Khalid Addi
email: khalid.addi@univ-reunion.fr

Samir Adly
email: samir.adly@unilim.fr

Daniel Goeleven
email: daniel.goeleven@univ-reunion.fr

Hassan Saoud
email: hassan.saoud@etu.unilim.fr

A sensitivity analysis of a class of semi-coercive variational inequalities using recession tools

Keywords: Finite variational inequalities, Recession analysis, Convex analysis, Positive semidefinite matrices, Clipping circuit, Ideal diode model, Practical diode model, coulomb friction

Using the recession analysis we study necessary and sufficient conditions for the existence and the stability of a finite semi-coercive variational inequality with respect to data perturbation. Some applications of the abstract results in mechanics and in electronic circuits involving devices like ideal diode and practical diode are discussed.

Introduction and position of the problem

The theory of variational inequalities, with its wide range of applications in engineering, economics, finance, industy and mechanics, has become a well-established and fruitful area of research. After the fundamental work of Lions and Stampacchia [START_REF] Lions | Variational inequalities[END_REF], this theory have been studied intensively. With the contributions of Brézis [START_REF] Brezis | Equations et inéquations non-linéaires dans les espaces vectoriels en dualité[END_REF][START_REF] Brezis | Problèmes unilatéraux[END_REF], Duvaut Lions [10], Browder [START_REF] Browder | Nonlinear maximal monotone mappings in Banach spaces[END_REF], (H 1 ) M ∈ R n×n is a symmetric and positive semidefinite matrix; (H 2 )

: R n → R ∪ {+∞} is a proper, convex, lower semicontinuous and bounded from below; (H 3 ) K ⊂ R n is a closed and nonempty convex set (H 4 ) 0 ∈ Dom ∩ K .

Several theoretical existence results for VI(M, q, , K ) in general reflexive Banach spaces and governed by a general operator M (not necessarily linear) are well known when a coerciveness condition hold for the operator M. We can cite for instance the contributions of Lions [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non-linéaires[END_REF], Brézis [START_REF] Brezis | Equations et inéquations non-linéaires dans les espaces vectoriels en dualité[END_REF][START_REF] Brezis | Problèmes unilatéraux[END_REF], Browder [START_REF] Browder | Nonlinear maximal monotone mappings in Banach spaces[END_REF] etc …. However, the variational formulation of many engineering problems leads generally to non-coercive variational inequalities (e.g., problems in mechanics which admits nontrivial virtual rigid body displacement). These problems are formulated by semi-coercive variational inequalities and was studied first by Fichera [START_REF] Fichera | Boundary value problems in elasticity with unilateral constraints[END_REF] and Lions and Stampacchia [START_REF] Lions | Variational inequalities[END_REF], Duvaut and Lions [START_REF] Duvaut | Les inéquations en mécanique et en physique[END_REF] (for problems with frictional type functionals). Recently many mathematicians and engineers has focused their attention on non-coercive unilateral problems, using several different approaches such as the critical point theory, the Leray-Schauder degree theory, the recession analysis or the regularization method by approximating non-coercive problems by coercive ones (see e.g., [START_REF] Adly | Recession mappings and noncoercive variational inequalities[END_REF][START_REF] Ang | Noncoercive variational inequalities: some applications[END_REF][START_REF] Baiocchi | Some existence results on noncoercive variational inequalities[END_REF][START_REF] Baiocchi | General existence theorems for unilateral problems in continum mechanics[END_REF][START_REF] Tomarelli | Noncoercive Variational Inequalities for Pseudomonotone Operators[END_REF][START_REF] Vy | Frictional contact of an elastic body with a rigid support[END_REF] and references cited therein). The main concern of these contributions is the obtainment of necessary or sufficient conditions for the solvability of such problems in a general setting by imposing some compacity conditions and some compatibility conditions on the right hand term q. More recently, Adly et al. [START_REF] Adly | Stability of non-coercive variational inequalities[END_REF][START_REF] Adly | Well-positioned closed convex sets and well-positioned closed convex functions[END_REF] has considered the situations in which the existence of the solution is stable with respect to small uniform perturbations of the data of the problem. This result should be of great interest for problems in finance and engineering where the data are known only with a certain precision and it is desired that further refinement of the data should not cause the emptiness of the set of solutions.

Our aim is thus to characterize the sensitivity of VI(M, q, , K ) with respect to the perturbations of the data M, q, and K . In this paper, we only discuss the case of finite variational inequalities. Some applications of our main results in electronics and mechanics are given in Sect. 4.

Preliminaries and notations

Let us first recall some background results from convex analysis which will be used later.

Let K be a closed convex subset of R n , the recession cone of K is the closed convex cone

K ∞ := t>0 K -x 0 t ,
with x 0 arbitrarily chosen in K . We denote by 0 (R n ) the set of all proper, convex and lower semicontinuous extended real valued functions : R n → R ∪ {+∞}.

Let ∈ 0 (R n ) be given. The recession function ∞ of is defined by:

∞ (x) := lim λ→+∞ (x 0 + λx) -(x 0 ) λ , (1)
where x 0 ∈ Dom is an arbitrary element. We set

ker ∞ = {x ∈ R n : ∞ (x) = 0}, which is clearly a closed convex cone in R n . The Fenchel conjugate * : R n → R ∪ {+∞} of is defined by: * (x * ) = sup x∈R n x * , x -(x) .
We recall also that the convex subdifferential of at a point x ∈ Dom is defined by:

∂ (x) = {w ∈ R n : w, y -x ≤ (y) -(x), ∀y ∈ R n }.
The effective domain of the multivalued mapping ∂ is defined by

D(∂ ) = {x ∈ R n : ∂ (x) = ∅}.
We note that

D(∂ ) ⊂ Dom .
The range of the multivalued mapping ∂ is defined by

R(∂ ) = x∈R n ∂ (x).
We have the following Fenchel correspondence:

w ∈ ∂ (x) ⇐⇒ x ∈ ∂ * (w). Hence, R(∂ ) = D(∂ * ) ⊂ Dom * . ( 2
)
The indicator function to a convex set K is given by:

I K (x) = 0 if x ∈ K +∞ if x ∈ K .
If K is a closed cone, its polar is defined by

K • = {x * ∈ R n : x * , x ≤ 0, ∀x ∈ K }.
The support function to K is defined by:

σ K (w) = (I K ) * (w) = sup x∈K w, x .
We recall that the barrier cone of K is defined by:

B(K ) = {w ∈ R n : sup x∈K w, x < +∞} = Dom σ K . ( 3
)
It is well-known that if K is a non-empty closed and convex subset, then

B(K ) • = K ∞ . ( 4
)
Therefore,

B(K ) = (K ∞ ) • . ( 5
)
We recall that for 1 , 2 ∈ 0 (R n ), the infimal convolution (or the epigraphical sum) is defined by:

( 1 2 )(x) = inf y+z=x { 1 (y) + 2 (z)} . ( 6
)
We say that the infimal convolution is exact provided that the infimum appearing in ( 6) is achieved.

We note that Dom

( 1 2 ) = Dom ( 1 ) + Dom ( 2 ). (7)
We recall that if 1 (or 2 ) is continuous on R n , then

( 1 + 2 ) * = * 1 * 2 (8)
and the infimal convolution * 1 * 2 is exact. Let us finally recall the following proposition Proposition 1 Let ∈ 0 (R n ) and p ∈ R n be given. We have:

(i) p ∈ Dom * ⇐⇒ ∞ (w) ≥ p, w , ∀w ∈ R n ; (ii) p ∈ Int (Dom * ) ⇐⇒ ∞ (w) > p, w , ∀w ∈ R n \ {0}.
Proof For a proof see Corollary 13.3.4 in [START_REF] Rockafellar | Convex Analysis[END_REF].

Characterization results

The solutions set of VI(M, q, , K ) will be denoted by

Sol(M, q, , K ) := {u ∈ K : Mu + q, v -u + (v) -(u) ≥ 0, ∀v ∈ K }.
The following resolvent set will also play an important role

R(M, , K ) = {-q ∈ R n : Sol(M, q, , K ) = ∅}.
Let us introduce the following function : R n → R ∪ {+∞} defined by

(u) = 1 2 Qu 2 + (u) + I K (u), (9)
where Q = I -P ker(M) and P ker(M) denotes the orthogonal projector from R n to ker(M).

Remark 1 We first remark that VI(M, q, , K ) is equivalent to the following variational inclusion: find u ∈ K such that

0 ∈ Mu + q + ∂( + I K )(u),
where I K denotes the indicator function of K . Hence,

-q ∈ Mu + ∂( + I K )(u) ⊂ u∈R n (Mu + ∂( + I K )(u)) ⊂ R(M) + R(∂( + I K )).
Using (2), we get

-q ∈ R(M) + Dom ( + I K ) * . ( 10
)
We have the following lemma.

