Nouha Chaoued

Anne Laurent

Amel Borgi
email: amel.borgi@insat.rnu.tn

Symbolic Approximate Reasoning within Unbalanced Multi-Sets: Autism Diagnosis

Keywords: Symbolic Approximate Reasoning Within Unbalanced Multi-sets: Application to Autism Diagnosis, Amel Borgi, Unbalanced Multi-Sets, Approximate reasoning, Generalized Modus Ponens, Generalized Symbolic Modifiers, Autisme diagnosis

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Introduction

In Artificial Intelligence, an important field is to develop machines with the human-like ability to reason with linguistic terms. Knowledge representation by computers must consider the fact that the information used in human thinking is often imprecise.

Zadeh proposed the fuzzy sets [START_REF] Zadeh | Fuzzy sets[END_REF] to deal with the complexity of this representation based on fuzzy logic. Thereafter, De Glas introduced, in the multi-valued logic context, the multi-sets representation model [START_REF] Glas | Knowledge representation in a fuzzy setting[END_REF][START_REF] Akdag | A qualitative theory of uncertainty[END_REF].

These two logics propose modeling membership degrees with words in the reasoning process. These words correspond to linguistic variables that take their values in a set of linguistic terms [START_REF] Zadeh | The concept of a linguistic variable and its application to approximate reasoning -I[END_REF]. Each linguistic variable is associated to adverbs qualifying the belief degree of the veracity of a proposition.

In most Knowledge-Based Systems (KBS), the manipulated multi-sets include terms that are uniformly and symmetrically distributed (Fig. 1). Nevertheless, in other cases, linguistic information is represented with terms neither uniformly nor symmetrically distributed (Fig. 2).

Fig. 2. Unbalanced Multi-set of 7 Linguistic Terms

KBSs include an inference process that can be based on approximate reasoning. It deduces meaningful outputs from imprecise inputs by means of the Generalized Modus Ponens (GMP) model.

In this work, we introduce an adaptation of the GMP model proposed in [START_REF] Kacem | Generalized modus ponens based on linguistic modifiers in a symbolic multi-valued framework[END_REF][START_REF] Kacem | On some properties of generalized symbolic modifiers and their role in symbolic approximate reasoning[END_REF][START_REF] Kacem | Extended symbolic approximate reasoning based on linguistic modifiers[END_REF] to infer within unbalanced multi-sets. It is based on Generalized Symbolic Modifiers (GSM).

In the context of multi-valued logic, Bel Haj Kacem et al. [START_REF] Kacem | Ramoli: A generic knowledge-based systems shell for symbolic data[END_REF] proposed a KBS designed for uniform multi-sets, named RAMOLI. We implement a new version that takes into account unbalanced multi-sets, named ARUMS (Approximate Reasoning within Unbalanced Multi-Sets). This tool integrates our inference model considering unbalanced mlti-sets. We test our proposal to autism diagnostis based on the Diagnostic and Statistical Manual of Mental Disorders (DSM) 5 [START_REF] Association | Diagnostic and statistical manual of mental disorders : DSM-5[END_REF]. This paper is organized as follows. Section 2 introduces the basic concepts of multi-valued logic, Generalized Symbolic Modifiers, and the unbalanced multisets. Then, we present existing works treating the approximate reasoning in the multi-valued logic context in section 3. In section 4, we propose our GMP model dealing with unbalanced multi-sets. The section 5 presents the tests results for autism diagnosis.

Preliminaries

Multi-valued logic is based on De Glas's theory [START_REF] Glas | Knowledge representation in a fuzzy setting[END_REF]. In this approach, each linguistic variable is represented by an ordered and finite list of M linguistic symbols. They constitute a multi-set denoted by [START_REF] Glas | Knowledge representation in a fuzzy setting[END_REF][START_REF] Akdag | Incertitude et logique multivalente, première partie : Etude théorique[END_REF]:

L M = {τ 0 , τ 1 , ..., τ M -1 }; M ∈ N * \{1} ( 1
)
τ i is the membership degree to the multi-set L M with i ∈ [0, M -1]. The membership relation in multi-valued logic is partial:

x ∈ α A ⇔ x belongs to A at a degree α (2)
It should be noted that symbolic degrees are connected only by the total order relation ≤ defined by [START_REF] Akdag | A qualitative theory of uncertainty[END_REF]:

τ α ≤ τ β ⇔ α ≤ β; ∀ α and β ∈ [0, M -1] (3)
Each degree τ i of L M is assimilated to a linguistic term corresponding to the precision degree of a proposition. It is a qualifier ϑ α that express the imprecision of a predicate.

x is ϑ α A ⇐⇒ (x is ϑ α A) is true ⇐⇒ (x is A) is τ α true (4)
For example saying that the communication is little impaired means that the communication satisfies the predicate impaired with the degree τ 1 (Fig. 3).

Fig. 3. Uniform Multi-set of 5 truth-degrees

The authors in [START_REF] Akdag | A symbolic approach of linguistic modifiers[END_REF] associate a numerical intensity level prop(τ i ) to each linguistic degree τ i . It is understood as a proportion associated with τ i : prop(τ i ) = i M -1 ; with M the size of the multi-set.

For the previous example the communication is little impaired means that the communication is considered by 25% ( 1 4 = prop(τ 1 )) of asked persons as impaired.

In multi-valued context, the authors in [START_REF] Akdag | A symbolic approach of linguistic modifiers[END_REF][START_REF] Akdag | Linguistic modifiers in a symbolic framework[END_REF][START_REF] Truck | Approches symbolique et floue des modificateurs linguistiques et leur lien avec l'agrégation: Application: le logiciel flous[END_REF] believe that any multi-valued symbol can be considered as a modification of another multi-valued symbol. A modifier allows to build a new term from an initial one, or to compare two values by finding out the modification that transforms from one to the other.

The membership relation corresponds to a symbolic degree τ i from a multi-set L M represented with [0-1] scale. Thus, the data modification is the transformation of the degree, the scale of the multi-set or the both. Indeed, some modifiers preserve the same multi-set but change the membership degree. Others erode or expand the multi-set. Akdag et al. [START_REF] Akdag | A symbolic approach of linguistic modifiers[END_REF][START_REF] Akdag | Linguistic modifiers in a symbolic framework[END_REF] proposed symbolic linguistic modifiers. They were generalized and formalized in [START_REF] Truck | Approches symbolique et floue des modificateurs linguistiques et leur lien avec l'agrégation: Application: le logiciel flous[END_REF][START_REF] Truck | Generalized modifiers as an interval scale: Towards adaptive colorimetric alterations[END_REF] as Generalized Symbolic Modifiers (GSM). Considering that the modifier is denoted by m, the result of applying m to τ i is τ i : m(τ i ) = τ i .

These modifiers are classified as:

-The weakening modifiers lead to the proportion decrease, i.e. prop(τ i ) < prop(τ i ). The four weakening modifiers defined in [START_REF] Truck | Approches symbolique et floue des modificateurs linguistiques et leur lien avec l'agrégation: Application: le logiciel flous[END_REF] are EW ρ , DW ρ , DW ρ and CW ρ . -The reinforcing modifiers involve the proportion increase, i.e. prop(τ i ) > prop(τ i ). The four reinforcing modifiers proposed in [START_REF] Truck | Approches symbolique et floue des modificateurs linguistiques et leur lien avec l'agrégation: Application: le logiciel flous[END_REF] are ER ρ , DR ρ , DR ρ and CR ρ . -The central modifiers preserve the proportion prop(τ i ) unchanged. Such modifiers act as a zoom on the base. The four central modifiers presented in [START_REF] Truck | Approches symbolique et floue des modificateurs linguistiques et leur lien avec l'agrégation: Application: le logiciel flous[END_REF] are EC ρ , EC ρ , DC ρ and DC ρ . The authors in [START_REF] Kacem | Generalized modus ponens based on linguistic modifiers in a symbolic multi-valued framework[END_REF] proposed another central modifier denoted by CC that conserves both the degree and the multi-set unchanged.

Kacem et al. [START_REF] Kacem | Extended symbolic approximate reasoning based on linguistic modifiers[END_REF] proposed a tool to aggregate modifiers. It takes into account the modifier mode, i.e. reinforcing or weakening. The authors introduce two operators:

-The M-norm A T for modifiers conjunction based on T-norm.

-The M-conorm A S based on a T-conorm for modifiers disjunction.

The authors consider three cases:

-Aggregation with CC, -Aggregation of two modifiers of the same type, -Aggregation of two dual modifiers.

In their proposition only the modifiers CC, CW and CR are used. Tables 1 and2 summarize the aggregation results for modifiers conjunction and disjunction respectively.

