Didier Dubois

Allel Hadj-Ali

Henri Prade

Paul Sabatier

Fuzzy Qualitative Reasoning with Words

Introduction

Computing in its traditional sense involves manipulation of numbers for the most part. Symbolic computation has been developed in the meantime especially in Artificial Intelligence, but often without preserving the capability of interfacing the symbolic processing of information with numerical data. By contrast, humans employ mostly words and qualitative descriptions when providing assessments of situations, or reasoning about complex physical or human systems, even if a part of the data refers to numerical scales. For a long time, fuzzy sets have been advocated by [START_REF] Zadeh | Outline of a new approach to the analysis of complex systems and decision processes[END_REF] as a methodology for interfacing numerical data about the world (which are often imprecisely perceived), with linguistic categories or classes used in reasoning. In particular, it is the case for fuzzy rules or fuzzy algorithms where words are interpreted as the labels of fuzzy sets. There are different kinds of situations where the underlying reasoning process can be handled within the framework of a limited vocabulary. When developing his approach to approximate reasoning, [START_REF] Zadeh | A theory of approximate reasoning[END_REF] was already separating the combination/projection machinery acting at the levels of the fuzzy set membership functions, from the knowledge representation and linguistic approximation steps which respectively turn the linguistic input information into possibility distributions and restate the obtained conclusions into a prescribed vocabulary understandable by the user. The fuzzy set framework should supply the right tools for mixing a symbolic treatment to be performed at the level of the linguistic labels with facilities for permanently interfacing them with their fuzzy set semantics along the reasoning process. Indeed, fuzzy sets restore gradual transitions between categories on continuous universes, which would be lost if the interface between labels and data were provided by crisp sets only. In this paper, we investigate the feasibility of such a view on a particular type of reasoning: the order of magnitude reasoning in terms of closeness and negligibility relations. The same line of research has been explored on more general patterns of approximate reasoning by [START_REF] Dubois | Two views of approximate reasoning[END_REF].

Qualitative reasoning, handled at the level of comparative relations, may be, viewed as a simple illustration of the idea of computing with words recently emphasized by [START_REF] Zadeh | Fuzzy logic = Computing with words[END_REF][START_REF] Zadeh | From computing with numbers to computing with words-From manipulation of measurements to manipulation of perceptions[END_REF]. As the expression suggests, 'computing with words' is a methodology in which words are used in place of numbers for computing and reasoning. As pointed out by [START_REF] Zadeh | Fuzzy logic = Computing with words[END_REF], there are two major imperatives for computing with words. "First, computing with words is a necessity when the available information is too imprecise to justify the use of numbers, and second, when there is a tolerance for imprecision which can be exploited to achieve tractability, robustness, low solution cost, and better report with reality. Exploitation of the tolerance for imprecision is an issue of central importance in computing with words", [START_REF] Zadeh | Fuzzy logic = Computing with words[END_REF] wrote. A key idea underlying computing with words is that it involves a fusion of symbolic processing and computation with fuzzy variables. The following summarizes Zadeh's main points. Computing with words is based on a granulation process, which generally serves as a way of achieving data compression. A granule g is a fuzzy set of points drawn together by similarity. A word w (e.g., young) is a label of granule g. In computing with words, a granule g which is a denotation of a word w is viewed as a fuzzy constraint on a variable. The rules of inference in fuzzy logic are employed to propagate fuzzy constraints from premises to conclusions. Thus, the techniques of computing with words underlie, in one way or another, almost all the apparatus of fuzzy logic.