Lemma 1 Suppose that the assumptions (H) hold. We have

Dom * = R(M) + Dom ( + I K ) * .
Proof Let us note first that the function defined in (9) can be rewritten

(u) = 1 2 (dist ker(M) (u)) 2 + ( + I K )(u).
Using [START_REF] Brezis | Problèmes unilatéraux[END_REF] 2 (which is convex and continuous) and 2 = (

for 1 = 1 2 (dist U (•))
+ I K ) ∈ 0 (R n ), we get * = 1 2 dist ker(M) (•) 2 * [ + I K ] * . Hence Dom * = Dom 1 2 dist ker(M) (•) 2 * + Dom [ + I K ] * .
On the other hand, we have:

1 2 dist ker(M) (•) 2 = 1 2 • 2 I ker(M) .
Using (8) again, we obtain

1 2 dist ker(M) (•) 2 * = 1 2 • 2 + I ker(M) ⊥ . Hence Dom dist ker(M) (•) 2 * = ker(M) ⊥ . Consequently, Dom * = R(M) + Dom [ + I K ] * . ( 11
)
Proposition 2 Suppose that assumptions (H) hold. Then a necessary condition for the existence of a solution of VI(M, q, , K ) is that

q, w + ∞ (w) ≥ 0, ∀w ∈ ker(M) ∩ K ∞ . ( 12
)
Proof Using Remark 1 and Lemma 1, we have

R(M, , K ) ⊂ Dom ( * ).
Using Part (i) of Proposition 1, we have

Dom ( * ) = {g ∈ R n : g, w ≤ ∞ (w), ∀w ∈ R n }.
It can also easily be checked that the recession function ∞ associated to in ( 9) is given by

∞ (w) = I ker(M) (w) + ∞ (w) + I K ∞ (w). ( 13
)
Consequently if Sol(M, q, , K ) = ∅ then -q ∈ R(M, , K ) and thus:

q, w + ∞ (w) ≥ 0, ∀w ∈ ker(M) ∩ K ∞
which competes the proof of Proposition 2.

Proposition 3 Let C be a non-empty closed convex subet of R n . We have

Int B(C) = ∅ ⇐⇒ C ∞ ∩ (-C ∞ ) = {0}, ( 14
)
where B(C) denotes the barrier cone of C defined in [START_REF] Adly | Well-positioned closed convex sets and well-positioned closed convex functions[END_REF].

Proof Since B(C) = Dom (I C ) * , we have Int B(C) = ∅ ⇐⇒ Int Dom (I C ) * = ∅. Using Part (ii) of Proposition 1 for = I C , we have Int Dom (I C ) * = {g ∈ R n : g, w < I C ∞ (w), ∀w ∈ R n , w = 0}.
Hence,

Int B(C) = {g ∈ R n : g, w < 0, ∀w ∈ C ∞ , w = 0}.
Therefore, [START_REF] Adly | Stability of non-coercive variational inequalities[END_REF]). An other characterization of this class of convex subsets in infinite dimensional spaces is also given in Lemma 2.4 [START_REF] Adly | Stability of non-coercive variational inequalities[END_REF] and Proposition 2.1 [START_REF] Adly | Well-positioned closed convex sets and well-positioned closed convex functions[END_REF]. Note that in finite dimensional spaces every subset which contains no line is well-positioned.

Int B(C) = Int (C ∞ ) • . Consequently, if Int B(C) = ∅ then there exists g ∈ R n such that: g, w < 0, ∀w ∈ C ∞ , w = 0, which implies that necessarily C ∞ ∩ -C ∞ = {0}. Conversely, suppose that C ∞ ∩ -C ∞ = {0}. Arguing by contradiction, let us sup- pose that Int (C ∞ ) • = ∅. Since (C ∞ ) • is cone, then there exists a linear subspace E of R n with 0 ≤ dim R (E) ≤ n -1 such that: (C ∞ ) • ⊂ E (see e.g. [14], p. 33). Therefore, E ⊥ ⊂ (C ∞ ) •• = C ∞ , where E ⊥ is the orthogonal of E. This contradicts the assump- tion C ∞ ∩ -C ∞ = {0} (since dim R E ⊥ ≥ 1 and E ⊥ ⊂ C ∞ ). Hence, Int (C ∞ ) • = ∅. Consequently, Int B(C) = ∅, which completes the proof of Proposition 3. Remark 2 A subset C satisfying the condition Int B(C) = ∅ is called well-positioned (see Proposition 2.1

Definition 1

The set defined by

q ∈ R n : q, w + ∞ (w) > 0, ∀w ∈ ker(M) ∩ K ∞ \ {0} ,
is called the compatibility set of the variational inequality VI(M, q, , K ).

The following proposition shows that the topological interior of the resolvent set coincides with the compatibility set of the variational inequality VI(M, q, , K ). Proposition 4 Suppose that assumptions (H) hold. We have

Int R(M, , K ) = q ∈ R n : q, w + ∞ (w) > 0, ∀w ∈ ker(M) ∩ K ∞ \ {0} . Proof Using Remark 1 and Lemma 1, we have Int R(M, , K ) ⊂ Int Dom * . By part (ii) of Proposition 1, we have Int Dom ( * ) = g ∈ R n : g, w < ∞ (w), ∀w ∈ R n \ {0} . Using (13), we get Int R(M, , K ) ⊂ q ∈ R n : q, w + ∞ (w) > 0, ∀w ∈ ker(M) ∩ K ∞ \ {0} .
Let us prove now the converse, i.e., let q ∈ R n such that

q, w + ∞ (w) > 0, ∀w ∈ ker(M) ∩ K ∞ \ {0}. (15)
We prove that -q ∈ R(M, , K ). Let (ε n ) n be a sequence of non-negative real numbers such that ε n → 0 + as n → +∞. Let u n ∈ K be the unique solution of problem VI(M + ε n I, , q, K ). We claim that the sequence (u n ) is bounded. Arguing by contradiction, suppose that there exists a subsequence, still denoted (u n ) such that u n → +∞ as n → +∞. We set w n = u n u n and along a subsequence, we may suppose that w n → w = 0.

It is clear that w ∈ K ∞ . We have

Mu n + q, v -u n + ε n u n , v -u n + (v) -(u n ) ≥ 0, ∀v ∈ K . ( 16
)
Setting v = 0 in ( 16), we get

Mu n + q, u n + ε n u n 2 + (u n ) -(0) ≤ 0.
Dividing by u n 2 and passing to the limit as n → +∞, we get Mw, w ≤ 0. Since M is symmetric and positive semidefinite, we have w ∈ ker(M) and consequently w ∈ ker(M) ∩ K ∞ \ {0}. On the other hand dividing ( 16) by t n = u n and using the fact that Mw, w ≥ 0, we obtain

q, w n + (t n w n ) t n - (0) t n ≤ 0.
Passing to the limit, we get

q, w + ∞ (w) ≤ 0,
which is a contradiction to the condition [START_REF] Hlavacek | Semi-coercive variational inequalities with uncertain input data. Applications to shallow shells[END_REF]. Thus the sequence (u n ) is bounded and there exists a subsequence, again denoted (u n ) such that u n → u as n → +∞. Passing to the limit as n → +∞ in ( 16), we show that u is a solution of VI(M, q, , K ). Hence -q ∈ R(M, , K ). Therefore,

Int Dom ( * ) ⊂ R(M, , K ), which implies that Int Dom ( * ) ⊂ Int R(M, , K ). Consequently, Int R(M, , K ) = Int Dom ( * ) = q ∈ R n : q, w + ∞ (w) > 0, ∀w ∈ ker(M) ∩ K ∞ \ {0} ,
which completes the proof of Proposition 4.

The following result characterizes the non-emptyness of the topological interior of the resolvent set R(M, , K ) associated to the variational inequality VI(M, q, , K ).

Proposition 5 Suppose that assumptions (H) hold. We have

Int R(M, , K ) = ∅ ⇐⇒ -(ker(M) ∩ K ∞ ∩ ker( ∞ )) (ker(M) ∩ K ∞ ∩ ker( ∞ )) = {0}, i.e., the cone (ker(M) ∩ K ∞ ∩ ker( ∞ )) is pointed. Proof We have Int R(M, , K ) = ∅ ⇐⇒ Int Dom ( * ) = ∅. (17)
On the other hand, we have

Dom ( * ) × {-1} = B(epi ) ∩ R n × {-1} , ( 18
)
where B(epi ) is the barrier cone to epi ( ). Indeed, let p ∈ Dom * be given. Then there exists M p ∈ R such that

p, x -(x) ≤ M p , ∀x ∈ R n ,
which is equivalent to

( p, -1), (x, (x)) R n ×R ≤ M p , ∀x ∈ R n .
Therefore,

( p, -1), (x, λ) R n ×R ≤ M p , ∀(x, λ) ∈ epi ( ).
Hence,

( p, -1) ∈ B(epi ). Now let ( p, -1) ∈ B(epi ) ∩ (R n × {-1}) . Then there existe M p ∈ R n such that ( p, -1), (x, λ) R n ×R ≤ M p , ∀(x, λ) ∈ epi ( ).
In particular for (x, λ) = (x, (x)), we have

p, x -(x) ≤ M p , ∀x ∈ R n .
Consequently,

( p, -1) ∈ Dom ( * ) × {-1}.
Using ( 17) and ( 18), we have

Int R(M, , K ) = ∅ ⇐⇒ Int B(epi ) = ∅. (19)
Using Proposition 3 for C = epi , we have

Int B(epi ) = ∅ ⇐⇒ ((epi ) ∞ ) ∩ (-(epi ) ∞ ) = {0}. (20)
Since, (epi

) ∞ = epi ∞ we get Int R(M, , K ) = ∅ ⇐⇒ ((epi ∞ )) ∩ (-(epi ∞ )) = {0}. (21)
Since is bounded from below, then is also bounded from below. Hence, ∞ ≥ 0 and it is then easy to check that:

((epi ∞ )) ∩ (-(epi ∞ )) = ((ker ∞ ) × {0}) ∩ (-(ker ∞ ) × {0}) .
Therefore,

Int R(M, , K ) = ∅ ⇐⇒ ((ker ∞ ) × {0}) ∩ (-(ker ∞ ) × {0}) = {0}. (22)
Consequently,

Int R(M, , K ) = ∅ ⇐⇒ ((ker ∞ )) ∩ (-(ker ∞ )) = {0}. (23)
Using [START_REF] Goeleven | Variational and Hemivariational Inequalities -Theory[END_REF], it is easy to see that

Int R(M, , K ) = ∅ ⇐⇒ -(ker(M) ∩ K ∞ ∩ ker( ∞ )) (ker(M) ∩ K ∞ ∩ ker( ∞ )) = {0}, (24)
which completes the proof of Proposition 5.

