to a function f : A → D.

-An element of the indexed disjoint sum a∈A D(a) is given by a pair (a, d) where a ∈ A and d ∈ D(a). When the set D(a) doesn't depend on a, this is simply the cartesian product A × D. -We write List(S) for the set of "lists" over set S. A list is simply a tuple (s 1 , s 2 , . . . s n ) of elements of S. The empty list is denoted ().

-The collection M f (S) of finite multisets over S is the quotient of List(S) by permutations. We write [s 1 , . . . s n ] for the equivalence class containing (s 1 , . . . s n ). We write "+" for the sum of multisets. It simply corresponds to concatenation on lists.

Concerning the product and sum operators, it should be noted that they have a computational content if one works in a constructive setting: an element of a∈A D(a) is an algorithm with input a ∈ A and output f (a) ∈ D(a); and an element of a∈A D(a) is simply a pair as above. This is in fact the basis of dependent type theory frameworks like Martin-Löf's type theory or the calculus of construction. Remark: even if it was an important motivation for this work, we do not insist too much on the "constructive mathematics" part. Readers familiar with constructive frameworks should easily see that everything makes computational sense; and classical readers can skip the comments about computational content.

Constructions

We now define the connectives of multiplicative exponential linear logic. With those, making Int into a denotational model of intuitionistic multiplicative exponential linear logic more or less amounts to showing that it is symmetric monoidal closed, with a well behaved comonad.

Constant. A very simple, yet important interaction system is "skip", the interaction system without interaction. Following the linear logic convention, we call it ⊥: Definition 4. Define ⊥ (or skip) to be the following interaction system on the Singleton set { * }:

A ⊥ ( * ) = { * } D ⊥ ( * , * ) = { * } n ⊥ ( * , * , * ) = { * } .
Depending on the context, this interaction system is also denoted by 1.

Note that it is very different from the two following interaction systems (on the same set of states) which respectively deadlock the Angel and the Demon:

A a ( * ) = ∅ A d ( * ) = { * } D a ( * , ) = D d ( * , * ) = ∅ n a ( * , , ) = n d ( * , * , ) = .
Those two systems play an important rôle in the general theory of interaction systems (the first one is usually called abort, while the second one is usually called magic) but they do not appear in the model presented below.

Synchronous Product. There is an obvious product construction reminiscent of the synchronous product found in SCCS (synchronous calculus of communicating systems, [START_REF] Milner | Calculi for synchrony and asynchrony[END_REF]):

Definition 5. Suppose w 1 and w 2 are interaction systems on S 1 and S 2 . Define the interaction system w 1 ⊗ w 2 on S 1 × S 2 as follows:

A w1⊗w2 (s 1 , s 2 ) = A 1 (s 1 ) × A 2 (s 2 ) D w1⊗w2 (s 1 , s 2 ), (a 1 , a 2 ) = D 1 (s 1 , a 1 ) × D 2 (s 2 , a 2 ) n w1⊗w2 (s 1 , s 2 ), (a 1 , a 2 ), (d 1 , d 2 ) = s 1 [a 1 /d 1 ], s 2 [a 2 /d 2 ] .
This is the synchronous parallel composition of w 1 and w 2 : the Angel and the Demon exchange pairs of actions/reactions. For any sensible notion of morphism, skip should be a neutral element for this product. It is indeed the case, for the following reason: the components of w ⊗ skip and w are isomorphic by dropping the second (trivial) coordinate:

w ⊗ 1 w S × { * } S A (s, * ) = A(s) × { * } A(s) D (s, * ), (a, * ) = D(s, a) × { * } D(s, a) n (s, * ), (a, * ), (d, * ) = s[a/d], * s[a/d]
This implies trivially that {((s, * ), s) | s ∈ S} is an isomorphism. For similar reasons, this product is transitive and commutative.

Lemma 1. " ⊗ " is a commutative tensor product in the category Int. Its action on morphisms is given by:

(s 1 , s ′ 1 ), (s 2 , s ′ 2 ) ∈ r ⊗ r ′ ⇔ (s 1 , s 2 ) ∈ r and (s ′ 1 , s ′ 2 ) ∈ r ′ Checking that r ⊗ r ′ is indeed a simulation is easy.
Note that not every isomorphism (in the category Int) is of this form: is is quite simple to find isomorphic interaction systems with non-isomorphic components. 3Linear Arrow. The definition of the interaction system w 1 ⊸ w 2 is not as obvious as the definition of the tensor (⊗): Definition 6. If w 1 and w 2 are interaction systems on S 1 and S 2 , define the interaction system w 1 ⊸ w 2 on S 1 × S 2 as follows:

A (s 1 , s 2 ) = f ∈A1(s1)→A2(S2) a1∈A1(s1) D 2 s 2 , f (a 1 ) → D 1 (s 1 , a 1 ) D (s 1 , s 2 ), (f, G) = a1∈A1(s1) D 2 s 2 , f (a 1 ) n (s 1 , s 2 ), (f, G), (a 1 , d 2 ) = s 1 [a 1 /G a1 (d 2 )] , s 2 [f (a 1 )/d 2 ] .
It may seem difficult to get some intuition about this interaction system; but it is a posteriori quite natural: (see Proposition 1) -An action in state (s 1 , s 2 ) is given by:

(1) a function f (the index for the element of the disjoint sum) translating actions from s 1 into actions from s 2 ;

(2) for any action a 1 , a function G a1 translating reactions to f (a 1 ) into reactions to a 1 . -A reaction to such a "translating mechanism" is given by:

(1) an action a 1 in A 1 (s 1 ) (which we want to simulate);

( It thus looks like the interaction system w 1 ⊸ w 2 is related to simulations from w 1 to w 2 . It is indeed the case:

Proposition 1. In Int, " ⊗ " is left adjoint to " ⊸ ".
Proof. The proof is not really difficult, but is quite painful to write (or read).