AS CC CRα CWα CC CC CRα CC CR β -CRγ, τγ = S(τα, τ β ) CR β CW β - - CWγ, τγ = ¬S(¬τα, ¬τ β )
We recall that computing with words is an important issue in many domains dealing with imperfect data. In fact, researchers may describe their knowledge with linguistic terms. Linguistic term sets are usually assumed to be uniformly distributed. The position of a term τ i is i M -1 , which is its proportion. The distance between each pair of successive terms is the same, i.e. 1 M -1 with M the number of terms in the set.

Nevertheless, in some cases, a particular sub-domain may be more informative than the remaining reference domain. Thus, the density of terms in this sub-domain is greater then in the others.

These linguistic term sets are not uniformly and symmetrically distributed. They are named unbalanced sets. Some authors [START_REF] Wang | A new version of 2-tuple fuzzy linguistic representation model for computing with words[END_REF][START_REF] Herrera-Viedma | A model of an information retrieval system with unbalanced fuzzy linguistic information[END_REF][START_REF] Herrera | A fuzzy linguistic methodology to deal with unbalanced linguistic term sets[END_REF][START_REF] Dong | Computing the numerical scale of the linguistic term set for the 2-tuple fuzzy linguistic representation model[END_REF][START_REF] Dong | Linguistic computational model based on 2-tuples and intervals[END_REF][START_REF] Dong | Consistency-driven automatic methodology to set interval numerical scales of 2-tuple linguistic term sets and its use in the linguistic GDM with preference relation[END_REF] proposed examples from real life using these sets. Truck [START_REF] Truck | Comparison and links between two 2-tuple linguistic models for decision making[END_REF] indicates that such sets "are commonly used to describe erratic phenomenons and irregular progression in the patterns such as weather events or behavior under the influence of alcohol". Xia and Xu [START_REF] Xia | An approach to multiplicative linguistic group decision making based on possibility degrees[END_REF] give the example of diminishing marginal utility in economics. They indicate that the investment of a same amount is different if the company has a bad performance or a good one. In this case, we need more precise bad information than good one. Hence, the gap between the grades expressing bad information are smaller than the one between the grades expressing good information.

In our work, we deal with unbalanced terms set in the context of multi-valued logic. This multi-set is defined as:

L M = {τ 0 , τ 1 , ..., τ (M -1) }; M ∈ N * \{1}; with M its cardinality.
Each term τ i represents a possible value for a linguistic variable. It is defined by its position π(τ i ) (Figure 4-a):

π : L M → [0, 1]
This function represents the distribution of the terms τ i using real values between 0 and 1.

These terms are not necessarily symmetrically distributed or equidistant from each others, i.e. ∆(τ i , τ i+1 ) are not equal for all i ∈ [0, M -1] (Figure 4-b): In this paper, we propose an inference model for approximate reasoning within unbalanced multi-sets. This model is implemented in a tool for autism diagnosis.

∆ : L M × L M → [0, 1] ∆(τ i , τ j ) = π(τ j ) -π(τ i ) (a)

Approximate Reasoning with Multi-Valued Logic

KBSs are able to perform reasoning without any exact measurement. Zadeh [START_REF] Zadeh | The concept of a linguistic variable and its application to approximate reasoning -I[END_REF][START_REF] Zadeh | The concept of a linguistic variable and its application to approximate reasoning -II[END_REF][START_REF] Zadeh | The concept of a linguistic variable and its application to approximate reasoning -III[END_REF][START_REF] Zadeh | A theory of approximate reasoning[END_REF] has introduced, in the fuzzy logic context, the concept of approximate reasoning. It allows inference with imprecise inputs to obtain meaningful outputs based on the Generalized Modus Ponens (GMP). GMP is an extension of the classic Modus Ponens for exact reasoning where new facts are obtained only when the observation and the rule premise are equal.

The schema of GMP in the multi-valued logic context is the following:

If (X is A) then (Y is B) (X is A ) (Y is B ) With:
-A is the rule premise predicate; -A is the observation predicate; -B is the rule conclusion predicate; -B is the inferred conclusion predicate;

-A is approximately equal to A; -B is approximately equal to B; -A, A , B and B are represented by multi-sets L M A , L M A , L M B and L M B respectively.
A KBS leads to deduce new knowledge from the facts and rules existing in its knowledge base. Thus, new facts are obtained from ones that are approximately similar to the rule premise. The similarity between the premise and the observation allows determining the conclusion belief degree.

The approximate reasoning is composed of two steps. The first one aims to find out which modification the premise undergoes to obtain the observation. While, the second concerns the deduction of the inferred conclusion. Its membership degree depends on the rule conclusion degree and the relation between the premise and the observation.

The GMP model previously presented includes a free rule as its premise and its conclusion are completely true. While in the general case truth degrees are associated with the premise and the conclusion. The observation is represented by the same multi-set as the premise but with a different membership degree. These rules are named strong rules. Khoukhi [START_REF] Khoukhi | Approche logico-symbolique du traitement des connaissances incertaines et imprécises dans les systèmes à base de connaissances[END_REF] proposed a GMP with these rules as follows:

If (X is ϑ α A) then (Y is ϑ β B) (X is ϑ γ A) (Y is ϑ λ B)
ϑ α , ϑ β , ϑ γ , and ϑ λ are linguistic qualifiers associated with the degrees τ α , τ β , τ γ and τ λ respectively (Eq.4).

The weakness of this type of reasoning is that it is concerned only with the degree of similarity between A and A , i.e. the modification undergone by the premise to obtain the observation. However, the type of the modification is not considerated.

In fuzzy logic context, Bouchon-Meunier [START_REF] Bouchon-Meunier | Fuzzy modus ponens as a calculus of logical modifiers: Towards zadeh's vision of implication calculus[END_REF] introduced a model of approximate reasoning based on linguistic modifiers. These later evaluate the similarity between the premise and the observation. The corresponding inference diagram adapted to the multi-valued logic context is the following:

If (X is τ α A) then (Y is τ β B) (X is m(τ α A)) (Y is m (τ β B))
With m and m are linguistic modifiers. Hence, deducing the inferred conclusion implies determining the modifier m . It is obtained from the modifier m and the causality between the observation and the premise.

In the context of multi-valued logic, Bel Haj Kacem et al. [START_REF] Kacem | Generalized modus ponens based on linguistic modifiers in a symbolic multi-valued framework[END_REF][START_REF] Kacem | On some properties of generalized symbolic modifiers and their role in symbolic approximate reasoning[END_REF][START_REF] Kacem | Extended symbolic approximate reasoning based on linguistic modifiers[END_REF] proposed an extended GMP model with GSM. The authors consider that the modifiers m and m are equal. They suggested determining the modifier m by the means of the det_mod algorithm [START_REF] Kacem | On some properties of generalized symbolic modifiers and their role in symbolic approximate reasoning[END_REF]. The modifier can be: CR, CW, ER, DW, CC [START_REF] Kacem | Generalized modus ponens based on linguistic modifiers in a symbolic multi-valued framework[END_REF][START_REF] Truck | Approches symbolique et floue des modificateurs linguistiques et leur lien avec l'agrégation: Application: le logiciel flous[END_REF] or the composition of two of them. Its identification depends on the granularities of the premise and the observation multi-sets (M A and M A ) and the truth degrees τ α and τ γ such that m(τ α A) = τ γ A .

The rule used in the previous GMP is called simple rule as the premise includes only one proposition. In the literature, the general case is the complex rule whose premise is a conjunction or a disjunction of propositions. The GMP with complex rules is as follows:

If

(X 1 is τ α1 A 1 ) op (X 2 is τ α2 A 2 ) ... op (X n is τ αn A n ) Then (Y is τ β B) (X 1 is τ γ1 A 1 ) op (X 2 is τ γ2 A 2 ) ... op (X n is τ γn A n ) (Y is τ λ B) With:
-op is one of the logical operators and, or or; -A 1 ,A 2 , ..., A n are the premise and observation predicates; n is the number of elements in the rule premise.

The inference engine aims to deduce the inferred conclusion τ λ B. It corresponds to m(τ β B). The modifier m is the aggregation of the modifiers m i [START_REF] Kacem | Extended symbolic approximate reasoning based on linguistic modifiers[END_REF]. Each modifier m i transforms an element of the premise (X i is τ αi A i ) to an element of the observation (X i is τ γi A i ).

Approximate Reasoning within Unbalanced Multi-sets

Our aim in this paper is to extend the model proposed by Bel Haj Kacem et al. [START_REF] Kacem | Generalized modus ponens based on linguistic modifiers in a symbolic multi-valued framework[END_REF][START_REF] Kacem | On some properties of generalized symbolic modifiers and their role in symbolic approximate reasoning[END_REF][START_REF] Kacem | Extended symbolic approximate reasoning based on linguistic modifiers[END_REF] to consider unbalanced multi-sets. To achieve this purpose, we use our algorithms for the representation of unbalanced multi-sets [START_REF] Chaoued | Linguistic modifiers with unbalanced term sets in multi-valued logic[END_REF].

