Salem Benferhat

Didier Dubois

Jérôme Lang

Henri Prade

HYPOTHETICAL REASONING IN POSSIBILISTIC LOGIC: BASIC NOTIONS, APPLICATIONS AND IMPLEMENTATION ISSUES

Possibilistic ATMS are truth maintenance systems oriented towards hypothetical reasoning where both assumptions and justifications can bear an uncertainty weight. Uncertainty is represented in the framework of possibility theory. In possibilistic logic uncertain clauses are handled as such and then in possibilistic ATMS the management of uncertainty is not separated from the other classical capabilities of the ATMS. The main interest of a possibilistic ATMS is to take advantage of the uncertainty pervading the available knowledge so as to rank-order environments in which a given statement is true. The approach is illustrated on a fault diagnosis problem. Basic algorithms associated with possibilistic ATMS are given.

Introduction

Assumption-based truth-maintenance systems (ATMS) ([9, 10, 11]; see also [START_REF]Reason Maintenance Systems and Their Applications[END_REF]) are automated reasoning systems oriented towards hypothetical reasoning since they are able to determine under which set of assumption(s) a given proposition is true. This set is called the "label" of the proposition. In this paper we present an extension of the ATMS, called "possibilistic ATMS" (or !-ATMS for short), where the management of uncertainty is integrated inside the basic capabilities of the ATMS. Uncertainty pervading justifications or grading assumptions is represented in the framework of possibility and necessity measures ( [START_REF] Zadeh | Fuzzy sets as a basis for a theory of possibility[END_REF], [START_REF] Dubois | Possibility Theory -An Approach to Computerized Processing of Uncertainty[END_REF]); these measures agree with the ordinal nature of what we wish to represent (it enables us to distinguish between what is plausible and what is less plausible). The certainty of each granule in the knowledge base (represented by a clause in possibilistic logic [START_REF] Dubois | Necessity measures and the resolution principle[END_REF]) is evaluated under the form of a lower bound of a necessity measure. This uncertainty in the deduction process is propagated by means of an extended resolution principle. Uncertainty degrees are then naturally attached to the configurations of assumptions in which a given proposition is true; one can also evaluate to what degree a given configuration of assumptions is inconsistent or compute the more or less certain consequences of a configuration of assumptions. This approach enables us to handle disjunctions and negations of assumptions without particular problem. Moreover, by rank-ordering configurations according to the degrees attached to them, !-ATMS provides a way of limiting combinatorial explosion when using ATMS in practice.

We present the basic definitions and results of possibilistic logic first. In Section 4 we give the basic definitions and functionalities of the !-ATMS, illustrated by a fault diagnosis problem previously introduced in Section 3. Section 5 presents an algorithm for the computation of labels and contradictory environments ("nogoods") based on an extension of the so-called CAT-correct resolution, initially developed by Cayrol and Tayrac [START_REF] Cayrol | Les résolutions CAT-correcte et CCT-correcte, la résolution CAT-correcte dans l'ATMS[END_REF][START_REF] Tayrac | ARC: an extended ATMS based on directed CAT-correct resolution[END_REF]. In Section 6, we will show how the !-ATMS can be used to revising inconsistent knowledge bases. An example of a data fusion problem will be described in this section. Lastly in Section 7 the proposed approach is briefly compared with existing uncertainty-handling ATMS's.

Possibilistic Logic

Modeling Uncertainty with Possibility and Necessity Measures

Possibilistic logic [START_REF] Dubois | Resolution principles in possibilistic logic[END_REF][START_REF] Dubois | Possibilistic logic[END_REF][START_REF] Lang | Logique possibiliste: aspects formels, déduction automatique[END_REF] is an extension of classical logic where one manipulates propositional or first-order calculus closed formulas weighted by lower bounds of possibility or necessity degrees which belong to [0,[START_REF] Baldwin | Evidential support logic programming[END_REF]. In this paper we restrict ourselves to a fragment of possibilistic logic, the "clausal possibilistic propositional logic", where the considered formulas are exclusively conjunctions of possibilistic propositional clauses weighted by necessity degrees [START_REF] Dubois | Necessity measures and the resolution principle[END_REF][START_REF] Dubois | Advances in automated reasoning using possibilistic logic[END_REF].

A possibility measure ! satisfies the following axioms [START_REF] Zadeh | Fuzzy sets as a basis for a theory of possibility[END_REF][START_REF] Dubois | Possibility Theory -An Approach to Computerized Processing of Uncertainty[END_REF]:

(i) !(") = 0; !(T) = 1 (ii) #p, #q, !(p $ q) = max(!(p),!(q))

where " and T denote respectively contradiction and tautology. We emphasize that we only have !(p % q) & min(!(p),!(q)) in the general case. A necessity measure is associated by duality with a possibility measure by #p, N(p) = 1 -!(¬p). Axiom (ii) is then equivalent to #p, #q, N(p % q) = min(N(p),N(q)) and, as a consequence, N(p) > 0 ' !(p) = 1. We adopt the following conventions:

• N(p) = 1 means that, given the available knowledge, p is certainly true; conversely, if p is said to be true we can consider p as certain. • 1 > N(p) > 0 that p is somewhat certain and ¬p not certain at all (since the axioms imply that #p, min (N(p),N(¬p)) = 0). • N(p) = N(¬p) = 0 (equivalent to !(p) = !(¬p) = 1) corresponds to the case of total ignorance; it expresses that, from the available knowledge, nothing enables us to say if p is rather true or rather false.

• 0 < !(p) < 1 (equivalent to 1 > N(¬p) > 0) that ¬p is somewhat certain and p not certain at all. • !(p) = 0, that p is certainly false.

Possibilistic logic is well-adapted to the representation of states of incomplete knowledge, since we can distinguish between the complete lack of certainty in the falsity of a proposition p (N(¬p) = 0) and the total certainty that p is true (N(p) = 1). N(p) = 1 entails N(¬p) = 0 but the converse is false. It contrasts with probability measures where Prob(p) = 1 is equivalent to Prob(¬p) = 0. Possibilistic logic contrasts also with usual multiple-valued logics which are fully truth-functional and deal with fuzzy propositions, while possibility degrees apply to standard propositions and are truth-functional only with respect to disjunction (see [START_REF] Dubois | Necessity measures and the resolution principle[END_REF] on this point) .

Possibilistic Clauses and Possibilistic Resolution

An uncertain clause will be a first-order logic clause c to which a valuation is attached, taken as a lower bound of its necessity measure.Thus, in the following we shall write (c () as soon as the inequality N(c) ) ( is taken for granted.

Resolution has been extended to possibilistic logic [START_REF] Dubois | Necessity measures and the resolution principle[END_REF][START_REF] Dubois | Resolution principles in possibilistic logic[END_REF]. The classical rule for propositional clauses are generalized by The refutation method can be generalized to possibilistic logic [START_REF] Dubois | Theorem proving under uncertainty -A possibility theory-based approach[END_REF][START_REF] Dubois | Necessity measures and the resolution principle[END_REF]. Indeed if we are interested in proving that a formula f is true, necessarily to some degree, we add to the knowledge base + the assumption N(¬f) = 1, i.e. that f is false (with total certainty). Let ' be the new knowledge base obtained by adding to + the clauses issued from ¬f, with the weight 1. Then it can be proved that any valuation attached to the empty clause produced by the extended resolution pattern from ' is a lower bound ( of the necessity measure of the conclusion f. It entails the existence of "optimal refutations", i.e. derivations of an empty clause with a maximal valuation.

Semantics and Partial Inconsistencies

A semantics has been defined for clausal possibilistic logic [START_REF] Dubois | Advances in automated reasoning using possibilistic logic[END_REF][START_REF] Dubois | Possibilistic logic[END_REF][START_REF] Lang | Logique possibiliste: aspects formels, déduction automatique[END_REF]. Here we only consider the case of necessity-valued propositional clauses. If c is a clause, M(c) the set of the models of c, then the models of (c () will be defined by a fuzzy set M (c () with a membership function

µ M(c () (I) = 1 if I , M(c) = 1 -( if I , M(¬c),
where I is a (classical) interpretation.

Then the fuzzy set of models of a set of weighted clauses + = {C 1 , C 2 , …, C n }, where C i stands for (c i ( i ), is the intersection of the fuzzy sets M(C i ), i.e.

µ M(+) (I) = min i=1,…,n µ M(C i ) (I).
The consistency degree of + will be defined by cons(+) = max I µ M(+) (I); it estimates the degree to which the set of models of + is not empty. The quantity Inc (+) = 1 -cons(+) will be called degree of inconsistency of +.

Finally we say that is a logical consequence of + if and only if #I , µ M( ) (I) ) µ M(+) (I), which will be written + . Let us note that all these definitions recover those of classical logic and are in accordance with Zadeh's theory of approximate reasoning [START_REF] Zadeh | A theory of approximate reasoning[END_REF]. We take the notation + (c *) if and only if there exists a *'-refutation with *' ) *, i.e. a deduction of (" *'), from the set of weighted clauses +' obtained by adding N(¬c) = 1 to +. Then we have the following completeness theorem ( [START_REF] Dubois | Advances in automated reasoning using possibilistic logic[END_REF][START_REF] Dubois | Possibilistic logic[END_REF][START_REF] Lang | Logique possibiliste: aspects formels, déduction automatique[END_REF]): let + be a set of weighted clauses; then + (c *) if and only if + (c *).

Hypothetical Reasoning

As it was pointed out in [START_REF] Dubois | A possibilistic assumption-based truth maintenance system with uncertain justifications, and its application to belief revision[END_REF], the weighted clause (¬p $ q () is semantically equivalent to the weighted clause (q min((, v

v(¬p $ q () = max(v(¬p $ q), 1 -() = max(1 -v(p), v(q), 1 -() = max(v(q), 1 -min(v(p), ()) = v(q min(v(p), ())
The equivalence between the weighted clauses (¬p $ ¬s $ q () and (¬p $ q min((, v(s)) expresses that the rule "if p and s are true then q is certain to the degree (" means that "in an environment where s is true, if p is true then q is certain to the degree (", when we decide to consider s as an assumption. This equivalence leads to the following modified presentation of the resolution rule

(¬p $ q min((, v(s)) (p $ r min(*, v(t))) ----------------- (q $ r min((, v(s), *, v(t)))
This enables us to express that if the clause ¬p $ q is certain to the degree ( in an environment where s is true and if the clause p $ r is certain to the degree * in an environment where t is true, then the resolvent clause q $ r is certain to the degree min ((,*) in an environment where s and t are true (since v(s % t) = min (v(s),v(t))). It turns out that when ( = * = 1, the above resolution rule is very close to the CAT-correct resolution rule [START_REF] Cayrol | Exploitation de la méthode du consensus dans les ATMS: la résolution CAH-correcte[END_REF][START_REF] Cayrol | Les résolutions CAT-correcte et CCT-correcte, la résolution CAT-correcte dans l'ATMS[END_REF][START_REF] Tayrac | ARC: an extended ATMS based on directed CAT-correct resolution[END_REF], which separates the assumptions from the other literals by sorting them, and where resolution is restricted so as to get rid of the non-assumption literals. Hence possibilistic logic can capture not only the propagation of uncertainty, but at the same time the propagation of the reasons for uncertainty.