A stability result

Let us now study the stability of the variational inequality VI(M, q, , K ) in the following sense.

Definition 2 One says that the variational inequality VI(M, q, , K ) is stable provided that there exists ε > 0 such that for any symmetric and positive semidefinite matrix M ε , any vector q ε ∈ q + εB n (here B n denotes the open unit ball in R n ), any proper lower semicontinuous bounded from below convex function ε , and any non-empty closed convex set K ε satisfying the following conditions

0 ∈ Dom ε ∩ K ε and ker(M)∩ker( ∞ )∩ K ∞ = ker(M ε ) ∩ker(( ε ) ∞ ) ∩(K ε ) ∞ , ( 25
)
the perturbed problem V I (M ε , q ε , ε , K ε ) has at least one solution.

Before starting our study, let us first give some simple examples to motivate the notion of stability of VI(M, q, , K ) with respect to small perturbations.

Example 1 Set

M = 0 0 0 0 , K 1 = {(x, y) ∈ R 2 : y ≥ 0}, ≡ 0.
In this case the resolvent set is given by R(M, 0, K 1 ) = {(q 1 , q 2 ) ∈ R 2 : q 1 = 0 and q 2 ≤ 0}.

We note that in this case Int R(M, 0, K 1 ) = ∅. Hence problem VI(M, q, 0, K 1 ) is not stable with respect to small perturbations of the right hand term q. More precisely, if we replace q by q ε such that qq ε ≤ ε for a given ε > 0, the solution set Sol(M, q ε , 0, K 1 ) of the perturbed problem may be empty. We note that in this case, we have

ker(M) ∩ K 1∞ ∩ ker( ∞ ) = R × R + ,
which is not a pointed cone.

Example 2 Let M and as in Example 1 Consider now, the convex and closed subset K 2 given by

K 2 = {(x, y) ∈ R 2 : y ≥ x 2 }.
In this case the resolvent set is given by

R(M, 0, K 2 ) = {(q 1 , q 2 ) ∈ R 2 : q 2 < 0} ∪ {(0, 0)}.
Here Int R(M, 0, K 2 ) = ∅ and problem V I (M, q, 0, K 2 ) is stable with respect to small perturbations of the right hand term q. We note that in this case, we have

ker(M) ∩ K 2∞ ∩ ker( ∞ ) = {0} × R + ,
which is a pointed cone.

We have the following existence and stability result related to the linear variational inequality VI(M, q, , K ).

Theorem 1 Assume that assumptions (H) are satisfied. Then the variational inequality VI(M, q, , K ) is stable in the sense of Definition 2 if and only if

q, w + ∞ (w) > 0, ∀w ∈ ker(M) ∩ K ∞ , w = 0. Proof We know from Proposition 4 that if q, w + ∞ (w) > 0, ∀w ∈ ker(M) ∩ K ∞ \ {0} then -q ∈ Int R(M, , K ) = ∅.
Therefore, there exists ε > 0 such that

-q + εB n ⊂ Int R(M, , K ). ( 26
)
Let now M ε be a symmetric and positive semidefinite matrix, q ε ∈ q + εB n , ε be a proper lower semicontinuous bounded from below convex function and K ε be a non-empty closed convex set satisfying

0 ∈ Dom ε ∩ K ε and ker(M) ∩ ker( ∞ ) ∩ K ∞ = ker(M ε ) ∩ ker(( ε ) ∞ ) ∩ (K ε ) ∞ .
Using Proposition 4, we have

Int R(M, , K ) = Int R(M ε , ε , K ε ).
Therefore,

-q ε ∈ -q + εB ⊂ Int R(M ε , ε , K ε ).
Consequently, Sol(M ε , q ε , ε , K ε ) = ∅. This ensures that the variational inequality VI(M, q, , K ) is stable in the sense of Definition 2. Suppose now that the variational inequality VI(M, q, , K ) is stable in the sense of Definition 2. Then there exists ε > 0 such that (taking

M ε = M, ε = and K ε = K ) for every q ε ∈ q + εB n , Sol(M, q ε , , K ) = ∅. Therefore, -q ∈ Int R(M, , K ).
Then using Proposition 4, we obtain

q, w + ∞ (w) > 0, ∀w ∈ ker(M) ∩ K ∞ , w = 0, ( 27
)
which completes the proof of Theorem 1.

Remark 3 It is clear that any symmetric and positive semidefinite matrix M ε such that ker(M) = ker(M ε ) and any function ε ∈ 0 (R n ) bounded from below such that ∞ = ( ε ) ∞ and any non-empty closed convex set K ε such that K ∞ = (K ε ) ∞ satisfy condition (25). This the case for example if we take

M = M ε , -ε ≤ ε ≤ + ε, K ⊂ K ε + εB n and K ε ⊂ K + εB n .
We have the following consequence of Theorem 1.

Corollary 1 Assume that assumptions (H) are satisfied. If the following compatibility con

- dition q, w + ∞ (w) > 0, ∀w ∈ ker(M) ∩ K ∞ , w = 0, ( 28
)
is satisfied, then VI(M, q, , K ) has at least one solution.

Proof Take M ε = M, ε = , q ε = q and K ε = K in Theorem 1.

Some applications

We give in this section some applications of Theorem 1 and Corollary 1.

Example 3 (Clipping circuit 1/Ideal diode) Let us consider the circuit of Fig. 1 involving a load resistance R > 0, an input-signal source u and corresponding instantaneous current i, an ideal diode as a shunt element and a supply voltage E. Figure 2 illustrates the ampere-volt characteristic of an ideal diode. This is a model in which the diode is a simple switch. If V < 0 then i = 0 and the diode is blocking. If i > 0 then V = 0 and the diode is conducting. We first see that the ideal diode is described by the complementarity relation The electrical superpotential of the ideal diode is

V ≤ 0, i ≥ 0, V i = 0
ϕ D (x) = I R + (x), (x ∈ R) Then ϕ * D (z) = I R -(z), (z ∈ R)
and the recession function of the electrical superpotential is:

(ϕ D ) ∞ (x) = ϕ D (x), (x ∈ R).
We have also

∂ϕ D (x) := ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ IR -if x = 0 0 if x > 0, ∅ if x < 0 (x ∈ R)
and

∂ϕ * D (z) := ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ IR + if z = 0 0 if z < 0, ∅ if z > 0 (z ∈ R).
The complementarity relation can be written as

V ∈ ∂ϕ D (i) ⇐⇒ i ∈ ∂ϕ * D (V ) ⇐⇒ ϕ D (i) + ϕ * D (V ) = i V. Kirchoff's voltage law gives u = U R + V D + E
where U R = Ri denotes the difference of potential across the resistor and

V D ∈ ∂ I IR + (i) is the difference of potential across diode. Thus E + Ri -u ∈ -∂ I IR + (i) (29)
which is equivalent to VI(R, Eu, 0, IR + ), i.e.,

i ∈ R + : (Ri + E -u)(v -i) ≥ 0, ∀v ∈ R + . ( 30
)
Here R > 0 and for each E, u ∈ R, we may apply Theorem 1 to assert that (30) is stable in the sense of Definition 2. Moreover:

(30) ⇐⇒ E R + i - u R ∈ -∂ I IR + (i) ⇐⇒ - E R + u R ∈ i + ∂ I IR + (i) ⇐⇒ i = (id R + ∂ I IR + ) -1 ( u -E R ) = 1 R max{0, u -E}.
If u ≤ E then the diode is blocking while if u > E then the diode is conducting.

Let us now consider a driven time depending input t → u(t) and define the output-signal

t → V o (t) as V o (t) = E + V (t).
The time depending current t → i(t) is given by

i(t) = 1 R max{0, u(t) -E} (31)
and thus

V o (t) = V (t) + E = u(t) -Ri(t) = u(t) + min{0, E -u(t)} = min{u(t), E}. (32) Fig. 3 Practical diode model V i + - V1 V2 1 -100 V (Volts) i (mA)
This shows that the circuit in Fig. 1 can be used to transmit the part of a given input-signal u which lies below some given reference level E.

Example 4 (Clipping circuit 1/concrete diode) Let us now consider the circuit in Fig. 1 with a concrete diode. Figure 3 illustrates the ampere-volt characteristic of a practical diode model. There is a voltage point, called the knee voltage V 1 , at which the diode begins to conduct and a maximum reverse voltage, called the peak reverse voltage V 2 , that will not force the diode to conduct. When this voltage is exceeded, the depletion may breakdown and allow the diode to conduct in the reverse direction. Note that usually

| V 1 | | V 2 |
and the model is locally ideal.