The first algorithm, named UnbalancedToUniform, allows to express an unbalanced term τ pos , of a multi-set L M , within a new uniform multi-set L M . This algorithm gives us the ability to use existing tools initially designed for uniform terms as aggregation tools and modifiers.

Each term τ k of L M can be represented within a specific uniform multi-set L M K . Hence, the granularity M of the uniform multi-set, used to represent all unbalanced terms, equals the LCM of these sets granularity, i.e M K .

The inputs of this algorithm are the M terms of the unbalanced set L M , the normalized distances between each pair of successive terms ∆(τ i , τ i+1 ), and the pair (τ pos , α) with α a proportion error. α is a numerical value representing a symbolic translation associated with the term τ pos . If α is positive, it is a reinforcement of τ pos , otherwise it is a weakening. A pair (τ γpos , L M ) is given as the algorithm output. It corresponds to (τ pos , α) in L M .

The second algorithm, named UniformToUnbalanced, aims to represent a uniform term τ i from L M with a term τ pos from an unbalanced set L M . It tends to identify the term τ pos whose position is the closest to π(τ i ), with τ i ∈ L M . τ pos is specified by comparing the position of τ i with the sum of the distances

∆(τ i , τ i+1 ) for all τ i ∈ L M .
This algorithm inputs are the position of τ i , i.e. π(τ i ), the M terms of the unbalanced multi-set L M and the normalized distances ∆(τ i , τ i+1 ) between each pairs of successive terms. As output, we obtain the pair (τ pos , α), τ pos ∈ L M and α the proportion error. α exists when no term τ i corresponds to the exact position of τ i .

In this paper we treat complex strong rule using the following GMP model:

If (X 1 is τ α1 A 1 ) op (X 2 is τ α2 A 2 ) ... op (X n is τ αn A n ) then (Y is τ β B) (X 1 is τ γ1 A 1 ) op (X 2 is τ γ2 A 2 ) ... op (X n is τ γn A n ) (Y is τ λ B ) With:
-op is one of the logical operators and, or or. -A 1 , A 2 , ..., A n are predicates representing the premise -A 1 , A 2 , ..., A n are predicates representing the observation -A i and A i are semantically equivalent(i ∈ {1,..n}) -B and B are the predicates representing the conclusion. They are semantically equivalent -The unbalanced multi-sets

L M A i (i ∈ {1,..n}) (resp. L M B ) and L M A i (i ∈ {1,..n}) (resp. L M B )
do not have necessarily the same granularity and terms distribution τ γi = m i (τ αi ) (i ∈ {1,..n}); τ λ = m(τ β ) with m i and m are GSM We tend to deduce the inferred conclusion τ λ . Our inference process is as follows:

-Transform unbalanced terms τ αi , τ β and τ γi from the multi-sets

L M A i , L M B and L M A i respectively into the uniform pairs (τ α i , L M A i ), (τ β ,L M B ) and
(τ γ i ,L M A i
) with the UnbanlancedToUniform algorithm [START_REF] Chaoued | Linguistic modifiers with unbalanced term sets in multi-valued logic[END_REF].

-Express all uniform terms within L M such as:

M = 1 + LCM (M A1 -1, .., M An -1, M A 1 -1, .., M A n -1, M B -1, M B -1) • τ α i = DC M -1 M A i -1 (τ α i ) • τ γ i = DC M -1 M A i -1 (τ γ i ) • τ β = DC M -1 M B -1 (τ β ) -Deduce the modifiers m i : m i = det_mod((τ α i , L M ), (τ γ i , L M )
) -Aggregate the modifiers m i to determine the modifier m [START_REF] Kacem | Extended symbolic approximate reasoning based on linguistic modifiers[END_REF].

-Apply the modifier m to the rule uniform normalized conclusion τ β :

τ λ = m(τ β )
-Express the inferred conclusion by means of the uniform multi-set

L M B : τ λ = EC M -1 M B -1 (τ λ )
-Represent the uniform inferred conclusion τ λ on the unbalanced multi-set L M B by means of the UniformToUnbalanced algorithm [START_REF] Chaoued | Linguistic modifiers with unbalanced term sets in multi-valued logic[END_REF].

To illustrate our method, we consider the context of autism diagnosis. The GMP schema in the multi-valued logic context is as follows:

If (social interaction is mildly altered) and (stereotyped character is mildly present)

Then (child is mildly autistic) (social interaction is severely altered) and (stereotyped character is moderately present)

(child is ϑ λ autistic)
We aim to determine the inferred conclusion τ λ autistic corresponding to ϑ λ . The linguistic variable present indicates the degree of presence of an autism symptom. Some characters are altered for autistic child, they are represented within the linguistic variable impaired. The multi-set used to express these linguistic variables are: L 7 ={not at all, very mildly, mildly, mildly to moderately, moderately, moderately to severely, severely} (Fig 2).

The linguistic variable autistic indicates the severity of the child affection by autism. The terms are the same as proposed in CARS: L 4 = {not at all, mildly, moderately, severely } (Fig 5). In CARS [START_REF] Schopler | Toward objective classification of childhood autism: Childhood Autism Rating Scale (CARS)[END_REF], it is indicated that:

-A score below 30 does not allow a diagnosis of autism; -Results between 30 and 37 represent a diagnosis of autism from mild to moderate; -A score above 37 means a diagnosis of severe autism. We apply the UnbanlancedToUniform algorithm [START_REF] Chaoued | Linguistic modifiers with unbalanced term sets in multi-valued logic[END_REF] to the unbalanced terms sets L 7 and L 4 . The granularities of the uniform multi-sets are:

M = 1 + LCM (9 -1, 9 -1, 9 -1, 5 -1, 5 -1) = 9. M au = 1 + LCM (4 -1, 7 -1, 3 -1) = 7.
Hence, the GMP model with uniform multi-sets is as follows:

If (social interaction is (τ 2 , L 9 )) and (stereotyped character is (τ 2 , L 9 )) Then (child is (τ 2 , L 7 )) (social interaction is (τ 8 , L 9 )) and (stereotyped character is (τ 4 , L 9 )) (child is (τ λ , L 7 ))
The granularity of the normalized multi-set is:

M = 1 + LCM (9 -1, 9 -1, 7 -1) = 25
To represent the terms within the normalized multi-set. Normally, we apply the modifier DC M -1

M A i -1
to every term of the GMP:

-τ α 1 = DC 25-1 9-1 (τ 2 ) = DC 3 (τ 2 ) = τ 6 -τ α 2 = DC 25-1 9-1 (τ 2 ) = DC 3 (τ 2 ) = τ 6 -τ β = DC 25-1 7-1 (τ 2 ) = DC 4 (τ 2 ) = τ 8 -τ γ 1 = DC 25-1 9-1 (τ 8 ) = DC 3 (τ 8 ) = τ 24 -τ γ 2 = DC 25-1 9-1 (τ 4 ) = DC 3 (τ 4 ) = τ 12
Afterwards, we determine the modifiers m i that allows transforming each element of the premise:

m 1 = det_mod((τ 6 , L 25 ), (τ 24 , L 25 )) = CR 18 m 2 = det_mod((τ 6 , L 25 ), (τ 12 , L 25 )) = CR 6
These modifiers are aggregated as follows: A T (CR 18 , CR 6 ) = CR 0

We use this modifier with the rule conclusion:

τ λ = CR 0 (τ 8 ) = τ 2
This term is represented in the normalized uniform set L 25 . To transform it into the initial unbalanced multi-set L 4 , we should first express it into the uniform multi-set, in this case L 7 :

τ λ = EC 25-1 7-1 (τ 8 ) = EC 4 (τ 8 ) = τ 2
The last step consists in finding the closest matching term back in L 4 . We use for that the UniformToUnbalanced algorithm proposed in [START_REF] Chaoued | Linguistic modifiers with unbalanced term sets in multi-valued logic[END_REF]. The position of this term is the same of τ 1 in L 4 . Thus, the inferred conclusion is the child is mildly autistic.

Autism Diagnosis

We implement the tool ARUMS (Approximate Reasoning within Unbalanced Multi-Sets) to take into account unbalanced multi-sets. It is based on the KBS designed for uniform multi-sets, RAMOLI [START_REF] Kacem | Generalized modus ponens based on linguistic modifiers in a symbolic multi-valued framework[END_REF].

Our tool integrates the inference model considering unbalanced multi-sets presented in the previous section. We test our proposal to diagnostic autism.