Here is an attempt. Note that the following form of the axiom of choice is constructively valid:

4 AC : ∀a ∈ A ∃d ∈ D(a) ϕ(a, d) ⇔ ∃f ∈ a∈A D(a) ∀a ∈ A ϕ a, f (a)
When the domain D(a) for the existential quantifier doesn't depend on a ∈ A, we can simplify it into:

AC : ∀a ∈ A ∃d ∈ D ϕ(a, d) ⇔ ∃f ∈ A → D ∀a ∈ A ϕ a, f (a)
In the sequel, the part of the formula being manipulated will be written in bold. That r is a simulation from w 1 ⊗ w 2 to w 3 takes the form5

(s 1 , s 2 , s 3 ) ∈ r ⇒ ∀a 1 ∈ A 1 (s 1 ) ∀a 2 ∈ A 2 (s 2 ) ∃a 3 ∈ A 3 (s 3 ) ∀d 3 ∈ D 3 (s 3 , a 3 ) ∃d 1 ∈ D 1 (s 1 , a 1 ) ∃d 2 ∈ D 2 (s 2 , a 2 ) s 1 [a 1 /d 1 ], s 2 [a 2 /d 2 ], s 3 [a 3 /d 3 ] ∈ r .
Using AC on the ∀a 2 ∃a 3 , we obtain:

(s 1 , s 2 , s 3 ) ∈ r ⇒ ∀a 1 ∈ A 1 (s 1 ) ∃f ∈ A 2 (s 2 ) → A 3 (s 3 ) ∀a 2 ∈ A 2 (s 2 ) ∀d 3 ∈ D 3 (s 3 , f (a 2 )) ∃d 1 ∈ D 1 (s 1 , a 1 ) ∃d 2 ∈ D 2 (s 2 , a 2 ) s 1 [a 1 /d 1 ], s 2 [a 2 /d 2 ], s 3 [f (a 2 )/d 3 ] ∈ r .
We can now apply AC on ∀d 3 ∃d 2 :

(s 1 , s 2 , s 3 ) ∈ r ⇒ ∀a 1 ∈ A 1 (s 1 ) ∃f ∈ A 2 (s 2 ) → A 3 (s 3 ) ∀a 2 ∈ A 2 (s 2 ) ∃g ∈ D 3 (s 3 , f (a 2 )) → D 2 (s 2 , a 2 ) ∀d 3 ∈ D 3 (s 3 , f (a 2 )) ∃d 1 ∈ D 1 (s 1 , d 1 ) s 1 [a 1 /d 1 ], s 2 [a 2 /g(d 3 )], s 3 [f (a 2 )/d 3 ] ∈ r
and apply AC one more time on ∀a 2 ∃g to obtain:

(s 1 , s 2 , s 3 ) ∈ r ⇒ ∀a 1 ∈ A 1 (s 1 ) ∃f ∈ A 2 (s 2 ) → A 3 (s 3 ) ∃G ∈ a2∈A2(s2) D 3 (s 3 , f (a 2 )) → D 2 (s 2 , a 2 ) ∀a 2 ∈ A 2 (s 2 ) ∀d 3 ∈ D 3 (s 3 , f (a 2 )) ∃d 1 ∈ D 1 (s 1 , d 1 ) s 1 [a 1 /d 1 ], s 2 [a 2 /G a2 (d 3 )], s 3 [f (a 2 )/d 3 ] ∈ r which is equivalent to (s 1 , s 2 , s 3 ) ∈ r ⇒ ∀a 1 ∈ A 1 (s 1 ) ∃(f, G) ∈ f ∈A2(s2)→A3(s3) a2∈A2(s2) D 3 (s 3 , f (a 2 )) → D 2 (s 2 , a 2 ) ∀(a 2 , d 3 ) ∈ A2(s2) D 3 (s 3 , f (a 2 )) ∃d 1 ∈ D 1 (s 1 , d 1 ) s 1 [a 1 /d 1 ], s 2 [a 2 /G a2 (d 3 )], s 3 [f (a 2 )/d 3 ] ∈ r
By definition, this means that r is a simulation from w 1 to w 2 ⊸ w 3 .

Once more, all this formal manipulation keeps the computational content of the simulations. (Because AC is constructively valid.) ⊓ ⊔

The notion of safety property from [START_REF] Hyvernat | Synchronous games, deadlocks and linear logic[END_REF] corresponds to simulations from 1 to w, or equivalently, subsets x of S such that:

s ∈ x ⇒ ∃a ∈ A(s) ∀d ∈ D(s, a) s[a/d] ∈ x .
The analogy with strategies should be obvious: if x is a safety property, and s ∈ x then the Angel has a strategy to avoid deadlocks, starting from s.

Multithreading. We now come to the last connective needed to interpret the λ-calculus. Its computational interpretation is related to the notion of multithreading, i.e. the possibility to run several instances of a program in parallel. Let's start by defining synchronous multithreading in the most obvious way: Definition 7. If w is an interaction system on S, define L(w), the multithreaded version of w to be the interaction system on List(S) with:

L.A (s 1 , . . . s n ) = A(s 1 ) × . . . A(s n ) L.D (s 1 , . . . s n ), (a 1 , . . . a n ) = D(s 1 , a 1 ) × . . . D(s n , d n ) L.n (s 1 , . . . s n ), (a 1 , . . . a n ), (d 1 , . . . d n ) = s 1 [a 1 /d 1 ], . . . s n [a n /d n ] .
This interaction system is just an "n-ary" version of the synchronous product. To get the abstract properties we want, we need to "quotient" multithreading by permutations. Just like multisets are list modulo permutation, so is !w the multithreaded L(w) modulo permutations. This definition is possible because L(w) is "compatible" with permutations: if σ is a permutation, we have

σ • (s 1 , . . . s n ) (a 1 , . . . a n )/(d 1 , . . . d n ) = σ • (s 1 , . . . s n ) [σ • (a 1 , . . . a n )/σ • (d 1 , . . . d n )] .
The final definition is: Definition 8. If w is an interaction system on S, define L(w), define !w to be the following interaction system on M f (S):

!A(µ) = s∈µ L.A(s) !D µ, (s, a) = L.D(s, a) !n µ, (s, a), d = S • L.n(s, a, d) .

Unfolded, it gives:

an action in state µ (a multiset) is given by an element s of the equivalence class µ (a list) together with an element a in L.A(s) (a list of actions); -a reaction is given by a list of reactions d in L.D(s, a); -the next state is the equivalence class containing the list s[a/d] (the orbit of s[a/d] under the action of the group of permutations).