For general purpose diodes used in low frequency/speed applications,

| V 1 | 0.7 -2.5 V and | V 2 | 5 kV; for high voltage rectifier diodes, | V 1 | 10 V and | V 2 |
30 kV; for fast diodes used in switched mode power supply and inverter circuits,

| V 1 | 0.7 -1.5 V and | V 2 | 3 kV and for Schottky diodes used in high frequency applications, | V 1 | 0.2 -0.9 V and | V 2 | 100 V.
The electrical superpotential of the practical diode is

ϕ P D (x) = ⎧ ⎨ ⎩ V 1 x if x ≥ 0 V 2 x if x < 0 , (x ∈ R).
Then and the recession function of the electrical superpotential is given by:

ϕ * P D (z) = I [V 2 ,V 1 ] (z), (z ∈ R)
(ϕ P D ) ∞ (x) = ϕ P D (x), (x ∈ R).
We see that

∂ϕ P D (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ V 2 if x < 0 [V 2 , V 1 ] if x = 0 V 1 if x > 0 , (x ∈ R)
recovers the ampere-volt characteristic (i, V ) while

∂ϕ * P D (z) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ R -if z = V 2 0 if z ∈]V 2 , V 1 [ R + if z = V 1 ∅ if z ∈ R\ [V 2 , V 1 ]
, (z ∈ R).

recovers the volt-ampere characteristic (V, i). The ampere-volt characteristic of the practical diode (Fig. 4) can thus be written as

V ∈ ∂ϕ P D (i) ⇐⇒ i ∈ ∂ϕ * P D (V ) ⇐⇒ ϕ P D (i) + ϕ * P D (V ) = i V.
We may follow the same steps as in the previous example to see that Kirchoff's law reduces to VI(R, Eu, ϕ PD , IR), i.e.,

i ∈ K := R : (Ri + E -u)(v -i) + ϕ P D (v) -ϕ P D (i) ≥ 0, ∀v ∈ R.
Here R > 0 and for each E, u ∈ R, we may apply Theorem 1 to assert that (4) is stable in the sense of Definition 2. Moreover:

i(t) = (id R + ∂ϕ P D ) -1 u(t) -E R = argmin x∈R 1 2 x - u(t) -E R 2 + ϕ P D (x) .
(33) and

V o (t) = u(t) -Ri(t). ( 34
) Fig. 5 Double-diode clipper u(t) R D 1 µ 1 µ 2 D 2
Example 5 Let us consider the double-diode clipper circuit in Fig. 5 involving a load resistance R > 0, two ideal diodes, an input-signal source and two supply voltages E 1 and E 2 .

It is assumed that E 1 < E 2 . We denote by i the current through the resistor R and we set i = i 1 +i 2 where -i 1 denotes the current through diode D 1 and i 2 is the current through diode D 2 . We denote by µ 1 the difference of potential across diode D 1 and by µ 2 the difference of potential across diode D 2 . Using Kirchoff's voltage laws, we get the system:

⎧ ⎨ ⎩ E 1 + R(i 1 + i 2 ) -u = +µ 1 E 2 + R(i 1 + i 2 ) -u = -µ 2 (35)
The ideal diodes D 1 and D 2 are simple switch. If µ 1 < 0 (resp. µ 2 < 0) then -i 1 = 0 (resp. i 2 = 0) and the diode is blocking while if -i 1 > 0 (resp. i 2 > 0) then µ 1 = 0 (resp. µ 2 = 0) and the diode is conducting. Thus

µ 1 ≤ 0, -i 1 ≥ 0, µ 1 i 1 = 0 and µ 2 ≤ 0, -i 2 ≥ 0, µ 2 i 2 = 0.
These last complementarity relations can also be written as:

µ 1 ∈ ∂ I R + (-i 1 ) = -I R -(i 1 )
and

µ 2 ∈ ∂ I R + (i 2 ) Setting K = R -× R + , M = ⎛ ⎝ R R R R ⎞ ⎠ , q = ⎛ ⎝ E 1 -u E 2 -u ⎞ ⎠ , I = ⎛ ⎝ i 1 i 2 ⎞ ⎠ , ( 36
)
we see that the system in (35) is equivalent to the variational inequality VI(M, q, 0, K), i.e.,

I ∈ K : M I + q, v -I ≥ 0, ∀v ∈ K . ( 37
)
Here the matrix M is positive semidefinite and symmetric,

K ∞ ≡ K , ker{M} = {v ∈ R 2 : v 2 = -v 1 } and K ∞ ∩ ker{M} = {(-α, α); α ≥ 0}. Moreover, for all v ∈ K ∞ ∩ ker{M}, v = (0, 0) there exists α > 0 such that v = (-α, α) and q, v = (E 1 -u)v 1 + (E 2 -u)v 2 = α(E 2 -E 1 ) > 0. ( 38
)
We may thus apply Theorem 1 to assert that a double-diode clipper involving ideal diodes is stable with respect to data perturbations in the sense of Definition 2. Moreover, using the relations in (35) we see that:

i * 1 + i * 2 = min i * 2 , u -E 1 R = max i * 1 , u -E 2 R
from which we deduce, after elementary calculations, that:

i * = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ u-E 1 R if u < E 1 0 if E 1 ≤ u ≤ E 2 u-E 2 R if u > E 2 .
So, for a driven time depending input t → u(t) the time depending current t → i * (t) through the resistor R is given by

i * (t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ u(t)-E 1 R if u(t) < E 1 0 if E 1 ≤ u(t) ≤ E 2 u(t)-E 2 R if u(t) > E 2 (39)
and the output-signal t → V o (t) defined by

V o (t) = V 2 (t) + E 2 = u(t) -Ri * (t)
is then given by the expression:

V o (t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ E 1 if u(t) < E 1 u(t) if E 1 ≤ u(t) ≤ E 2 E 2 if u(t) > E 2 . ( 40
)
This shows that the circuit can be used to transmit the part of a given input-signal u that lies above some level E 1 and below some level E 2 (Fig. 6).

Example 6 Let us again consider the circuit in Fig. 5 and suppose that the electrical superpotential of each diodes D 1 and D 2 is given by (practical diode model): where V 2 < 0 < V 1 . We suppose also that

ϕ P D (x) = ⎧ ⎨ ⎩ V 1 x if x ≥ 0 V 2 x if x < 0 , (x ∈ R)
| V 2 | > E 2 -E 1 2 . ( 41
)
We set

E 2 + R(i 1 + i 2 ) -u = -µ 2 ∈ -∂ϕ P D (i 2 ) ( 43
)
which is equivalent to the variational inequality VI(M, q, , IR 2 ), i.e.,

ϒ ∈ IR 2 : Mϒ + q, v -ϒ + (v) -(ϒ) ≥ 0, ∀v ∈ R 2 , ( 44
)
with M and q as in (36) and as in (42). Here

ker{M} = {v ∈ R 2 : v 2 = -v 1 }.
Let v ∈ ker{M}, v = 0, be given. Then:

q, v + ∞ (v) = v 2 (E 2 -E 1 ) + ϕ P D (-v 1 ) + ϕ P D (v 2 ) = v 2 (E 2 -E 1 ) + 2ϕ P D (v 2 ).
It results that if v 2 > 0 then

q, v + ∞ (v) = v 2 (E 2 -E 1 ) + 2V 1 v 2 while if v 2 < 0 then q, v + ∞ (v) = -v 2 (2 | V 2 | -(E 2 -E 1 )) > 0.
We may then apply Theorem 1 to assert that the variational inequality VI(M, q, , IR 2 ) is stable in the sense of Definition 2. Moreover, the function is strictly convex and the solution ϒ * of (44) is unique and given by: (47) and the output-signal V o can then be determined by the formula:

ϒ * =
V o (t) = u(t) -Ri * (t).
Example 7 Let us consider the system of two rigid bodies M 1 and M 2 interconnected by a spring of stiffness k > 0 and constrained to move only in the horizontal direction. The position of M 1 relative to the origin is represented by x 1 while the position of M 2 relative to the origin is determined by x 2 (Fig. 8).

The mass M 1 is subjected to some external force q 1 and static Coulomb friction force F 1 while the mass M 2 is subjected to some external force q 2 and static Coulomb friction F 2 .

Fig. 1 Fig. 2

 12 Fig. 1 Clipping circuit 1: diode as shunt element

Fig. 4

 4 Fig.[START_REF] Ang | Noncoercive variational inequalities: some applications[END_REF] Clipping circuit 1: general diode as shunt element using,V 1 = 0.1, V 2 = -90, E = 1

Fig. 6

 6 Fig. 6 Double-diode clipper: ideal diode, E 1 = 0.1, E 2 = 0.6

E 1 +

 1 φP D (x) = ϕ P D (-x), ∀ x ∈ R and (x) = φP D (x 1 ) + ϕ P D (x 2 ), ∀ (x 1 , x 2 ) ∈ R 2 . R(i 1 + i 2 )u = +µ 1 ∈ -∂ φP D (i 1 )

Fig. 7 Fig. 8

 78 Fig. 7 Double-diode clipper: practical diode Fig. 8 Two rigid bodies interconnected by a spring

The research of S. Adly has been supported by the "Fondation EADS" and the ANR project "Guidage".

The equilibrium states of the system are characterized by the equilibrium equations:

Static friction Coulomb models for F 1 and F 2 are

and

where µ > 0 denotes Coulomb friction coefficient. These set-valued relations can also be written as

and

where

we see that our equilibrium system is equivalent to the variational inequality VI(M, -q, , IR 2 ), i.e.,

Here the matrix M is positive semidefinite and symmetric. Moreover

Suppose now that

For all v ∈ ker{M}, v = (0, 0) there exists α = 0 such that v = (α, α) and

We may thus apply Theorem 1 to assert that our system is stable with respect to data perturbations in the sense of Definition 2.


