Maxence Grand
email: maxence.grand@univ-grenoble-alpes.fr

Damien Pellier
email: damien.pellier@imag.fr

IncrAMLSI: Incremental Learning of Accurate Planning Domains from Partial and Noisy Observations

Hand-encoding PDDL domains is generally considered difficult, tedious and error-prone. Many different machine learning techniques have been proposed to deal with these issues. Recent novel approaches such as AMLSI (Action Model Learning with System Interaction) achieve high level of accuracy, i.e. the learnt domains are accurate enough to solve planning problems without human proofreading. However, in most of the real world applications, training datasets are difficult and costly to acquire and become available gradually over time. To tackle this issue, we present IncrAMLSI, which is a version of AMLSI for incremental training datasets. We show that IncrAMLSI outperforms AMLSI to learn IPC benchmarks and converges with few learning iterations from partial and noisy data.

I. INTRODUCTION

Hand-encoding PDDL domains is generally considered difficult, tedious and error-prone. The reason is that the experts of the domains to model are not always PDDL experts and vice versa. To overcome this issue, two main approaches have been proposed. One is to develop knowledge engineering tools facilitating PDDL writing, e.g., GIPO [START_REF] Simpson | Planning domain definition using GIPO[END_REF], EUROPA [START_REF] Barreiro | Europa: A platform for AI planning, scheduling, constraint programming, and optimization[END_REF], itSimple [START_REF] Vaquero | itsimple 2.0: An integrated tool for designing planning domains[END_REF], PDDL Studio [START_REF] Plch | Inspect, edit and debug pddl documents: Simply and efficiently with pddl studio[END_REF]. These tools provide support for consistency and syntactic error checking, domain visualisation etc. An inconvenient of these tools is that they require PDDL expertise, or common knowledge in software engineering [START_REF] Shah | Knowledge engineering tools in planning: State-of-theart and future challenges[END_REF].

Another approach consists in developing machine learning algorithms to automatically produce PDDL domains, and avoid PDDL expertise: for instance, ARMS [START_REF] Yang | Learning action models from plan examples using weighted MAX-SAT[END_REF], SLAF [START_REF] Shahaf | Learning partially observable action schemas[END_REF], Louga [START_REF] Kucera | LOUGA: learning planning operators using genetic algorithms[END_REF], LSONIO [START_REF] Mourão | Learning STRIPS operators from noisy and incomplete observations[END_REF], LOCM [START_REF] Cresswell | Acquiring planning domain models using LOCM[END_REF], IRale [START_REF] Rodrigues | Incremental learning of relational action models in noisy environments[END_REF], PlanMilner [START_REF] Segura-Muros | Learning numerical action models from noisy and partially observable states by means of inductive rule learning techniques[END_REF]. In this approach, training data are (possibly noisy and partial) intermediate states, and plans previously generated by a planner, or randomly generated action sequences. The challenge is then to rebuild the PDDL domains that were used to generate these states and plans.

Learning techniques for PDDL domains are promising but they are still in the preliminary stages. One major drawback of these techniques is the accuracy of the generated domains, which measures the capacity to solve planning problems that were not in the training datasets. Thus these domains are not directly usable by planners, and a stage of domain proofreading by an expert is almost always necessary to correct errors or inaccuracies in the predicates of the actions.

Recently, [START_REF] Grand | Amlsi: A novel and accurate action model learning algorithm[END_REF] have proposed a novel approach called AMLSI (Action Model Learning with System Interaction) to address the accuracy problem. AMLSI is based on grammar induction [START_REF] Gold | Language identification in the limit[END_REF]. It takes as input noisy and partial state observations and learns planning domains accurate enough to be used by planners to solve new planning problems despite high levels of noise in observations. A specific feature of AMLSI is that it works with both feasible and infeasible action sequences generated from random walks while most approaches use only feasible action sequences or plan traces as training datasets.

However, another important drawback of the learning techniques including AMLSI is that, in pratice, dataset acquisition is a long term evolutive process: in real world applications, training data become available gradually over time, are difficult and costly to obtain, as for instance, Mars Exploration Rover operations [START_REF] Bresina | Activity planning for the mars exploration rovers[END_REF] or robot fleets for offshore missions [START_REF] Carreno | Task allocation strategy for heterogeneous robot teams in offshore missions[END_REF]. Moreover, in practice, it is important to be able to update learnt planning domains to new incoming data without restarting the learning process from scratch.

In this paper, we propose an incremental version of AMLSI called IncrAMLSI. To our best knowledge, IncrAMLSI is the only approach in the literature able to incrementally learn accurate planning domains with noisy and partial observations: other approaches learn in batch mode from a given set of states/plans, while IncrAMSLI learns incrementally from incoming data, and does not restart from scratch when new data become available.

The rest of the paper is organized as follows. In section II we propose a problem statement. In section III we give some backgrounds on the AMLSI approach and, in section IV, we detail the incremental extension of AMLSI. Then, the section V evaluates the performance of IncrAMLSI on IPC benchmarks. We show that IncrAMLSI outperforms AMLSI in terms of accuracy when training data are noisy and partially observable, and converges with very few iterations. Finally, in section VI we present the related works.

II. FORMAL FRAMEWORK

A STRIPS planning problem is a tuple P = (L, A, S, s 0 , G, , ), where L is a set of propositions using to describe the world, S is a set of state labels, s 0 2 S is the label of the initial state and G ✓ S is a set of goal state labels. Note we will often just refer to its elements as state, though these are just the labels of the states. is an observation function : S ! 2 L that assigns to each state label the set of atomic propositions true in that state. A is a set of action labels or action names. In a similar way to the states, we will often just refer to its elements as actions, though these are just the labels of the actions that do not have the often-used tuple form. Their preconditions and positive as well as negative effects are given by the functions prec, add and del that are included in , i.e. = (prec, add, del). prec is defined as prec : A ! 2 L . The functions add and del are defined in the same way. Without loss of generality, we chose this unusual formal framework inspired by [START_REF] Höller | Assessing the expressivity of planning formalisms through the comparison to formal languages[END_REF] for defining S and A in order to facilitate the statement of the STRIPS learning problem.

The function ⌧ : A⇥S ! {true, f alse} returns whether an action is applicable to a state, i.e. ⌧ (a, s) , prec(a) ✓ (s). Whenever an action is applicable, the state transition function : A ⇥ S ! S returns the state resulting from applying an action to a state:

(s i , a) = s i+1 such that (s i+1 ) = [ (s i ) \ del(a)] [ add(a).
A sequence (a 0 a 1 . . . a n ) of actions is applicable to a state s 0 when each action a i with 0  i  n is applicable to the state s i . Given an applicable sequence (a 0 a 1 . . . a n ) in state s 0 , (s 0 , (a 0 a 1 . . . a n )) = ( (s 0 , a 0 ), (a 1 . . . a n )) = s n+1 . It is important to note that this recursive definition of entails to the implicit generation of a sequence of states (s 0 s 1 . . . s n+1 ). A goal state is a state s such that (g) ✓ (s) with g 2 G. s |= g, i.e. s satisfies g if and only if s is a goal state. An action sequence is a plan (also solution) to a planning problem P if and only if it is applicable to s 0 and results in a goal state when applied to s 0 .

In formal languages, a set of rules is given that describe the structure of valid words and the language is the set of these words. For STRIPS planning problem P = (L, A, S, s 0 , g, , ), this language is defined as (0  i  n):

L(P ) = {! = (a 0 a 1 . . . a n )|a i 2 A, (s 0 , !) |= g} (1)
We know that the set of languages generated by STRIPS planning problems are regular languages [START_REF] Höller | Assessing the expressivity of planning formalisms through the comparison to formal languages[END_REF]. In other words, a STRIPS planning problem P = (L, A, S, s 0 , G, , ) generate a language L(P ) that is equivalent to a Deterministic Finite Automaton (DFA) ⌃ = (S, A, ). S and A are respectively the nodes and the arcs of the DFA and the transition function.

For any arc a 2 A, we call pre-set of a the set µ Ante = {s 2 S | (s, a) = s 0 } and post-set of a the set µ P ost = {s 0 2 S | (s, a) = s 0 }. µ Ante (s, a) (resp. µ P ost (s 0 , a)) gives the intersection of the observed state set before (resp. after) the execution of a in state s: a is an outcoming (resp. incoming) edge of s (resp. s 0 ) in the DFA.

A STRIPS learning problem is as follow: given a set of observations ⌦ ✓ L(P ), is it possible to learn ⌃ and deduce P ?

For instance, suppose ⌦ = {a, ab, ba, bab, abb, . . .