Autism is a pervasive developmental disorder characterized by the alterations of three principal areas [START_REF] Association | Diagnostic and statistical manual of mental disorders : DSM-5[END_REF]: verbal and nonverbal communication; social interaction; and behavior, interests and activities. Specialists notice that child behaviors are restricted and stereotyped.

The diagnosis is made by child psychiatrists. They observe the behavior of a child and ask its parents according to some standard protocols as DSM5 [START_REF] Association | Diagnostic and statistical manual of mental disorders : DSM-5[END_REF]. An evaluation questionnaires exist based on CARS [START_REF] Schopler | Toward objective classification of childhood autism: Childhood Autism Rating Scale (CARS)[END_REF]. These questionnaires help psychiatrists to detect the autism symptoms.

In [START_REF] Kacem | A diagnosis aid system of autism in a multi-valued framework[END_REF], the authors proposed to deal with autism diagnosis within uniform multi-sets. They extracted autism symptoms from the decision-making algorithm of DSM5 [START_REF] Association | Diagnostic and statistical manual of mental disorders : DSM-5[END_REF]. With the help of child psychiatrists of Razi Hospital, they built a rules base by means of these symptoms.

The predicates used in our tool for autism diagnosis are:

- We present in this section an implementation of our inference engine to diagnosis autism within unbalanced set. The multi-set representing the linguistic variable autistic is unbalanced in the CARS definition but not for the description of child behaviors, i.e. impaied and present. In fact, a note between 1 and 4 with a step of 0.5 is given to each symptom. For this reason, we use the uniform multi-set L 7 (Fig. 1) for the linguistic variables impaired and present and the unbalanced multi-set L 4 (Fig. 5) for autistic variable. 36 child with different autism severity degrees are treated by psychiatrist of Razi Hospital. The given questionary responses are integrated as observed facts in our knowledge base. Each patient will be described with 15 facts, i.e. the 15th first predicates indicated bellow, corresponding to the autism symptoms. For our tests, the treated child are:

-11 not autistic; -6 mildly autistic; -13 moderately autistic; -6 severely autistic.

The inference engine applies rules to the knowledge base to deduce new knowledge. This process is iterated when a new fact in the knowledge base could trigger additional rules in the inference engine. This process allows to diagnosis if the child is autistic or not. It it is the case which is its severity degree.

The results of our tests are presented in the table 3. We notice that our diagnosis tool gives good result to distinguish between autistic and not autistic child. In our test, only one child mildly autistic is diagnosed as not autistic. However, we obtain wrong diagnosis of autism severity. In particular moderately autistic child are considered as severely autistic.

Conclusion

Our contribution concerns approximate reasoning within unbalanced multi-sets. We propose to refine the model proposed by Kacem et al. [START_REF] Kacem | Generalized modus ponens based on linguistic modifiers in a symbolic multi-valued framework[END_REF][START_REF] Kacem | On some properties of generalized symbolic modifiers and their role in symbolic approximate reasoning[END_REF][START_REF] Kacem | Extended symbolic approximate reasoning based on linguistic modifiers[END_REF]. The authors introduced an approximate reasoning based on GSM in the multi-valued context. We aim to extend their GMP model to consider unbalanced multi-sets. It allows inferring with complex strong rules.

We implement a KBS ARUMS that takes into account unbalanced multisets. To illustrate our method for approximate reasoning within unbalanced multi-sets, we treat the autism diagnosis based on DSM5 and CARS.

As future work, we propose to use our tool with more patient to check the obtained results. We also aim to ameliorate the identification of autism severity degree.

Moreover, researchers propose, in the literature, aggregation operators in multi-valued logic context. Therefore, we aim to elaborate a tool to aggregate unbalanced terms without transforming them to a uniform multi-set.

Fig. 1 .

 1 Fig. 1. Uniform Multi-set of 7 Linguistic Terms

Fig. 4 .

 4 Fig. 4. Terms Definitions

Fig. 5 .

 5 Fig. 5. Representation of Autistic Multi-set

Table 1 .

 1 Aggregation results by AT[START_REF] Kacem | Extended symbolic approximate reasoning based on linguistic modifiers[END_REF]

	AT CC	CRα	CWα
	CC CC	CC	CWα
	CR β -CRγ, τγ = T (τα, τ β )	CWα
	CW β -	-	CWγ, τγ = ¬T (¬τα, ¬τ β )

Table 2 .

 2 Aggregation results by AS[START_REF] Kacem | Extended symbolic approximate reasoning based on linguistic modifiers[END_REF]

  If Ability to develop peer relationships is mildly impaired then Social interactions are mildly impaired -If Willingness to share is mildly impaired then Social interactions are mildly impaired -If Reciprocity is mildly impaired then Social interactions are mildly impaired -If Social reciprocity is mildly impaired then Reciprocity is mildly impaired -If Emotional reciprocity is mildly impaired then Reciprocity is mildly impaired -If Development of spoken language is mildly impaired then Communication is mildly impaired -If Conversation is mildly impaired then Communication is mildly impaired -If Stereotyped or idiosyncratic language is mildly present then Communication is mildly impaired -If Imitative play is mildly impaired then Communication is mildly impaired -If Patterns of interest are mildly impaired then Stereotyped, restricted or repetitive characters are mildly present -If Stereotyped patterns of interest are present then Patterns of interest are mildly impaired -If Patterns of interest abnormal in intensity are present then Patterns of interest are mildly impaired -If Patterns of interest abnormal either in focus are present then Patterns of interest are mildly impaired -If Inflexible or ritual,nonfunctional adherence are present then Stereotyped, restricted or repetitive characters are mildly present -If Stereotyped and repetitive motor mannerisms are present then Stereotyped, restricted or repetitive characters are mildly present -If Persistent preoccupation with parts of objects are present then Stereotyped, restricted or repetitive characters are mildly present

	-Patterns of interest
	The rules base, used in the work of Bel Haj Kacem et al. [30] and the same
	as in our tool, includes:
	-If Social interactions are mildly impaired and Communication is mildly im-
	paired and Stereotyped, restricted or repetitive characters are mildly present
	then Patient is mildly autistic
	-If Nonverbal behaviors are mildly impaired then Social interactions are
	mildly impaired
	-
	Nonverbal behaviors
	-Ability to develop peer relationships
	-Willingness to share
	-Social reciprocity
	-Emotional reciprocity
	-Development of spoken language
	-Conversation
	-Stereotyped or idiosyncratic language
	-Imitative play
	-Stereotyped patterns of interest
	-Patterns of interest abnormal in intensity
	-Patterns of interest abnormal either in focus
	-Inflexible or ritual,nonfunctional adherence
	-Stereotyped and repetitive motor mannerisms
	-Persistent preoccupation with parts of objects
	-Social interactions
	-Communication
	-Stereotyped, restricted or repetitive characters
	-Patient
	-Reciprocity

Table 3 .

 3 Precision Recall F-measure for Autism Diagnosis tests

		Not autistic	mildly	Autistic moderately	severely
			autistic	autistic	autistic
	Accuracy	0.97		0.97
	Precision	0.92	0.67	1	0.35
	Recall	1	0.8	0.08	1
	F-measure	0.96	0.73	0.15	0.52


Ansem Ben Cheikh

Ben Cheikh

Yoann Blein
email: yoann.blein@univ-grenoble-alpes.fr

Salim Chehida

German Vega

Yves Ledru
email: yves.ledru@univ-grenoble-alpes.fr

Lydie Du Bousquet

Yves Ledruu

An Environment for the ParTraP Trace Property Language (Tool Demonstration)

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Many applications, such as software intensive medical devices, require high quality software but their criticality does not mandate the use of formal proofs.

Therefore, the formal methods community has promoted a lightweight approach to formal methods [START_REF] Jackson | Lightweight formal methods[END_REF]. The ParTraP language [START_REF] Blein | Extending specication patterns for verication of parametric traces[END_REF] goes into that direction.

Most computer systems can easily be augmented to produce traces of their activity. ParTraP allows to express properties of these traces which are evaluated by monitoring. ParTraP and its associated toolset are designed to support software engineers not trained in formal methods. The language supports a unique combination of features: it is parametric and provides temporal operators to increase expressiveness; it is declarative and uses descriptive keywords to favour user-friendliness. It reuses Dwyer's specication patterns [START_REF] Dwyer | Patterns in property specications for nite-state verication[END_REF] to express intuitive properties. Properties of parameters can be expressed in Python, a language familiar to our target users. Its toolset includes a syntax-directed editor.