It has been shown that if, adding to the knowledge base the clauses corresponding to N(¬p) = 1, we obtain by resolution the empty clause with a strictly positive weight, then this weight is a lower bound of the necessity degree of p. The exploration of all proof paths leading to the empty clause will thus help determine the set of environments in which p is somewhat certain if all assumptions are carried over to the weight side, before running the refutation. This enables us to define a possibilistic ATMS in Section 4 and to develop the associated basic procedures.

An Example

This example, inspired from Siegel [START_REF] Siegel | Représentation et utilisation de la connaissance en calcul propositionnel[END_REF], relates to the functioning of an engine. This example may appear somewhat fancy, for it is not easy to satisfy the triple requirement of a short, realistic and pedagogical example in order to show the different capabilities of possibilistic ATMS. The clauses describing how the system works contain three kinds of atoms:

-the assumptions, associated with each component which may be faulty: by convention, assumptions will be understood as "absence of fault", assumption A being true if component A works, false if it is faulty. These atoms are OIL (there is still some oil) BELT (the engine belt is not broken) RECTIFIER (the rectifier is not cut off) BATTERY (the battery is not faulty) FUSE (the fuse is not melt)

-the "sensors" atoms, which describe observable facts whose truth value may be known by the user, but with a possibly non total certainty: indicator (the temperature indicator is red) ammeter (the charge ammeter is positive) engine (the engine is running) headlights (the headlights are working) horn (the horn is working)

-two intermediary atoms coil (the coil is supplied) secondary-circuits (the secondary circuits are supplied)

Uncertainty is involved at three different levels:

-on clauses describing how the system works: for instance, "if the secondary circuits are supplied then it is moderately certain (0.7) that headlights are working"; -on "sensors" atoms (we might be unsure of a result given by a sensor) -on assumptions: by default, components are not faulty (but some faults are more plausible than others); it leads to the clauses (A i ( i ) expressing that we are certain to the degree ( i that A i will not be faulty.

The system is described by the following statements:

(1) if the temperature indicator is red then there is no oil left or the belt is broken (certainty: 0.9) (2) if the temperature indicator is red then secondary circuits are supplied (certainty: 1) (3) if there is no oil left and if secondary circuits are supplied then the temperature indicator is red (certainty: 0.7) (4) if secondary circuits are supplied then the headlights are working (certainty: 0.7) (5) if the battery is not faulty then the headlights are working (certainty: 0.9) (6) if the engine is running then the coil is supplied (certainty: 1) (7) if the battery is not faulty then secondary circuits are supplied (certainty: 0.7) (8) if the engine is running, the belt is not broken and the rectifier is not cut off then the charge ammeter is positive (certainty: 0.8) (9) if the engine is running and the rectifier is not cut off then the charge ammeter is positive (certainty: 0.3) (10) if the coil is supplied and the fuse is not blown then the charge ammeter is positive (certainty: 0.6) (11) if the fuse is not blown and the battery is working then the horn is working (certainty: 0.9)

Then we write that by default, the components are not faulty:

(12) the fuse is not melt (certainty: 0.2) (13) the belt is not broken (certainty: 0.

Basic Principles and Definitions of a Possibilistic A.T.M.S.

Classical ATMS require that the clauses contained inside the knowledge base (justifications and disjunctions of assumptions) are certain; we may wish to handle more or less uncertain information without losing the capacities of the ATMS. The basic principle of the !-ATMS introduced in [START_REF] Dubois | A possibilistic assumption-based truth maintenance system with uncertain justifications, and its application to belief revision[END_REF] is to associate to each clause a weight ( which is a lower bound of its necessity degree 1 . Assumptions may also be weighted, i.e. the user or the inference engine may decide at any time to believe an assumption with a given certainty degree. A !-ATMS is able to answer the following questions:

(i) Under what configuration of the assumptions is a fact d certain to some degree? (i.e., what assumptions shall we consider as true, and with what certainty degrees in order to have d certain to degree (?)

(ii) What is the inconsistency degree of a given configuration of assumptions ? (iii) In a given configuration of assumptions, to what degree is each observed fact certain ?

The kind of classical ATMS extended here is Cayrol and Tayrac's [START_REF] Cayrol | Exploitation de la méthode du consensus dans les ATMS: la résolution CAH-correcte[END_REF][START_REF] Cayrol | Les résolutions CAT-correcte et CCT-correcte, la résolution CAT-correcte dans l'ATMS[END_REF][START_REF] Tayrac | ARC: an extended ATMS based on directed CAT-correct resolution[END_REF] generalized ATMS, where each piece of information is represented by a (general) propositional clause. It make it possible to have:

-a uniform representation for all pieces of knowledge (no differentiated storage and treatment between justifications and disjunctions of assumptions). -the capability of handling negated assumptions as assumptions, i.e. environments and nogoods may contain negations of assumptions (they are called generalized environments and nogoods by Cayrol and Tayrac); this approach differs from DeKleer "NATMS" [START_REF] Kleer | A general labeling algorithm for assumption-based truth maintenance[END_REF] where negated assumptions do not appear inside the environments. -a simple and uniform algorithm for the computation of labels, based on a restricted form of resolution (see Section 5).

Environments

First, the basic notions attached to the classical ATMS can be generalized. Let + be a set of necessity-valued clauses. Let E be a set of valued assumptions; the following definitions are useful:

Labels

In order to define the label of a fact d, we consider only non-weighted assumptions (i.e. they will have the implicit weight 1). It can be shown that it is useless to weight the assumptions inside the labels (this remark holds also for the base of no-goods). The label of the fact d, L(d) = {[E i ( i ], i,I} is the unique fuzzy subset of the set of environments for which the four following properties hold:

-(weak) consistency: #[E i ( i ] , L(d), E i U + is *-inconsistent,with * < ( i (i.e.
Inc(E i -+)< ( i in the sense of Section 2, the certainty degree associated to the E i 's being 1); it guarantees that either E i is consistent (i.e. * = 0), or its inconsistency degree is anyway strictly less than the certainty with which d can be deduced from E i -+ (i.e. we are sure to use a consistent sub-base of E i -+ to deduce d, see [START_REF] Dubois | Advances in automated reasoning using possibilistic logic[END_REF][START_REF] Dubois | Possibilistic logic[END_REF]). Ranking environments according to their weight in the label of each fact provides a way for limiting the consequences of combinatorial explosion (the problem was already pointed out by Provan [START_REF] Provan | An analysis of ATMS-based techniques for computing Dempster-Shafer belief functions[END_REF] and Raiman [START_REF] Raiman | Two heuristic integrating probabilities and logic: a preliminary report on parsimonious search[END_REF]): indeed when a label contains too many environments, the !-ATMS can help the user by giving the environments with the greatest weight(s) only.

These notions are now illustrated on the example introduced in Section 3. The only nogoods existing in the system are negations of all "non-fault" assumptions, i.e. nogood 0.4 {¬BELT}, nogood 0.3 {¬BATTERY}, etc. A nogood containing only a positive "non-fault" atom would mean that it is somewhat certain that a given component has to be faulty whatever the results given by the sensors, which would not be too realistic; a nogood containing several "non-fault" atoms (positive or negative) would mean that some faults are not independent (it is not the case in our example). Once the system has been described, let us introduce information given by the sensors. Let us first introduce "the engine is running (certainty: 1)".

C17 engine (1)

The label of the statement "the charge ammeter is positive" is: { {} 0.2 , {RECTIFIER} 0.3 , {RECTIFIER, BELT} 0.8 , {FUSE} 0.6 }, meaning that the charge ammeter is positive with a high certainty degree (0.8) if the rectifier is not cut off and is the belt is not broken, with a moderate certainty degree (0.6) if the fuse is not blown, and with a low certainty degree (0.3) if rectifier is not cut off (and no extra assumption). Moreover, the presence in the label of the empty environment weighted by 0.2 means that if no extra information is given by the sensors, it is already certain to the degree 0.2 that the charge ammeter is positive. { {} 0.2 , {RECTIFIER} 0.3 , {RECTIFIER, BELT} 0.9 , {FUSE} 0.6 } would not be a correct label for the same fact since adding to the clauses already present in the system the two clauses (RECTIFIER 1) and (BELT 1) does not enable to deduce (ammeter 0.9). { {} 0.2 , {RECTIFIER} 0.3 , {RECTIFIER, BELT} 0.8 , {FUSE} 0.6 , {FUSE, BELT} 0.5 } would not be minimal since {FUSE,BELT } 0.5 is a redundant environment, being subsumed by {FUSE} 0.6 . Contrarily, having in the label the (-environnements {RECTIFIER} 0.3 and {RECTIFIER, BELT} 0.8 keeps the label minimal, since none of these two (-environments is redundant. { {RECTIFIER} 0.3 , {RECTIFIER, BELT} 0.8 , {FUSE} 0.5 } would not be complete since (FUSE 1) enables to deduce that N(ammeter) ) 0.6, which is not expressed in the above set of environments since the highest certainty degree with which the fact can be deduced is only 0.5. { {} 0.2 , {RECTIFIER} 0.3 , {RECTIFIER, BELT} 0.8 , {FUSE} 0.6 , {¬BATTERY} 0.2 } would not be weakly consistent since {¬BATTERY} is a 0.3-nogood.

Besides, the label of "indicator" is { {¬ OIL, BATTERY} 0.7 }, meaning that the temperature indicator is red with a high certainty degree if there is no oil left and if the battery is working.

Contexts

To extend the ATMS notion of context, we now consider weighted assumptions. A weighted assumption is a couple (H () where H is an assumption and ( , [0,1] is the a priori certainty degree assigned to H.

The context associated with the set of weighted assumptions is the set of all couples (d, val The proof of the theorem is obvious using the results of classical ATMS. This theorem gives an immediate algorithm to compute contexts, given the label of every fact:

(_max 1 0; For every [E i ( i ] , Label(d) do if * . E i then (_max 1 max((_max, min(( i , * 1 , * 2 , …, * n ))
where * 1 , …, * n are the weights attached to H i,1 , …, H i,n in . end Intuitively, for each environment of the label of d included in *, the algorithm computes the degree of certainty with which this environment entails d; this degree depends upon the justifications used in deriving d (via ( i ) and the weights of the assumptions in this environment.

Example (continued):

Let us consider the system {C1, …, C17} and let E be the environment E = {RECTIFIER, BELT, BATTERY}. The context associated with E is: { RECTIFIER (1), BELT (1), BATTERY (1), FUSE (0.4), OIL (0.1), engine (1), coil (1), horn (0.4), secondary-circuits (0.7), headlights (0.9), ammeter (0.8) } (the nonunderlined atoms were already deducible before adding the three assumptions), which means that in the configuration of assumptions where it is certain that the rectifier, the belt and the battery are working, then it is certain to the degree 0.7 that the secondary circuits are supplied, etc.

Let now the environment contain weighted assumptions ' = {(RECTIFIER 0.5), BELT (0.4), BATTERY (0.6)}. The context associated with ' is { RECTIFIER (0.5), BELT (0.4), BATTERY (0.6), FUSE (0.4), OIL (0.1), engine (1), coil (1), horn (0.4), secondary-circuits (0.6), headlights (0.6), ammeter (0.4) }.