} such that s 0 a ! s 2 , s 0 a ! s 2 b ! s 2 , s 0 b ! s 1 a ! s 2 , s 0 b ! s 0 start s 2 s 1 b a a b µ P ost (a)
µ Ante (a)

Fig. 1: An example of DFA with pre-set µ Ante and post-set

µ P ost s 1 a ! s 2 b ! s 2 , s 0 a ! s 2 b ! s 2 b ! s 2 .
. . Can we learn ⌃ (see Figure 1) with actions {a, b}, the initial state s 0 and some states mark as goal G = {s 2 } so that all observations in ⌦ are a path from the initial state to a goal state to deduce P ?

III. BACKGROUND ON AMLSI

In practice, AMLSI generates ⌦ as input using random walk to learn ⌃ = (S, A, ) and deduce P = (L, A, S, s 0 , G, , ). AMLSI assumes L, A, S, s 0 known and the observation function possibly partial and noisy (a partial observation is a state where some propositions are missing and a noisy obsevation is a state where the truth value of a proprosition is erroneous). No knowledge of the goal states G is required. Once ⌃ is learnt, it remains for AMLSI to deduce from the transition function . Concretely, can be represented as PDDL planning domain containing all the actions of the problem P and by induction the classical PDDL operators.

The AMLSI algorithm consists of 4 steps: (1) generation of the observations, (2) learning the DFA corresponding to the observations, (3) induces the PDDL operator from the learnt DFA; (3) finally, refining these operators to deal with noisy and partial state observations. The first step consisting to generate ⌦. AMLSI uses random walk by applying an action from the initial state of the problem. If the action is applicable in the current state the sequence of actions from the initial state is valid and is added to I + (the set of positive samples) otherwise the random walk is stopped and the sequence is added to I (the set negative samples).

The second step consisting in learning the DFA ⌃ = (S, A, ) is carried out by using a variant of the classic algorithm for learning regular grammar called RPNI [START_REF] Oncina | Inferring regular languages in polynomial update time[END_REF] proposed by [START_REF] Grand | Retro-engineering state machines into pddl domains[END_REF] based on I + and I . The third step consists in generating the PDDL operators of the planning domain to learn. AMLSI begins by inducing the preconditions and effects of actions. To learn the preconditions prec of an action a, AMLSI find all propositions that are always present in the pre-set µ Ante (s, a) of all the state where a is feasible. Formally, p (resp ¬p) 2 prec(a) if and only if:

8(s, a) 2 ⌃ : p (resp ¬p) 2 µ Ante (s, a) (2)
Then, To learn the positive (resp. negative) effects add (resp. del) of an action a, AMLSI finds all propositions that are

8(s, a, s 0 ) 2 ⌃ : p 6 2 µ Ante (s, a) ^¬p 2 µ P ost (s 0 , a) (3) And ¬p 2 del(a) if and only if 8(s, a, s 0 ) 2 ⌃ : p 2 µ Ante (s, a) ^p 6 2 µ P ost (s 0 , a) (4)
Once actions preconditions and effects have been learnt, ALMSI generalize actions as PDDL operators and obtain a PDDL domain .

Finally, the last step consists in refining the PDDL operators induced at step 3 to deal with noisy and partial state observations. First of all, AMLSI starts by refining the operator effects to ensure that the generated operators allow to regenerate the induced DFA. To that end, AMLSI adds all effects allowing to ensure that each transition in the DFA are feasible. Then, AMLSI refines the preconditions of the operators. AMLSI makes the following assumptions as in [START_REF] Yang | Learning action models from plan examples using weighted MAX-SAT[END_REF]: the negative effects of an operator must be used in its preconditions. Thus, for each negative effect of an operator, AMLSI adds the corresponding proposition in the preconditions. Since effect refinements depend on preconditions and precondition refinements depend on effects, AMLSI repeats these two refinements steps until convergence, i.e., no more precondition or effect is added. Finally, AMLSI performs a Tabu Search to improve the PDDL operators independently of the induced DFA, on which operator generation is based. Once the Tabu Search reaches a local optimum, AMLSI repeats all the three refinement steps until convergence. 1) Update of the DFA (regular grammar) with the modified version of RPNI algorithm described in [START_REF] Grand | Retro-engineering state machines into pddl domains[END_REF] in order to accept I t + and to reject I t at iteration t; 2) Overhaul of the PDDL operators in order to remove preconditions and effects that are no longer compatible with I t + and I t , and the updated DFA at iteration t, 3) Refinement of the PDDL operators as in AMLSI to deal with noisy and partial states in I t + and I t , and to produce the new domain t+1 . The incremental learning phase is operated each time new training datasets are input and until convergence of the PDDL domain. In the rest of this section, we detail the overhaul step (sub-section IV-A) and the convergence criterion (sub-section IV-B). The steps 1 and 3 are identical to AMLSI.

IV. INCREMENTAL AMLSI

A. Operator overhaul

When at iteration t new positive and negative samples are integrated and the DFA is updated, it is possible that the domain previously learnt t is no longer compatible with it. There are two possibilities: either some effects and preconditions in t operators have to be removed to match the updated DFA, or it requires to add some preconditions and effects in the operators of t . Algorithm 1 describes the procedure to compute t+1 from t , I t + , and I t at iteration t. First of all, functions backtrackInfeasibleEffects and backtrackInfeasiblePreconditions (line 1, 2) remove the effects and preconditions infeasible in the DFA. More precisely, backtrackInfeasibleEffects removes positive and nega-tive effects that does not respect the conditions given by the equations 3 and 4 in section III. For instance, suppose we have (holding ?x 2 ) 2 add(unstack(?x 1 ?x 2 )), and there are no states in the current DFA where this effect appears, then (holding ?x 2 ) is removed. Likewise, backtrackInfeasiblePreconditions removes extra preconditions that does not validate equation 2. For instance, suppose we have (holding ?x 2 ) 2 prec(unstack(?x 1 ?x 2 )), if the action unstack(a, b) is feasible in state s, and (holding b) 6 2 µ Ante (s, stack(a b)), then

(holding ?x 2 ) is removed.
Then, as observed states are noisy and partial, it is possible that some extra preconditions and effects are not present in the current DFA. Therefore lines 3 -9 remove extra preconditions and effects independently of this DFA. First of all, backtrack-Preconditions (line 5) removes all the preconditions that are not compatible with I t + . For instance, suppose we have this positive sample: )), i.e. in the observed state before the last action unstack(a, b), and (holding ?x 1 ) 2 prec(unstack(?x 1 ?x 2 )). Thus we remove this unsatisfied precondition (holding ?x 1 ) from unstack(?x 1 ?x 2 ).

Then, with backtrackNegativeEffects (line 6), we remove all the negative effects that does not satisfy PDDL syntactical constraints [START_REF] Yang | Learning action models from plan examples using weighted MAX-SAT[END_REF] (e.g. negative effects must be in the preconditions etc.) We repeat these two functions until the domain accepts all the positive samples (line 7), i.e. are able to regenerate all the positive samples I t + . The next step (line 8) in the algorithm is to remove all the effects that are not compatible with the negative samples I t with function backtrackEffects. For instance, suppose that two samples share the same prefix unstack(a, b): Also, suppose that, wrongly, (holding ?x 2 ) 2 add(unstack(?x 1 ?x 2 )), and, correctly, (holding ?x) 2 prec(put down(?x)). And suppose that the current domain t can generate unstack(a, b). The wrong effect (holding b) 2 add(unstack(a, b)) allows to the precondition of the unfeasible action put down(b) to be satisfied, and (holding b) is not in the preconditions of the feasible action put down(a). Thus, (holding ?x 2 ) must be removed from add(unstack(?x 1 , ?x 2 )).

As the backtrackEffects function removes effects, it is possible that some positive samples can no longer be generated by the current domain t . We therefore have to repeat these three backtracking functions until the domain accepts all the positive samples (line 9). Indeed, repeating these functions until the domain accepts all the positive samples allows to ensure that the last removed effects do not impact the acceptation of the positive samples (termination is ensured by only allowing precondition and/or effect withdrawals).

Finally, the previous backtracking steps (line 3 -9) allow to remove all the preconditions and effects that are not compatible with the DFA and the samples: all the wrong preconditions/effects have been removed, as well as possibly some correct ones. The objective of lines 10 -11 is to integrate into t all the remaining information in the DFA. This cannot corrupt t because, by definition, the DFA accepts I t + and rejects I t . We build a new domain new (line 10) by generating the operators as in Phase 1. Then, the effects and preconditions of new are refined (line 11) as in AMLSI [START_REF] Grand | Amlsi: A novel and accurate action model learning algorithm[END_REF]. Finally, we merge new with the current domain t (line 12) by making the union sets of the preconditions and the effects of the operators.