It generates detailed explanations to help understand why a property evaluates to true or false on a given trace. We also prototyped a generator of examples and counter-examples, to help users understand the meaning of their properties. Section 2 gives an overview of the ParTraP language. Section 3 presents its associated toolset, and section 4 draws the conclusions and perspectives of this study. On the ParTraP web site 1 , you will nd a companion video demonstrat- ing the tool, links to reference documents describing the syntax and semantics of ParTraP, and instructions on how to download the ParTraP eclipse plugin. This work is funded by the ANR MODMED project (ANR-15-CE25-0010). 1 http://vasco.imag.fr/tools/partrap/ 00. [{"state": "Connect", "id": "EnterState", "time": 4042.7}, 01. {"state": "Connect", "id": "ExitState", "time": 4042.7}, ... 04. {"id": "CameraConnected", "time": 4747.34}, ... 13. {"ty": "P", "id": "TrackerDetected", "time": 4783.8}, 14. {"ty": "T", "id": "TrackerDetected", "time": 4798.0}, 15. {"v1": 41.44, "id": "Temp", "time": 4816.6}, 16. {"state": "Save", "id": "EnterState", "time": 4816.611}, ... 20. {"v1": 55.62, "id": "Temp", "time": 4847.6}] Together, we considered a medical system (TKA) that guides Total Knee Arthroplasty surgeries, i.e. the replacement of both tibial and femoral cartilages with implants. TKA produces traces of the sensors acquisitions and interactions with the surgeon. Over time, more than 10 000 traces of actual surgeries have been collected by Blue Ortho. Each trace counts about 500 signicant events. Fig. 1 gives a simplied excerpt of such a trace, in JSON format.

We identied 15 properties, listed in [START_REF] Ledru | Requirements for a trace property language for medical devices[END_REF], representative of such medical devices for assisted surgery. These properties specify the TKA traces. A careful analysis of the properties revealed that they express temporal relations between events, involve event parameters, and may apply to a restricted scope of the trace. A few properties also refer to physical time or involve 3D calculations.

Based on these properties, we designed ParTraP (Parametric Trace Property language) [START_REF] Blein | Extending specication patterns for verication of parametric traces[END_REF][START_REF] Blein | MODMED WP1/D1: Preliminary Denition of a Domain Specic Specication Language[END_REF]. The language is aimed at being used for oine trace checking.

In [START_REF] Ledru | Requirements for a trace property language for medical devices[END_REF], we discussed how the language can be used during development to express and check properties of traces produced by system tests, but also during exploitation in order to identify how the system is used or mis-used.

Structure of ParTraP Properties

A ParTraP temporal property is described by its scope in the trace, and a temporal pattern over events satisfying some predicate. For example, the following property expresses that once the camera is connected, the device temperature does not go below 45 o C .

VAlidTemp1 : after first CameraConnected, absence_of Temp t where t.v1 < 45.0

The rst line denes the scope of the property, here it is the sux of the trace starting after the rst event of type CameraConnected. The second line expresses a temporal pattern. Here, it is an absence pattern, stating that no event should be a Temp whose v1 parameter is less than 45.0. t is a local variable designating an event of type Temp. The where clause refers to this event and its parameter. The evaluation of this property on the trace of Fig. 1 yields false and returns the following message 2 :

[WARNING] False on trace unit-tests_Trace.json: -In the scope from 5 to 20 with the environment:{} found 1 event that should not occur: {trace_occ_index=15, time=4816.6, id=Temp, v1=41.44}

Actually, event CameraConnected appears on line 4 of Fig. ParTraP allows to express more complex properties. For example, property

ValidTrackers in Fig. 3 features nested scopes and universal quantication to express that all types of trackers have been detected before entering a state whose name includes `TrackersVisibCheck'.

ParTraP oers a variety of operators to express scopes and patterns. Scopes refer to events located before or after an event or between two events. The scope may consider the rst or the last occurrence of the event, but also be repeated for each occurrence of the event. Patterns may refer to the absence or occurrence of an event, but may also refer to pairs of events where one event enables or disables the occurrence of the other. Moreover, it is possible to express physical time constraints stating that a property holds within a time interval.

Finally, expressions occurring in the where clause may be written either in a basic language with support for numeric and string expressions, or in Python as in [START_REF] Barringer | Formal analysis of log les[END_REF]. Python allows to take advantage of software libraries to express complex or domain dependent properties, e.g. properties based on 3D calculations.

Basic ParTraP temporal properties can be combined using propositional logic connectors (and, or, implies, equiv) or quantiers (exists and forall).

ParTraP properties are dened on nite traces and evaluated after completion of the trace. A detailed description of the language is given in [START_REF] Blein | Extending specication patterns for verication of parametric traces[END_REF] or [START_REF] Blein | MODMED WP1/D1: Preliminary Denition of a Domain Specic Specication Language[END_REF].

Related Work

Several languages based on temporal logic have been proposed to express trace properties. In [START_REF] Blein | Extending specication patterns for verication of parametric traces[END_REF], we compared ParTraP to several temporal specication languages using multiple criteria. Table 1 summarizes this study and groups languages with similar features combination. Please refer to [START_REF] Blein | Extending specication patterns for verication of parametric traces[END_REF] for detailed explanations about this table. The Parametric column indicates whether a language 2 The message has been slightly simplied to t in the size of the paper. See the console in Fig. 3 for the actual message.

Style

Dwyer's patterns [START_REF] Dwyer | Patterns in property specications for nite-state verication[END_REF], Propel [START_REF] Smith | PROPEL: an approach supporting property elucidation[END_REF], LTL f [START_REF] Bauer | Runtime verication for LTL and TLTL[END_REF], CFLTL [START_REF] Regis | Specifying event-based systems with a counting uent temporal logic[END_REF] n/a n/a n/a decl.

RSL [START_REF] Reinkemeier | A pattern-based requirement specication language: Mapping automotive specic timing requirements[END_REF], Salt [START_REF] Bauer | SALT -structured assertion language for temporal logic[END_REF], TLTL f [START_REF] Bauer | Runtime verication for LTL and TLTL[END_REF] n/a n/a n/a decl.

Eagle [START_REF] Barringer | Rule-based runtime verication[END_REF] global decl. Stolz's Param. Prop. [START_REF] Stolz | Temporal assertions with parametrised propositions[END_REF] local decl. FO-LTL + [START_REF] Hallé | Runtime monitoring of message-based workows with data[END_REF] local decl. MFOTL/MONPOLY [START_REF] Basin | Monitoring metric rst-order temporal properties[END_REF][START_REF] Bversiooasin | MONPOLY: monitoring usage-control policies[END_REF] global decl. JavaMOP [START_REF] Jin | JavaMOP: Ecient parametric runtime monitoring framework[END_REF] global mixed QEA/MarQ [START_REF] Barringer | Quantied event automata: Towards expressive and ecient runtime monitors[END_REF][START_REF] Reger | MarQ: Monitoring at runtime with QEA[END_REF], Mun [START_REF] Decker | Runtime monitoring with union-nd structures[END_REF] global oper.

Ruler [START_REF] Barringer | Rule systems for run-time monitoring: from Eagle to RuleR[END_REF], Logre [START_REF] Havelund | Rule-based runtime verication revisited[END_REF] n/a oper.

LogScope [ supported at the language level, in which case specications involving timing constraints are easier to express. Finally, the specication style of a language can be declarative, operational, or mixed between the two and oers the choice to the user. As shown in Table 1, ParTraP supports a unique combination of these features, appropriate for our industrial context and motivated by the need for expressiveness. Other approaches, like [START_REF] Calafato | A controlled natural language for tax fraud detection[END_REF], have similar goals as ours, and use a controlled natural language. However, the resulting language is domain specic and can not be applied to our industrial application.

Associated Toolset

ParTraP-IDE is a toolset designed to edit and execute the ParTraP language directly on a set of trace les. Given a set of properties, the tool provides the set of traces violating them and an explanation of the error causes. events, with an average of 530 events. We evaluated the 6 ParTraP properties presented in chapter 5 of [START_REF] Blein | MODMED WP4/D1: Test assessment -preliminary study and tool prototype[END_REF]. These properties typically combine a scope with a temporal pattern. For each property, we constructed a variant whose where clause is expressed in Python (except property 3 which already has its assertion in Python). We led the experiments on a Windows 10 machine with an Intel(R)

Core(TM) i7-6600U CPU @ 2.60GHz, and 16 Go of RAM. Each experiment was performed 50 times and the average execution times are reported in table 2.

Column 2 reports the time in milliseconds to evaluate the property on 100 traces. The 100 traces are covered in a few seconds for each property. Property 4 takes longer because it features a complex scope involving pairs of events.

Column 3 reports on the same properties but with their where clause expressed in Python. Their evaluations are slower because of the extra cost of interactions between the java monitor and the Python interpreter.