Adrien Revault D'allonnes

Marie-Jeanne Lesot
email: marie-jeanne.lesot@lip6.fr

Dynamics of Trust Building: Models of Information Cross-Checking in a Multivalued Logic Framework

Dynamics of trust building: models of information cross-checking in a multivalued logic framework

Adrien Revault d'Allonnes, Marie-Jeanne Lesot

I. INTRODUCTION

Information evaluation ranges from fact-checking to source verification and spans various other topics along the way, among which information quality, search query ranking and algorithm performance analysis are but a few. One of the common objectives of these domains, however, is to determine how much faith to put into a piece of information [START_REF] Nato | Annex to STANAG2022[END_REF]- [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF]. One way of looking at this problem is to focus on trust building, the process through which belief is garnered. Numerous criteria can be considered in this task [START_REF] Capet | Information evaluation in the military domain: Doctrines, practices and shortcomings[END_REF], such as characteristics of the source providing the considered piece of information, e.g. reliability, competence or sincerity, or of the contents, e.g. plausibility with respect to the rater's background knowledge or credibility, understood here as cross-validation by other sources.

This crucial credibility dimension usually corresponds to the final step of the trust building process and aims at crosschecking the considered piece of information, i.e. finding confirmations or invalidations by other sources [START_REF] Nato | Annex to STANAG2022[END_REF], [START_REF] Lesot | Proposition of a semi-automatic possibilistic information scoring process[END_REF], [START_REF] Revault D'allonnes | An architecture for the evolution of trust: Definition and impact of the necessary dimensions of opinion making[END_REF], possibly independently from the actual truth of the underlying fact. Such homologous information lead the rater to update his current trust level: confirmations tend to increase it whereas invalidations lower it.

Trust building is linked to the art of persuasion or Aristotle's rhetoric. One particular aspect of interest in this system is the arrangement of arguments and its influence on persuasion. Indeed, the dynamics of trust building result from cross-checking: two confirmations followed by an invalidation may lead to a different level of trust than a confirmation followed by a contradiction and another confirmation might. This dependence on the order of the information stream is the source of many a theory on argumentation, some of which study the ideal moment to contradict an opponent in a political debate or in court.

Trust building in general and sensitivity to argumentation organisation are, of course, highly subjective processes. As such, different attitudes when confronted to an information stream may be observed. One rater may, for instance, be much more sensitive to contradictions than to confirmations, thus exhibiting a distrustful behaviour. Another may not favour invalidations over corroborations but be more unwilling to change his mind, showing a more mulish disposition. An efficient trust building model should be able to represent these different attitudes, called credulity stances hereafter. This paper extends the multivalued logic formalisation of the trust building process presented in [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF] by focusing on the influence of identified homologous information on the ensuing level of trust. It proposes a generic approach that makes it possible to exploit all available pieces of information, taking into account even dubious ones. It also discusses variants, thus offering a rich modelling framework which allows the representation of a wide range of realistic credulity stances, when faced with contradictory information stream.

The paper is organised as follows: Section II reviews existing models of credibility and, in particular, details the main features of the approach based on a multivalued logic framework. Section III describes the proposed extension of this approach, called cumulative credibility, and Section IV discusses some variants that further enrich its modelling capacities. Section V illustrates the richness of the proposed models and the variety of credulity stances they allow to represent, commenting on their respective properties.

II. EXISTING CREDIBILITY MODELS

This section discusses the notion of credibility, which occurs in several existing information scoring models, detailing and enriching with further comments the approach based on an extended multivalued logic that this paper proposes to extend.

A. Credibility Definition and Integration

In the framework of information scoring and trust building, credibility is understood as a degree of confirmation, resulting from an information cross-checking step [START_REF] Nato | Annex to STANAG2022[END_REF], [START_REF] Lesot | Proposition of a semi-automatic possibilistic information scoring process[END_REF], [START_REF] Revault D'allonnes | An architecture for the evolution of trust: Definition and impact of the necessary dimensions of opinion making[END_REF]: this step aims at finding possible corroboration, identifying complementary information backing up or undermining a considered piece of information to be rated. The trust put in it is indeed influenced by these possible confirmations and invalidations.

The importance of cross-checking for information evaluation is underlined from the original military model [START_REF] Nato | Annex to STANAG2022[END_REF]. Indeed, it scores any piece of information as a bi-gram, defined as the concatenation of two symbols measured on two discrete graded scales associated to linguistic labels. The first symbol, called reliability, characterises the source, the second one, called credibility, the information content. If this credibility dimension can be estimated, it is measured on a 5-level scale, associated to the linguistic labels improbable, doubtful, possibly true, probably true, confirmed by other sources [START_REF] Nato | Annex to STANAG2022[END_REF]. These labels mainly describe the information certainty; however the detailed description they are accompanied by relate them to a degree of confirmation of the considered piece of information, as also indicated by the maximal degree 'confirmed by other sources'. This underlines the crucial role of the cross-checking step in the evaluation of a piece of information: its result is directly represented in the final global score. However, this model does not define a practical procedure to measure credibility, leaving it to the subjectivity of the rater. A fortiori, the model gives no indication regarding the effect of confirmation/invalidation order on the trust level evolution and its dynamics.

The information scoring model proposed in [START_REF] Lesot | Proposition of a semi-automatic possibilistic information scoring process[END_REF], formalised in possibility theory, aims at estimating whether a considered event occurred or not, based on the evaluation of pieces of information that confirm or invalidate it. It includes a final corroboration step that combines the possibility distributions inferred from individual pieces of information, to update the a priori event estimation taking into account the homologous information. Among other aggregation operations, a temporal effect is considered, making the model sensitive to the information order, through a weighting scheme that allows to take into account information obsolescence to reduce the effect of older and possibly out-of-date pieces of information.

The trust building model proposed in [START_REF] Revault D'allonnes | An architecture for the evolution of trust: Definition and impact of the necessary dimensions of opinion making[END_REF], [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF] focuses on the process by which a rater builds his conviction on a hitherto unknown piece of information. Although it does not aim at measuring the certainty of a piece of information as the models mentioned above do, but how convinced one is by it, it shares with them the final cross-checking step that looks for corroboration to update a current trust level. Since this paper proposes to extend it, as described in Sections III and IV, the following subsections detail the formal framework it considers, its general characteristics and an implementation, enriching it with further comments.

B. Extended Multivalued Logic

The formal framework considered in [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF] to represent and manipulate trust scores is an extended multivalued logic framework. This model has trust defined on a single discrete graded scale, clarified with linguistic labels, thus improving the legibility of a unique degree with a semantic interpretation.

Formally, the multivalued logic paradigm [START_REF] Akdag | A qualitative theory of uncertainty[END_REF] that this model extends is based on a symbolic, discrete truth scale defined as a totally ordered set L M = {τ 0 , . . . , τ M -1 }, where τ α ≤ τ β ⇔ α ≤ β. These degrees span, at varying granularity with M , the different levels of truth from τ 0 , meaning 'false', to τ M -1 , for 'true'. The change from positive to negative truth appears around the middle value τ M -1

2

, usually forced into the chosen scale by choosing an odd M . Multivalued logics are equipped with formal tools to combine the truth degrees through logical operators, that generalise conjunction, confidence cannot be estimated credibility cannot be estimated disjunction or implication [START_REF] Akdag | A qualitative theory of uncertainty[END_REF], as well as arithmetical ones, defining symbolic addition and subtraction [START_REF] Seridi | Approximate Reasoning for Processing Uncertainty[END_REF].

The extended multivalued logic proposed in [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF] introduces an additional degree, denoted τ ? . It aims at allowing to distinguish between facts that are 'neither true nor false', i.e. that have a neutral truth value, and facts whose truth value cannot be evaluated. The neutral, or indifferent, value can then be mapped to τ M -1 2 , as usual, whereas the unknown one can be represented by τ ? , i.e. a distinct value.

Formally, τ ? is defined by the following properties [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF], where denotes any multivalued operator:

(P 1 ) τ ? ∈ L M (P 2 ) ∀τ α ∈ L M , τ ? τ α = τ α τ ? = τ α (P 3 ) ¬τ ? = τ ? (P 4 ) τ ? τ ? = τ ?
Property (P 1 ) takes τ ? out of the ordering constraints imposed on all τ α in L M . This is justified by the fact that even a very doubtful element cannot be compared to one whose truth value is unknowable. For similar reasons, property (P 2 ) turns τ ? into a neutral element for all multivalued operators. Properties (P 3 ) and (P 4 ) describe the rules for combining τ ? with itself.

The extended multivalued logic is then defined as L e M = L M ∪ {τ ? }, where the usual degree manipulation operations, both logical and arithmetical, are extended according to properties (P 2 ) to (P 4 ) to handle τ ? . Characteristics and properties of L e M are discussed further in [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF].

C. Extended Multivalued Credibility Model

The model proposed in [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF] defines the trust building process as the sequential integration, into an initially indeterminate level of trust, of relevant dimensions, namely reliability, competence, plausibility and credibility. It is formalised in the extended multivalued logic framework briefly described in the previous subsection: the trust level and the activation of all involved dimensions are represented and manipulated in L e M . Table I illustrates, in the case of L e 5 , possible linguistic labels associated to trust and credibility [START_REF] Revault D'allonnes | An architecture for the evolution of trust: Definition and impact of the necessary dimensions of opinion making[END_REF]. As in the models briefly reviewed in Section II-A, credibility is understood as a degree of confirmation of the rated piece of information. As a consequence, in the case where the homologous piece of information it is compared to is an invalidation, credibility has a degree in the bottom half of the scale, τ < τ M -1

2

; if it it is a confirmation, the degree is greater than τ M -1 2 .