Fault diagnosis

Let us now add to the system, not only that the engine is running, but also the two pieces of information: C18 indicator (1) C19 ¬ ammeter (0.9)

The system is now 0.3-inconsistent: indeed, from the clauses C1, C13, C16 et C18, a contradiction of degree 0.3 can be deduced. It was somewhat certain at the beginning that the belt was not broken (C13) and that there was some oil left (C16). The extra information (C18) enables us to deduce that these two "non-fault" assumptions are mutually exclusive, and with a greater certainty (0.9). The new minimal nogoods are nogood 0.6 {FUSE}, nogood 0.8 {RECTIFIER, BELT} and nogood 0.9 {OIL, BELT}, nogood 0.4 {RECTIFIER}, nogood 0.9 {OIL} expressing that:

-it is certain to the degree 0.6 that the fuse is blown; -it is certain to the degree 0.8 that the rectifier is cut off or that the belt is broken; -it is certain to the degree 0.9 that there is no oil left or that the belt is broken.

Moreover we have, like in the initial system, nogood 0.4 {¬BELT}; contrarily, {¬FUSE}, {¬RECTIFIER}, {¬BATTERY} and {¬OIL} are not minimal nogoods anymore since their membership degree to the nogood base is now less than the inconsistency degree of the knowledge base, which can be translated by nogood 0.3 {}.

In classical logic, in order to provide the user with an answer under the form of a set of alternative fault configurations, the nogoods ¬FUSE % (¬RECTIFIER $ ¬BELT) % (¬OIL $ ¬BELT) are expressed in the equivalent disjunctive form:

(¬FUSE % ¬RECTIFIER % ¬OIL) $ (¬FUSE % ¬BELT).
In possibilistic logic, the transformation is trickier since, if from (¬FUSE 0.6) % (¬RECTIFIER $ ¬BELT 0.8) % (¬OIL $ ¬BELT 0.9) it can be deduced that N(¬FUSE % (¬RECTIFIER $ ¬BELT) % (¬OIL $ ¬BELT)) ) 0.6, and then that N((¬FUSE % ¬RECTIFIER % ¬OIL) $ (¬FUSE % ¬BELT)) ) 0.6, however nothing can be immediately concluded on N(¬FUSE % ¬RECTIFIER % ¬OIL) and N(¬FUSE % ¬BELT), which does not enable us to rank the fault configurations. On the contrary, one can try to find a lower bound of the certainty of these configurations of faults, by refutation, by adding their negation (weighted by 1) to the system. Thus we obtain that N(¬FUSE % ¬RECTIFIER % ¬OIL) ) 0.4 (but no non-trivial lower bound can be found for N(¬FUSE % ¬BELT)), which suggests to the user that the configuration (fuse blown, rectifier cut off, no oil left) is more plausible than (fuse blown, belt broken).

Interpretations in possibilistic logic

As said above an environment in the classical ATMS is a set of assumptions, and every set of assumptions characterizes a context. An environment E is said to be an interpretation (in the classical ATMS) if it is consistent and if adding any assumption to it make it inconsistent, namely E satisfies the two following conditions;

1. E is consistent (with +) 2. # H 2 E (H is an assumption), E -{H} is inconsistent.
Interpretations are complete descriptions of a given situation in terms of assumptions. One approach for computing interpretations consists in building a lattice of environments (2 T ,3) where T is the set of all assumptions, and nodes in the lattice represent subsets of assumptions. This lattice is obtained by drawing an edge from each environment E to every subset of this environment. The top of the lattice is the set of all assumptions and the bottom is the empty set. Figure 1 gives an example of such a lattice for the set of assumptions T = {A,B,C,D}.

Notice that if a given environment E is a nogood then all of its supersets are contradictory environments as well. So, once nogoods are computed, we remove their elements (and hence their supersets) from the lattice, and therefore only the consistent environments remain and hence any maximal set of assumptions (relatively to the inclusion relation 3) represents an interpretation. For example, in Figure 1, where the set of nogoods is {{A,B}, {C,D}}, the set of interpretations is {{A,C}, {A,D}, {B,C}, {B,D}}. In possibilistic logic, an interpretation E may be partially inconsistent, namely it may contains a nogood which does not however violate weak consistency. For the need of the application, we say that an interpretation E is totally consistent if it satisfies the two following requirements:

1. Incons(+ -E) = 0 2. # H 2 4 (H is an assumption), Incons(+ -E -{H}) > 0.
where Incons(X) is the degree of the inconsistency of X. We now give another approach for computing totally consistent interpretations using the notions of candidates which are environments containing at least one assumption in every nogood. A candidate C is said to be minimal if it does not exist a candidate C such that C' 0 C. The following proposition holds: Proposition: the complement of a minimal candidate is a totally consistent interpretation.

Proof

Let C be a minimal candidate, and I its complement.

Assume that I is inconsistent (partially or then there exists a nogood which is included in I. By definition, a candidate C contains at least one element of every nogood, and particularly it contains an element of , so it contains also an element of I which contradicts our assumption that I is the complement of C. Let us show now that I is maximal; suppose that I -{H} is consistent, then it does not contain any nogood and therefore for every nogood there exist an element which belongs to C -{H}, which means that C -{H} is a candidate and contradict therefore our assumption that C is minimal. In this section we have seen that the associated to nogoods are not used to build minimal candidates. We will see in Section 6. that the use of the valuations is very important for revising inconsistent possibilistic knowledge bases and reducing the complexity of the revision process by selecting a smaller number of maximal consistent sub-bases.

Basic Algorithms of a Possibilistic ATMS

The type of ATMS that we extend to possibilistic logic is the one proposed by Cayrol and Tayrac [START_REF] Cayrol | Exploitation de la méthode du consensus dans les ATMS: la résolution CAH-correcte[END_REF][START_REF] Cayrol | Les résolutions CAT-correcte et CCT-correcte, la résolution CAT-correcte dans l'ATMS[END_REF][START_REF] Tayrac | ARC: an extended ATMS based on directed CAT-correct resolution[END_REF]. Contrary to De Kleer's [START_REF] Kleer | An assumption-based TMS[END_REF][START_REF] Kleer | Extending the ATMS[END_REF] As in the classical ATMS, the set of data (called also "literals" or "atomic propositions") is divided into two groups: T and NT containing respectively the assumption data and the non-assumption ones.

The computation of labels and nogoods requires some further definitions.

. 1 . Definition of a 7-CATI

A clause is of the form A 8 B where A is a conjunction of positive literals and B is a disjunction of positive literals. Each propositional formula has a unique clausal representation. A is called antecedent and B consequent. A weighted clause (C () is a 7-CATI, if and only if:

1. every literal in the antecedent side of C is an assumption, and, 2. it contains at most one non-assumption literal in the consequent side of C.

If (C () satisfies only the first condition, call it a 7-CAT (Possibilistic Clause with Typed Antecedent).

Example:

Let T = {A,B}, and NT = {a,b,c,d}

The weighted clause (A 8 aB (1) is a 7-CATI, because A belongs to T and it contains exactly one non-assumption literal (a) in his consequent side, in contrast with the following clauses (a 8 b (2) and (A 8 cd (3) which are not 7-CATI because the first one contains a literal in his antecedent side which is not an assumption (a), and the latter has two non-assumption data in its consequent side (c and d).

7-RCO Possibilistic Resolution

The basic idea of the 7-RCO (Oriented CAT-correct Possibilistic is to use the possibilistic resolution rule in order to deduce every 7-CATI which is logically entailed from the initial set of clauses.

In order to minimize the number of applications of possibilistic resolution, an arbitrary but complete ordering O is introduced on the set of non-assumption data NT.

The 7-RCO consists in applying possibilistic resolution between two weighted clauses C1 and C2 of the form: --------------------

C1: A1 8 {x} -B1 (1 C2: A2 -{x} 8 B2 (2
-- 9(C1,C2): A1 -A2 8 B1 -B2 min((1,(2)
where 9(C1,C2) denotes the resolvent of C1 and C2, under one of the two following conditions:

1. C1 and C2 are 7-CATI such that the consequent side of 9(C1,C2) contains at most one non-assumption literal. 2. the three following conditions are satisfied: 2.1. C1 is a 7-CAT, 2.2. x is the smallest (relatively to O) non-assumption datum in the antecedent side of C2. 2.3. x is one of the two smallest non-assumption data in the consequent side of C1.

Example:

Let T = {A,B}, NT = {a,b,c}, O: a < b < c and the initial set of clauses is:

(A 8 c (1) (B 8 abc (2) (c 8 a (3)
The resolution between the clauses (B 8 abc (2) and (c 8 a (3) is not a 7-RCO resolution because c does not belong to the two smallest non-assumption data in the consequent side of the clause (B 8 abc (2). Thus, only the resolution between the clauses (A 8 c (1) and (c 8 a (3) is a 7-RCO resolution.

It can be proved that the use of 7-RCO is complete for deduction of all possible 7-CATI. A proof of 7-RCO completeness in the classical case can be found in (Tayrac [START_REF] Tayrac | ARC: an extended ATMS based on directed CAT-correct resolution[END_REF]).

The Algorithm

As in the classical ATMS, the maintenance system 7-RCO is incremental, provided that the set of clauses + is saturated, which means that before the introduction or the deduction of a new clause, all possible possibilistic resolutions 7-RCO are executed, thus only the possibilistic resolutions which are enabled by the newly introduced clause are processed.

On the other hand, after the introduction or the inference of the new clause, the set of clauses + must be minimized by removing every tautology and every subsumed clause.

We recall that a clause (A 8 B (1) is a tautology, if and only if, the intersection of the sets of literals A and B is not empty. The clause (A1 8 B1 (1) subsumes the clause (A2 8 B2 (2), if and only if, A2 : A1, B2 : B1 and (1 ) (2.

Each time a new clause is introduced in the system, it is checked if it is a tautology or a subsumed clause; then, if none of these holds, all clauses which are subsumed by the newly introduced clause are removed, and lastly all possible 7-RCO resolutions are triggered with this new clause.

This cycle is repeated for every deduced clause, and we give below an algorithm summarizing these operations: procedure introduce (cl) begin ; cl is the newly introduced clause, + represents the knowledge base. J:={cl}; J contains all deduced or introduced clauses. while (J56) do .choose a weighted clause c of J.

.remove c from J. .if c is not subsumed and c is not a tautology then .remove all clauses from + which are subsumed by c.

.add c to +.

.K:=all-7-RCO(c) ; K contains all possible clauses which allow us to apply 7-RCO ; resolution with c. while (K56) do .remove a weighted clause c1 from K .add to J the 7-RCO resolvent 9(c1,c) end introduce

Using 7-RCO in a Possibilistic ATMS

We show in this section how possibilistic resolution 7-RCO can be used to compute every minimal nogood and the label of each datum.

Computing labels

We recall that the label of a literal d, L(d) {[E i ( i ] / i , I}, is the only fuzzy subset of the set of environments which is weakly consistent, sound, complete and minimal, where E i denotes a set of assumptions and negated assumptions.