This operation enjoys a very strong algebraic property:

Proposition 2. " ! " is a comonad in Int.
Proof. We need to find two operations:

ε w : !w → w defined as ε w = [s], s | s ∈ S ; -and δ w : !!w → !w defined as the graph of the "concat" function:

δ w = [µ i ] i∈I , i∈I µ i | ∀i ∈ I µ i ∈ M f (S)
For any w, those operations are indeed simulations: for ε w , it is quite obvious, and for δ w , it is quite painful to write. Let's only give an example from which the general case can easily be inferred:

1. we have ([[s 1 , s 2 , s 3 ], [t 1 ], []], [s 1 , s 2 , s 3 , t 1 ]) ∈ δ w 2. for any command ((a 1 , a 2 , a 3 ), (b 1 ), ()) in state [[s 1 , s 2 , s 3 ], [t 1 ], []],
we need to find an action in [s 1 , s 2 , s 3 , t 1 ]: simply take (a 1 , a 2 , a 3 , b 1 ); 3. for any reaction (d 1 , d 2 , d 3 , e 1 ) to this action, we need to find a reaction to the original command, i.e. to ((a 1 , a 2 , a 3 ), (b 1 )()): take ((d 1 , d 2 , d 3 ), (e 1 ), ()); 4. the next states are respectively

-[[n(s 1 , a 1 , d 1 ), n(s 2 , a 2 , d 2 ), n(s 3 , a 3 , d 3 )], [n(t 1 , b 1 , e 1 )], []] -and [n(s 1 , a 1 , d 1 ), n(s 2 , a 2 , d 2 ), n(s 3 , a 3 , d 3 ), n(t 1 , b 1 , e 1 )
]. They are indeed related through δ w .

To be really precise, one would need to manipulate lists of states (representative of the multisets); but this only makes the proof even less readable.

Checking that the appropriate diagrams commute is immediate. It only involves the underlying sets and relations, and not the interaction systems or simulation conditions. (In fact , finite multisets form a comonad in the category of sets and relations...) ⊓ ⊔

Interpreting the λ-Calculus

We now have all the ingredients to give a denotational model for the typed λ-calculus: a type T will be interpreted by an interaction system T * ; and a judgement "x 1 : T 1 , . . . x n : T n ⊢ t : T " will be interpreted by simulation from

!T * 2. Γ ⊢ t : ω → ω ′ Γ ⊢ u : ω Γ ⊢ (t)u : ω ′ ; 3. Γ, x : ω ⊢ t : ω ′ Γ ⊢ λx.t : ω → ω ′ .
We follow Krivine's notation for the application and write "(t)u" for the application of t to u.

Interpretation of Types

We assume a set of type variables ("propositional variables"): X, . . . Nothing depend on the valuation we give to those type variables, so that we are almost interpreting Π1 λ-calculus. 6For a valuation ρ from type variables to interaction systems, the interpretation of types is defined in the usual way: Definition 9. Let ω be a type. Define the interpretation ω * of ω as:

-X * = ρ(X); -(ω → ω ′ ) * = !ω * ⊸ ω ′ * .

Interpretation of Terms

If ω is a type, write |ω| for the set of states of its interpretation:

-|X i | = S i (set of states of ρ(X i )); -|ω → ω ′ | = M f |ω| × |ω ′ |.
A valuation is a way to interpret typed variables from the context:

Definition 10. If Γ = x 1 : ω 1 , . . . x n : ω n is a context, an environment for Γ is a tuple γ in M f |ω 1 | × . . . M f |ω n |.
To simplify notation, we may write the tuple γ = (µ 1 , . . . µ n ) as "x 1 := µ 1 , . . . x n := µ n ". We may also write γ(x) for the projection of γ on the appropriate coordinate. Sum of tuples of multisets is defined pointwise.

We now interpret judgements: if we can type Γ ⊢ t : ω ′ and if γ is an environment for Γ , the interpretation [[t]] γ of term t in environment γ is a subset of |ω| defined as follows: Definition 11. We define [[t]] γ by induction on t:

1. if we have Γ ⊢ x : ω with x : ω in Γ , then [[x]] γ = {s} if γ(x) = [s] and γ(y) = [] whenever x = y ∅ otherwise ; 2. if we have Γ ⊢ t : ω → ω ′ Γ ⊢ u : ω Γ ⊢ (t)u : ω ′ , then s ∈ [[(t)u]] γ iff (µ, s) ∈ [[t]] γ0 for some µ = [s 1 , . . . s n ] ∈ M f |ω| s.t. s i ∈ [[u]
] γi for all i = 1, . . . n and γ = γ 0 + γ 1 + . . . γ n ;

3. if we have Γ, x :

ω ⊢ t : ω ′ Γ ⊢ λx.t : ω → ω ′ , then [[λx.t]] γ = {(µ, s) | µ ∈ M f |ω|, s ∈ [[t]] γ,x:=µ }.
It is immediate to check that this definition is well formed.

If Γ = x 1 : ω 1 , . . . x n : ω n , write !Γ for !ω * 1 ⊗ . . . !ω n ; similarly, we omit the superscript * and write ω for ω * . The interpretation of terms is correct in the following sense:

Proposition 3. Suppose that Γ ⊢ t : ω ′ , then the relation " ∈ [[t]] " is a simulation relation from !Γ to ω ′ . In other words, if s ∈ [[t]] γ , then s (in ω ′ ) simulates γ (in !Γ ).
This is quite surprising because the interpretation of t doesn't depend on the interaction systems used to interpret the types but only the underlying set of states. 71.

Γ ⊢ 0 : ω and Γ ⊢ t : ω Γ ⊢ u : ω Γ ⊢ t + u : ω ; 2. Γ ⊢ t : ω → ω ′ Γ ⊢ u : ω Γ ⊢ D t • u : ω → ω ′ .
The intuitive meaning is that "D t • u" is the result of (non-deterministically) replacing exactly one occurrence of the first variable of t by u. We thus obtain a sum of terms, depending on which occurrence was replaced. This gives a notion of differential substitution (or linear substitution) which yields a differentialreduction. The rules governing this reduction are more complex than usual β-reduction rules. We refer to [START_REF] Ehrhard | The differential lambda calculus[END_REF] for a detailed description. We extend the interpretation of terms in the following way:

Definition 12. Define the interpretation of a typed differential λ-term by induction on the type inference:

1. if we have Γ ⊢ 0 : ω , then we put

[[0]] γ = ∅; 2. if we have Γ ⊢ t : ω Γ ⊢ u : ω Γ ⊢ t + u : ω , then we put [[t + u]] γ = [[t]] γ ∪ [[u]] γ ; 3. if we have Γ ⊢ t : ω → ω ′ Γ ⊢ u : ω Γ ⊢ D t • u : ω → ω ′ , then we put (µ, s ′ ) ∈ [[D t • u]] γ iff (µ + [s], s ′ ) ∈ [[t]] γ1 for some s ∈ [[u]] γ2 s.t. γ = γ 1 + γ 2 .
Proposition 3 extends as well:

Proposition 5. Suppose that Γ ⊢ t : ω ′ where Γ is a context and t a differential λ-term. The relation " ∈

[[t]] " is a simulation relation from !Γ to ω ′ .
Proof. The proof for the sum and the 0 are contained in proposition 2.