Given a current evaluation of the trust score, denoted τ α , the integration of the credibility dimension aims at updating τ α with respect to a homologous piece of information: the output trust level depends on the degree of confirmation it provides, i.e. activation of the credibility dimension, denoted τ β , as well as its current evaluation, denoted τ γ . Indeed, if a homologous piece of information completely confirms the element, it obviously increases the trust level. However, if this absolute confirmation is doubtful, its effect should be less than it would be, were it nearly certain.

Therefore the cross-checking operator is formalised as a mapping G : L e M × L e M × L e M -→ L e M , that satisfies the following properties [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF]:

-G is increasing in its first 2 arguments -G is: -increasing in its third argument if τ β ≥ τ M -1 2 -decreasing in its third argument if τ β < τ M -1 2 -∀τ α , τ β , τ γ ∈ L M G(τ α , τ β , τ γ ) ≥ τ α if τ β ≥ τ M -1 2 G(τ α , τ β , τ γ ) ≤ τ α if τ β < τ M -1 2 -τ ? is a neutral element for all 3 arguments
The first property corresponds to a double statement: an increase in the first argument τ α imposes that the higher the initial trust level, the higher the updated one, whatever the correction. Similarly, for any current score, the updated score increases with how much the homologous piece of information confirms the current evidence.

The second property emanates from this two-sided scale. Indeed, for a fixed current level of trust and credibility, two cases appear. The case of a confirmation is briefly discussed above: the higher the trust level τ γ of the compared piece of information, the higher the updated score. The opposite case of a contradiction has a similar effect, that is a certain invalidation lowers the score further than a doubtful one would.

The third property describes the respective effects of confirmations, when τ β ≥ τ M -1

2

, for which G shows an emphasising effect on the trust level, and invalidations which have an abating effect: for confirmations, the initial trust score is a lower bound of the updated score, whereas it is an upper bound in the case of invalidations.

The final property deals with undetermined values: if the current score τ α is unknown, then the updated score is determined by the degree of confirmation and the trust level of the compared piece of information. If the degree of confirmation τ β or the trust level of the compared piece of information τ γ is unknown, the current score is not modified.

D. Implementation as Immediate Credibility

Many functions can be considered in order to satisfy the desired properties above. We propose to name immediate credibility the one described in [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF] and briefly recalled in this section, as opposed to the model proposed in the next section, called cumulative credibility.

1) Principle based on Transition Thresholds: In the general case where τ β = τ ? , i.e. when the degree of confirmation offered by the homologous piece of information has a determined value, the cross-checking operator is based on the definition of transition thresholds, specified as parameters κ γ α ∈ L M : they express constraints for the transition between a current score τ α and an updated value τ γ . More precisely, they allow to update the current score from τ α to τ γ if the influence of the considered homologous information is beyond κ γ α . The notion of influence, defined as weighted credibility, is described in the next subsection.

τ 0 τ 1 τ 2 τ 3 τ 4 κ 0 1 κ 0 2 κ 1 2 κ 0 3 κ 1 3 κ 2 3 κ 0 4 κ 1 4 κ 2 4 κ 3 4
Fig. 1. Graphical representation of Fc(τα, τ β ) ignoring τ ? [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF] This principle is illustrated on Figure 1 in the case of invalidation: the shaded disks represent the current level of trust in L e 5 and the arrows show the transitions from a current score to an updated one. All arrows are oriented towards the left, as invalidations can only lower the trust level. A decrease of the trust level from τ 3 to τ 1 can for instance only occur if the homologous piece of information has an influence above κ 1 3 , as formalised below.

2) Weighted Credibility: The influence that a given homologous piece of information has on the amount of change the rated piece of information undergoes both depends on the degree of confirmation it provides, as measured by the credibility dimension, and on its own trust level. As previously stated, for a given level of corroboration, a dubious homologous piece of information should influence the trust update to a lesser extent than a highly trusted one; similarly, an absolute confirmation should have a higher impact than a partial one.

The weighted credibility operator, written g(τ β , τ γ ), that implements this principle is a mapping g : L e M × L e M → L e M satisfying the following properties [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF]: (i) g is increasing (resp. decreasing) in its first argument for confirmations (resp. invalidations) and increasing in its second argument, (ii) τ ? is an absorbing element for both arguments, and (iii) g is bounded above by τ β for confirmations and below by ¬τ β for invalidations. The negation operator serves to allow a symmetric behaviour.

As a consequence, a homologous piece of information that is dubious or has a low cross-checking effect has a low influence on the evolution of the trust level.

3) Abating and Emphasising Functions: Two functions, denoted Fc and Fi , are defined to represent the emphasising and abating influence [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF], respectively required for invalidating and confirming pieces of information. Both are mappings L e M × L e M → L e M that take as arguments the score to be updated, denoted τ α , and the weighted credibility, i.e. the influence strength, τ ω = g(τ β , τ γ ); they output the updated score. Both are increasing in their first argument, but, due to their abating and emphasising influences, Fi is decreasing in the second one whereas Fc is increasing. Moreover, Fi is bounded above by its first argument, i.e. Fc (τ α , τ ω ) ≤ τ α , whereas Fc is bounded below.

Using the transition thresholds, Fi and Fc can for instance

Fc τ 0 τ 1 τ 2 τ 3 τ 4
Fig. 2. Graphical representation of the immediate credibility operator G, in L 5 [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF] be defined as [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF]:

Fi (τ α , τ ω ) = min{τ γ ∈ L M |τ ω ≥ κ γ α } if τ α , τ ω ∈ L M τ α if τ ω = τ ? τ ω if τ α = τ ? Fc (τ α , τ ω ) = max{τ γ ∈ L M |τ ω ≥ κ γ α } if τ α , τ ω ∈ L M τ α if τ ω = τ ? τ ω if τ α = τ ?
Both functions follow the most extreme transition that can be triggered by the considered influence τ ω according to the transition thresholds, respectively in an abating or emphasising way. Also, τ ? is a neutral value for all arguments. Figure 1 illustrates the general architecture of Fi omitting the cases where τ ? .

Note that the monotonicity property of Fi and Fc imposes constraints on the parameters κ γ α : in the case of Fi for instance, they must be ordered so that if τ γ < τ δ , then κ γ α ≥ κ δ α . In general, the second inequality is strict, so as to avoid multiple outcomes for one τ ω .

4) Immediate Credibility Operator:

The global crosschecking operator satisfying the desired behaviour stated in Section II-C is finally defined as the combination of these two functions, whose application depends whether the homologous information is a confirmation or an invalidation [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF]:

G(τ α , τ β , τ γ ) =    τ α if τ β = τ ? Fi (τ α , g(τ β , τ γ )) if τ β < τ M -1 2 Fc (τ α , g(τ β , τ γ )) otherwise
It is illustrated in Figure 2, where the lower part represents the Fi function, similar to the one shown on Figure 1, and the upper part Fc , omitting the transition thresholds for clarity's sake. Specific examples are given and commented in Section V-B.

5) Transition Threshold Setting and Resulting Properties:

The proposed immediate credibility implementation of the cross-checking operator is defined by transition thresholds, allowing to model both conviction dynamics and a large variety of credulity stances.

Indeed, the ordering constraints on κ γ α are imposed for a given value of α but constraints of the form κ β α = κ γ α ⊕ κ β γ , where τ γ is an intermediate value between τ α and τ β and ⊕ a symbolic addition operator [START_REF] Seridi | Approximate Reasoning for Processing Uncertainty[END_REF], are not required. As a consequence, going directly from τ α to τ β need not be equivalent to successive transitions from τ α to τ γ and from τ γ to τ β .

Moreover the definitions of the Fi and Fc functions are independent: in particular, it is not required that κ γ α = κ α γ . This property allows the immediate cross-checking operator G to represent different credulity stances, modelling different sensitivities to contradictory arguments: a dubious invalidation may for instance lower the trust level to a lesser extent than a dubious confirmation may increase it.

Besides, except in the special case where the considered transition parameters make the operators associative, the proposed function does not lead to the same result for an identical set of homologous information presented in a different order: integrating two confirmations followed by an invalidation does not necessarily provide the same result as a confirmation followed by an invalidation in turn followed by a confirmation. This property grants the proposed model the desired ability to model the dynamics of conviction.

III. CUMULATIVE CREDIBILITY

This section describes the proposed extension of the immediate cross-checking operator, offering an alternative implementation of credibility integration that satisfies the desired properties as well as a refined usage of all available homologous information.

A. Motivation

The immediate model for cross-checking given above only takes into account pieces of information whose current weighted impact exceeds transition thresholds. What this means is that any piece of information whose level of trust combined to how comparable it is is lower than these thresholds has no influence over the evaluation and the final score is the same as had they not occurred.

One may consider that this behaviour is too radical. Indeed, an abundant succession of confirmations, even if they are doubtful, may have an impact, such as that of persistent rumours. Even when what they report seems doubtful, their incessant stream of confirmations usually has an impact. Consider, for instance, a mildly believable piece of information, with score τ 3 . Suppose it is confirmed by a succession of highly doubtful pieces of information, each of weighted credibility τ 1 . If the strategy is such that only confirmations with weighted credibilities above τ 2 have a positive influence, that is κ 4 3 = τ 2 , applying G does not increase trust, be there dozens such -possibly independent-confirmations.

Even though strategies aim at limiting drastic behaviours, it seems worthwhile to keep track of these confirmations, or invalidations, to fully exploit all information. The cumulative strategy proposed in this section aims at offering such a behaviour.