B. Convergence criterion

IncrAMLSI stops when, after a given number of iterations, the domain t checks three conditions: (1) t accepts all the positive samples I t + , (2) t rejects all the negative samples I t , and (3) t = t+1 during iterations. We will show experimentally (see section V) that IncrAMLSI with partial and noisy state observations converges to a domain D. However, it is possible to show formally that it is true when state observations are complete and noiseless and ! 1. The sketch of the proof is as follows: RPNI algorithm, which is at the core of IncrAMLSI, converges if the training dataset is a "characteristic sample" [START_REF] Oncina | Inferring regular languages in polynomial update time[END_REF], i.e. the dataset covers all the states and transitions of the targeted grammar.

V. EXPERIMENTS

In order to compare AMLSI with IncrAMLSI, we use the same experimental framework as in [START_REF] Grand | Amlsi: A novel and accurate action model learning algorithm[END_REF]. The experiments are based on 10 IPC domains: Blocksworld, Gripper, Hanoi, N-Puzzle, Peg Solitaire, Parking, Zenotravel, Sokoban, Neg-Visit-All and Neg-Elevator. Neg-Visit-All and Neg-Elevator are modified versions of Visit-All and Elevator with negative preconditions to show AMLSI ability to learn them. All the used benchmarks are STRIPS domains 1 .

To deal with the fact that the initial state chosen may impact the learning process, we test each IPC domain with 3 different initial states over five runs and we used five seeds randomly generated for each run. Then, the length of the random walk sequences generated is randomly chosen between 10 and 20. Finally we generate partial observations by randomly removing a fraction of the propositions of the states and we generate noise changing the value of a fraction of observable proposition of the states. All tests were performed on an Ubuntu

A. Evaluation Metrics

We evaluate IncrAMLSI with three different metrics: the syntactical error [START_REF] Zhuo | Learning complex action models with quantifiers and logical implications[END_REF] that computes the distance between the original domain and the domain learnt, the accuracy [START_REF] Zhuo | Refining incomplete planning domain models through plan traces[END_REF] that express the capability of a planner to use the domain learnt to solve new problems. Even though the syntactical error is the most used metric in the literature, we argue that the accuracy is the most important metric in pratice for planning because it measures to what extent a learnt domain is useful. Indeed, it often happens that one missing precondition or effect, which amounts to a small syntactical error, makes them unable to solve planning problems. Finally, the last metric is the FScore that express the capability of the learnt domain to regenerate the regular grammar. Formally, the syntactical error error(a) for an action a is defined as the number of extra or missing predicates in the preconditions ⇢ a , the positive effects ✏ + a and the negative effects ✏ a divided by the total number of possible predicates. By extension, the syntactical error for a domain composed of a set of actions A is:

E = 1 |A| X a2A error(a)
Formally, the accuracy Acc = N N ⇤ is the ratio between N , the number of correctly solved problems with the learnt domain, and N ⇤ , the total number of problems to solve. In the rest of this section the accuracy is computed over 20 problems. The problems are solved with Fast Downward v19.06 [START_REF] Helmert | The Fast Downward planning system[END_REF]. Plan validation is realized with the automatic validation tool used in the IPC competition VAL [START_REF] Howey | Val's progress: The automatic validation tool for pddl2. 1 used in the international planning competition[END_REF].

Formally, the FScore is computed as follows: FScore = and P is the precision, i.e. the rate of sequences e accepted by the learnt domain that are sequences accepted by the ground truth domain, computed as follow:

P = |{e2E + | accept(D,e)}| |{e2E + | accept(D,e)}[{e2E | accept(D,e)}| .
The test set used to compute this metric is generated by a random walk using the observation generation method described in Section -III.

B. Results

In order to study the performance of IncrAMLSI with respect to noisy and partial state observations, we use four different experimental scenarios: (1) complete intermediate First of all, Table I gives a comparison between AMLSI and IncrAMLSI. The results of the first scenario (complete intermediate observations and no noise) show that AMLSI and IncrAMLSI perfectly learn the preconditions and the effects of the operators of the IPC domains. For all algorithms and for all domains FScore and accuracy are optimal. Note that for 4 IPC domains, Hanoi, N-Puzzle, Parking and Sokoban, the syntactical error is not equal to 0. This is because algorithms learn implicit preconditions that are not encoded in the IPC domain. The results of the second scenario (complete intermediate observations and high level of noise (20%)) show that IncrAMLSI is generally more robust to noise than AMLSI when observations are complete. IncrAMLSI learns optimal domains for Blocksworld and Zenotravel and outperforms AMLSI. For Peg Solitaire, the IncrAMLSI algorithm outperforms AMLSI and obtains optimal accuracy, however the syntactical distance is deteriorated. The results of the third scenario (partial intermediate observations (25%) and no noise) are similar to the results of the first scenario. Finally, the results of the fourth scenario (partial intermediate observations (25%) and high level of noise (20%)) show that IncrAMLSI is generally the most robust algorithm. Indeed, in 3 IPC domains (Blocksworld, Hanoi, and Sokoban), IncrAMLSI learns better domains than AMLSI, while only 2 IPC domain (PegSOliatire and NegElevator) is better for AMLSI. To conclude, we observe that for all the domains, whatever the experimental scenario, IncrAMLSI always obtains a higher FScore than AMLSI. This is explained by the convergence criterion used by IncrAMLSI: the convergence criterion tends to learn a domain able to perfectly regenerate the grammar. However, we notice that a better FScore does not always guarantee a better accuracy. For instance, in the Peg Solitaire domain of the fourth experimental scenario, the FScore of IncrAMLSI is 99% while the FScore of AMLSI is 97.9%. But the accuracy of IncrAMLSI is 95% while the accuracy of AMLSI is 96.3%. However, we note that, in general, a better FScore is correlated with a better accuracy.

Then, Table II shows the average number of iterations for IncrAMLSI convergence. We note that for the first and the third scenario, IncrAMLSI converges in less than 20 iterations for each domain. For the second scenario we note that for each domain, except Peg Solitaire, IncrAMLSI converges in 15 iterations or less. Finally for the last scenario, IncrAMLSI needs more than 30 iterations for 4 IPC domains (Blocksworld, Peg Solitaire, Parking and Sokoban). However, for these domains, IncrAMLSI generally outperforms AMLSI.

Finally, Figure 3 shows the performance in terms of accuracy of the IncrAMLSI approach while varies between 2 and 10. For each domain, increasing makes it possible to improve the accuracy. We can notice that the Gripper, Hanoi, N-Puzzle, Zenotravel and NegVisitAll domains always converge towards the optimal domain whatever the scenario if is high enough. For Blocksworld, when the observations are noisy and partial, the convergent algorithm seems converge if > 10. Finally, the domains Peg Solitaire, Sokoban and

NegElevator seems to have trouble to converge, however learnt domains are always accurate (Acc > 50%) whatever the level of noise and observability and the value of for these domains.

VI. RELATED WORKS

Many approaches have been proposed to learn planning domains. We propose below a classification according to the input data of the learning process. The input data can be plan "traces" obtained by resolving a set of planning problems, partial planning domains to complete or "random walks". The input data can be complete (states and actions), partial, completely blind, or noisy.

A first group of approaches takes as input a set of plan traces and a partial domain, and tries to incrementally refine this domain to complete it, as for instance, OpMaker [START_REF] Mccluskey | An interactive method for inducing operator descriptions[END_REF] or RIM [START_REF] Zhuo | Refining incomplete planning domain models through plan traces[END_REF]. In practice, RIM constructs sets of "soft" and "hard" constraints between observed states and actions, which are solved with weighted MAX-SAT solvers to generate refined actions. In all of these approaches, it is assumed that the observations are complete and noiseless. Opmaker works differently. It induces operators with user interactions. The partial domain is built by the user with the GIPO tools [START_REF] Simpson | Planning domain definition using GIPO[END_REF], then the user gives plan traces and intermediate state observations, and OpMaker induces pre-conditions and effects.