By definition, [4 (] is an environment of a literal d, relatively to the set of clauses +, if and only if, E -+ (d (); this is equivalent to + (E 8 d () [START_REF] Dubois | Possibilistic logic[END_REF][START_REF] Lang | Logique possibiliste: aspects formels, déduction automatique[END_REF]. Thus an environment of a literal d is characterized by a clause containing exactly the literal d in his consequent side, and only assumptions or negated assumptions in his antecedent side. Obviously, this clause is a 7-CATI. Thus, the label of a non-assumption literal d is:

L(d) = {(P -{¬x / x , Q} (] / (P 8 {d} -Q () , + is a 7-CATI}.

Computing nogoods

An environment [4 (] is a nogood if and only if (" () is a logical consequence of E -+, i.e + (E 8 " (). Thus a nogood is denoted by a clause containing only assumptions in their antecedent part; these clauses are also 7-CATI. Therefore, the nogoods are computed as below: Nogood = {[P -{¬x / x , Q} (] / (P 8 Q () , + is a 7-CATI and T : Q}.

Revising Knowledge Bases

In classical logic, when the inconsistency appears in a given knowledge base, then any formula is a logical consequence of this base, and therefore we can infer from this base that both p and its denial ¬p are simultaneously true. In possibilistic logic, the situation is less trivial since it is possible to use such bases considering as valid all formulas having valuations strictly higher than the degree of inconsistency of the knowledge base (the others being inhibited). This strategy verifies the revision rationality postulates of Alchourron, Gärdenfors and Makinson [START_REF] Gärdenfors | Knowledge in Flux-Modeling the Dynamic of Epistemic States[END_REF]. However, at the syntactic level it is not entirely satisfactory because some of the inhibited clauses have nothing to do with the inconsistency. For instance the knowledge base + = {(a .5), (b .6), (c .4), (¬a $ ¬b 1)} has a level of inconsistency equal to .5 due to the contradictory clauses (a. 5), (b. 6) and (¬a $ b 1). Hence (c. 4) will be inhibited. However it seems more satisfactory to delete (a. 5) from + in order to restore consistency rather than deleting both (a. 5) and (c. 4). Indeed a more parsimonious revision is obtained by deleting only the least certain formulas involved in a contradiction. It suggests to choose the revised knowledge base among the largest ones which preserve consistency.

Preferred consistent sub-bases

Definition 6.1: A sub-base + of + is said to be maximal consistent if it is consistent and if adding any formula ; of (+ -+ i ) to + i produces an inconsistent knowledge base.

In practice, the number of maximal consistent sub-bases of an inconsistent knowledge base increases proportionally with the size of the base and in general, it is not possible to take into account all the maximal consistent sub-bases. The idea is then to take advantage of the weights to rank-order the maximal consistent sub-bases and keep only the best ones.

The wish to delete only the least certain formulas which cause an inconsistency leads to consider maximal consistent sub-bases that obey the following definition: Definition 6.2: A sub-base + i is said to be strongly maximal consistent sub-base if and only if:

(i) + i is totally consistent; i.e. / / ( > 0, + (" () (ii) #(< () , + -+ i , Incons(+ i -{(< ()}) = ( > 0.
There is another way of considering a strongly maximal consistent sub-base. Namely, + can be viewed as a layered knowledge base, i.e. + = B 1 -B -… -B n where B j contains only clauses of necessity weight ( j , and ( 1 = 1 > ( 2 >… > ( n . Then + i is a strongly maximal consistent sub-base of + if and only if it can be constructed as follows

+ i = E 1 -E 2 -… -E n where # j = 1,n, E 1 -E 2 -… -E j
is a maximal consistent sub-base of B 1 -B 2 -… -B j . The proof of this equivalence is in the annex. Such maximal consistent sub-bases are exactly what Brewka [START_REF] Brewka | Preferred subtheories: an extended logical framework for default reasoning[END_REF] calls "preferred sub-theories" (see also Testemale [31]). An algorithm for computing the strongly maximal consistent sub-bases can be found in [START_REF] Lang | Logique possibiliste: aspects formels, déduction automatique[END_REF].

We can also introduce an ordering among the maximal consistent sub-bases of an incoherent knowledge base +. A maximal consistent sub-base + i is said to be lexicographically preferred to + j, denoted by + i & + j if and only if (+ -+ i ) & (+ -+ j ) where (X) = {( i / (; i ( i ),X} is a multiset of valuations of formulas and "&" is defined as:

-+ i ) & (+ -+ j ) = (i) (+ -+ i ) = 6 or (ii) Max( (+ -+ i )) < Max( (+ -+ j )) or (iii) Max( (+ -+ i )) = Max( (+ -+ j )) and (+ -+ i ) -Max( (+ -+ i )) & (+ -+ j ) -Max( (+ -+ j )).
It has be shown [START_REF] Dubois | Inconsistency in possibilistic knowledge bases -to live or not live with it[END_REF] that the minimal element of +'s sub-bases of inconsistent knowledge base + relatively to the ordering "& " are strongly maximal consistent subbases.

It is important to distinguish between strongly maximal consistent sub-bases and lexicographically maximal consistent sub-bases. Although each lexicographically maximal consistent sub-base is a strongly maximal consistent sub-base, the contrary is not true. Indeed, consider an example where our knowledge base contains the following set of incertain clauses: + = {C 1 : (a 0. The next section shows how the 7-ATMS can be used to compute the maximal consistent sub-bases of an inconsistent knowledge base.

Using the !-ATMS

The computation of maximal consistent sub-bases of an inconsistent knowledge base can easily be done using the functionalities of the !-ATMS. We first point out the close relationship between totally consistent interpretations and maximal consistent sub-bases.

Let + = {(C i ( i )} be an inconsistent uncertain knowledge base to be revised. Let +' be a new knowledge base built from + by adding to the antecedent side of every clause C i an assumption H i , we obtain then: +' = {(C' i ( i )}, where C' i = H i 8 C i , we will say that the assumption H i controls the clause C i . We assume that only the set of data {H i } are assumptions in +'. The !-ATMS will work on +', and we can show besides what the different notions known in the !-ATMS represent for +. Notice first that an environment E = {H 1 , …, H n } represents the set of clauses = {C 1 , …, C n } in +. The following property is easy to check:

# = {(C 1 ( 1 ), …, (C n ( n )} 3 +, is inconsistent iff / 3
= {H 1 ,…, H n } such that belongs to the nogood set of +'. The nogoods of +' then represents the set of clauses of + which allows us to deduce the contradiction when we substitute every assumption H i by the clause C i it controls. Since the nogood has the completeness property, then all possible sets of clauses which are responsible for an inconsistency of + are computed by the !-ATMS. An immediate application is to check the consistency of the knowledge base +. Indeed if the nogood base of +' is empty then + is consistent .

It is possible to relate the notion of totally consistent interpretations with the notion of the maximal consistent sub-bases, since, as it can be easily seen, # 3 +, is a maximal consistent sub-base of + iff = {H 1 , …, H n } is a totally consistent interpretation of +'.

An immediate result of this fact is that the candidates in +' represent the smallest set of clauses such that if we remove their elements from +, then the knowledge base + becomes inconsistent.

The algorithm presented in Section 4 computes the set totally consistent interpretations. And we have seen that each totally consistent interpretation represents a maximal consistent sub-base. Then the problem of selecting the lexicographically maximal consistent sub-bases require only to sort the set = { (+ -+ i )} using the ordering "< " defined above where + i is a maximal consistent sub-base and (+ -+ i ) is a set of the weights of clauses which are in + -+ i .

Example

Assume that our knowledge base contains the following set of uncertain clauses: Let +' the possibilistic knowledge base be constructed from + by associating to each formula C i in + a specific assumption H i , then we obtain : Deciding which sub-base is lexicographically maximal consistent requires to sort the set = { (+ -+ i )} = {{0.8}, {0.7 0.8}, {0.8 0.6}} using the ordering "< ". we only find + 1 as a lexicographically consistent maximal sub-base.

+ =
+' = {(

Application to data fusion

An application to data fusion problems has been carried out (Monai et al. [START_REF] Monai | Logique possibiliste et gestion d'hypothèses multiples: application à un problème de fusion de données[END_REF]). It consists in finding a structure for a particular object from observations of elementary objects by aggregating the partial observations in a coherent manner.

Objects handled in the system are layered in different levels. Each level contains different kinds of objects. Every object of level i is either observed (then is said to be elementary) or constructed only from objects of level i -1 (it is then said to be composite). We assume that objects of level 0 are all elementary objects. The construction of a complex object must satisfy the following constraint: "Two objects of level i containing a common object of level -1) must not be aggregated together"

The knowledge base of the system is organised into n independent parts. Each part of the knowledge base gives the conditions for aggregating objects of level i to objects of higher levels.

B1 C1

The hierarchy shown in Figure 2 summarizes the way objects are aggregated. Here is an example of a rule of level 0:

#x 1 , #x 2 , S 0 (x 1 ) % B 0 (x 2 ) 8 B 1 (t) ((t).

This rule means that if we observed two objects x 1 and x 2 of level 0 belonging respectively to classes S 0 and B 0 , then these two objects can be aggregated to give an object of level 1, which belongs to the class B 1 . ((t) is a real number between 0 and 1 which estimates the compatibility of aggregating the objects S 0 (x 1 ) and S 0 (x 2 ). For instance ((t) may be all the greater as the locations of x 1 and x 2 are close to each other. If ((t) = 0 then S 0 (x 1 ) and S 0 (x 2 ) cannot be aggregated together (and in this case the rule will not be instanciated). In a similar manner, a base of facts is organised into n independent parts. Each part i brings together all observed objects of level i.

Processing data

Processing data consists in aggregating observed objects, starting from the lower level in order to deduce the structure of objects of higher levels. The system executes, in a recursive manner (for every level i), the two following steps:

1 -Triggering rules (the rules produce potential objects) 2 -Computing maximal consistent sub-bases (the best objects are selected).

The first step consists in triggering rules relating facts of level (i -1) to objects of level i. Two objects A i (t 1 ) and A i (t ) are said to be contradictory if they contain at least one common sub-object (of level i -1). This constraint will be expressed by the following clause:

A i (t 1 ) A i (t 2 ) 8 " 1 A clause of this type is added each time two rules having a common instanciated variable have been triggered to form composite objects t 1 and t 2 . The knowledge base resulting from step 1 is composed of facts F i+1 (objects newly constructed from observed or constructed objects of level i) and constraints G i+1 on these objects. This knowledge base is generally inconsistent (except if the set of constraints is empty). The second step is added in order to limit the computational complexity due to the number of possible combinations of aggregating objects of level i. Its main role is then to only select the best sub-bases F' i+1 of the base F i+1 of potential composite objects. The elements of F' i+1 will be added to the base of facts of level i+1. The system will start again by applying the two steps described above to part i+1 of the knowledge base. The schema of Figure 3 sums up the two steps and their iteration.

Example

Assume our knowledge base contains only rules for aggregating objects of level 0 to level 1 (for the sake of simplicity, we restrict ourselves to the lowest level of hierarchy shown in Figure 2). Let suppose that initially the set of facts F0 contains only the following elementary (observed) objects x 1 , x 2 , x 3 , x 4 and x 5 which belong respectively to classes S 0 , S 0 , S 0 , S 0 and B 0 . The set of constraints G 0 is empty since all the objects in F 0 are elementary and they do not share the same sub-object. Let the numbers (0.9, 0.8, 0.8, 0.7, 0, 0.8, 0.7, 0) be the estimates of the compatibility between objects respectively in the aggregations (x 1 x 2 x 3 ), (x 1 x 2 x 4 ), (x 1 x 3 x 4 ), (x 2 x 3 x 4 ), (x 1 x 5 ), (x 2 x 5 ), (x 3 x 5 ), (x 4 x 5 ).