This paper focuses on a particular type of reasoning where computation can be handled at the word level, as it will be shown: qualitative reasoning about relative orders of magnitude, where fuzzy constraints expressing closeness or negligibility relations have to be propagated. Building inference systems endowed with the ability to reason about physical systems in the same way as humans -or at least engineers-do, has led to the development of qualitative reasoning [START_REF] Kuipers | Qualitative Reasoning -Modeling and Simulation with Incomplete Knowledge[END_REF]MQ&D Project, 1995;[START_REF] Travé-Massuyès | Le Raisonnement Qualitatif pour les Sciences de l'Ingénieur[END_REF]. The main aim of qualitative reasoning is both to address the need to deal with physical systems where some magnitudes are not easy to quantify (numerical data are not available), and to be able to reason at a qualitative or symbolic level (for example, reasoning directly in terms of orders of magnitude). The last decade has seen significant progress towards the development of formal methods for qualitative reasoning about the behavior of physical systems. The simplest formalism used in qualitative reasoning is based on the sign algebra (-, 0, +); see for instance [START_REF] Dormoy | Nouvelles méthodes de calcul qualitatif[END_REF][START_REF] Kuipers | Qualitative Reasoning -Modeling and Simulation with Incomplete Knowledge[END_REF]. Such models are enough to represent the sign of quantities and the direction in which each quantity affects another quantity. Information about magnitudes, or even relative orders of magnitude is not represented. As a consequence, the sign-based approach has too limited an expressive power in some practical cases to be widely applicable. A major limitation lies in the fact that often the sign of the result of an operation can be determined only if the order of magnitude of the involved parameters are considered. The only knowledge of the sign of the quantities involved in a mathematical expression often leaves completely indeterminate the sign of the result. Reasoning with relative orders of magnitude is an attempt towards overcoming these limitations. See [START_REF] Raiman | Order of magnitude reasoning[END_REF] for a general discussion. This type of qualitative reasoning corresponds to a particular form of commonsense reasoning where ideas of closeness, comparability and negligibility are involved. The first attempt to formalize such reasoning appeared with the formal system FOG, proposed by [START_REF] Raiman | Order of magnitude reasoning[END_REF][START_REF] Raiman | Le raisonnement sur les ordres de grandeur[END_REF]. FOG is based on three basic relations expressing the relations 'negligible in relation to' (Ne), 'close to' (Cl), and 'has the same sign and order of magnitude as', i.e., 'is comparable to' (Co). FOG includes one axiom and 31 inference rules, which allow for deduction from pieces of knowledge expressed in terms of relative orders of magnitude. This set of rules was proved to be consistent by giving an interpretation to the three relations in the framework of Non-Standard Analysis. Note that FOG handles relative orders of magnitude through a purely symbolic computation process. Nevertheless, no numerical interpretation is provided for the symbolic computation of orders of magnitude in the FOG's approach which thus does not allow for the incorporation of numerical values. This drawback is remedied in the system O(M) proposed by Mavrovouniotis andStephanopoulos (1987, 1988), where the primitive relations ('much smaller than', 'moderately smaller than', 'slightly smaller than', 'exactly equal', 'slightly larger than', 'moderately larger than', 'much larger than') are interpreted in terms of the location of the quotient of the two compared quantities in some interval. All these intervals are disjoint, and defined via a unique parameter which is fixed according to the application context. The fact that this parameter is fixed creates some problem since the result of the (repeated) composition of the primitive relations cannot always be expressed by a primitive relation if we want to perfectly adhere to the interval semantics. Indeed for instance, if x/y ∈ [a, b] and y/z ∈ [a, b], with a > 0, then x/z ∈ [a 2 , b 2 ], and [a 2 , b 2 ], or more generally [a n , b n ], are not necessarily included into an interval corresponding to the semantics of a primitive relation. [START_REF] Dague | Symbolic reasoning with relative orders of magnitude[END_REF] also provides an attempt to give a numerical interpretation to FOG-like relations manipulated in a formal way. However, FOG or O(M) do not take into account the fact that the satisfaction of relations such as Ne, Cl, or "much larger than", by two numerical values is often a matter of degree. Moreover FOG does not cope with the fact that it is also context-dependent. FOG does not acknowledge either that a relation of closeness (resp. negligibility) is usually transitive in a weak (resp. strong) sense (which will be made precise in the following), rather than just transitive (Dubois and Prade, 1989a). The same type of limitation applies to O(M) as well. The modeling of relations Ne, Cl and Co by means of fuzzy relations constitutes an appropriate framework for solving these problems.