For differentiation, suppose we have (µ, s

′ ) ∈ [[D t • u]] γ , i.e. (µ + [s], s ′ ) ∈ [[t]] γ1 for some s ∈ [[u]] γ2 , with γ = γ 1 + γ 2 . We need to show that (µ, s ′ ) (in ω → ω ′ ) simulates γ (in !Γ ). Since γ = γ 1 + γ 2 , it is enough to show that we can simulate (γ 1 , γ 2 ) (in !Γ ⊗ !Γ ).
By proposition 1, this is equivalent to showing that s ′ (in ω ′ ) simulates (γ 1 , γ 2 , µ)

(in !Γ ⊗ !Γ ⊗ !ω).
Let a γ1 ∈ !A Γ (γ 1 ), a γ2 ∈ !A Γ (γ 2 ) and a µ ∈ !A ω (µ); we need to find an action in A ω ′ (s ′ ) to simulate (a γ1 , a γ2 , a µ ):

(1) by induction hypothesis, we know that s (in ω) simulates γ 2 (in !Γ ); so that we can find an action a ∈ A ω (s) simulating a γ2 ;

We thus obtain reactions d γ1 , d γ2 and d µ as desired. That the resulting next states are still related is quite obvious...

⊓ ⊔

We now need to check that the interpretation is invariant by β-reduction and differential reduction. Proposition 6. For all differential terms t and u and environment γ, we have: Just like for Proposition 4, the proof is quite easy but tedious. The interested reader can find it at http://iml.univ-mrs.fr/∼hyvernat/academics.html.

Conclusion

Technically speaking, this work is not very different from [START_REF] Hyvernat | Synchronous games, deadlocks and linear logic[END_REF], which is itself quite close to [START_REF] Hyvernat | Predicate transformers and linear logic: yet another denotational model[END_REF]. The main reasons for producing it are:

first, it shows that we can give a computational content to the notion of simulation if we do not try to interpret all of linear logic; -second, it shows that some of the additional structure of interaction systems and simulation does have a logical significance. We showed that by interpreting the differential λ-calculus.

Even if we haven't done it formally, it is quite easy to extend the model to full intuitionistic linear logic while keeping the computational content of simulations.

To define the additive, we use the definition of ⊕ from [START_REF] Hyvernat | Synchronous games, deadlocks and linear logic[END_REF].

It is in principle possible to formalise all the above in a proof assistant (Agda [START_REF]The Agda proof assistant[END_REF] or Coq [START_REF]The Coq proof assistant[END_REF] come to mind).9 From such a system, one could extract the simulations. For example, a term of type T → T ′ would give an algorithm simulating many synchronous occurrences of T by a single occurrence of T ′ .

  [[(λx.t)u]] γ = [[t[u/x]]] γ [[D(λx.t) • u]] γ = [[λx . (∂t/∂x) • u]] γ

  2) and a reaction d 2 in D 2 (s 2 , f (a 1 )) (which we want to translate back). -Given such a reaction, we can simulate a 1 by a 2 ∈ A 2 (s 2 ) obtained by applying f to a 1 ; and translate back d 2 into d 1 ∈ D 1 (s 1 , a 1 ) by applying G a1 to d 2 . The next state is just the pair of states s 1 [a 1 /d 1 ] and s 2 [a 2 /d 2 ].

In the finite case, one can duplicate a command a into a1 and a2 to obtain sets of commands of different cardinality.

This form of the axiom of choice is provable in Martin-Löf's type theory or in the calculus of construction...

modulo associativity (S1 × S2) × S3 ≃ S1 × (S2 × S3) ≃ S1 × S2 × S3...

⊗ . . . !T * n to T * .

2.1 Typing rulesThe typing rules for the simply typed λ-calculus are given below:1.Γ ⊢ x : ω if x : ω appears in Γ ;

System-F in which all the quantifiers appear at the beginning of the term. To get an idea on how to get a real model of system-F , refer to[START_REF] Hyvernat | Predicate transformers and linear logic: second order[END_REF].

The interpretation is called the relational interpretation: it can be defined in the category of sets and relations...

One needs to be careful to be able to deal with the notion of equivalence classes used in the definition of !w. The idea is to use interaction systems on "setoids", where the equivalence relation is a simulation...

Proof. We work by induction on the structure of the type inference. -

(b) we need to find an action in A ω ′ (s) simulating (a 1 , a 2 , a 3 ):

(1) by induction hypothesis,

Thus, we can find an action a ∈ A ω ′ (s) simulating (a 2 ), (b 1 , b 2 ) . By composing the above two simulations on the right ((b 1 , b 2 ) simulates (a 1 , a 3 )), we thus obtain that a simulates (a 1 , a 2 , a 3 ). We now need to translate the reactions back: let d ∈ D ω ′ (s, a), (3) by induction, we can translate d into a reaction (d 2 ), (e 1 , e 2 ) to (a 2 ), (b 1 , b 2 ) ; (2) we can translate e 2 into a reaction () to b 2 ;

(1) and finally we can translate e 1 into a reaction (d 1 , d 3 ) to (a 1 , a 3 ). Thus, we obtain reactions

The new states we get from those actions/reactions are: s[a/d] on one side; and

] on the other side. They are indeed related because:

3. Abstraction: this is immediate. Suppose (µ, s) ∈ [[λx.t]] γ ; we need to show that (µ, s) (in ω → ω ′ ) simulates γ (in !Γ ). By proposition 1, this is equivalent to showing that s (in ω ′ ) simulates (γ, µ) (in !Γ ⊗ !w). This is exactly the induction hypothesis.

⊓ ⊔

To summarise all this, here is a tentative rewording of the above: if Γ ⊢ t : ω,

(1) each type represent a process;

(2) each process in the context can be run in parallel multiple times;

(3) the environment γ represents the initial states for the context;

(4) if s ∈ [[t]] γ then s can be used as an initial state to simulate γ;

(5) the algorithm for the simulation is contained in t.