Let us show now how the algorithm presented above proceeds in aggregating the elementary objects. The first step of the algorithm (triggering step) allows us to deduce six complex objects C (t 1 ), C 1 (t 2 ), C 1 (t 3 ), C 1 (t 4 ), B 1 (t 1 ) and B 1 (t 2 ) of level 1, constructed respectively from (x 1 x 2 x 3 ), (x 1 x 2 x 4 ), (x 1 x 3 x 4 ), (x 2 x 3 x 4 ), (x 2 x 5 ), (x 3 x 5 ). In the six complex objects constructed in the first step, there are objects which share the same elementary object(s); therefore the system generates the following set of constraints G 1 :

C 1 (t Note that the knowledge base + composed of the set of facts F 1 = {(C 1 (t 1 ) 0.9), (C 1 (t 2 ) 0.8), (C 1 (t 3 ) 0.8), (C 1 (t 4 ) 0.7), (B 1 (t 1 ) 0.8), (B 1 (t 2 ) 0.7)} and the previous set of constraints G 1 is partially inconsistent. The inconsistency of the knowledge means that not all of the aggregated objects can be formed.

As it was shown in the last section, the aim of the second step of the algorithm (computing maximal consistent sub-bases) is to select the best set of objects which will be added to the base of facts of level 1. This task corresponds to compute the best maximal sub-base of the inconsistent knowledge base +. Using the 7-ATMS, we can obtain four possibilities + 1 = {C 1 (t 1 )} ; + 2 = {C 1 (t 2 ), B 1 (t 2 )} ; + 3 = {C 1 (t 3 ), B 1 (t 1 )} ; + 4 = {C 1 (t 4 )}.

Each sub-base represents a maximal set of complex objects which are made of distinct sub-objects. Finally from the ordering "< " defined above, only the object C 1 (t 1 ) is remained, and will be added to the base of facts of level 1. See [START_REF] Benferhat | Raisonnement hypothétique en logique possibiliste[END_REF] and Monai et al. [START_REF] Monai | Logique possibiliste et gestion d'hypothèses multiples: application à un problème de fusion de données[END_REF] for a complete example.

This abstract example suggests the application of this technique to the reconstruction of scenarii from partial information. De Kleer and Williams [START_REF] Kleer | Diagnosing multiple faults[END_REF], in a multiple faults diagnosis problem, compute the probability of a candidate (i.e. a configuration of faults) given the outcomes of the system and assuming that faults are mutually independent. Falkenhainer [START_REF] Falkenheiner | Towards a general-purpose belief maintenance system[END_REF] has incorporated Dempster-Shafer theory into Doyle's T.M.S. [START_REF]A truth maintenance system[END_REF]. Provan [START_REF] Provan | An analysis of ATMS-based techniques for computing Dempster-Shafer belief functions[END_REF], D'Ambrosio [START_REF] D'ambrosio | A hybrid approach to reasoning under uncertainty[END_REF], Laskey

and Lehner [START_REF] Laskey | Belief maintenance; an integrated approach to uncertainty management[END_REF] have independently incorporated belief functions into the ATMS; their approaches are somewhat different but share the same basic features: first, a function is associated with each assumption; then, for each datum x we compute the belief Bel(x) of x by first computing the label of x symbolically (which can be done in the classical way), then we compute Bel(x) from Label (x); Provan computes Bel(x) by expressing logically Label(x) under the form of a formula containing only independent sub-formulas, assuming mutual independence of assumptions. D'Ambrosio's approach is based on Baldwin's Support Logic Programming method [START_REF] Baldwin | Evidential support logic programming[END_REF] and does not use the full Dempster-Shafer theory; besides, his way of propagating uncertainty is closer to a rule-based system than to a pure logical system (as he uses conditional beliefs and not beliefs of logical implications). Laskey-Lehner's and Provan's approaches both incorporate the full Dempster-Shafer theory into the ATMS, and differ essentially by their algorithm for the computation of Bel(x). See also Pearl [START_REF] Pearl | Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference[END_REF] whose proposal is close to Provan's (without normalizing Dempster rule). Bigham [START_REF] Bigham | Computing numerical beliefs using propositional inference as a basis[END_REF] incorporates Dempster-Shafer theory into the ATMS in order to compute belief functions. Lastly, Bernasconi et al. [START_REF] Bernasconi | On the notion of uncertain belief revision systems[END_REF] have outlined the definition of an ATMS based on fuzzy logic in Lee and Mukaidono's style. Our approach differs from the preceding ones on several points: First, the ATMS with embedded belief functions enable us to rank interpretations (i.e. maximal consistent environments), as they intend to do, but they cannot handle the uncertainty pervading the knowledge base if any, i.e. the justifications. They hold the justifications as classical knowledge (i.e. certain) and, putting weights on the assumptions, they deduce the weights attached to the other data. Our approach does not attach any a priori weight to each datum (assumptions or non-assumptions) but assigns a certainty degree to the clauses contained in the knowledge base (not only justifications, but also disjunctions of assumptions and nogoods specified by the user); then, taking into account this uncertain knowledge, we establish a mapping from certainty degrees assigned to assumptions to the certainty degrees of the data (i.e. the consequences of the assignment).

Secondly, in the other approaches, truth maintenance and uncertainty management are completely separated, i.e. the label of a datum x is first computed, then Bel (x) is computed (of course, taking into account Label(x)). Our approach completely integrates the uncertainty management into the truth maintenance system, by assigning a weight to each environment of a datum. Besides, the handling of weights will be done during the computation of labels, since certainty weights are present inside the label.

Thirdly, the main purpose of introducing uncertainty into A.T.M.S. is the capability of ranking solutions (i.e. interpretations) and eliminating solutions which are too uncertain. Hence the precise values of certainty degrees are not as important as their ordering. Possibility theory offers a framework where the ordering of the uncertainty degrees is more important than their precise values, and which requires easier computations (only min and max operations are used) than the general Dempster-Shafer approach. Furthermore, the weights assigned to nogoods are always available and no extra computations are required in order to renormalize results.

Fourthly, in Provan's model, the belief of the datum x is first computed formally by rewriting the label of x into an "independent form" and then computed numerically; if decide to change the weight of an assumption, the formal expression of the label of x remains unchanged and the re-computation of Bel(x) is immediate; however this is not the case if the label of x changes (which may happen each time we add a new justification): then we have to transform again the expression of the label of x; in the possibilistic model, both changes of weights or labels lead to almost no more computations than in De Kleer's original model.

Lastly, the other models cannot handle disjunctions of assumptions; besides, Provan's model needs to consider the assumptions as independent as long as they are not mutually exclusive (i.e. not containing any nogood). Possibilistic ATMS do not require this assumption, and can handle disjunctions of assumptions, as well as negated assumptions.

Conclusion

Possibilistic ATMS enable a joint handling of assumptions and uncertainty relative to a knowledge base, in the framework of possibilistic logic. The fact that possibilistic logic remains on many points close to classical logic facilitates the extension of efficient procedures, such as those based on Oriented CAT-correct resolution for the computation of labels, nogoods and contexts, by ordering the environments of a datum according to the certainty with which it can be deduced from each of them.

The truth maintenance system based on the 7-RCO resolution seems to have a number of advantages over the classical ATMS. Firstly, it allows the use of general clauses without requiring the creation of new data, justifications and disjunctions of assumptions as in [START_REF] Kleer | Extending the ATMS[END_REF]. The second advantage is that all justifications of the knowledge base are stored under the form of general weighted clauses, while in the classical ATMS, the justifications are stored in four different areas depending on the clause type: Horn clauses, disjunctions of assumptions, nogoods and environments. Finally, we have seen that negated assumptions are naturally taken in account, which is not possible in the classical ATMS, and they may appear in environments, nogoods and labels.

Possibilistic ATMS can be applied to truth maintenance problems in presence of uncertainty, to default reasoning, to fault diagnosis, or to the generation of (the most plausible) explanations.

  (c () (c' *) ------(c" min((,*)) where c" is a classical resolvent of the classical clauses c and c'.

  (p))) where v(p) is the truth value of p, i.e. v(p) = 1 if p is true and v(p) = 0 if p is false. Indeed, for any uncertain proposition (p () we can write µ M(p () (I) under the form v(p () = max(v(p), 1 -(), where v(p) is assigned by interpretation I. Then obviously:

  -) [E (] is an environment of the fact d if and only if N(d) ) ( is a logical consequence of E -+, where the assumptions of E are considered as certainly true (the certainty degree of the associated clauses is 1); -) [E (] is an (-environment of d if and only if [E (] is an environment of d and if #('>(, [E ('] is not an environment of d (( is maximal); -) [E (] is an (-contradictory environment, or (-nogood if and only if E -+ is (-inconsistent (i.e. E -+ (" ()), with ( maximal. The notation nogood ( E is used. The (-nogood [E (] is said to be minimal if there is no *-nogood [E' *] such that E . E' and ( & *.

-

  soundness: L(d) is sound if and only if # [E i ( i ] , L we have E i U + (d ( i ) where has been defined in Section 2; i.e., L(d) contains only environments of d. completeness: L(d) is complete if and only if for every environment E' such that E' -+ (d (') then / i , I such that E i 0 E' and ( i ) ('. I.e., all minimal (environments of d are present in L(d). minimality: L(d) is minimal if and only if it does not contain two environments (E 1 ( 1 ) and (E 2 ( 2 ) such that E 1 0 E 2 and ( 1 ) ( 2 . It means that L(d) only contains the most specific (-environments of d (i.e. all their assumptions are useful).

  (d)), where d is a fact or an assumption, and val (d) = sup{(, -+ (d ()}. Let us now give the following theorem: Let be a set of valued assumptions. Let d be a fact; it can be shown that -+ (d () in possibilistic logic if and only if / [E i ( i ] , Label(d), E i = {H i,1 , H i,2 , …, H i,n } such that (i) * . E i where * is the classical set of assumptions obtained from by ignoring the weights. (ii) ( & min(( i , * 1 , * 2 , …, * n ) where * 1 , * 2 , …, * n are the weights attached to H i,1 , H i,2 , ..., H i,n in . (iii) ( > Inc(E -+).