A second group of approaches takes as input only plan traces. Most of them deals with partial observations (except Observer [START_REF] Wang | Learning by observation and practice: An incremental approach for planning operator acquisition[END_REF] that needs complete observations). Among these approaches are ARMS [START_REF] Yang | Learning action models from plan examples using weighted MAX-SAT[END_REF], SLAF [START_REF] Shahaf | Learning partially observable action schemas[END_REF], or Louga [START_REF] Kucera | LOUGA: learning planning operators using genetic algorithms[END_REF]. ARMS gathers knowledge on the statistical distribution of frequent sets of actions in the plan traces. It then forms a weighted propositional satisfiability problem (weighted SAT) and solves it with a weighted MAX-SAT solver. Unlike ARMS, SLAF is able to learn actions with conditional effects. To that end, SLAF relies on building logical constraint formula based on a direct acyclic graph representation. Louga takes also as input plan traces and work with partial noiseless observations. However, Louga is able to learn actions with static properties and negative preconditions. Louga uses a genetic algorithm to learn action effects and an ad-hoc algorithm to learn action preconditions. Plan-Milner uses a classification algorithm based on inductive rule learning techniques: it learns action models with discrete numerical values from partial and noisy observations. Finally, the LOCM family of action model learning approaches [START_REF] Cresswell | Acquiring planning domain models using LOCM[END_REF], [START_REF] Cresswell | Generalised domain model acquisition from action traces[END_REF], [START_REF] Gregory | Domain model acquisition in the presence of static relations in the LOP system[END_REF], [START_REF] Gregory | Domain model acquisition in domains with action costs[END_REF] works without information about initial, intermediate and final states. These algorithms extract, from plan traces, parameterized automata representing the behaviour of each object of the planning problems. Then preconditions and effects are generated from these automata.

The last group of approaches takes as input random walks, that is, sets of action sequences randomly generated. Random walk approaches like IRALe [START_REF] Rodrigues | Incremental learning of relational action models in noisy environments[END_REF] deal with complete but noisy observations. IRALe is based on an online active algorithm to explore and to learn incrementally the action model with noisy observations. Others approaches such as LSO-NIO [START_REF] Mourão | Learning STRIPS operators from noisy and incomplete observations[END_REF] deal with both partial and noisy observations. LSO-NIO uses a classifier based on a kernel trick method to learn action models. It consists of two steps: (1) it learns a state transition function as a set of classifiers, and (2) it derives the action model from the parameters of the classifiers.

VII. CONCLUSION

In this paper we have addressed the problem of learning PDDL domains from plan traces with noisy and partial observations. To deal with this problem, we have presented an incremental learning approach called IncrAMLSI that achieves a high level of accuracy, i.e. the learnt domains are accurate enough to solve planning problems without human proofreading. A specific feature of IncrAMLSI is that it is able to deal with data that becomes available gradually over time. We have experimentally shown that IncrAMLSI (1) outperforms the most accurate approach existing in the literature, and (2) converges to the experimental domains with a few iterations from partial and noisy observations. In future works, we will extend our approach to learn more expressive planning domains such as Temporal domains and HTN domains by building on the breakthroughs of grammar induction algorithms.

[

  pick up(a), put down(a), pick up(a), stack(a, b), unstack(a, b)] 2 I t + and, when we execute it with the current domain t , (holding a) 6 2 ( (s 0 , [pick up(a), put down(a), pick up(a), stack(a, b)]

[

  unstack(a, b), put down(a)] 2 I t + and [unstack(a, b), put down(b)] 2 I t

P

  +R where R is the recall, i.e. the rate of sequences e accepted by the ground truth domain that are successfully accepted by the learnt domain, computed as follows: R = |{e2E + | accept(D,e)}| |E|

Fig. 3 :

 3 Fig. 3: Convergent learning while varies

  An overview of incremental AMLSI (IncrAMLSI) is shown in Figure 2. It learns incrementally from incoming data and does not restart from scratch when new data become available at each iteration t. It consists of two phases: Algorithm 1: Overhaul Operator( t , I t + , I t ) Phase 1. (Initialising phase) consists in generating the initial PDDL domain 0 from the initial training datasets I 0 + and I 0 by applying the AMLSI algorithm as described in section III, • Phase 2. (Incremental learning phase) consists in incrementally updating the PDDL domains t with the new incoming training datasets I t + and I t available at iteration t to produce the domain t+1 . This phase is made up of three steps:

	1 backtrackInfeasibleEffects( t	);
	2 backtrackInfeasiblePreconditions( t	);
	3 repeat
	4	repeat
	5		backtrackPreconditions( t	);
	6		backtrackNegativeEffects( t	);
	7 8	until acceptAll( t , I t + ); backtrackEffects( t )
	9 until acceptAll( t , I t + ); 10 new operatorGeneration();
	11	new	DFARefinement( new	);
	12	t+1	merge( t , new	);
	13 return t+1
	•

  [START_REF] Gold | Language identification in the limit[END_REF].04 server with a multi-core Intel Xeon CPU E5-2630 clocked at 2.30 GHz with 16GB of memory.

	Observability			100%					25%
	Noise		0%			20%			0%			20%
	Domain	Algorithm	FScore E	Acc	FScore E	Acc	FScore	E	Acc	FScore E	Acc
	Blocksworld	AMLSI IncrAMLSI	100 1000	0 0	100 100	94.6 100 100 100	0.6 0 0 0	93.7 100 100 100	100 100	0 0	100 100	77 100 100 100	1.2 0.4 0.4 0.4 94.3 94.3 94.3 76.3
	Gripper	AMLSI IncrAMLSI	100 100	0 0	100 100	100 100	0 0	100 100	100 100	0 0	100 100	100 100	0 0	100 100
	Hanoi	AMLSI IncrAMLSI	100 100	0.9 0.9	100 100	100 100	0.9 0.9	100 100	100 100	0.9 0.9	100 100	91.6 100 100 100	2 0.9 0.9 0.9	86.7 100 100 100
	N-Puzzle	AMLSI IncrAMLSI	100 100	5.6 5.6	100 100	100 100	5.6 5.6	100 100	100 100	5.6 5.6	100 100	100 100	5.6 5.6	100 100
	Peg Solitaire	AMLSI IncrAMLSI	100 100	4.2 4.2	100 100	96 100	7.1 5.2 5.2 5.2	93.3 100	100 100	4.2 4.2	100 100	97.9 99 99 99	6.9 96.3 96.3 96.3 6.3 6.3 6.3 95
	Zenotravel	AMLSI IncrAMLSI	100 100	0 0	100 100	95 100 100 100	0.2 0 0 0	99.3 100 100 100	100 100	0 0	100 100	100 100	0 0	100 100
	Parking	AMLSI IncrAMLSI	100 100	3.9 3.9	100 100	100 100	3.9 3.9	100 100	100 100	4.1 3.9	100 100	93.5 100	4.2 4.2 4.2 4.3	100 100
	Sokoban	AMLSI IncrAMLSI	100 100	3.9 3.9	100 100	100 100	3.9 3.9	100 100	100 100	3.9 3.9	100 100	67.1 86.1 86.1 86.1	6.1 5.5 5.5 5.5	60 73 73 73
	NegVisitAll	AMLSI IncrAMLSI	100 100	0 0	100 100	100 100	0 0	100 100	100 100	0 0	100 100	100 100	0 0	100 100
	NegElevator	AMLSI IncrAMLSI	100 100	0 0	100 100	100 100	0 0 0 0.2	100 100	100 100	0 0	100 100	100 100	0 0 0 0.1	100 100 100 93.3

TABLE I :

 I Comparison between AMLSI and IncrAMLSI with = 10. Best results for each metrics are in bold.

	Observability	100%	25%
	Noise	0%	20%	0%	20%
	Domain	Number of iterations
	Blocksworld	11.5	16.9	13.2	32.5
	Gripper	11	11.2	11.3	14.4
	Hanoi	11	13	13.2	22.2
	N-Puzzle	11	11	11	11
	Peg Solitaire	11.7	26.7	20.5	45.6
	Zenotravel	11	14.5	12.8	19.2
	Parking	11	13.4	20.3	34.7
	Sokoban	11	15	14.8	55.5
	Neg Visit All	11.7	12.3	11.2	11.9
	Neg Elevator	11.1	13.5	13.9	24.7

TABLE II :

 II Average number of iterations to converge when

	= 10

The experimental setup and results including additional experimental results are available at https://gitlab.com/AMLSI/amlsi.

ACKNOWLEDGEMENTS

This research is supported by the French National Research Agency under the "Investissements d'avenir" program (ANR-15-IDEX-02) on behalf of the Cross Disciplinary Program CIRCULAR.