Columns 4 and 5 report on the time needed to evaluate a property on a single trace. Actually, the initialisation of the monitor involves some overhead independently of the number of traces. Hence, we arbitrarily selected one of the traces whose length, 521 events, was close to the average length of our set. As expected, the average time to evaluate each property is signicantly longer than one hundredth of the time of columns 2 or 3.

In summary, these experiments show that ParTraP monitors perform well on traces provided by our partner. Most results are computed in less than one second, even if they involve Python assertions. These performances match the needs of our industrial partner. But further experiments should be led to evaluate how these performances scale up for much longer traces.

Example and Counter-Example Generator

To help software engineers understand or write ParTraP formulae, we are working on a prototype that generates examples and counter-examples and lets engineers check that they match their intuition of the meaning of the formula. We use the Z3 SMT solver 6 . Its input is a script composed of declarations (constants or functions) and assertions. Z3 computes whether the current assertions are satisable or not, and gives a valuation of the variables, for satisable formulae. 6 https://github.com/Z3Prover/z3

We have dened the semantics of ParTraP operators as Z3 functions. Par-TraP properties are translated as Z3 assertions which use these functions. Z3 then checks these assertions for satisability and, if possible, produces a trace satisfying the property. For example, the following Z3 assertion expresses property VAlidTemp1 (absence of temperature below 45 o C after the camera is connected).

(assert (afterFirst "CameraConnected" (absence_of_where "Temp" t (< (v1 t) 45.0))))

This property is satisable and the solver generates the following example, which trivially satises the property by avoiding Temp events.

[{"id": "CameraConnected", "time": 5263, "v1": 2.0}, {"id": "C", "time": 5264, "v1": 0.0}, {"id": "CameraConnected", "time": 5853, "v1": 4.0},]

Counter-examples are generated by evaluating satisability of the negation of the property. In our example, it produces the following trace where a Temp event with low temperature (0.0) is generated after the CameraConnected event.

[{"id": "C", "time": 8, "v1": 5.0}, {"id": "a", "time": 9, "v1": 7.0}, {"id": "CameraConnected", "time": 2436, "v1": 2.0}, {"id": "Temp", "time": 2437, "v1": 0.0},]

This part of the tool is currently at a prototyping stage. It will be included in the ParTraP distribution in the coming months. A major limitation of this tool is that it does not support Python expressions.

Conclusion

This paper has briey presented ParTraP and its associated toolset. Par-TraP is aimed at software engineers with poor knowledge of formal methods. Hence, we designed a keyword oriented language based on intuitive constructs such as Dwyer's patterns. We also integrated Python expressions in Par- The companion video of this paper illustrates the main constructs of the language and shows how the toolset helps to edit them, generates Java monitors, evaluates properties and explains the result of their evaluation. The tool was successfully experimented on 6 properties evaluated on 100 traces of surgical operations, provided by our partner. Work in progress applies the tool to home automation traces. Future work will apply the tool to other medical devices.

Fig. 1 .

 1 Fig. 1. A trace excerpt in JSON Format

  ParTraP-IDE relies on the Eclipse IDE and the XText framework 3 . Xtext provides a complete infrastructure including: parser, lexer, typechecker and a compiler generator. Fig. 2 shows the ParTraP-IDE architecture. Part A presents how XText generates the toolset. Part B presents the usage of the tool.

  TraP properties to give access to domain specic libraries. Moreover, the evaluation of ParTraP expressions produces detailed logs to explain why a property was veried or failed. An examples and counter-examples generator is currently prototyped to help engineers understand the meaning of their formulae.

  1. So the scope of the property ranges from lines 5 to 20 which corresponds to the last event of the trace. The error message tells that line 15 features a Temp event which violates the property because its v1 has value 41.44 which is actually below 45 o C.

Table 1 .

 1 Comparison of ParTraP with several temporal specication languages

	Language	Para-	Comp.	Quan-	Ref.	Wall-
		metric	values	tica-	past	clock
				tion	data	time

Table 2 .

 2 Performance evaluation (times in seconds)

	Property	100 traces		1 trace with 521 events
		with Python		with Python
	1	0.307	2.988	0.064	0.105
	2	0.326	2.378	0.064	0.088
	3	n/a	1.736	n/a	0.070
	4	17.605	51.639	0.255	0.728
	5	0.674	2.170	0.065	0.116
	6	1.517	4.969	0.074	0.118

The ParTraP Editor (1) helps the user to write syntactically correct properties. The congured editor provides syntax coloring according to concepts (name, keyword, python script,..) as shown in Fig. 3 under the editor window. Python expressions are delimited by dollars signs (`$') as featured by property VAlidTemp2 in Fig. 3. Moreover, some validation constraints are enforced by the editor in order to forbid undesired language expressions like double use of property names or recursivity when referencing properties. Saving the le automatically calls the ParTraP compiler (2) and produces the set of Java classes under package `src-gen' (see project explorer in Fig. 3). The example property presented in section 2.1 is typed in the editor of Fig. 3 and named VAlidTemp1. When applied on the trace of Fig. 1, the console reports that one event having the temperature value `41.44' violates this property.

Python Expressions Property VAlidTemp2 is an alternate expression of the same property whose where clause is expressed in Python and uses the Python math module. As it is important for our envisioned users to dene complex calculations in properties expressions, ParTraP properties support the integration of Python expressions using declared Python libraries. As a consequence, the designed IDE allows the import of Python modules and the execution of Python scripts. This is made possible by the use of JEP (Java Embedded Python) 4 which is a Python package generating a jar le `jep.jar' added in the Java build path to exchange data and scripts between the JVM and the Python Interpreter.

Performance Although we traded performance o against expressiveness of the language, we carried out several experiments to check that the generated monitors featured sucient performance in the context of our industrial partner, who typically collects and analyzes several dozens of traces every day. Therefore, we collected 100 traces from our partner 5 . The traces range from 304 to 1163 4 https://github.com/ninia/jep 5 These traces are not publicly available for condentiality reasons.


Michel Jébrak

Noémie Fayol

The Late-Archean Magmatic Gold Event: Toward a Non-Uniformitarian Approach

Keywords: Gold deposits, Geodynamic, Archean

Gold is exceptionally abundant at the end of Archean. Three main steps are recognized: (1) a late TTG event at the core of calderas, producing small porphyries and gold-rich VMS deposits; (2) a sanukitoïd event, producing alkaline and felsic subvolcanic plutons with large gold hydrothermal systems, and (3) a hydrothermal/structural event. A large part of the gold mineralization may be directly related to the unique geodynamic situation, including the first subduction with two generations of gold-enriched intrusions.

Introduction

The end of Archean was a key period for crustal construction; more than 50 % of the volume of the continental crust were formed. It is also an exceptional period for gold concentrations as more than 50 % of the world's gold resources are hosted by sedimentary and volcanic rocks of late-Archean age [START_REF] Goldfarb | Distribution, Character, and Genesis of Gold Deposits in Metamorphic Terranes[END_REF]. Gold has been mainly exploited from clastic sedimentary units, shear zones (orogenic gold deposits), gold-rich polymetallic volcanogenic massive sulfide deposits and disseminated zones around pluton.

Until now, most of the interpretations related to this major metallogenic period have favoured an uniformitarian approach [START_REF] Kerrich | The geodynamics of the world-class gold deposits: characteristics, space-time distribution, and origins[END_REF]. Most of the Archean gold deposits have strong similarities with Phanerozoic deposits of the same class, and their interpretation does not imply major differences in the volcanic or structural environment. The only exception would have been the Witswaterand gold-rich basin where almost no recent equivalents are known [START_REF] Phillips | Hydrothermal alteration in the Witwatersrand goldfields[END_REF].

However, most of these interpretations rely on the assumption that the geodynamic processes during the Archean were similar to the present time plate tectonics regime [START_REF] Kerrich | The geodynamics of the world-class gold deposits: characteristics, space-time distribution, and origins[END_REF]. Most of the interpretations do recognize however that Earth was different: the atmosphere was lacking of oxygen, the ocean composition was reduced, life was not a major component of the geochemical cycle. Moreover, several key elements of the present time plate tectonic regime were lacking such as true ophiolite and high grade metamorphic belts. Mantle dynamics were also different, due to the higher geothermal gradient and its primitive composition. Ongoing debates argue about the first occurrences of recognizable plate tectonics style processes, from 3.2 to 2.6 [START_REF] Bédard | How many arcs can dance on the head of a plume? A 'Comment' on: a critical assessment of Neoarchean 'plume only' geodynamics: evidence from the Superior Province, by Derek Wyman[END_REF][START_REF] Wyman | A critical assessment of Neoarchean "plume only" geodynamics: evidence from the Superior Province[END_REF].

It is therefore far from certified that the metallogeny of gold at the end of Archean would have been similar to the gold mineralisation episodes in the Proterozoic and Phanerozoic. The gold cycle could have been very different because of the high geothermal gradient that was changing the mechanical properties of the crust, because of the different composition of the lithospheric mantle and therefore because of a different magmatism.