  Figure 1

  The following algorithm computes minimal candidates from the set of nogoods:ProcedureComputing-minimal-candidates begin /* let C : the set of minimal candidates initialized to {{}} Nog : the set of nogoods. */ -loop while Nog 5 6: let an element of Nog; remove from Nog -C := {c i -H j / c i , C and H j , } remove from C all not minimal candidates. end and therefore totally consistent Interpretations can be computed from the following relation: I = {T -c / T is the set of all assumptions, and c is a minimal candidate}. Example (continued) Let us consider the system {C1, …, C16} where the nogoods are: Nogoods = {[{¬FUSE} 0.2], [{¬BELT} 0.4], [{¬BATTERY} 0.3], [{¬OIL} 0.3], [{¬RECTIFIER} 0.1] } The algorithm described below computes the following minimal candidates: Candidates = {{¬FUSE, ¬BELT, ¬BATTERY, ¬OIL, ¬RECTIFIER} } and therefore by complementation, we find only one totally consistent interpretation: I = { {FUSE, BELT, BATTERY, OIL, RECTIFIER} } Remark:

  8), C 2 : (a 8 b 0.7), C 3 : (¬a 0.8), C 4 : (¬a 8 ¬b 0.6), C 5 : (¬b 0.6)}. It is easy to check that + is partially inconsistent and admits two strongly maximal consistent sub-bases + 1 = {C 1 , C 2 , C 4 } and + 2 = {C 2 , C 3 , C 4 , C 5 } but only + 2 is lexicographically maximal consistent since (+ -+ 1 ) = {0.8, 0.6}, (+ -+ 2 ) = {0.8} then + 2 is preferred then + 1 .

  {C 1 : (a 0,8), C 2 : (a 8 b 0.7), C 3 : (¬a 0.8), C 4 : (¬a 8 ¬b 0.6), C 5 : (¬b 0.6)}.

H 1 8

 8 a 0.8), ((H 2 % a) 8 b 0.7), (H 3 8 ¬a 0.8), (H 4 % ¬a 8 ¬b 0.6), (H 5 8 ¬b 0.6}. The 7-ATMS computes the base of nogoods which is equal to: Nogood(+') = {(H1 H2 H5}, {H1 H3}}. Then applying the algorithm presented in Section 4 we find three totally consistent interpretations I 1 = {{H 2 H 3 H 4 H 5 }, I 2 = {H 1 H 4 H 5 } and I 3 = {H 1 H 2 H 4 } which correspond to the maximal consistent sub-bases + 1 = {C C 3 C 4 C 5 }, + 2 = {C 1 C 4 C 5 } and + 3 = {C 1 C 2 C 4 }.

Figure 2 Xn

 2 Figure 2

  Figure 3

7 .

 7 Possibilistic A.T.M.S. vs. Other Existing "Uncertainty-Handling" A . T . M . S .

It is not difficult to define possibilistic ATMS with both lower bounds of necessity degrees and possibility degrees; however, for the sake of clarity, we restrict ourselves in this paper to necessity-valued possibilistic ATMS.

Acknowledgements

This work has been supported by the European ESPRIT Basic Research Action number 3085 entitled Defeasible Reasoning and Uncertainty Management Systems (DRUMS), and by a grant from Thomson CSF, Colombes, France.

Annex

Let + = B 1 -B 2 -… -B n be a possibilistic knowledge base where B j contains clauses of the (a ji , ( j ), and ( 1 > ( 2 >… > ( j . Then +' 3 + is a strongly maximal consistent sub-base of + if and only if

Proof

Assume +' is not strongly maximal consistent. Then there is an ( j and a clause (a ij , ( j ) such that the level of inconsistency of +' -{(a ij , ( j )} is smaller than ( j , say Inc(+' -{(a ij , ( j )}) = ( k < ( j . This implies that E 1 -E 2 -… -E j -{(a ij , ( j )} is consistent. Indeed the inconsistency of +' -{(a ij , ( j )} can disappear by taking away clauses in E k

} is consistent; then +' -{(a ij , ( j )} is inconsistent with a level of inconsistency that is smaller than ( j , since it involves some clause in E j+1 -… -E n . Q.E.D.


domain: spi phys

text:
S Segondy

• C Rio

• S Landais

M.-P Bacos

• C Guyon

• F Rousseau

Andre ´mcdonald

General Electric Yuk-Chiu Lau

Impact of Precursor Thermal Decomposition Reactions on YSZ Coatings Deposited with a Low-Power Plasma Torch at Short Spray Distance

Keywords: atmospheric low-power plasma torch, solution precursor spraying, TBC, YSZ

Yttria-stabilized zirconia (YSZ) deposition at short spray distance is of great interest for the aeronautic industry, especially to perform a thermal barrier coating (TBC) local repair. Typical thermal spray techniques usually employed to deposit such coatings are not designed for this type of application due to the powers involved. This work focuses on a plasma torch operating at atmospheric pressure, at low power (\ 1 kW) and at short spray distance (4 mm). Precursor solutions are directly injected in the plasma afterglow to be sprayed and deposited onto a substrate. During flight time, precursor conversion is initiated. However, because of the low afterglow temperature (600 °C), droplets are not fully converted into YSZ when they impact the substrate. Typically, posttreatment is applied to improve the conversion of precursors into oxides, but this operation is very time-and therefore moneyconsuming. This study is focused on exploring a way to maximize the heat transfer to droplets already deposited by generating exothermicity using a redox reaction between zirconium and yttrium nitrates with ammonium acetate. This work highlights how exothermic precursor decomposition reactions as well as the gel-like consistency of the deposited clusters can improve both chemical conversion and coating microstructure.

Introduction

Combustion engines, such as turbojets, are built with nickel-based superalloys that are very resistant to deformation. However, given the high temperatures, mechanical stresses and oxidizing atmospheres that superalloys are exposed to, a thermal barrier coating (TBC) is required to ensure proper operation over a longer period of time (Ref 1, 2). TBCs are composed of an alumina-forming metallic bond coat (usually NiCrAlY), which promotes adherence and ensures better corrosion resistance of the substrate, and a ceramic coating (generally yttria-stabilized zirconia (YSZ)), which is a thermal insulator (Ref 3).

The ceramic coating is generally deposited by atmospheric plasma spray (APS) for the non-moving parts of the turbojet, such as the combustion chamber, and by electron beam physical vapor deposition (EBPVD) for the rotating parts, such as the turbine blades ( Ref 4,[START_REF] Feuerstein | Technical and Economical Aspects of Current Thermal Barrier Coating Systems for Gas Turbine Engines by Thermal Spray and EBPVD: A Review[END_REF]. These two techniques result in two different coating morphologies: decomposition reactions and more precisely their capacity to absorb or release heat (endo or exothermic processes). Typically, the strategy is to add a posttreatment step to the deposition process to help complete the decomposition of particles already deposited (Ref 13,[START_REF] Rousseau | Deposition of Thick and 50% Porous YpSZ Layer by Spraying Nitrate Solution in a Low Pressure Plasma Reactor[END_REF][START_REF] Rousseau | A New Injection System for Spraying Liquid Nitrates in a Low Power Plasma Reactor: Application to Local Repair of Damaged Thermal Barrier Coating[END_REF]. The strategy presented hereafter consists of adding a reducing agent (acetate group in the form of ammonium acetate) to the solution (the zirconium and yttrium nitrate groups being the oxidizer) to produce an exothermic redox reaction as it was proposed by Muoto et al. (Ref 18,[START_REF] Muoto | Phase Homogeneity in Y2O3-MgO Nanocomposites Synthesized by Thermal Decomposition of Nitrate Precursors with Ammonium Acetate Additions[END_REF].

Experimental Procedure Precursor Solution Preparation

Three different precursor solutions were produced, and the respective concentrations are presented in Table 1. The first solution did not include a reducer and was only composed of zirconium oxynitrate hydrate ZrO(NO 3 ) 2 Á6H 2 O, abbreviated to ZrO[n] (Sigma-Aldrich, USA) and yttrium nitrate hexahydrate Y(NO 3 ) 3 Á6H 2 O, abbreviated to Y[n] (Sigma-Aldrich, USA). Ammonium acetate CH 3 COONH 4 , abbreviated to NH [START_REF] Hardwicke | Advances in Thermal Spray Coatings for Gas Turbines and Energy Generation: A Review[END_REF] [a] (Sigma-Aldrich, USA), was added to the other two solutions. In the second solution (ZrO[n]Y[n] ? 50% NH 4 [a]), 50 mol% ammonium acetate of the total molar quantity of nitrates in solution was added. The third solution (ZrO[n]Y[n] ? 100% NH 4 [a]) contained 100 mol% of ammonium acetate in addition to the nitrate mixture. Net oxidizing and reducing valencies were calculated for each reactant by adding the net valencies of every atom (Ref 19,[START_REF] Nair | Sr-Doped LaCoO3 through Acetate-Nitrate Combustion: Effect of Extra Oxidant NH4NO3[END_REF]. The valencies of oxygen, nitrogen, hydrogen, yttrium, carbon and zirconium were considered equal to 2 -, 0, 1 ? , 3 ? , 4 ? and 4 ? , respectively. The net valencies of zirconium oxynitrate, yttrium nitrate and ammonium acetate were therefore 10 -/mol, 15 -/mol and 11 ? /mol, respectively. In the case of a 92/8 mol% ratio between ZrO(NO 3 ) 2 and Y(NO 3 ) 3 , the net valency was 10.4 -/mol. Therefore, in regard to valencies to balance the redox reaction, the reducing agent was added in a slight excess, as the ratios between reducers (NH 4 [a]) and oxidizers (ZrO[n]Y[n]) corresponded to 52.9 mol% and 105.8 mol%. For each solution, the reactants were dissolved and mixed in deionized water (ZrO[n]Y[n] concentration: 0.05 M) to obtain a final composition of 7.4 wt% Y 2 O 3 -ZrO 2 .

Solution Characterization

Each solution was evaporated to analyze its thermal behavior by differential scanning calorimetry (DSC). Solutions were evaporated on a heating plate (programmed at 230 °C in order to reach 100 °C in the beaker) until the columnar for EBPVD coatings and lamellar for APS coatings (Ref 3,[START_REF] Karger | Atmospheric Plasma Sprayed Thermal Barrier Coatings with High Segmentation Crack Densities: Spraying Process[END_REF]. Columnar coatings have a slightly higher thermal diffusivity (4 9 10 -7 m 2 /s at 1100 °C (Ref 7)) than lamellar coatings but are more resistant to mechanical stress due to column presence, which is required for rotating parts. Lamellar coatings promote better thermal insulation, represented by a lower thermal diffusivity (2.3 9 10 -7 m 2 /s at 1100 °C (Ref 8)). The turbojet's combustion chamber is the most exposed to high temperatures due to fuel combustion; as such, good thermal insulation is essential (Ref 4,[START_REF] Feuerstein | Technical and Economical Aspects of Current Thermal Barrier Coating Systems for Gas Turbine Engines by Thermal Spray and EBPVD: A Review[END_REF].

TBCs exposed to difficult environments can damage (cracks, delamination, etc.) requiring repairs. Generally, repair work involves a two-step process: TBC removal (by sandblasting or a high-pressure water jet, for example (Ref 9, 10)) followed by the entire TBC redeposition (usually with the same deposition techniques depending on the component type: APS, EBPVD). This process implies engine immobilization and turbojet disassembly and reassembly after the repair. This process is time-consuming and not very cost-effective, which is why local repair of the ceramic coating is of interest. Moreover, given the size, component geometry, accessibility and thermal spray process specificities (stand-off distance ranging between 50 and 150 mm (Ref 11, 12)), local repair of combustion chamber parts with commonly used plasma spray processes is impossible. To repair a combustion chamber locally, the plasma process developed needs to be equipped with a relatively small plasma torch that could operate close to the substrate.