Céline Becquet
email: celine.becquet@unicaen.fr

Fausto Viader
email: fausto.viader@unicaen.fr

Francis Eustache
email: francis.eustache@unicaen.fr

Peggy Quinette
email: peggy.quinette@unicaen.fr

pre-existing vulnerability factors

Keywords: transient global amnesia, objective self-awareness, self-focused attention, episodic memory, anxiety, hippocampus PA_03 PA_04 PA_06 PA_07 PA_08 acute

come

Introduction

Transient global amnesia (TGA) is a neurological syndrome characterized by a transient and massive loss of memory with iterative questioning (Hodges & Warlow, 1990). During the acute phase, patients present memory dysfunction affecting both its anterograde side, particularly impressive because of iterative questioning, and, to a lesser extent, its retrograde side (Guillery-Girard et al., 2004[START_REF] Viader | Transient global amnesia[END_REF]. The disorder affects exclusively the episodic memory system, making TGA a unique model for the study of its functioning. Because of its transient character and the absence of cognitive reorganization and/or compensatory mechanisms in patients, it is especially adapted to the investigation of cognitive components related to episodic memory, such as self-awareness (SA), a high-level function allowing individuals to integrate themselves into experiences they are living and adopt a reflexive position about themselves. Self-awareness is strongly integrated into the episodic memory concept defined by [START_REF] Tulving | Memory and Consciousness[END_REF] which described a first-person memory processed with an inner feeling of reliving or projecting him/her self toward subjective time. Then, the ability to process information orientated toward the self can be consider as involved in the feeling that events belong to the self peculiar to the episodic memory functioning.

SA is a polysemic concept which refers to multiple processes. The objective SA theory defines objective SA as a particular state in which "the person takes himself to be an object" [START_REF] Duval | A theory of objective self-awareness[END_REF][START_REF] Wicklund | Objective self-awareness[END_REF]. According to these authors, attention is a dichotomic function that may be selectively oriented either on the self or on the environment.

Objective SA can thus be described as a process in which the focus of attention is oriented towards the self. According functional imaging studies, the process of information related to the self is assumed by the default mode network (DMN), involving medial prefrontal cortices, posterior cingulate cortex, precuneus and lateral parietal cortex, that is deactivated when the attention is engaged toward an external task, at the opposite of the central executive network implicated when the individual is focused on the outside world [START_REF] Raichle | The Brain's Default Mode Network[END_REF].

In TGA, a study exploring metacognition, a high-level process requesting objective SA, showed that patients had poor explicit knowledge of their amnesia during the acute stage [START_REF] Hainselin | Awareness of disease state without explicit knowledge of memory failure in transient global amnesia[END_REF]. This study also showed that these difficulties appeared concomitantly with a state of high anxiety, characteristic in acute TGA patients (Noël et al., 2008). TGA often takes place around precipitating events. These events can be physical (swimming in cold water, sexual intercourse, intensive gardening…), psychological (bad news disclosure…) or both (a medical test can both be painful and psychologically dreaded by the patient). According to Bartsch et al (2010), such precipitating events result in a physiological stress, that in turn triggers a cascade of metabolic reactions, finally giving rise to a temporary dysfunction of the CA1 hippocampal neurons, that are especially vulnerable to metabolic stress [START_REF] Bartsch | Selective affection of hippocampal CA-1 neurons in patients with transient global amnesia without long-term sequelae[END_REF]. In other pathological conditions linked to stress reactions, such as post-traumatic stress disorder (PTSD) or situational depression, modifications of self-focused attention, generally evidenced by questionnaires, have sometimes been observed (for review: [START_REF] Ingram | Self-Focused Attention in Clinical Disorders: Review and a Conceptual Model[END_REF], as well as impaired episodic memory [START_REF] Dere | Emotion and episodic memory in neuropsychiatric disorders[END_REF]. Nevertheless, while the relationships between self-focused attention and negative affect or between stress and memory have been thoroughly investigated, the way by which SA interacts with memory has not been studied so far in amnesic patients [START_REF] Mor | Self-focused attention and negative affect: A meta-analysis[END_REF].

Self-focused attention can depend on pre-existing personal characteristics, that could influence vulnerabilities to develop pathological stress reactions [START_REF] Ingram | Self-Focused Attention in Clinical Disorders: Review and a Conceptual Model[END_REF]. The presence of psychogenic vulnerabilities in TGA patients has been suggested. [START_REF] Inzitari | Emotional arousal and phobia in Transient Global Amnesia[END_REF] observed phobic attitudes in about half of the 51 patients evaluated in their study. In a retrospective study including 142 cases, [START_REF] Quinette | What does transient global amnesia really mean? Review of the literature and thorough study of 142 cases[END_REF] also found phobic attitudes, and showed that 21.2% of TGA patients had a medical history of depression or anxiety episodes. In [START_REF] Döhring | Stress-related factors in the emergence of transient global amnesia with hippocampal lesions[END_REF], acute TGA patients used less effective coping strategies than did control subjects, with a tendency to focus on internal process (emotion-centered coping), which is often considered as less adaptive. This study, including 21 acute TGA patients, failed to find association between stressful events, hippocampal CA1 lesions and behavioral patterns during the TGA attack, but led the authors to identify a subcategory of "emotional" TGA patients that could affect some predisposed people. This idea was already evoked by [START_REF] Quinette | What does transient global amnesia really mean? Review of the literature and thorough study of 142 cases[END_REF] who distinguished with a hierarchical cluster analysis, three categories of TGA patients: 1) a first category associating emotional variables and the psychopathological variables with women patients; 2) a second category associating men with an absence of medical history and physical precipitating events; 3) a third category gathered a medical history of migraine and the younger patients.

The present study focused on two considerations: 1) the relationship between selffocused attention (objective SA) and episodic memory functioning during the acute phase of TGA; 2) the pre-existing disposition in TGA patients to be self-focused.

In the first part of this study, we focused on situational effect of episodic memory dysfunction on SA by comparing patients seen during the acute phase of TGA to those during the recovery phase or follow-up. In the second part, we considered pre-existing characteristics of SA in patients according to the kind of precipitating event at the origin of the TGA episode (psychological, physical, and other or none). The objective was to learn if pre-existing SA tendencies might predict the kind of precipitating event related to TGA. Thus, we compared scores in trait-SA in patients seen during periacute stages compared to follow-up patients according to their different precipitating event groups, to better understand interactions between situational and pre-existing dimensions.

Materials and methods

Population

Data of 27 patients presenting or who had experienced TGA were collected during their admission to the emergency or the neurological departments of Caen University Hospital between 2016 and 2019. All the patients met the clinical diagnostic criteria for TGA (Hodges & Warlow, 1990) and had normal neurological and brain CT scan examinations. The study was approved by the local ethics committee of the medical faculty of Caen university and conformed to the Declaration of Helsinki. All participants were informed and gave their consent to take part to the research.

To assess episodic memory functioning, we used a verbal episodic memory task, the ESR-r (encoding, storage, retrieval) [START_REF] Noël | The Still Enigmatic Syndrome of Transient Global Amnesia: Interactions Between Neurological and Psychopathological Factors[END_REF]. We used the strategic-learning part of this task consisting in a verbal learning of a list of words involving a semantic processing of the information by asking the participants to formulate a sentence with the target word (ie.

"turnip"?"). A cued-recall was proposed every other item to make sure that deep encoding had been done (Craik & Tulving, 1975). Three indicators were calculated: the encoding score (on 16), the immediate free recall score (on 16) and the delayed 20 minutes recognition score (on 16).

Patients seen within the first 24 hours of the episode formed the "acute" and the "recovery" groups. Those who had impaired anterograde memory at the 20 min recognition score of the ESR test (z-score inferior to -1.65 in comparison to the normative values in [START_REF] Noël | The Still Enigmatic Syndrome of Transient Global Amnesia: Interactions Between Neurological and Psychopathological Factors[END_REF] were classified into the acute group, and those presenting no anterograde memory impairment into the recovery group (see table in Appendix for complete description of patients). Patients seen within the first 24 h but for whom these data were missing were classified into the acute group if they had iterative questioning and into the recovery group if not.

Six patients were seen during the acute phase, 8 during recovery and the remaining 13 at a follow-up appointment between 1 and 15 months (mean=3.88; SD=3.88) after the episode.

Mean TGA duration was 2 hours and 40 minutes (SD=2 hours and 09 minutes). In the second part of the study, we further classified the patients according to the kind of precipitating events potentially linked to the occurrence of the TGA. This classification of events (from 1 to 8) is inspired from [START_REF] Quinette | What does transient global amnesia really mean? Review of the literature and thorough study of 142 cases[END_REF]: 1=emotional stress the day of the episode; 2=physical effort the day of the episode; 3=severe pain; 4=unusual contact with water; 5=sexual intercourse; 6=other; 7=context of stress during the weeks preceding the episode; 8=no identified precipitating event. Three groups of patients were defined as follows: PSY= TGA following to psychological events (events 1 and 7); PHYS=physical event having occurred in the 24 h before the episode ( events 2,3,4 and5); NONE=no or other precipitating event (6,8) (see further information about precipitating events in table in Appendix). When several events seemed to be present at the same time, e.g. for PA_02 who had a sea bath the day of the TGA (4), while she was confronted to a context of stress ( 7), we chose the event closer to the episode (in this case, the sea bath, so we classified this patient into the PHYS group). Patients were compared with 11 normal-control (NC) subjects matched for age and education level.