To finish the justification that we have a denotational model, we now need to check that the interpretation is invariant by β-reduction. Proposition 4. For all terms t and u and environment γ, we have

The proof works by induction and is neither really difficult nor very interesting. It can be found on http://iml.univ-mrs.fr/∼hyvernat/academics.html.

Interpreting the Differential λ-calculus

Simulation relations from w to w ′ enjoy the additional property that they form a complete sup-lattice: Lemma 2. The empty relation is always a simulation from any w to w ′ ; and if (r i ) i∈I is a family of simulations from w to w ′ , then i∈I r i is also a simulation from w to w ′ .

The proof is immediate... Unfortunately, this doesn't reflect any property of λ-terms. The reason is that (1) not every type is inhabited, and (2) we do not see a priori how to take the union of two terms. For example, what is the meaning of λxλy.x ∪ λxλy.y in the type X → X → X? 8 Ehrhard and Regnier's differential λ-calculus ( [START_REF] Ehrhard | The differential lambda calculus[END_REF]) extends the λ-calculus by adding a notion of differentiation of λ-terms. One consequence is that we need to have a notion of sum of arbitrary terms, interpreted as a non-deterministic choice. It is not the right place to go into the details of the differential λ-calculus and we refer to [START_REF] Ehrhard | The differential lambda calculus[END_REF] for motivations and a complete description.

In the typed case, we have the following typing rules: It is however difficult to apply this to obtain real-life simulations. The problem is that we only get "purely logical" simulations. Simulations of interest for application rely heavily on the different interaction systems used. One way to get more interesting simulations (from a practical point of view) might be to use constant interaction systems (booleans, natural numbers, or more practical ones like stacks, memory cells, etc..) as ground types, together with specific simulations (the values true and false, successor function, or more practical simulations) as inhabitant of specific types.

In pretty much the same way as [START_REF] Hyvernat | Synchronous games, deadlocks and linear logic[END_REF] makes Int into a denotational model for classical linear logic, we can make interaction systems into a denotational model for "classical differential linear logic": differential interaction nets [START_REF] Ehrhard | Differential interaction nets[END_REF]. This system doesn't make much sense logically speaking, but seems to enjoy relationship with process calculi. This is an encouraging direction of research.


Amaratou Mahamadou Saley

Jérémie Marchand

Aicha Sekhari

Vincent Cheutet

Jean-Baptiste Danielou

State-of-art and maturity overview of the nuclear industry on predictive maintenance

Keywords: lifecycle management, predictive maintenance, nuclear system

In order to ensure as continuous as possible nuclear energy production, it is necessary to guarantee the availability of all equipment involved in the production chain, in all its complexity. The safety of all these equipments is based on a good command of the maintenance policy with very specific conditions related to the demanding regulations of the nuclear field. It is about having facilities that are safe, available with a good quality and a limitation to radiological risk. Today, with the advancement of digital technology, substantial improvements have occurred in the tools that can be applied in the maintenance and monitoring of Structures, Systems and Components (SSCs), enabling an understanding of equipment performance far beyond that available only a few decades ago. Therefore, predictive maintenance becomes a subject of prior interest for the nuclear industry. In this paper, we will emphasize the incentives and obstacles of predictive maintenance deployment in a nuclear context. The objectives here are to draw a clear picture of what can be the practices of predictive maintenance in the nuclear industry context and to identify the requirements and the needs to implement a predictive maintenance model on the lifecycle management of a nuclear facility.

Introduction

Nuclear energy is one main source of electricity worldwide and the safe operation of nuclear industry is generally acknowledged as contributing to society's success and promoting economic performance. However, managing nuclear energy production is very complex. In order to maintain a stable energy, it is necessary to guarantee the availability of all the equipment involved in this production chain. IAEA (International Atomic Energy Agency) provides a PLiM (Plant Life Management) program for long-term operation, which aims at identifying all of the factors and requirements for the overall life cycle of the plant for all [START_REF]IAEA: Implementation Strategies and Tools for Condition Based Maintenance at Nuclear Power Plants[END_REF]. One of the important factor of this program is the maintenance prioritization. Maintenance in nuclear industry remains a major challenge for operators: a majority of nuclear power plant (NPP) systems are aged. The inherent risk of the activity imposes strict regulations and compliance. This consumes money time and hard efforts. We need to put in front of it reliable means, methods and tools to respond to it. The regulatory activities on maintenance are defined by various authorities and institutions working for nuclear safety. Among the standards referred to the maintenance, we have NF EN 13306 which defines the types of maintenance and organization of maintenance, the IAEA safety standards especially for the operation, maintenance and safety of NPP [START_REF]IAEA: Application of Reliability Centred Maintenance to Optimize Operation and Maintenance in Nuclear Power Plants[END_REF][START_REF]IAEA: Implementation Strategies and Tools for Condition Based Maintenance at Nuclear Power Plants[END_REF].

The research interest regarding maintenance strategy to optimizing availability/reliability, minimizing risk and cost has been growing up time by time. According to Web of Science, since 1975, 18857 journals and articles discussing maintenance in nuclear context are found using the keywords "Maintenance" and "Nuclear".

However, there is a wide gap concerning the implementation of maintenance in research and practice. Although many models, methods, and strategies have been proposed, most of the nuclear industries in the world are still using traditional approaches [START_REF] Ngarayana | Nuclear Power Plant Maintenance Optimisation: Models, Methods & Strategies[END_REF]. These traditional approaches help in minimizing the risk of failure. Nevertheless, they do not guarantee a full availability of all the critical equipments. At the same time, we are now facing the development of industry 4.0 strategy linked with the emergence and maturity of digital. Known as the fourth Industrial Revolution, it is characterized by the widespread use of digital and physical environments called Cyber Physical Systems. This convergence achievement of information technologies, Internet Of Things (IoT), big data, cloud, communication systems (SCADA), Cybersecurity, data analytic, with additional accelerators, such as Artificial Intelligence and cognitive science makes it possible to improve the traditional maintenance strategies with the Predictive Maintenance (PdM) [START_REF] Huang | Application of rcm analysis based predictive maintenance in nuclear power plants[END_REF][START_REF] Zwingelstein | La maintenance préventive -Méthodes et technologies[END_REF]. Regarding NF EN 13306 standard, the PdM is a "Condition-based maintenance carried out following a forecast derived from repeated analysis or known characteristics and evaluation of the significant parameters of the degradation of the item". The condition based maintenance is a "preventive maintenance which include assessment of physical conditions, analysis and the possible ensuing maintenance actions". Its implementation requires a large number of concepts and tools such as Industrial IoT [START_REF] Compare | Challenges to IoT-Enabled Predictive Maintenance for Industry 4.0[END_REF], data mining, cloud, big data, and artificial intelligence [START_REF] Zwingelstein | La maintenance préventive -Méthodes et technologies[END_REF].