Our work consists in developing a YSZ deposition technique functioning at low power (\ 1 kW), at atmospheric pressure and at a short spray distance of 4 mm. Because low power is involved, the temperature in the afterglow reaches 600 °C which is too low to melt oxide particles (Ref 13) (ZrO 2 melting point: 2677 °C (Ref 14)). That is why precursor solutions are preferred, similarly to what is used in solution precursor plasma spray (SPPS). In our case, it implies that the only way to get ''splat'' or ''granule'' formation is to oxidize the precursors after they have reached the substrate under a liquid or a gel form. In previous works (Ref [START_REF] Segondy | Development of a Low Power Plasma Reactor for the Local Deposition of YSZ Thermal Barrier Coatings at Atmospheric Pressure[END_REF][START_REF] Segondy | De ´po ˆt Local de Zircone Yttrie ´e Par Plasma Atmosphe ´rique Basse Puissance: Applications Aux Turbines Ae ´ronautiques[END_REF], we highlighted that heat transfer is important to reach good precursor conversion values. This work's aim is to explore ways to improve the conversion of precursors into YSZ and the microstructure obtained from water-based solutions by boosting the precursor thermal decomposition during the coating process (Ref 13). It involves testing strategies maximizing heat transfer to droplets already deposited. The goal is to obtain a morphology similar to that produced by APS (lamellar microstructure). Consequently, this study focuses on understanding the impact of precursor thermal reactants reached a dried gel-like consistency. The evaporation process took approximately one hour to be completed. Dried gel enthalpies of reaction (endo or exothermic) resulting from their decomposition were evaluated with a DSC instrument (DSC131 EVO, SETARAM Instrumentation, France). The analyses were performed by heating 53 ± 1 mg of each sample from 25 to 450 °C at 10 °C/min rate in aluminum crucibles.

Coating Deposition

The plasma torch used in this study was designed by AcXys Plasma Technologies (France), and its principle was described in a previous paper (Ref [START_REF] Segondy | Development of a Low Power Plasma Reactor for the Local Deposition of YSZ Thermal Barrier Coatings at Atmospheric Pressure[END_REF]). The experimental setup is visible in Fig. 1. The plasma discharge was generated by AC electrical power with a typical frequency of 80 kHz between two electrodes. This process operated at a high gas flow rate of 33 SLPM, which pushed the plasma discharge beyond the electrodes and formed the plasma afterglow. Air was used as the discharge gas and plasma power ranged between 700 and 800 W. The temperature of the plasma afterglow was previously estimated at 600 °C (Ref 13). The coating scanning pattern followed multiple scans on the same single line across the substrate at a speed of 1 mm/s, which resulted in coating depositions covering approximately a 25 9 5 mm 2 surface area and maximizing heat transfer between the plasma afterglow and the substrate. The substrate, a superalloy substrate (Hastelloy-X), with a NiCrAlY bond coat (32 9 17 9 1.5 mm) was placed at a distance of 4 mm from the nozzle exit. Before deposition, the substrate was preheated by 6 scans (2 min) of the plasma torch to reach a temperature of around 200 or 250 °C, which promoted coating adhesion to the substrate.

An injector connected to a syringe pump (BIOSEB 8000/9000, France) was used to spray the precursors solution directly into the plasma afterglow at a liquid flow rate of 600 lL/min. It was positioned at 1 mm from the torch nozzle exit and was centered based on the afterglow. The total solution volume injected was fixed to 16 mL. Coatings were therefore obtained after 86 scans (6 scans of substrate preheating and 80 scans of deposition, 28min40s total).

In some cases, a posttreatment step was added to the deposition process. Posttreatment corresponded to a step where no additional precursor solution was injected. The plasma afterglow only treated the droplets that had been previously deposited onto the substrate's surface. The deposition cycle was as follows: precursor injection for one minute followed by one minute of posttreatment. This cycle was repeated 26 times to reach the targeted volume of precursor solution. Coatings with posttreatment were The first solidified droplets were studied by SEM (Quattro S, Thermo Fisher Scientific, USA) to compare their structure depending on the precursor type. Two scans were performed on the same line on stainless steel instead of superalloy to facilitate droplet localization. The substrate was first preheated by multiple scans of the plasma torch. Before every SEM analysis, a metallic coating was sprayed on top of the sample (Cressington sputter coater 108 under argon, Germany).

[n] Y[n] NH 4 [a] ZrO[n] Y[n] NH 4 [a] ZrO[n]Y[n] 4.

Results and Discussion

Study of Precursor Thermal Decomposition Reactions

For all solutions, a dried gel consistency is obtained as shown in Fig. 2(a)-(c). It is important to note that both solutions containing a nitrate/acetate mixture showed a gel precipitation halfway through the evaporation process. Also, the dried gels look different from the one with ZrO[n]Y[n] only. This correlates to the heterogenous gel formation observed by just mixing zirconium oxynitrate hydrate and ammonium acetate powders. This can be explained by the interaction between hydrated nitrate and hygroscopic ammonium acetate molecules. It is likely that counterions are swapped between zirconium nitrate and ammonium acetate leading to the formation of zirconium acetate, which forms a hydroxy-bridged polymer which is known to be a hydrophobic compound (Ref 21,[START_REF] Deville | Ice-Structuring Mechanism for Zirconium Acetate[END_REF].

DSC thermographs of these dried gels are presented in Fig. 3. The ZrO[n]Y[n] thermal decomposition is confirmed to be an overall endothermic process from 25 to 450 °C through the presence of three broad peaks at 143, 226 and 268°C of -53, -40 and -39 mW, respectively. The first one starting at 100 °C can be associated with hydration water evaporation while the other can be correlated to reactions leading to zirconium and yttrium oxidation and nitrate decomposition. As expected, solutions containing a nitrate/acetate mix show two exothermic peaks for temperatures over 250 °C, which corresponds to combustion reactions correlated to precursor decomposition and metal oxidation (Ref [a], no endothermic trough is clearly visible, but two exothermic peaks at 270 °C (? 244 mW) and at 334 °C (? 69 mW) are noticeable. The produced exothermic heat reaches higher values when the acetate is more concentrated in the solution. This is expected because more reducer is available to react with the oxidizing agent obtained after 166 scans (55min20s total): 6 scans of substrate preheating, 80 scans of deposition and 80 scans of posttreatment. An annealing step corresponding to the first heating cycle of the engine was performed on some of the deposits (even for posttreated ones) to replicate the conditions of use. These samples were annealed at 1100 °C for 1 h in an open-air furnace (Pyrox Thermique Mate ´riaux, France). The heating rate was set at 200 °C/h.

Coating Characterization

Fourier transformed infrared spectroscopy (FTIR) instrument (Agilent Technologies Cary 600 Series FTIR spectrometer, USA) equipped with a diamond ATR (PIKE Technologies GladiATR, USA) was used to assess the precursor conversion into oxides. Two bands were specifically tracked: the first one centered at around 1250 cm -1 corresponding to the stretching vibration of N-O from NO 3

-(visible when nitrates were remaining in the coating and coordinated to a metal) and the second peak at 400 cm -1 corresponding to the stretching vibration of Zr-O, both visible in the precursor (ZrO(NO 3 ) 2 ) and the oxide (ZrO 2 ), as described in a previous work (Ref 13). Precursor conversion to oxides was estimated by tracking the 1250 cm -1 band intensity, which was the most intense nitrate band, as well as the 400 cm -1 one. It is important to note that FTIR analyses were done at multiple locations on each coating and corresponded to the entire thickness. This allowed estimating an average composition. However, there could be conversion disparities between coating's top and bottom parts and the absorbance value was maybe not homogenous through the coating's thickness.

Coating crystallinity was analyzed through an x-ray diffractometer (Malvern PANalytical XPERT Pro, UK) to obtain XRD patterns and determine YSZ crystallographic phase (diffractograms ranging between 20° and 80° at a scan speed of 0.042°/s).

Cross-sectional morphologies were studied through scanning electron microscopy (TESCAN MIRA3, Czech Republic). Beforehand, the samples were embedded in an epoxy resin (Araldite, ESCIL, France), cut, and then ground with abrasive paper (gradation 350, 1200, 4000 for 5 min each) and polished with diamond suspensions on corresponding pads (6, 3, lm for 15, 5 and 5 min, respectively) on a manual grinding and polishing machine (Struers Planopol Polisher, Denmark). Coating porosity was assessed through image analysis with ImageJ software (free open-source software), by converting the image to black and white pixels depending on a manually set threshold. The ratio between black and white pixels represented the porosity of the image studied. To calculate the porosity, five images were analyzed three times for each sample.

(nitrate molecules) leading to a more energetic combustion reaction. Thermal decomposition of nitrate and acetate groups coordinated to a metal leads to the formation of the corresponding oxide as well as secondary compounds, which are released as gases (such as NO x or CO 2 ).

Study of First Droplets Deposited on the Surface

SEM images of the first droplets deposited on the substrate surface are presented in Fig. 4. As shown in images (a)-(c), granules have different sizes, ranging between less than 1 lm and above 5 lm for all solutions tested. Compared to what is obtained with SPPS, no large splats that would correspond to flatten spherical molten droplets are visible (Ref 23).The temperature is not high enough to melt the particles and therefore to produce such structures in our case. It is likely that the ''clusters'' are made of a mixture of remaining precursors and partly amorphous, partly crystallized 7.4YSZ if the thermal decomposition process is not complete.

Figure 4(d)-(f) is close-ups of the biggest granules deposited on the surface from solutions containing 0 mol, 50 mol and 100 mol% acetate, respectively. The cluster porosity difference is clearly visible on these three images. In the case of the solution ZrO[n]Y[n], the biggest particles exhibit a very porous morphology. The ones from solutions ZrO[n]Y[n] ? 50 or 100 mol% NH 4 [a] clearly show the gel-like consistency expected for a nitrate/acetate mix and porosities are less visible. The clusters' density depends on the reducer/oxidizer ratio. This implies that remaining precursors are composed of zirconium acetate and that the particles need to be further treated by multiple scans to boost the thermal decomposition/combustion process. Flight time is therefore too short to fully support precursor conversion

Precursor Conversion

As shown in the DSC analysis, nitrate thermal decomposition is exclusively an endothermic process. A typical way to tend toward a full precursor conversion is to increase the contact time between deposited droplets and the plasma afterglow by adding a posttreatment step to the coating process. Figure 5 compares FTIR spectra from coatings produced with and without a posttreatment step or with two different ammonium acetate amounts. When comparing spectra corresponding to coatings with and without a posttreatment step, we can see that a lower nitrate amount is remaining with a posttreatment step, as made evident by the low absorbance value at 1250 cm -1 corresponding to the N-O stretching band compared to the absorbance of Zr-O stretching vibration at 400 cm -1 . Peaks at 1020 and 1550 cm -1 are, respectively, associated with bands from the NO stretching and the asymmetric NO 3 stretching (Ref 24). This confirms that adding a posttreatment step to the coating cycle can improve the precursor conversion.