In this paper, we will review some specific aspects of gold deposits in the Abitibi greenstone belt, the world's largest Archean greenstone belt, emphasizing the magmatic connections of a large number of deposits. This concept has gain in importance as the largest producing gold deposits presently mined in Canada and Australia since 2012 are respectively Malartic (Abitibi belt) and Boddington (Yilgarn block), both associated with late Archean intrusive rocks.

Late Archean gold deposits in the Abitibi greenstone belt

Gold occurs in a variety of deposit styles in the Abitibi Greenstone Belt (AGB). Archean gold (coppermolybdenum) porphyry deposits have long been described in the Superior Craton (Ayers and Cerny 1982). Volcanogenic Massive Sulphide (VMS) deposits show major enrichment in gold. Gold shear zones are archetypal of the AGB, and several gold deposits are associated with alkaline intrusions. Minor deposits may show also an association with BIF (Chimo), black shales (Telbel) or within an epithermal environment (Selbaie B zone, Géant Dormant). Geological environment and detailed geochronological studies allow recognition of three main periods of gold deposition, associated with the volcanic arc construction, and with two main deformation events.

Volcanic arc construction

The magmatic arcs of the Abitibi greenstone belt have been constructed between 2735 and 2696 Ma [START_REF] Mueller | Subaqueous calderas in the Archean Abitibi greenstone belt: An overview and new ideas[END_REF]. Two main lithologies dominate, thick bimodal volcanic series and subvolcanic tonalite-trondjemitegranite (TTG) suites forming subconcordant bodies of limited extension [START_REF] Galley | Composite synvolcanic intrusions associated with Precambrian VMS-related hydrothermal systems[END_REF]. The felsic volcanism that hosts VMS mineralization are chemically and temporally related to the TTG [START_REF] Huston | Gold in volcanic-hosted massive sulfide deposits: distribution, genesis and exploration[END_REF]. Gold is associated with both environments, and TTG have been considered as the possible source of metal for VMS [START_REF] Poulsen | Copper and gold mineralization in an Archean trondhjemitic intrusion, Sturgeon Lake, Ontario[END_REF].

In the Northern Zone of the AGB, the Chibougamau district displays many styles of mineralization [START_REF] Pilote | Géologie et Métallogénie du district minier de Chapais -Chibougamau. Nouvelle vision du potentiel de découverte[END_REF] from zoned porphyries to more distal vein systems. The main Cu-Au vein-hosted ores present an assemblage of pyrrhotite, chalcopyrite, pyrite, magnetite and gold that reflect an oxidising environment of deposition. The porphyry-style Mop2 deposit shows disseminated facies with a more reduced assemblage and an alteration zoning similar to Phanerozoic equivalents [START_REF] Lépine | Le porphyre tonalitique à or de Mop2, district de Chibougamau, Abitibi, Québec[END_REF]. Several other Au or Cu-Au prospects with porphyritic facies have been encountered in the Northern Zone of the AGB, such as in the Selbaie district where a pyritic massive sulphide deposit is associated with a rhyolitic volcanic sequence characterized by porphyritic dykes and epithermal mineralisation [START_REF] Faure | Structural evolution of Les Mines Selbaie, northern Abitibi belt, Québec, Canada[END_REF].

In the Southern Zone of the AGB, polyphased calderas of the Blake River district also display several porphyry-style occurrences, [START_REF] Pearson | An Archean megacaldera complex: The Blake river Group, Abitibi greenstone belt[END_REF]. At Don Rouyn (QC), chalcopyrite-chalcocite concentrations are surrounded by a hematitic halo. The St-Jude breccia is an example of breccia pipe dated at 2696.6 ± 0.9 Ma [START_REF] Mcnicoll | New U-Pb geochronology from the Blake River Group, Abitibi Subprovince, Québec: implications for geological interprettations and base metal exploration[END_REF]) indicating that they were emplaced at the very end of the Blake River Group volcanism. In the Doyon-Bousquet-LaRonde camp (QC), the intrusion-hosted Au ± Cu (5 Moz) Doyon deposit is associated with the Mooshla Intrusive Complex [START_REF] Dubé | Gold-rich Volcanogenic Massive Sulphide Deposits in Goodfellow[END_REF]). The recent large Coté Lake (ONT) discovery is associated with magmatichydrothermal breccia and disseminated pyrite and chalcopyrite. Re-Os dating indicate that it is coeval with an early tonalitic event at 2737 Ma [START_REF] Kontak | Magmatic-Hydrothermal Gold Systems in the Archean: Examples of Porphyry Gold System and Syenite-Associated Au-Cu Deposits; Presented at Metallogeny of the Brazilian, Canadian and African Shields Seminar during the SIMEXMIN Symposium[END_REF] These synvolcanic gold porphyry deposits in the AGB share several characteristics: an association with surficial magmatic chambers in tonalitic volcanic complexes; a relation with post-cauldron resurgence; the presence of Cu and Mo and an oxidizing depositional environment characterized by the presence of magnetite and hematite; a poorly developed potassic alteration; and a shallow level of emplacement in an underwater environment, without large boiling and fluid concentration;

Most of these synvolcanic intrusions are associated with bimodal volcanic sequences that also host gold-rich VMS deposits. The most important ones are located in the Southern Abitibi Zone such as the Horne and Laronde districts, in the Black River Group. In this latter district, gold is present in Au-rich sulphide stockworks and disseminations (e.g., Bousquet 1 deposit), Au-rich VMS systems (LaRonde Penna, Bousquet 2-Dumagami) and Au-Cu quartz-sulphide veins that are possibly genetically related to the Mooschla intrusion (2698-2697 Ma), where several stages of volatiles exsolution have been observed (e.g., Doyon deposit; Galley and Lafrance 2014).

Although largely debated, field evidence (Mercier-Langevin et al. 2014) and isotopic dating indicate that tonalitic porphyries and VMS deposits are subcontemporaneous and reflect two expressions of magmatic-hydrothermal activity related to the late uplift of sub-marine calderas.

First Deformation Event

After the formation of large calderas, uplift continued and the first detrital sedimentation occurred in sedimentary basins along the major fault zones. The interpretation of this uplift remains uncertain as evidence of normal, strike-slip and reverse movements have been observed along the major breaks.

Syntectonic gold-rich porphyries are related to this tectonic phase (2686-2661 Ma), synchronous with the formation of detrital basins along major breaks (Porcupine and Temiskaming sediments). Most of the deposits are located near or within sediments and are associated with small alkaline intrusions of monzonitic to syenitic composition. They may form multiple plugs and breccia pipes, more or less elongated [START_REF] Legault | Discrimination des syénites associées aux gisements aurifères de la Sous-Province de l'Abitibi[END_REF]. They present a large spectrum of composition, from oxidized mafic (pyroxene-amphibole) to felsic intrusions. Fe-enrichment occurs both at the magmatic and hydrothermal stages. Trace element signatures allow clear distinction between a sanukitoid clan [START_REF] Laurent | The diversity and evolution of late-Archean grabitoids: Evidence for the onset of "modern-style" plate tectonics between 3.0 and 2.5 Ga[END_REF], represented by mafic intrusion with a bull-eye magnetic signature [START_REF] Hattori | Magnetic felsic intrusions associated with Canadian Archean gold deposits[END_REF], from a more felsic/alkalic clan, with a doughnut shape magnetic anomaly. These two styles derive from the same oxidized magmatism, with different degree of evolution (Fayol et al. submitted).

Numerous occurrences have been discovered and mined along the major breaks of the AGB [START_REF] Robert | Syenite-associated disseminated gold deposits in the Abitibi greenstone belt, Canada[END_REF]. Along the Cadillac Larder-Lake Fault zone, syenite-related gold mineralization has been recognized in the Young Davidson, Kirkland Lake, Upper Canada, Upper Beaver and Kerr Addison districts in Ontario, and in Francoeur-Wasamac, Baie Renault, Canadian Malartic and Camflo districts in Québec. These deposits total more than 1200 t of gold. Along the Destor Porcupine Fault Zone, syenite-related gold mineralization has been recognized in West Timmins, Black Fox, Canadian Arrow (Ontario) and Beattie (Quebec). More than 200 t of gold are related to the alkaline magmatism along these fault zone. Gold with alkaline intrusion have also been recognized in other large deformation zones such as the Casa-Berardi E-W corridor (Douay) and the Wedding-Lamarck NE-SW deformation zone with the Lac Bachelor and Lac Short deposits.