Nevertheless, deploying such technologies in nuclear industry is not an easy task due to the regulatory constraints. Also, the maturity of PdM in nuclear facilities is not the same compared to other industries. The nuclear industry can be seen as a conservative one that is usually strictly regulated [16]. Also, the equipment involved in the production chain are very specifics, with very long lifespans. The current fleet of NPP is old, with a majority of plants built and commissioned during the 70s and 80s and an initial lifespan of around 40 years. In addition, of this ageing effect, the equipments that make up the facility are diverse, varied and do not have the same level of criticality. Some systems are very critical, with highly regulated owing to the risks associated with plant accidents and radiation exposure to the public. The objective of this paper is to establish an overview of maintenance practices in the nuclear industry to evaluate its maturity and to identify opportunities and threats for PdM deployment.

The paper is structured as follow: section 2 is dedicated to the related works of maintenance in nuclear facilities. Section 3 introduces the PdM as a whole. Section 4 focuses on PdM in a nuclear context by identifying the current evolution. Section 5 makes a critical analysis of the maturity of PdM in the nuclear industry and introduces further research directions.

Maintenance in nuclear facilities

As seen in the introduction, 18857 articles were found in the Web of Science database considering "Maintenance" and "Nuclear" keywords. The process for relevant papers collections and selections consisted of three main phases. First, we have filtered 18857 articles by categories, considering only papers that are closed to our research area. Then, we keep papers written in English that are journals and proceedings and removed duplicated articles. At least, a total of 8882 research papers were short-listed for classification. In order to deepen our investigation of the short-listed papers, VOSviewer software [START_REF] Van Eck | Software survey: Vosviewer, a computer program for bibliometric mapping[END_REF] was used to carry out an analysis of the co-occurrence of authors keywords (Figure 1a).

Firstly, we can highlight a cover of keywords around maintenance, CBM, PdM, fault diagnosis in the NPP. This cluster shows the scientific orientation on the NPP over the last ten years. Then a second group of keywords around ITER remote maintenance, remote handing is observed. ITER (International Thermonuclear Experimental Reactor) is an international nuclear fusion research and engineering megaproject aimed at replicating the fusion processes of the sun to create energy on the Earth. A final keyword overlay around the maintenance is visible. This cluster is linked to all the other ones and is very representative. It can be understood that the focus on the maintenance in nuclear industry is not only on nuclear fission (NPP) but also on nuclear fusion, and the topics are the same and centered on the PdM, optimization, remote maintenance, equipments availability, fault diagnosis.

From this state of the art, we observe that whatever the industry, the challenges remain the same on the maintenance strategies and methods. IAEA highlight the importance of the Reliability Centered Maintenance (RCM) in a nuclear facility optimization [START_REF]IAEA: Application of Reliability Centred Maintenance to Optimize Operation and Maintenance in Nuclear Power Plants[END_REF] as demonstrated by EDF in France [START_REF] Despujols | Optimisation de la maintenance par la fiabilité (OMF)[END_REF]. The RCM was introduced in the late 1960 by the airline industry [START_REF]IAEA: Application of Reliability Centred Maintenance to Optimize Operation and Maintenance in Nuclear Power Plants[END_REF]. The core of the RCM philosophy is that maintenance will be performed only after evaluating the consequences of failures at component level. It deals with optimization of preventive maintenance activities considering failure consequences. The RCM approach is system-oriented and may be implemented free of a company's organizational culture. It helps in establishing system boundaries and developing a critical equipment list. An equipment is said to be critical if it performs a function or if its failure can affect functioning of equipment that impacts nuclear safety, prevention of release of radioactivity to the environment, personnel safety The maintenance in nuclear industry are how to perform to the optimum level of the capability of equipments and plants. The literature describe several aspects of maintenance. They may be grouped into the technical aspect, commercial one and management one. This implies that maintenance decisions need to be made in a framework that takes into account these issues from an overall business perspective.

The overall maintenance strategies can be categorize in two types. Corrective maintenance is carried out after failure detection and aims to restore an asset to a condition in which it can perform its intended function [START_REF] Mobley | An introduction to predictive maintenance[END_REF]. In the literature, authors identify two types of corrective maintenance: the planned and the unplanned corrective maintenance [START_REF] Mobley | An introduction to predictive maintenance[END_REF]. Preventive maintenance aims at doing the maintenance just before a component fails so the system downtime and the emergency will not happen, by scheduling machine repairs or rebuilds [START_REF] Zwingelstein | La maintenance préventive -Méthodes et technologies[END_REF].

Two types of preventive maintenance are discussed in the literature [START_REF]IAEA: Implementation Strategies and Tools for Condition Based Maintenance at Nuclear Power Plants[END_REF]: the time-based maintenance (TBM) aiming at performing periodic maintenance on the equipment and the condition-based maintenance (CBM) carried out in response to a significant deterioration in a machine as indicated by a change in a monitored parameter of the machine condition.

The procedure for performing corrective or preventive maintenance in a nuclear environment is complex. Indeed, it requires a lot of preparation up, given the regulatory safety aspects. First of all, the maintainers must assess the radiological environment, the chemical risks of the process and facility concerned by the maintenance. Then, it is necessary to examine the operating procedure. This allows to optimize the intervention time since this time is very limited and strongly correlated to the radiation exposure. Often, given the complexity of some processes, the failure is difficult to detect.

Maintenance is thus becoming more complex and detecting failures is becoming harder and more time-consuming. The labor cost to carry out maintenance has also been increasing. As a result, maintenance will continue to evolve and the two main drivers for this are (i) technology and (ii) management.