Assessments

Situational self-awareness

The French version of the Situational Self-Awareness Scale (SASS) [START_REF] Govern | Development and validation of the situational self-awareness scale[END_REF] traduced by Auzoult (2013) was used to assess subjective levels of patients' immediate SA. For each of the 9 items, patients had to respond on a 5-point Lickert scale ranging from 1 (strongly disagree) to 5 (strongly agree). Three different scores were calculated and expressed in mean: private SA, public SA and surroundings (Auzoult, 2013). Private SA corresponds to the awareness of personal feelings, thought or actual state ("Right now, I am conscious of my inner feelings"), while public SA corresponds to the attention of the subject of his/her own image reflected toward other people ("Right now, I am concerned about what other people think of me"). Both of these two scales concern personal SA. On the other hand, the surroundings scale concerns the attention of the subject dedicated toward the external world ("Right now, I am conscious of what is going on around me").

Dispositional self-awareness

We used the French version of the Self-Consciousness Scale (SCS) [START_REF] Fenigstein | Public and private self-consciousness: Assessment and theory[END_REF] traduced by [START_REF] Rimé | Le concept de conscience de soi et ses opérationnalisations[END_REF] to assess the dispositional or trait SA on three scales (private SA: eg. "I'm always trying to figure myself out", public SA: eg. "I'm usually aware of my appearance", social anxiety: eg. "I get embarrassed very easily"). For each of the 23 propositions supposed to describe themselves, participants have to pick on a 5-point scale between: 0=false, 1=more or less false, 2=I don't know, 3=more or less true, 4=true. Scores were expressed in mean of propositions for each of the three scales.

Anxiety

In order to assess anxiety, we used the Spielbergers' State-trait anxiety inventory (STAI). Both scales (state and trait) were proposed to patients at the same time.

Statistical methodology

We used non-parametric tests. Group comparisons were processed with Kruskal-Wallis ANOVAs by Ranks (H test). Post-hoc tests were realized with Mann & Whitney U tests. Intragroup comparisons were processed with Friedman ANOVAs test and intra-group post-hoc analyses were performed with Wilcoxon matched pair tests. Effect sizes were expressed in Spearman's r coefficient of concordance.

Results

Part 1: Self-awareness and episodic memory at different stages of TGA

In this part, we explored the relationship between episodic memory dysfunction and situational self-awareness ("TGA stage" factor) (Table 1). We measured trait SA to control for a possible bias linked to intrapersonal characteristic.

[Table 1 near here]

Patients in acute phase obtained lower scores than patients in recovery group in every part of the ESR-r (all p<.01, all r>0.82), same compared to follow-up and NC groups (all p<.001, all r>0.77) (Fig. 1.).Patients in recovery phase were not different than follow-up and NC groups for the encoding and the 20' recognition scores (all p>0.79) but tended to obtain lower score than follow-up patients (Mann & Whitneys'U test= 10.5, p<.05, r=0.57) and NC group (p=0.08) for the free recall score. Follow-up patients did not differ from NC group in none ESR score (all p>0.42).

State-anxiety (STAI A) decreased significatively with TGA phase (Kruskal-Wallis' H test=9.80, p<.05) (Table .1). The level of state-anxiety was higher in the acute than in the follow-up (Mann & Whitneys' U test = 4.5, Z=2.22, p<.05, r=0.58) and the NC groups (Mann & Whitneys' U test = 3.5, Z=2.16, p<.05, r=0.60). There was no significant difference between the acute and recovery groups (p=0.14). Patients from the recovery group had a higher level of state-anxiety than patients from the follow-up group (Mann & Whitneys' U test = 12, Z=2.07, p<.05, r=0.50) but did not differ from NC subjects (p=0.12). Patients from the followup group did not differ from NC subjects (p=0.54). Kruskal-Wallis ANOVA did not show group effect for trait-anxiety (STAI B) (p=0.28).

Self-awareness

Kruskal-Wallis ANOVA for SASS found no effect of the TGA stage on any scale between patients (all p>0.31). Friedman's ANOVA showed differences between SASS' subscales in all (all p<.05) but the acute TGA group (ANOVA Chi²= 1.37; p=0.50). In every group except the acute group (Wilcoxon's Z test: all p>0.22 in the acute group), participants were more focused toward surroundings than toward private and public self-representations (Fig. 1).

[Figure 1 near here] Kruskal-Wallis test did not reveal differences between groups for the self-awareness trait scale (SCS) (all p>0.07).

Part 2: Personality particularities in TGA according to the precipitating event group

Here, patients were compared for dispositional self-consciousness or anxiety-trait to the same control group as in part 1, according to the kind of precipitating event (psychological = PSY, physical = PHYS, and absent = NONE) (Table 2). As in part.1, we controlled for situational SA between groups to avoid a bias linked to the actual memory status of patients. Afterwards, we clustered patients seen the day of the TGA (acute and recovery patients) into the periacute group to compare with patients seen during follow-up (Table 3.). We excluded the NONE group from these analyses because it includes only 3 patients. Mann & Whitney analyses were carried out. We compared PSY periacute and PHYS periacute patients, PSY follow-up and PHYS follow-up patients to explore the effect of the precipitating event group in patients seen at the same phase of TGA. We compared PSY periacute and PSY follow-up patients, and PHYS periacute and PHYS follow-up patients to explore more specifically the effect of the temporality in each group separately. Anxiety trait (Stai_B) was comparable for patients of any groups and with healthy subjects (H=0.47, p=0.93). Same, state anxiety (Stai_A) did not differ significantly between none group (H=2.32, p=0.51).

[Table 2 near here] [Table 3 near here]

Of our 27 patients, 10 had a history of depression or anxiety. Another one, PA_13, did not report depression or anxiety history but she evoked several major psychic traumas. PA_27 did not report anxiety disorders. Nevertheless, she followed class against aquaphobia. TGA occurred during one of these sessions of aquaphobia class. Pearson's Chi-square tests did not reveal any dependency in TGA patients between the precipitating event group (PSY versus PHYS) and the presence of psychiatric medical history (Chi²=1.60; df=1; p=0.21) nor between the gender and the kind of precipitating event (Chi²=0.18; df=1; p=0.67).

Self-awareness

There was no significant group effect in Self-Consciousness scale (SCS) for "private self-awareness" (H=5.68, p=0.13) and "social anxiety" (H=3.36, p=0.34) scores (Fig. 2), with only a trend for "public self-awareness" score (H=7.58, p=0.06). Mann & Whitney test showed that patients from PHYS group reported higher public self-awareness than patients from PSY group (U=28.5, Z=2.28, p<.05, r=0.48) an NC subjects (U=30.5, Z=1.94, p<.05, r=0.41).

[Figure 2 near here]

Friedman ANOVAs revealed no difference between the three scales of the Self-Consciousness scale SCS in PSY group (Friedman's Anova Chi²=3.5; p=0.17), NONE group (Friedman's Anova Chi²=0.67; p=0.72) and NC group (Friedman's Anova Chi²=0.97; p=0.61).

Nevertheless, we found an effect of the scale in the PHYS group (Friedman's Anova Chi²=9.45; p<.01) with a public SA score superior to the social anxiety score (Wilcoxon's Z=2.67; p<.01; r=0.57) and a trend toward higher public SA score compared to private SA score (p=0.07).

Kruskal-Wallis ANOVA did not show difference between groups in situational selfawareness scale for private (p=0.28) and public (p=0.29) SA. We observed a trend for surrounding scale (p=0.07), but post-hoc tests were not significant (all p > 0.15) except between PSY and PHYS groups, with a higher level of surrounding awareness for the PSY group (U=39.5, Z=1.85, p<.05, r=0.45).

Comparison of periacute and follow-up patients in function of their precipitating event group

About situational self-awareness, we found no difference between groups in SASS scores (all p>0.11).

Regarding dispositional self-awareness, as previously observed, we found a difference between PSY and PHYS patients in the "public SA" score of the SCS, in patients seen during the periacute phase (Mann & Whitney U=6.5, p<.05, r=0.53). We found no such difference for this scale in PHYS patients seen in follow-up compared to PSY follow-up patients (p=0.17), nor in other scales (all p>0.17). There was no difference in any scale for periacute versus follow-up patients in PSY and PHYS groups (all p>0.47).