B. Trust Updating Algorithm

1) General Idea: The proposed method enriches operator G by allowing it to keep track of past homologous information. In order to achieve this, two accumulators, written g + and g -, are introduced: the former stores the cumulative weighted

1: if τ β = ? then 2: τ ← τ 3: g -← g - 4: g + ← g + 5: else if τ β < τ M -1 2 then 6: g -← g -⊕ g(τ β , τ γ ) 7: if Fi (τ, g -) = τ then 8: τ ← Fi (τ, g -) 9:
g -← τ ? 10:

g + ← τ ? 11: end if 12: else if τ β ≥ τ M -1 2 then 13: g + ← g + ⊕ g(τ β , τ γ ) 14: if Fc (τ, g + ) = τ then 15: τ ← Fc (τ, g + ) 16: g -← τ ? 17: g + ← τ ? 18:
end if 19: end if Fig. 3. Cumulative cross-checking algorithm G * credibility of confirmations, the latter that of invalidations. Functions Fc and Fi then apply to the current score and, respectively, g + for confirmations and g -for invalidations, instead of just the latest g value.

The persistence of the accumulators g + and g -no longer allows credibility integration to be written as a function. It is, thus, presented below as a score updating algorithm G * which can return the current score value at any time.

2) Description: The algorithm G * , whose complete description is given in Fig. 3, is initially called on a piece of information -and its current trust score τ α -when it encounters its first homologous counterpart. As long as new instances emerge, the level of trust τ , initialised by τ α , is updated. At any time, the current value of τ can be returned to update τ α .

On the initial call, the two accumulators g + and g -, internal parameters of the algorithm, are initialised at τ ? since, before the first confirmation and invalidation are observed, their value is undetermined.

For each subsequent homologous information, of credibility τ β and trust value τ γ , the appropriate accumulator is updated: g -(line 6) for an invalidation, i.e. if τ β < τ M -1 2 , g + (line 13) for a confirmation. In both cases, the weighted credibility, representing the influence the considered homologous piece of information must play, is added to the current accumulator value. If either the trust score or the degree of confirmation is still unknown, i.e. if either τ β = τ ? or τ γ = τ ? , no internal variable (τ , g + or g -) is modified: for τ β , this is explicitly stated in lines 1 to 4; for τ γ , it results from τ ? being an absorbing element for g (see Section II-D2).

Any time the accumulated baggage influence the dynamics of conviction, that is when either Fi (τ, g -) or Fc (τ, g + ) modify the current trust value (lines 7 and 14 for invalidation and confirmation respectively), τ is updated and the two accumulators are reset.

3) Properties: It can be shown easily that the cumulative cross-checking algorithm G * satisfies all desired properties described in Section II-C, namely the monotonicity constraints, the behaviour of τ ? and the boundary conditions. The proofs, informally sketched below, rely on the properties of functions g, Fc and Fi and on that of the symbolic addition operator ⊕, which is increasing in both its arguments [START_REF] Seridi | Approximate Reasoning for Processing Uncertainty[END_REF].

The fact that G * is monotonically increasing in its first argument τ α , the current evaluation before credibility analysis begins, comes down to it only serving as an initialisation for its internal placeholder τ which, in turn, is used either with Fc or Fi , lines 8 and 15, both increasing in their first argument.

To prove G * 's monotonicity in its second argument, one must consider two values τ β and τ β , such that τ β ≤ τ β . Suppose both are less than τ M -1 2 , then they are parameters of g and participate in g -'s update, line 6, and thus potentially in τ's, line 8. Since g is decreasing in its first argument for invalidations and ⊕ is increasing, this leads to the inequality g -≥ g -. Finally, as Fi is decreasing in its second argument, it holds that Fi (τ, g -) ≤ Fi (τ, g -). If either both or neither updates are triggered, the required increase follows. If only one update is performed, the property is justified by Fi 's boundary property. The case where both τ β and τ β are greater than τ M -1 2 is proved similarly. Finally, if τ β < τ M -1 2 ≤ τ β , the required property derives from the boundary constraints on Fi and Fc : the update value based on τ β , defined in line 8 and denoted τ new , satisfies τ new ≤ τ and the one based on τ β , denoted τ new and defined in line 15, is such that τ ≤ τ new .

The monotonicity constraint on τ γ can be proved distinguishing by between confirmations and invalidations. If

τ β ≥ τ M -1 2
, the increasing monotonicity in τ γ can be proved using the fact that g, ⊕ and Fc are increasing in their second argument. The case of invalidations is proved similarly using Fi 's monotonicity.

The emphasising (resp. abating) behaviour for confirmations (resp. invalidations) emanates from operator definitions. Suppose, e.g., τ β ≥ τ M -1 2 , the possible update for τ is defined as Fc (τ, g + ) (line 8) which is greater than or equal to τ according to the boundary condition on Fc . If τ is not updated, the inequality trivially holds. The case of invalidations is similar, using the boundary condition on Fi .

Finally, the neutrality of τ ? for the first argument comes from its neutrality for the two functions Fc and Fi . For the second and third arguments, it is shown by the absence of modification τ β = τ ? and τ γ = τ ? lead to, discussed above.

IV. CUMULATIVE CREDIBILITY VARIANTS

This section discusses variants of the proposed cumulative cross-checking algorithm G * , considering alternatives for accumulator re-initialisation and trust score updating tests.

A. Trust Momentum Preservation

In the cross-checking algorithm above, every time the current trust level is updated the accumulators are reset (lines 9:10 or 16:17, depending on τ M -1

2

). When modelling the dynamics of trust building, such a drastic change limits expressiveness. Indeed, if, instead of having g + and g -go back to the undetermined τ ? value, some trace of the previous directions were to be kept, then the trust building momentum could be preserved.

Since any time either one exceeds a threshold κ β α both accumulators are reset, two options exist. Suppose that the trust level increases because g + ≥ κ β α . Instead of resetting g + one may wish to hang on to some potential surplus confirmation history by choosing g + κ β α as its next value, where denotes a symbolic subtraction operator [START_REF] Seridi | Approximate Reasoning for Processing Uncertainty[END_REF]. Another possibility to consider is that, even though a sufficient number of confirmations has been reached, one may wish to keep a memory of one's previous doubts. To do this, g -may need to stay unchanged.

These options therefore offer three alternative variants to the algorithm above: an enthusiastic attitude, where only positive momentum is preserved; a mistrustful one, in which only the doubts remain; and an eager sceptic, where both traces are kept. Algorithmically they respectively correspond to replacing lines 16 and 17 with the lines indicated in the lower part of Table II.

Obviously, had the transition been reversed (i.e. g -≥ κ β α ), all opposite options would have been considered, as indicated by the first two lines of the same Table II.

B. Order Independent Credibility

Even though the momentum preservation and the rest of the dynamics of trust modelling presented above increase the expressiveness of the model, there are some cases where they are not acceptable. Suppose, for instance, that one wishes to evaluate trust in the production of an automatic knowledge extraction system. Suppose further that the extraction system is non-deterministic in the order it produces knowledge. In such a case the integration of credibility can neither be immediate nor cumulative. The stability of the evaluation supersedes the benefits of expressiveness.

One way of circumventing this problem is to accumulate all available homologous information before actually updating the trust value. Explicitly, this comes down to updating g + and g -, without evaluating either Fc nor Fi until all homologous information have been taken into account.

The last remaining option becomes whether to inspect invalidations or confirmations first, i.e. Fc ( Fi (τ α , g -), g + ) and Fi ( Fc (τ α , g + ), g -), either along credulity stance or relative importance of the dynamics. As long as these settings are fixed for a user, the output remains stable.

V. ILLUSTRATIVE EXAMPLE

This section illustrates the richness of the model by comparing the credulity stances of two users on a common stream of information. All evaluations (trust, credibility, weighted credibility and transition thresholds) are in L e 5 . This section first describes the considered example, presenting the given information stream and credulity stances. It then discusses the results obtained by three cross-checking operators: the immediate, the cumulative and the order independent ones.

↓ ↑ ↓ ↑ κ 0 1 τ3 κ 1 0 τ3 κ 0 1 τ2 κ 1 0 τ4 κ 0 2 τ4 κ 2 0 τ4 κ 0 2 τ3 κ 2 0 und κ 1 2 τ3 κ 3 0 und κ 1 2 τ2 κ 3 0 und κ 0 3 und κ 4 0 und κ 0 3 τ3 κ 4 0 und κ 1 3 τ4 κ 2 1 τ3 κ 1 3 τ2 κ 2 1 τ3 κ 2 3 τ3 κ 3 1 τ4 κ 2 3 τ1 κ 3

A. Information Stream

The example we use is defined only in terms of the model: only evaluations are given, the actual facts and story are left out. The evaluation is that of a piece of information which, before the credibility step, is rated τ 2 , i.e. possible. A contradictory stream containing six homologous pieces of information is then observed, as indicated in Table III. Each comes with its credibility score, that is its confirmation degree to the original piece of information, named τ β and its own level of trust, written τ γ . To simplify further, Table III does not give τ γ , only the result of its aggregation with τ β , i.e. the weighted credibility value g(τ β , τ γ ).

The information stream is made of six pieces of information: the first a confirmation (τ β ≥ τ M -1

2

) with a low influence (g(τ β , τ γ ) = τ 1 ). The rest of stream alternates between invalidations and confirmations.

B. Considered User Credulity Stances

Table IV describes the credulity stances of the two users. For each of them, the thresholds for both abating and increasing operators, respectively Fi , in the columns headed with ↓, and Fc , in the ↑ columns, are given.