Predictive maintenance as a whole

The literature highlight many research areas related to PdM. As in the previous case, web of science is used for relevant articles collection. 3750 articles were found using the key word "Predictive maintenance". We filtered the most relevant ones using the same process as in the maintenance section. At least, a total of 2453 research papers were short-listed for classification. Figure 1b draws up three main clusters around PdM literature. The first one is on the maintenance management. It includes all the research studies related to the reliability, fault detection, fault diagnosis, maintenance engineering for maintenance optimization. The second cluster covers research topics on the PdM techniques and tools. This cluster is the most representative one, as it is linked to all the others clusters. It can be understood that PdM in the literature is an association of three main research field: maintenance management, IoT and artificial intelligence. The last cluster relies on the prognostics, health management and the remaining useful life (RUL). Very present in the literature, it is a part of PdM which aims at equipment heath monitoring.

PdM has the potential to make the production more reliable and improve service provisioning. However, the maturity for a proper implementation of PdM in research and in practice is not the same. That is why we conducted a multi-case study to underline the conditions and technological aspects for implementing a PdM system in practice. We found that PdM initiatives are triggered by severe impacts of failures on revenue and profit. Furthermore, successful PdM initiatives require that some pre-conditions are fulfilled: Data must be available and accessible. Also, the support by the management is very important. We identified four important factors for PdM implementation: Data integration (highly facilitated by Cloud-based mechanisms), failure detection (enabled by advanced analytics), PdM operations execution (supported by data-driven process automation) and the visualization [START_REF] Möhring | Predictive maintenance information systems: The underlying conditions and technological aspects[END_REF].

It is important to note that the failure modelling approaches for PdM are a bit common to the maintenance and fault computing. PdM also relies on two distinct approaches:

-The first approach is the probabilistic approach. It is based on a statistical and reliability analysis. The input data is retrieved from the Computerized Maintenance Management Information System (CMMIS) software. The CMMIS centralizes maintenance information and facilitates maintenance operations processes. Moreover, the CMMIS optimizes the use and equipment availability.

-The second approach is the deterministic approach. It is characterized by the use of some sensors data combined with data mining techniques and artificial intelligence algorithms to predict a machine failure.

The second approach was of great interest to the researchers because it made it possible to optimize maintenance costs better than the first approach. When CBM was introduced in 2002, many implementation methods were developed. These methods involved at measuring certain process parameters from sensor data. The most used methods at the beginning of CBM are: Vibration analysis, Acoustics analysis, Lubrication oils analysis, Corrosive analysis, and Thermal analysis or Infrared thermography [START_REF] Coandǎ | A state of the art of predictive maintenance techniques[END_REF][START_REF] Selcuk | Predictive maintenance, its implementation and latest trends. Proceedings of the Institution of Mechanical Engineers[END_REF].

As methods have evolved since 2010, new PdM techniques have emerged. A new method called Prognostic and Health Management (PHM) was introduced [START_REF] Cachada | Maintenance 4.0: Intelligent and predictive maintenance system architecture[END_REF][START_REF] Zwingelstein | La maintenance préventive -Méthodes et technologies[END_REF]. The PHM is an engineering process where algorithms are used to detect anomalies, diagnose faults and predict RUL. Moreover, the financial benefits such as operational and maintenance cost reductions and extended lifetime are achieved with this method [START_REF] Sutharssan | Prognostic and health management for engineering systems: a review of the data-driven approach and algorithms[END_REF].

To implement a PHM, many steps are required. Author of [START_REF] Cachada | Maintenance 4.0: Intelligent and predictive maintenance system architecture[END_REF] addresses the Open System Architecture for Condition-Based Maintenance (OSA-CBM), and draws up six steps for performing a PHM based on ISO 13374: Data acquisition (Access to digitized sensors), Data manipulation (perform single and/or multichannel signal transformations), State detection or Fault Detection, Diagnosis step, Prognostics assessment and Advisory generation [START_REF] Cachada | Maintenance 4.0: Intelligent and predictive maintenance system architecture[END_REF][START_REF] Coandǎ | A state of the art of predictive maintenance techniques[END_REF][START_REF] Das | Essential steps in prognostic health management[END_REF][START_REF] Zwingelstein | La maintenance préventive -Méthodes et technologies[END_REF]. The architecture system for the PHM includes a database which stores data at each step and a Human Machine Interface for visualization.

CBM and PHM are sometimes confused: CBM includes data acquisition, data processing, fault detection and isolation, and failure diagnostics. Whereas PHM is more a prognosis, an estimation of the RUL and a decision-making process for the implementation of PdM and associated logistics [START_REF] Compare | Challenges to IoT-Enabled Predictive Maintenance for Industry 4.0[END_REF][START_REF] Zwingelstein | La maintenance préventive -Méthodes et technologies[END_REF].

The predictive modelling techniques used in diagnostic and prognostic functionalities of PHM can be categorized into three: physical modelling (data-based models), artificial intelligence methods (machine learning and deep learning) and statistical-based methods [START_REF] Ayo-Imoru | A survey of the state of condition-based maintenance (CBM) in the nuclear power industry[END_REF][START_REF] Compare | Challenges to IoT-Enabled Predictive Maintenance for Industry 4.0[END_REF][START_REF] Davari | A survey on data-driven predictive maintenance for the railway industry[END_REF][START_REF] Zwingelstein | La maintenance préventive -Méthodes et technologies[END_REF].

Predictive maintenance in a nuclear context

The state of the art on maintenance has shown that sometimes, a maintenance intervention can be complex especially when a failure is not anticipated. That is why nuclear industry is today moving towards PdM to overcome this problem. The explosion of digital technologies has enabled the collection of sensor data to better model the failure of an equipment. However, is this industry enough mature in deploying such maintenance strategy?

In this sense, we made a literature review based on Web Of Science. 90 articles have been found considering the combination of the following keywords: "predictive maintenance" and "nuclear". By using the same process as seen previously for relevant papers collection, we finally get a total of 44 research papers. VOSviewer software makes it possible to analyze the co-occurrence of authors keywords.

Overall, Figure 1c draws up all the topics covered in the literature around PdM in the nuclear industry. We identify four main clusters. First, we observe a grouping of keywords around the PHM, characterizing thus the relevant topics. Those keywords (RCM, RUL, wavelet transform, condition monitoring, maintenance, availability, machine learning) show the research orientation of the PHM in nuclear industry. We can also see a significant link between the PHM and the RCM standard. It can be understood that PHM highly impact on the RCM. The second cluster covers various topics around PdM. The topics are: condition based monitoring, industrial IoTs and so one. It is the most representative cluster because it is highly correlated to all the other clusters. The third cluster gathers different topics around the NPP: optimization using fault diagnosis, remaining useful file. Finally, the last cluster includes all the machine learning techniques for PdM implementation. These are: genetic algorithm, support vector machine, neural network and anomaly detection.