Discussion

In studies of TGA, self-awareness (SA) is usually considered to be preserved during the episode [START_REF] Frederiks | Transient Global Amnesia[END_REF][START_REF] Noël | The Still Enigmatic Syndrome of Transient Global Amnesia: Interactions Between Neurological and Psychopathological Factors[END_REF][START_REF] Quinette | What does transient global amnesia really mean? Review of the literature and thorough study of 142 cases[END_REF] except for metacognitive abilities [START_REF] Hainselin | Awareness of disease state without explicit knowledge of memory failure in transient global amnesia[END_REF]. In these studies, however, the concept of SA is used as to mean the global preservation of semantic personal knowledge, which is independent from episodic memory functioning [START_REF] Noël | The Still Enigmatic Syndrome of Transient Global Amnesia: Interactions Between Neurological and Psychopathological Factors[END_REF][START_REF] Quinette | What does transient global amnesia really mean? Review of the literature and thorough study of 142 cases[END_REF]. Here, we focused on objective SA, exploring the ability to process self-related information as the object of the consciousness [START_REF] Duval | A theory of objective self-awareness[END_REF].

The present study sought to explore links between objective SA, episodic memory functioning and vulnerability factors in TGA. The first part focused on state dimension by comparing patients seen at different stages of the attack (acute, recovery, follow-up) while the second part explored pre-existing SA dimensions in patients according to the kind of precipitating event of the attack (psychological, physical or none).

Situational dimensions of self-awareness in TGA

Regarding situational dimension of SA, our results revealed that awareness of surroundings was higher than personal SA (public and private) in all but the acute group, in which awareness was equally distributed between self and surroundings. This result is consistent with another study we led, showing that, in the acute phase, TGA patients tended to use more often first-person pronoun than control subjects when asked to give autobiographical memories, supporting the idea that attention towards the self is increased during the acute phase of TGA (Becquet et al, in progress). This excess of attention towards the self in the acute stage of TGA parallels the high level of anxiety-state found in this group. The anxiety caused by the TGA condition, may lead patients to focus on themselves rather than on the environment. That might contribute to the characteristic iterative questioning during TGA: "Where they are, where are their relatives, why they do not remember having been to the grocery" …etc.

In TGA, poor metacognitive abilities (e.g. SA of cognitive deficits) was already observed, with an impaired awareness of memory disorder during the acute phase [START_REF] Hainselin | Awareness of disease state without explicit knowledge of memory failure in transient global amnesia[END_REF] suggesting that an intense self-focused attention could not necessary lead to efficient metacognitive abilities in this population. Numerous studies put in evidence that the episodic memory disorder observed during TGA are related to a transient hippocampus impairment (for a review see: Bartsch & Deuschl, 2010). Hippocampus structures dysfunction was already involved in SA disturbances in amnesic patients. [START_REF] Chavoix | Self-awareness and the medial temporal lobe in neurodegenerative diseases[END_REF] reviewed several functional imaging studies suggesting relationships between anomalies in medial temporal lobes structures, including hippocampus, and disabled metacognitive abilities in patients suffering from neurodegenerative diseases. More specifically, some of these studies put in evidence that the lack of SA in neurodegenerative diseases could be associated to disconnection between cortical midline structures, supporting SA (for review see: [START_REF] Northoff | Selfreferential processing in our brain-A meta-analysis of imaging studies on the self[END_REF], and hippocampus areas [START_REF] Perrotin | Anosognosia in Alzheimer Disease: Disconnection between Memory and Self-related Brain Networks[END_REF]. From an experimental study investigating healthy participants, [START_REF] Bergouignan | Out-of-body-induced hippocampal amnesia[END_REF] also put in evidence relationships between modified SA and activity changes in hippocampus leading to memory dysfunction, arguing from a strong relationship between these two processes. Moreover, during the acute phase of TGA, a lack of functional connectivity in executive network, including prefrontal areas, was also recently observed, suggesting that even the episodic memory disorder is the most characteristic feature of TGA, impaired connectivity in prefrontal areas could also be involved in SA modifications, an hypothesis that remain to be explored deeply [START_REF] Zidda | Resting-state connectivity alterations during transient global amnesia[END_REF]. Another hypothesis that could be proposed involves a reduced connectivity into the salience network, especially the fronto-insular connectivity, also reported in TGA patients in [START_REF] Zidda | Resting-state connectivity alterations during transient global amnesia[END_REF]. The salience network is known to be involved in switching between the DMN and the central executive network and then from the processing of self information toward external task [START_REF] Goulden | The salience network is responsible for switching between the default mode network and the central executive network: replication from DCM[END_REF]. In that case, the self-focused attention in acute TGA could result from difficulties to switch the locus of attention, which could be related to impairment in prefrontal areas functioning and/or salience network connectivity.

Taken together, these data suggest that SA variations found in TGA could be an effect of the episodic memory dysfunction more than pre-existing dimensions in SA, which did not differ between patients classified according the stage of TGA in the first part of the present study. However, several reports suggested that psychological predisposing factors could be involved in TGA and differ according to the kind of precipitating events linked to the occurring of the attack [START_REF] Döhring | Stress-related factors in the emergence of transient global amnesia with hippocampal lesions[END_REF][START_REF] Quinette | What does transient global amnesia really mean? Review of the literature and thorough study of 142 cases[END_REF]. These predisposing factors were explored deeply in the second part of the study.

Predisposing psychological dimensions in TGA

The physiopathological mechanisms of TGA are only partially known. They are thought to include an acute metabolic stress of hippocampus areas impairing its functioning (Bartsch & Deuschl, 2010). Since the 90's, it has been hypothesized that, in some cases, TGA could result from a physiological cascade led by emotional arousal or the occurrence of a phobogenic situation [START_REF] Inzitari | Emotional arousal and phobia in Transient Global Amnesia[END_REF][START_REF] Pantoni | Transient global amnesia: a review emphasizing pathogenic aspects[END_REF]. Espiridion et al ( 2019) recently presented a case-study of a patient who had experienced a TGA in a context of PTSD, a few days after a psychological appointment in which trauma was evoked. An enhanced secretion of cortisol has been found during the acute phase of TGA, correlating with symptoms of depression and anxiety and a persisting hyper-reactivity to an experimental stress (socially evaluated cold pressor test (SECPT)) in the recovery phase, again supporting the hypothesis that stress might play a significant part in the pathogenesis of TGA [START_REF] Griebe | Enhanced cortisol secretion in acute transient global amnesia[END_REF][START_REF] Schneckenburger | Serum cortisol levels in patients with a transient global amnesia[END_REF].

While the amnesic syndrome of TGA is stereotyped, the medical histories of TGA patients are extremely varied. In large cohort studies, [START_REF] Quinette | What does transient global amnesia really mean? Review of the literature and thorough study of 142 cases[END_REF] (142 cases) and Bartsch et al (2010), who compiled data from 631 TGA episodes in a review, showed that about 30% of patients presented TGA following emotional stress while in 30% TGA was associated to strenuous physical activity. In addition, [START_REF] Quinette | What does transient global amnesia really mean? Review of the literature and thorough study of 142 cases[END_REF] and [START_REF] Döhring | Stress-related factors in the emergence of transient global amnesia with hippocampal lesions[END_REF] evoked possible psychopathological vulnerabilities in TGA patients, especially in the emotional or "psychological" category of patients. Many patients from the present study had experienced major traumas (find a suicided father, mortal car accident, repeated abuses, violent decease of a child…) and/or medical history of depression, as has been already reported [START_REF] Inzitari | Emotional arousal and phobia in Transient Global Amnesia[END_REF]Leonardo Pantoni et al., 2005;[START_REF] Quinette | What does transient global amnesia really mean? Review of the literature and thorough study of 142 cases[END_REF]. Nevertheless, we found no relationship between the precipitating event group and the psychiatric history in our patients. This could suggest that the psychological event TGA group is not necessarily composed of psychologically vulnerable people, even if a psychopathological history seems to constitute a general risk factor for all kinds of TGA.

Regarding pre-existing SA dimensions, our results showed a tendency to a high level of public SA in patients from the PHYS group. In this group, TGA followed an intense physical effort (see description of patients in Appendix). In our study, most patients from the physical precipitating event group went over their physical condition by high demand activities such as "strong" gardening, gym, carrying alone a washing machine, etc. TGA episodes may also occur during an overwork period [START_REF] Fisher | Transient Global Amnesia Precipitating Activities and Other Observations[END_REF][START_REF] Quinette | What does transient global amnesia really mean? Review of the literature and thorough study of 142 cases[END_REF]. This is often studied in a professional context, but the term overwork is also applicable to "domestic overwork" or "physical overwork", as in some of our patients. Social psychology literature about overwork suggests that even if some particular personalities are more likely exposed to develop workaholic behaviors (work investment considered as pathological), these behaviors are most of all closely linked to social demands [START_REF] Schaufeli | Heavy work investment, personality and organizational climate[END_REF]. In this case, high public SA in PHYS TGA patients could be related to an increased understanding of social expectancies, that could lead to workaholic behaviors (in a broader sense including not only the professional sphere) and overwork situations contributing to the genesis of TGA. Further studies should be led in TGA to consider the social context of onset of this event rather than the personality of subjects.