Due to the constraints imposed to κ β α , some transitions are impossible. For instance, for user 1, the threshold to get from τ 4 to τ 2 is κ 2 4 = τ 4 . The constraints impose that κ 2 4 < κ 1

4

(see Section II-D3). Since τ 4 = τ M -1 , the maximal possible value, trust cannot directly go from τ 4 to τ 1 . Such cases with undefined transition are represented in the table as und. A more intuitive representation of the user credulity stances is provided by Fig. 4 and5. In order to preserve the expressiveness of the overall view of the underlying operator, undefined transitions are represented, with light arrows without thresholds. The two operators must be aligned for a full definition of G.

For user 1, the transition thresholds only take large values, either τ 3 or τ 4 , making it hard for him to change his mind. Only homologous information that either provide a clear confirmation or invalidation of the considered piece of information, i.e. with a high credibility activation, or with a high trust level, can lead him to update his trust level. Because his thresholds are symmetrical, i.e. κ γ α = κ α γ , he exhibits no inclination either way. For these traits, we call him mulish.

For user 2, the transition thresholds are not symmetrical in their increasing and abating versions. For instance, κ 4 3 = τ 3 , whereas κ 3 4 = τ 1 . The induced conviction dynamics can be characterised as distrustful: the increasing thresholds are all greater than the abating ones, meaning that user 2 requests higher confirmations, as measured by the weighted credibility, to grant trust, than to withdraw it. In other words, he is more easily discouraged than he is convinced.

C. Immediate Credibility

The left columns in Table V show the evolution of the information score for both users during the progressive integration of the information stream using the immediate credibility approach. The first line corresponds to the original considered piece of information, each following line is numbered with the piece of homologous information's identifier, from Table III.

For user 1, the first four pieces of information have too low an influence and are thus insufficient to trigger a modification in his trust level. Indeed, their weighted credibility is lower than all transition thresholds starting from τ 2 , i.e. κ γ 2 for all γ. Only the confirmation i 5 reaches the minimal value, leading the user to increase his evaluation to τ 3 , i.e. likely. User 2 is more sensitive and reacts to the invalidations i 2 and i 4 . Contrary to user 1, as he is easily put off and disinclined to trust, he does not take into account the confirmation i 5 . As a consequence, his final evaluation is the minimal value, i.e. τ 0 or unlikely.

D. Cumulative Credibility

The right columns in Table V show the evolution of the information score for both users during the progressive integration of the information stream with a cumulative credibility approach. Two additional columns for each user show the evolution of the accumulators.

Contrary to the immediate approach, even pieces of information with low credibility play a role in the evolution of the score. For this reason, user 1 does not wait for confirmation i 5 to update his level of trust, but starts after the second confirmation, i 3 . Since the considered implementation of the cumulative credibility resets accumulators at each evolution, the memory of the τ 2 doubts is lost. The next homologous information to have an impact being i 5 , the final score is the maximum, τ 4 , a higher score than in the case of immediate credibility. User 1 is less sceptical than previously.

User 2, more amenable to a change of heart than user 1 bur harder to convince reproduces the same behaviour as in the immediate case. In order for him to exhibit a different behaviour, the information stream has to be reordered or a momentum preserving variant chosen.

E. Total Cumulative Credibility

In the total cumulative case, described above as order independent credibility, the decision is taken once the entire stream has been considered, that is only once all 6 pieces of information have been processed. In the example above, since the accumulators are not reinitialised, adding the weighted credibility of confirmations and invalidations respectively, they end up with the same value g + = g -= τ 4 . Note that, since the decision is postponed and the accumulators augmented, transitions may skip more trust levels than with previous operators.

Because user 1 has a symmetrical strategy, the order in which Fc and Fi are applied does not influence the final result: in both cases, the output score is τ 2 , i.e. identical to the initial score, as g + = g -. User 2, more distrustful than user 1, is discouraged more easily than he is convinced. In the present case, where arguments in favour are as convincing as those against, the order in which they are taken into account determines his final trust level. If contradictions are integrated first, he finds the piece of information doubtful, with a final score of τ 1 . If, however, confirmations are considered first and invalidations revealed afterwards, he does not believe the information at all and attributes it the minimal score τ 0 .

F. Impact of the Order on Trust Building

This section highlights the influence of the order in which the information are considered on the evolution of trust, thereby underlining the dynamic component of the trust building process. The homologous pieces of information considered are the same as in the original stream, described in Table III, only their order differs. This reordering can be seen in the first column of Table VI, the other columns group all evolutions of trust, for both users and both credibility approaches.

Even though both users end with the same final evaluations, for the immediate credibility, than they did before, the reordering changes the levels of trust visited during the evaluation process. This hints at the possibility of different results if another piece of information had been inserted in the stream.

The impact of the reordering becomes clearer when looking at the cumulative credibility. Indeed, neither user then finishes with same trust level as he did before. User 1, whom we refer to as mulish, is initially favourable to this new order, reaching the maximum level earlier. However, the final increasing lot of contradictions, and the ensuing necessity for a change of mind, have a larger negative impact than ever before, leaving him thinking the piece of information just possible. Distrustful user 2 is less sensitive to the final contradictory closing arguments. Interestingly, he never thinks the piece of information unlikely, this time, but twice finds it maximally extremely likely. He changes his mind more often than user 1 does, yet manages to have a higher score than before.

As a conclusion of the various studied cases, User 1 appears to be persuaded, to degrees varying between τ 2 and τ 4 , whereas

VI. CONCLUSION AND FUTURE WORKS

This paper offers a multivalued formalisation of the crosschecking step of a trust building process. Various operators are proposed which allow to model the dynamics of the process, different stances with regards to persuasion and sensitivities to the order arguments are delivered. Different variants of the operators offer different tools for information evaluation, to take into account all levels of persuasion momentum. Future works will include an experimental comparison of the model and its variants, needing to address the difficult task of making an evaluation protocol. Another perspective lies in the study of the proposed model in the framework of argumentation theory, from the point of view of debaters. Such a study would look at the ideal moment to take part in a debate and provide contradictory arguments, in the aim of maximising the trust level of the audience.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Graphical representation of 1's credulity stance, Fi on the left, Fc(right)

TABLE I

 I

	.	EXAMPLE OF LINGUISTIC LABELS ASSOCIATED TO TRUST
		LEVEL AND CREDIBILITY IN L e 5
	L e 5	Trust label	Credibility label
	τ4	extremely likely	totally confirmed
	τ3	likely	partially confirmed
	τ2	possible	insufficiently confirmed
	τ1	doubtful	partially contradicted
	τ0	unlikely	totally contradicted
	τ ?

TABLE ALTERNATIVE

 ALTERNATIVE

	RE-INITIALISATION SCHEMES TO PRESERVE TRUST MOMENTUM, κ β α IS THE TRIGGERED THRESHOLD
	Line #	Enthusiastic	Mistrustful	Eager sceptic
	9: 10:	g-← τ ? g+ ← g+	g-← g-κ β α g+ ← τ ?	g-← g-κ β α g+ ← g+
	16:	g-← τ ?	g-← g-	g-← g-
	17:	g+ ← g+ κ β α	g+ ← τ ?	g-← g+ κ β α

TABLE V .

 V SCORE EVOLUTION USING BOTH CREDIBILITY APPROACHES

		Immediate			Cumulative
		User 1	User 2		User 1			User 2
	id.			τ	g+	g-	τ	g+	g-
		τ2	τ2	τ2	τ ?	τ ?	τ2	τ ?	τ ?
	i1	τ2	τ2	τ2	τ1	τ ?	τ2	τ1	τ ?
	i2	τ2	τ1	τ2	τ1	τ2	τ1	τ ?	τ ?
	i3	τ2	τ1	τ3	τ ?	τ ?	τ1	τ2	τ ?
	i4	τ2	τ0	τ3	τ ?	τ2	τ0	τ ?	τ ?
	i5	τ3	τ0	τ4	τ ?	τ ?	τ0	τ3	τ ?
	i6	τ3	τ0	τ4	τ ?	τ2	τ0	τ3	τ2

TABLE VI

 VI User 2 remains unconvinced: this example shows how the model makes it possible to represent distinct credulity stances, providing a global formalism allowing to express very different behaviours and different adjustments to sensitivities.

		.	SCORE EVOLUTION USING BOTH CREDIBILITY
	APPROACHES ON A REORDERED INFORMATION STREAM
		Immediate			Cumulative
		User 1	User 2		User 1			User 2
	id.			τ	g+	g-	τ	g+	g-
		τ2	τ2	τ2	τ ?	τ ?	τ2	τ ?	τ ?
	i5	τ3	τ4	τ3	τ ?	τ ?	τ4	τ ?	τ ?
	i2	τ3	τ2	τ3	τ ?	τ2	τ2	τ ?	τ ?
	i1	τ3	τ2	τ3	τ1	τ2	τ2	τ1	τ ?
	i3	τ3	τ3	τ4	τ ?	τ ?	τ4	τ ?	τ ?
	i4	τ3	τ1	τ4	τ ?	τ2	τ2	τ ?	τ ?
	i6	τ3	τ0	τ2	τ ?	τ ?	τ1	τ ?	τ ?


Abdelmoula Mustapha

Philippe Refait

Sidi El

Hassane Drissi

J P Mihe

Jean-Marie R Génin

Conversion electron Mossbauer spectroscopy and X-Ray diffraction studies of the formation of carbonate-containing green rust one by corrosion of metallic iron in NaHCO3 and(NaHCO3+NaCl) solutions