The specificity of PdM in nuclear industry lies in its implementation. First of all, equipment instrumentation is governed by regulations that are highly correlated to the radiation exposure and various risks associated with the facility. The location of the sensors on the equipment depends mainly on the risk associated with the process. This is why a risk analysis on the overall process must first be carried out before any intervention. Second, data is often subject to military regulations requiring authorization before any accessibility. Also, predictive models must be in line with the expectations of the process experts and their deployment should be in accordance with the confidentiality rules as defined by the operator and the various safety authorities.

It is important to note that before the introduction of PdM, the nuclear industry, as in all production industries, used traditional techniques such as Infrared Thermography, Ultra sound analysis, Motor current signature analysis, Oil analysis, and Partial discharge to maintain all the facilities. Also, the CMMIS data is mainly used to compute the risk associated with operating the facility, expressed in terms of various statistic metrics related to the different levels of failure to the facility (e.g. core failure frequency), or its environment (e.g. societal or individual risk) [START_REF]IAEA: Applications of probabilistic safety assessment ( PSA ) for nuclear power plants[END_REF][START_REF]IAEA: Implementation Strategies and Tools for Condition Based Maintenance at Nuclear Power Plants[END_REF]. However these techniques were not enough efficient in predicting a failure. This gave rise to the PdM methods proposed by [START_REF] Hashemian | State-of-the-art predictive maintenance techniques[END_REF] PdM has positively impacted the RCM standard in nuclear industry. This positive impact is justified by a good improvement in plant productivity and a reduction in maintenance costs. In [START_REF] Huang | Application of rcm analysis based predictive maintenance in nuclear power plants[END_REF], authors state that a correct implementation of RCM analysis based on PdM in NPP could bring great benefits in many aspects. Moreover, it is considered to be the best maintenance strategy [START_REF] Huang | Application of rcm analysis based predictive maintenance in nuclear power plants[END_REF]. Three main lines of research stand out when referring to PdM in the nuclear industry.

The first line is based on the appropriate methods and techniques used. The concepts are almost the same as what we saw in the previous section. Many researchers have made a state-of-art of PdM techniques in nuclear industry. Among them, Hashemian was more interesting. He describes three PdM techniques in identifying the onset of equipment failure in NPP: sensor-based technique, test-sensor-based technique and test-signal-based technique [START_REF] Coandǎ | A state of the art of predictive maintenance techniques[END_REF].

The second line focuses on the development of Machine learning algorithms for PdM. The topics were centered on the continuous machine learning for abnormality identification to aid condition-based maintenance [START_REF] Abdussalam Nuhu | Machine learning in predictive maintenance towards sustainable smart manufacturing in industry 4.0[END_REF][START_REF] Nuhu | Machine learning in predictive maintenance towards sustainable smart manufacturing in industry 4.0[END_REF] and prediction of the RUL of a component [START_REF] Abdussalam Nuhu | Machine learning in predictive maintenance towards sustainable smart manufacturing in industry 4.0[END_REF]. The models work as PHM models and the anomaly detection method is the most used in the reality.

The last line focuses on data acquisition through the Wireless sensor architecture and applications. This topic is of great interest to many researchers especially Hashemian [START_REF] Hashemian | Wireless sensors for predictive maintenance of rotating equipment in research reactors[END_REF][START_REF] Hashemian | State-of-the-art predictive maintenance techniques[END_REF]. He reported a research and development program to implement wireless sensors for equipment condition monitoring and other application. He proposed an architecture system which integrate a signal in the existing wireless sensors and news wireless sensors for a holistic view of the equipment health and process. He also proposed some wireless sensors for PdM of rotating equipment [START_REF] Hashemian | Wireless sensors for predictive maintenance of rotating equipment in research reactors[END_REF][START_REF] Hashemian | State-of-the-art predictive maintenance techniques[END_REF].

In the same way, authors of [START_REF] Gohel | Predictive maintenance architecture development for nuclear infrastructure using machine learning[END_REF] developed a PdM architecture for nuclear infrastructure using machine learning. They developed the architecture in two parts: a data acquisition part (Offline modelling) and a representation learning part (Online evaluation) with the integration of machine learning algorithms.

Critical analysis and conclusions

The research literature conducted in this paper enhance various research aspect of maintenance in nuclear. To come with clear and constructive outcomes, we moved from exploring the maintenance in the nuclear industry with special focus on PdM as a whole and PdM in nuclear industry.

The first finding of this literature study is the gap in the use of PdM technologies between industries. In fact, the maturity is not the same. All the papers that introduced PdM using AI in the nuclear industry are very recent in comparison with the airline industry. In order to maintain and optimize their facilities, nuclear operators use traditional methods based on failure probability analysis, failure modelling, most often computed using data retrieved from CM-MIS. These analyses contributed in computing some classic performance criteria such as MTBF, MTTR. All these methods help in minimizing the risk of failure. Nevertheless, they do not guarantee a full equipment availability. RCM has become a mandatory standard that has been implemented by the entire NPP in the world. It allowed them to have a solid methodology to optimize the efficiency of their facilities. PdM technologies are mainly used at NPP; not at the whole fuel cycle.

On the modelling part, most of researchers used some simulated data to implement machine learning algorithms for PdM. The problem here is that we are not sure to get the same performance on real data. Also many adhoc algorithms haven been proposed. The algorithms are very specific to each equipment of a nuclear plant; thus do not include a generality.

The information system for PdM has also taken an important part in this scientific literature. The proposed architecture do not integrate nuclear requirements. The architecture for deploying PdM in nuclear context is not well explained: there is no information about how or where the sensors are placed in the equipment, how they communicate with Supervisory control and data acquisition system, and how a PdM solution could be deployed in a manufacturing execution system. Furthermore there is no distinction between cloud or on premise PdM software solution. Some old wireless technologies for vibration analysis, acoustics analysis are mentioned in many studies. Not enough studies have been done on the new wireless technologies such as IIoT or 5G.

This study reveals also a methodological gap for the deployment and implementation of PdM in nuclear context. This methodology shall include all the nuclear stakeholders intended to create, manage and exploit PdM technologies. This methodology included in the lifecycle management will increase the speed of deployment and reduce the cost of this technology.

Fig. 1 :

 1 Fig. 1: Clustering of co-occurrence of authors keywords. source: VOSviewer Clustering