To conclude, in "physical-event" patients, the organism is stressed because it is pushed out of its limits, possibly without the patient being aware of the stress situation. Such patients are more attentive to their public image and possibly to social expectancies, which could lead them to exceed their body limits. "Psychological-event" patients, however, do perceive their stress and it could be the failure of defensive coping strategies that leads to the physiological cascade of TGA [START_REF] Döhring | Stress-related factors in the emergence of transient global amnesia with hippocampal lesions[END_REF]. These interpretations are consistent with the increased level of state-anxiety reported during the acute TGA [START_REF] Noel | Psychopathological factors, memory disorders and transient global amnesia[END_REF] also found in the present study, and the involvement of hippocampal CA1 lesion on one of the most possible pathogenesis of TGA, this latter region processing both episodic memory and stress-regulation responses [START_REF] Bartsch | Evolution of Hippocampal CA-1 Diffusion Lesions in Transient Global Amnesia[END_REF]Bartsch & Deuschl, 2010). Intragroup analysis for SASS scale. Boxplots were constructed from median and 1 rst and 3 rd quartile scores. Bars corresponds to 1.5*IQR. Friedman ANOVAs are significant for recovery group (ANOVA Chi²= 6,87;p<.05),88;p<.01), trend for NC group (ANOVA Chi²= 4,43;p =0.11), but not for the acute group (ANOVA Chi²=1.37;p=0.50). Post-hoc analysis showed that surroundings scale was superior than private and public (self-scales) SA in recovery group (private/surroundings : Wilcoxon's Z=1.89,p=.05,r=0.54;public/surroundings: Wilcoxon's Z=2.20,p<.05,r=0.64),p<.01,r=0.63;public/surroundings: Wilcoxon's Z=2.62,p<.01,r=0.56) and NC group (private/surroundings : Wilcoxon's Z=2.14,p<.05,r=0.48;public/surroundings: Wilcoxon's Z=2.45,p<.05,r=0.52). There was no difference between private and public SA scales in none group (all p>0.22).

Figure 2. Comparison of dispositional Self-consciousness scale scores in TGA groups according to the kind of precipitating event (PSY, PHYS or NONE) and NC group. Boxplots were constructed from median and 1 rst and 3 rd quartile scores. Bars correspond to 1.5*IQR. Differences are significant between PHYS and PSY groups (p<.05) and between PHYS and NC groups (p<.05) in "Public SA" subscores.

  None TGA precipitating event group differed from NC subjects in ESR encoding p=0.26) nor in ESR 20' recognition (H Kruskal-Wallis'test = 4.85, p=0.18) (Fig.2.). Nevertheless, there is an effect of the group for ESR free recall (H Kruskal-Wallis'test = 8.57, p<0.05). TGA patients from PSY group get lower scores than PHYS group (Mann-Whitney s' U test= 39, Z=-2.09, p<.05, r=0.42) and NC group (Mann-Whitney s'U test= 28.5, Z=-2.67, p<.01, r=0.53).

  FiguresFigure1

Figure

  Figure 2

  Figure 2

Figure captions

  Figure captions

Figure 1 .

 1 Figure1. Intragroup analysis for SASS scale. Boxplots were constructed from median and 1 rst and 3 rd quartile scores. Bars corresponds to 1.5*IQR. Friedman ANOVAs are significant for recovery group(ANOVA Chi²= 6,87; p<.05),88; p<.01), trend for NC group(ANOVA Chi²= 4,43; p =0.11), but not for the acute group(ANOVA Chi²=1.37; p=0.50). Post-hoc analysis showed that surroundings scale was superior than private and public (self-scales) SA in recovery group (private/surroundings : Wilcoxon's Z=1.89, p=.05, r=0.54; public/surroundings: Wilcoxon's Z=2.20, p<.05, r=0.64), p<.01, r=0.63; public/surroundings: Wilcoxon's Z=2.62, p<.01, r=0.56) and NC group (private/surroundings : Wilcoxon's Z=2.14, p<.05, r=0.48; public/surroundings: Wilcoxon's Z=2.45, p<.05, r=0.52). There was no difference between private and public SA scales in none group(all p>0.22).

Table 1 . Description of TGA patients and their matched control groups for the factor "TGA stage".

 1

		PA_17		PA_18	PA_19	PA_20		PA_21	PA_23	PA_24
		follow-up		follow-up	recovery	follow-up	acute	recovery	follow-up
	gender	M		M	M		M		F	F	M
	age	76		58	69		74		74	52	74
	level of education	11		14	17		20		12	12	9
	Memory score (z-score)*	0.29		0.45	0.29	0.29		-14	0.32	0.45
	number of TGA	1		1	1		2		1	1	2
	psychiatric history	-		several	-		-		recent	-	-
				depressive					depressive
				episodes					episode
	context of present TGA	repaired, lifted and	maybe after a	gym	played a public	gardening + felt	car issue +	very painful
		carried alone a washing		shower?			concert	unable to join	context of	constipation
		machine							her husband to	stressful moving
									visit their
									daughter's grave
	precipitating event code	2		8 (+ 4)	2		1		2 + 1	1 + 7	3
	self-estimation of health	3		4	4.5	4.5		-	2	3
	precipitating event group	PHYS		NONE	PHYS	PSY		PHYS	PSY	PHYS
		PA_25		PA_26		PA_27		PA_28	PA_29	PA_30
		follow-up		follow-up		follow-up		follow-up	follow-up	follow-up
	gender	F		F		F			F	F	F
	age	53		69		69			71	72	72
	level of education	12		11		9			11	15	15
	Memory score (z-score)*	0.32		0.29		0.45		0.29	0.29	0.29
	number of TGA	1		1		3			2	1	2
	psychiatric history	depressive episode 15		-		aquaphobia	depressive episode	-	-
		years ago						in young adult
									period
	context of present TGA	gardening: pull a big root	invasive medical	aquaphobia class	shower? new	Argument with	sexual intercourse
		off		examination, very	(swimming pool)	medication?	the neighbour
				stressful to her
	precipitating event code	2		1		4			4	1	5
	self-estimation of health	3		3		4			4	3.5	4
	precipitating event group	PHYS		PSY		PHYS		NONE	PSY	PHYS
	*z-score of the ESR 20'recognition score

Table 2 . Description of TGA patients and their matched control groups for the factor "precipitating event group".

 2

						Kruskal-
	PSY		PHYS	NONE	NC	Wallis s'
						ANOVA
	N (F/M)	12 (8/4)	12 (7/5)	3 (2/1)	11 (6/5)
	Mean age (SD)	64.58	66.17	66.67	61.91	H=3.46
		(±8.30)	(±8.66)	(±7.51)	(±3.42)	p=0.33
	Mean level of education	12.42	13.25	13	12.09	H=1.09
	(SD)	(±3.82)	(±2.99)	(±1.73)	(±2.59)	p=0.78
	Mean ESR (encoding	13.09	15.25	15.50	15.89	H=2.44
	score)	(±5.11)	(±1.42)	(±0.71)	(± 0.33)	p=0.49
	/16
	Mean ESR (free-recall	5	7.50	9.50	8.22	H=5.84
	score)	(±3.52)	(±3.80)	(±4.95)	(±2.33)	p=0.12
	/16
	Mean ESR (recognition	13.91	14.67	16	16	H=3.38
	score)	(±4.23)	(±2.50)	(±0.00)	(±0.00)	p=0.34
	/16
	STAI_A (state-anxiety)	28.38	30.90	33	26.11	H=2.32
	/80	(±9.29)	(±16.58)	(±8)	(±7.79)	p=0.51
	STAI_B (trait-anxiety)	38	35.89	37.33	31.11	H=0.47
	/80	(±6.45)	(±9.44)	(±4.51)	(±9.36)	p=0.93

Table 3 . Repartition of TGA patients according to TGA stages (columns) and precipitating event group (lines).

 3

Acknowledgments

We thank Esther Ivanusa, Antoine Dumas and Cécile Saget for collecting data from the healthy controls, and Julien Cogez, Alice Pélerin and Aurelija Juskenaite for their assistance in recruiting patients.