The emplacement of these alkaline rocks spans from the end of the tonalitic magmatism to the onset of crustal melting, represented by high-Al leucogranite (2669-2639 Ma). The Timmins district is slightly older than most of the other districts (2691-2687 Ma, Bleeker 2012) and could belong to the end of the TTG suites.

Au-rich alkaline intrusion-associated mines and prospects share numerous characteristics [START_REF] Robert | Syenite-associated disseminated gold deposits in the Abitibi greenstone belt, Canada[END_REF]: they consist of disseminated sulphides and stockworks of quartz, with zones of carbonate, albite, K-feldspar and sericite alteration; their metal association is characterized by Cu, As and Te with variable amounts of Pb, Zn, Ag, Hg, Mo, Bi, W and Sb. Anhydrite, hematite, fluorite, tourmaline and scheelite have also been recognized in several of these deposits. Both anhydrite and hematite are indicative of relatively high fO2, which is corroborated by the negative isotopic value of the sulphur in the pyrite.

The precise relationships between magmatic systems and gold is still strongly debated. On one hand, the strong spatial and temporal association, field studies and some isotopic results demonstrate a contemporaneity of magmatic and hydrothermal system such as in the Lac Bachelor deposit [START_REF] Fayol | Gold Associated with Neoarchean Alkaline Intrusion[END_REF] or in the Beattie mine [START_REF] Bigot | Mineralizations at the Syenite-hosted Beattie gold deposit at Duparquet, Archean Abitibi Belt, Quebec, Canada[END_REF]. On the other hand, the late emplacement of the richest gold veins and isotopic dating in Timmins and Malartic suggest that gold mineralization could have been up to 30 My younger than the intrusion [START_REF] Helt | Constraints on the genesis of the Archean oxidized intrusion-related Canadian Malartic Gold deposit, Québec, Canada[END_REF]). More works is therefore require to better understand the role of early and late events in the formation of these gold deposits.

Second Deformation Event

The second deformation event represent both a vertical (reverse and normal?) increment of deformation and a transcurrent dextral shear along the Cadillac break. This brittle ductile deformation is contemporaneous with the emplacement of numerous orogenic gold deposit such as Sigma-Lamaque, Lapa, McWatter in Quebec and Kirkland Lake, Larder Lake, Detour Lake, Dome, Pamour in Ontario. These deposits have been carefully studied since 30 years and have been attributed to the orogenic gold model [START_REF] Colvine | An empirical model for the formation of Archean Gold deposits: Products of Final Cratonization of the Superior Province, Canada[END_REF][START_REF] Groves | Orogenic gold deposits: A proposed classification in the context of their crustal distribution and relationship to other gold deposit types[END_REF].

A majority of workers have relate their ore genesis to processes of crustal devolatilization as indicated by the CO2-H2O composition of the hydrothermal fluids, the progressive decrease in concentration of elements enriched in the gold deposits with increasing metamorphic grade of the country rocks, and the common association of ores with medium-grade metamorphic environments. Yet, significant debate continues in order to understand the origin of these deposits [START_REF] Goldfarb | Distribution, Character, and Genesis of Gold Deposits in Metamorphic Terranes[END_REF].

Detailed studies both in the Abitibi and the Yilgarn block of Western Australia have shown that this class is rather heterogeneous, with multiple ages of emplacement [START_REF] Duuring | Granitoïd-associated orogenic, intrusion related, and porphyry style metal deposits in the Archean Yilgarn Craton, Western Australia[END_REF][START_REF] Rafini | Typologie des minéralisations aurifères le long de la Faille Cadillac -Phase 2 (Segment Cadillac -Kirkland Lake)[END_REF]. Some are associated with plutonic stocks whereas some others do not show any relation with magmatism. In the sedimentary environment, a continuity has been demonstrated from syn-sedimentary gold concentration to epigenetic deposits [START_REF] Large | Carbonaceous sedimentary source-rock model for Carlin-Type and orogenic gold deposits[END_REF]). The study of the Beattie deposit shows that the same continuity could be observed for magmatic gold concentration in the Abitibi [START_REF] Bigot | Mineralizations at the Syenite-hosted Beattie gold deposit at Duparquet, Archean Abitibi Belt, Quebec, Canada[END_REF]. These results suggest that the source of gold could be of different origins, at different crustal level, from deep metamorphic mobilization to local reconcentration. Late H2O-CO2 fluids have been observed in most of the orogenic belts, from Archean to present. For instance, in the Apennine belt, H 2 O-CO 2 fluids are strongly associated with the seismicity; their chemistry results from local equilibration with the host rock although their gas content come from the underlying mantle [START_REF] Agostinetti | Fluid migration in continental subduction; The Northern Apennines case study[END_REF]. Orogenic gold deposits may mainly indicate the presence of previous gold enrichment remobilized during subsequent deformation, metamorphism and hydrothermal fluid circulation. The distance of migration is therefore a key element to be addressed.

Geodynamics of gold in the Late Archean

Gold deposition occurs early during the AGB arcs construction processes, associated both with plutonic and volcanic rocks. The association of gold with copper and molybdenum and the similarities of the late Archean systems with modern ones suggest that gold enrichment is related to the magmatic evolution itself. The geochemical signatures of the TTG complexes (Galley and Lafrance 2014) are typical of I-type calc-alkaline magmas belonging to the magnetite series, indicating that they were formed by melting of igneous rocks. Their average content in traces elements indicates that these subvolcanic TTG belong to the low-pressure sodic family of TTG (<12 kbar), near the garnet-in limit, coexisting with a plagioclase+pyroxene(s) ± amphibole in which a moderate amount of magnetite crystallization has taken place [START_REF] Moyen | The composite Archaean grey gneisses: Petrological significance, and evidence for a non-unique tectonic setting for Archaean crustal growth[END_REF][START_REF] Galley | Setting and evolution of the Archean synvolcanic Mooshla Intrusive Complex, Doyon-Bousquet-LaRonde mining camp, Abitibi greenstone belt: emplacement history, petrogenesis and implication for Au metallogenesis[END_REF]. They appear very similar to plagiogranites /dacites found in a mid-ocean ridge or oceanic plateau setting. Such magmatism could be produced by partial melting of hydrous mafic rocks at the base of a thick crust or mafic plateaus, or by a flat subduction. The second step of concentration is associated with the sanukitoïd magmatism. These K-Mg enriched rocks probably result from mantle-crust interactions [START_REF] Martin | The sanukitoïd series: magmatism at the Archaean-Proterozoic transition[END_REF]. Sanukitoïds formation requires a mechanism to carry crustal rocks (TTG, sediments) down into the mantle (Van Huten and Moyen 2012), Isotopic data indicate that the subducting slab did not contain old cratonic material and that the mantle metasomatism took place shortly before the alkaline magmatism, probably during the tonalitic event [START_REF] Hattori | Melt and source mantle compositions in the Late Archaean: A study of strontium and neodynium isotope and trace elements in clinopyroxenes from shoshonitic alkaline rocks[END_REF]. The presence of ethane in the fluid could indicate a source from organic-rich sediments [START_REF] Gaboury | Does gold in orogenic deposits come from pyrite in deeply buried carbon-rich sediments?: Insight from volatiles in fluid inclusions[END_REF]. High temperature gradients would imply a flat subduction setting. Re-melting of previously gold-enriched magma is known to increase the gold content of the magma [START_REF] Jégo | Gold Solubility in Arc Magmas: Experimental Determination of the Effect of Sulfur at 1000C and 0.4 GPa[END_REF]. Therefore, the melting of gold-enriched previous TTG (and sediments?), appears to be the most probable mechanism for gold enrichment in late-Archean sanukitoïds.

The third step of concentration is illustrated by the numerous "orogenic" type gold deposits. Several systems have been recognized along the Cadillac break [START_REF] Rafini | Typologie des minéralisations aurifères le long de la Faille Cadillac -Phase 2 (Segment Cadillac -Kirkland Lake)[END_REF]) and show different styles of deformation and mineral association type. They are the reflections of the many syntectonic hydrothermal systems that record the circulation of H2O-CO2 fluid at the end of the orogeny.

Conclusions

The construction of the Abitibi gold belt results from the formation of different styles of deposits during three main events:

(1) an early juvenile tonalitic magmatic/ hydrothermal event, sharing numerous similarities with presenttime tonalitic gold porphyries; (2) a sanukitoïd magmatic-hydrothermal event, that could be the result of the first subduction event on Earth;

(3) a late structurally-controlled hydrothermal system.

Gold deposits of different type and age are located in the same area, defining a "provinciality" [START_REF] Huston | Gold in volcanic-hosted massive sulfide deposits: distribution, genesis and exploration[END_REF]. It could have come either by consanguinity or by inheritance from the same TTG.

Acknowledgements

This work was funded by a personal NSERC discovery grant to MJ and an FRQNT-MRN strategic program.