The nitrate conversion for precursor mixtures containing ammonium acetate reaches the same value as when a posttreatment step is added to the coating process. The N-O absorbance value for both ammonium acetate percentages is indeed equivalent to the posttreatment one. It means that increasing the heat available by generating an exothermic reaction is helping the conversion process just like a posttreatment does. The redox reaction is advantageous because the conversion is the same as with a posttreatment step, but it requires twice less time to form a coating. 6c), it appears relatively denser (porosity of 27 ± 3%). There is with the symmetric carboxylate band. The one at 1520 cm -1 is an overlap between the antisymmetric carboxylate band from the acetate and the asymmetric NO 3 stretching band from the nitrate (Ref 25). This implies that not all acetate groups are decomposed during the coating process. Adding 50 or 100 mol% of acetate does not seem to make any difference conversion-wise. Even if increasing the ammonium acetate amount is improving the exothermic reaction, it is also adding more precursors to decompose. That is maybe why the 100 mol% FTIR spectrum is the same as when only 50 mol% is added to the solution. These results confirm that giving more thermal energy to the coating can improve the conversion. Moreover, no large band between 2600 and 3600 cm -1 associated with O-H stretching vibrations is visible, meaning that all water molecules from the precursors and solution have been evaporated. a contrast between the top and the bottom parts. Close to the substrate, more porosities are visible, and the coating is like what is obtained with the nitrate-based solution. However, the coating's top part appears less porous and more homogenous. These differences are even sharper for the coating made with the solution containing 100 mol% of ammonium acetate, shown in Fig. 6(d). The bottom part is particularly porous and appears more columnar (porosity of 37 ± 7%). In contrast, the top part is showing very little porosities (porosity of 20 ± 4%). In some coating's regions, there are cracks parallel to the substrate splitting up the coating in two, as illustrated in the SEM picture. It appears that in all cases, the coating's microstructure is denser at the top than at the bottom. This phenomenon was already explained in (Ref 15) but it is shown here that it is valid for different precursor type and is not limited to water-based nitrate solutions.

As a reminder, for both nitrates and acetates, thermal decomposition implies releasing gases, such as NO x or CO 2 . As they are released from the coating, these gases can create bigger porosities, columnar structures, or cracks. That is why it is important to release them gradually while increasing the coating thickness. Adding 100 mol% of ammonium acetate to the nitrates forms a precursor solution that generates an important exothermic response during the thermal decomposition process. It means that the precursors are quickly converted and therefore release a large gas volume in a small amount of time. This phenomenon can explain the porosity formation as well as the coating delamination right at the center. Moreover, because the clusters deposited are denser with a higher oxidizer/reducer ratio, the gases can be entrapped by the overall top layer densification, leading to the coating delamination. When only 50 mol% of ammonium acetate is added to the solution, the microstructure is more stable. Because the conversion was evaluated equivalent by FTIR analysis, it means that a higher oxidizer/reducer ratio emphasizes the thermal decomposition process.

SEM pictures took from coating's regions close to the substrate at high magnification are presented in Fig. 6(e) to (h). These pictures illustrate microstructure differences. Figure 6(e) and (f) corresponds to coatings made with the ZrO[n]Y[n] precursor solution without or with a posttreatment, respectively. Nanoscale porosities are clearly visible, especially on the picture (e). These small porosities are almost completely gone in the case of precursor solutions including ammonium acetate (Fig. 6g andh). These differences can be linked to the first droplets microstructure as a function of the precursor type shown in the previous section. Because the intermediate compound formed (supposedly zirconium acetate) has a strong gel consistency, it helps getting a more compact microstructure. That is probably why the coating top part is particularly dense in Fig. 6(c) and (d).

The deposits were also observed after annealing. The clear difference observed at a microscopic scale between coatings produced using the nitrate-only solution with a posttreatment step and the ones made with nitrate/acetate solutions is still noticeable and not modified after annealing. As the deposited coating's thickness is approximately 130 lm, peaks corresponding to the substrate crystalline structure are visible and are indicated on the figure. XRD data obtained before the annealing step show that as-deposited coatings exhibit peaks coinciding to the expected YSZ tetragonal phase. However, the peaks are not all well-defined, especially for coatings deposited with a precursor solution containing ammonium acetate. As a reminder, FTIR analyses show that a certain precursor amount is remaining in the coating, which have an amorphous phase (see section precursor conversion). It is particularly true for coatings still including carbon-based compounds. That can explain annealing is shown in Fig. 7 as well. The peaks are more defined, demonstrating better crystallinity due to more elevated temperatures, and the pattern still only contains the peaks of the expected YSZ tetragonal phase. It is also seen that whatever the precursor solution used, the diffractograms obtained after the annealing step are all identical. It means that when the coating reaches higher temperatures, the precursor type employed to make the solution does not affect the coating's crystalline structure.

Coating Phase Composition

Conclusion and Perspectives

A YSZ deposition operated at atmospheric pressure, low power of 700-800 W and short spray distance of 4 mm involved specific operating conditions at temperature less than 800 °C, different than the ones employed in thermal spray techniques such as SPPS or APS. Compared to these methods, the phenomenon was different because precursors in solution cannot be converted into oxides and melted before their impact with the substrate as with thermal plasma. That is why maximizing heat transfer to droplets already deposited was crucial in a process like the one presented in this work. YSZ coatings were produced by injecting a water-based precursor solution directly in the plasma afterglow. All solutions were at least composed of a zirconium oxynitrate and yttrium nitrate mixture. The ratio was chosen so that the final composition of the coatings was 7.4 wt% Y 2 O 3 -ZrO 2 . As it was confirmed by DSC analyses, nitrate groups underwent endothermic thermal decomposition when exposed to heat, meaning that they required energy to go through their reaction process. Adding a posttreatment increased the contact time between deposited droplets and the plasma afterglow resulting in a conversion improvement due to the temperature as well as the presence of oxidative species in the plasma afterglow. Concerning the microstructure, the posttreatment step did reduce slightly the porosity value. However, this approach also multiplied by two the experiment time, which is not ideal. Ammonium acetate was added with a specific molar ratio in some solutions. Interestingly, nitrate and acetate together caused redox reactions, making the process exothermic. The release of heat during the decomposition generated enough energy to improve the precursor conversion. FTIR analyses showed that the conversion was indeed improved with acetate added to the mix. SEM pictures illustrated interesting microstructures. For 50 or 100 mol% of ammonium acetate added to the solution, the particles (granules) appeared denser at a microscale. This can explain why the microstructure was more homogenous with the ZrO[n]Y[n] ? 50% NH 4 [a] solution. It is, however, important to manage the exothermic reaction. In the case of the ZrO[n]Y[n] ? 100% NH 4 [a] solution, the reaction was supposedly too powerful and the coating was getting too dense entrapping the gases formed, leading to the coating delamination at the center.

Even though working with a reducer/oxidizer mixture improved the thermal decomposition, a full conversion was still not reached. This implies that a small number of carbon-based compounds were remaining in the coating at the end. Also, the XRD analysis showed that the coatings were more amorphous when produced with solutions containing ammonium acetate. However, after an annealing step at 1100 °C, all diffractograms were equal, corresponding to the expected YSZ tetragonal phase. These results showcased how the precursor conversion and coating microstructure can be improved through exothermic thermal decomposition reactions.

Author Contributions SS contributed to conceptualization, investigation, writing-original draft preparation, writing-reviewing and editing, CR contributed to investigation, SL contributed to conceptualization, methodology, investigation, M-PB contributed to conceptualization, methodology, investigation, writing-original draft preparation, writing-reviewing and editing, CG contributed to conceptualization, methodology, investigation, supervision, writing-reviewing and editing, and FR contributed to conceptualization, methodology, investigation, supervision, writing-original draft preparation, writing-reviewing and editing, project administration, funding acquisition.

25 9 10 -Fig. 1

 101 Fig. 1 (a) Photograph of the plasma torch in operation (b) schematic of the experimental setup

Fig. 2

 2 Fig. 2 Photographs of evaporated solutions (a) ZrO[n]Y[n], (b) ZrO[n]Y[n] ? 50 mol% NH 4 [a] and (c) ZrO[n]Y[n] ? 100 mol% NH 4 [a]

Fig. 3

 3 Fig. 3 DSC thermographs obtained from evaporated solutions

Figure 6 (

 6 Figure 6 illustrates coating cross-sectional SEM images deposited with solutions with or without a posttreatment as well as with or without ammonium acetate. The SEM images presented here correspond to deposits before annealing. When comparing pictures (a), (c) and (d) corresponding to coatings produced with solutions with 0 mol, 50 mol and 100 mol% acetate, respectively, differences in microstructure are visible. The coating made with the ZrO[n]Y[n] precursor solution, shown in Fig. 6(a), exhibits a columnar-like microstructure and is overall quite porous. Its porosity was evaluated at 30 ± 5%. Its microstructure was further discussed in (Ref 15).Figure 6(b) is illustrating a coating produced with the ZrO[n]Y[n] precursor solution but with a posttreatment step. The overall microstructure is considered denser than without a posttreatment (porosity of 28 ± 5%). However, there are still quite porous regions similar to what is observed in Fig. 6(a), especially close to the substrate. In the case of the coating produced with the solution made with ZrO[n]Y[n] ? 50% NH 4 [a] (Fig.6c), it appears relatively denser (porosity of 27 ± 3%). There is

Fig. 4

 4 Fig. 4 SEM images of particles deposited after two scans at two different magnifications from solutions containing (a); (d) ZrO[n]Y[n] (b); (e) ZrO[n]Y[n] ? 50 mol% NH 4 [a] and (c); (f) ZrO[n]Y[n] ? 100 mol% NH 4 [a]

Figure 7

 7 Figure7showcases diffractograms obtained for coatings displaying the best conversion according to FTIR analysis, before and after an annealing step at 1100 °C for one hour. The experimental conditions are:ZrO[n]Y[n] solution with a posttreatment step, ZrO[n]Y[n] ? 50% NH 4 [a] solution and ZrO[n]Y[n] ? 100% NH 4 [a] solution.As the deposited coating's thickness is approximately 130 lm, peaks corresponding to the substrate crystalline structure are visible and are indicated on the figure. XRD data obtained before the annealing step show that as-deposited coatings exhibit peaks coinciding to the expected YSZ tetragonal phase. However, the peaks are not all well-defined, especially for coatings deposited with a precursor solution containing ammonium acetate. As a reminder, FTIR analyses show that a certain precursor amount is remaining in the coating, which have an amorphous phase (see section precursor conversion). It is particularly true for coatings still including carbon-based compounds. That can explain

Fig. 5

 5 Fig. 5 FTIR spectra obtained for coatings produced from the solutions containing ZrO[n]Y[n] with and without a posttreatment step, ZrO[n]Y[n] ? 50 mol% NH 4 [a] and ZrO[n]Y[n] ? 100 mol% NH 4 [a]

Fig. 6

 6 Fig. 6 Cross-sectional SEM images of coatings deposited by injecting solutions containing (a) ZrO[n]Y[n], (b) ZrO[n]Y[n] with posttreatment, (c) ZrO[n]Y[n] ? 50% NH 4 [a], (d) ZrO[n]Y[n] ? 100% NH 4 [a], (e), (f), (g) and (h) close-up SEM images in respective order

Fig. 7

 7 Fig. 7 XRD diffractograms obtained for coatings produced from solutions as indicated (before and after an annealing step at 1100 °C for one hour)

Table 1

 1 Composition of the solutions producedMolar concentration, mol/L Mass concentration, g/L

	ZrO