The paper is organized as follows. Section 2 gives the necessary background on the formal system FOG. Section 3 introduces fuzzy equality and inequality relations (such as 'approximately equal', 'much larger than') modelled in terms of difference, and presents a set of inference rules based on these relations. We show that the composition of such relations reduces to simple fuzzy arithmetic operations. However, a simple example suggests that the modelling in terms of difference may be more appropriate for temporal relations. In Section 4, we discuss the modeling of the basic relations of FOG using fuzzy relations in terms of ratios of values. New inference rules, based on relations Ne and Cl, are established. The composition of the fuzzy relations representing relations Ne and Cl also reduces to simple arithmetic operations on fuzzy numbers which play the role of parameters underlying the semantics of Ne and Cl. Results of inferences are always expressed in terms of the relations Ne and Cl, but based on fuzzy parameters which are symbolically computed from the fuzzy parameters underlying the relations appearing in the premises. This machinery contributes to provide a standard numerical semantics for the symbolic computation performed by the FOG's rules, as well as by the new inference rules which are proposed. Illustrative examples for solving equations, in an approximate way, are presented in Section 5, while the concluding remarks point out some problems which are faced for the development and the efficient implementation of the approach.

The Formal System FOG

The FOG system is based on three relations, which capture closeness, negligibility and comparability and which are respectively denoted by Cl, Ne and Co. These three operators are:

x Ne y, which stands for x is negligible in relation to y; x Cl y, which stands for x is close to y, understood as (x -y) is negligible in relation to y; x Co y, which stands for x has the same sign and order of magnitude as y.

The underlying idea is that if x Ne z then y Ne z (when x Co y).

FOG has one axiom (A1: x Cl x) and 31 inference rules. Hereafter, we give some of these rules ([x] denotes the sign (+, -, 0) of the quantity x):

1) x Cl y ⇒ y Cl x 8) x Ne y ⇒ (x+y) Cl y 2) x Cl y ⇒ x Co y 9) (x+y) Cl z, y Ne x ⇒ x Cl z 3) x Cl y, y Cl z ⇒ x Cl z 10) x Ne y, z Cl t ⇒ x • z Ne y • t 4) x Cl y, y Ne z ⇒ x Ne z 11) x Ne y, z Ne x ⇒ x • z Ne y • t 5) x Co y, y Co z ⇒ x Co z 12) x • y Cl z • t, x Cl z, [x] ≠ 0 ⇒ y Cl t 6) x Ne y, y Ne z ⇒ x Ne z 13) x • y Cl z • t, x Ne z, [z] ≠ 0 ⇒ t Ne y 7) x Co y, y Cl z ⇒ x Co z 14) x • y Ne z • t, z Ne x, [x] ≠ 0 ⇒ y Ne t.
Note that both Cl and Co are equivalence relations (i.e., reflexive, symmetric and transitive), with Co coarser than Cl. The FOG rules, which can be justified from the point of view of Non-Standard Analysis, describe how the three relations work together and allow for the propagation of initial qualitative information. Most of the rules have an explicit intuitive interpretation and need no explanation. Note that they are not independent (e.g., rules 2 and 5 ⇒ rule 7). Note that FOG has been used successfully in the DEDALE system of analog circuit diagnosis [START_REF] Dague | Troubleshooting when modeling is the trouble[END_REF], and in macroeconomics [START_REF] Bourgine | Economics as reasoning on a qualitative model[END_REF].

Nevertheless, FOG has several limitations, as announced in the introduction:

• No standard numerical interpretation of the three relations Ne, Cl and Co is provided in this approach. It may create interface problems with genuine numerical values. It does not allow for the incorporation of quantitative information such as the assessment of numerical values. However, such information is often available and useful when solving engineering problems. FOG cannot use this type of information because it does not relate numbers to order of magnitude relations. Moreover FOG is unable to express a gradual change from one order of magnitude to another in the computation process, due to the absence of gradual transitions between the orders of magnitude [START_REF] Dague | Symbolic reasoning with relative orders of magnitude[END_REF].

• FOG does not take into account the fact that the extent to which two numerical values satisfy the relations Ne, Cl and Co is often a matter of degree and is also context-dependent (for example, the relation 'negligible' does not mean the same thing in a preliminary-design context and a detailed-design context).

• FOG cannot express that the relations are "weakly", or "strongly", transitive rather than just transitive in the usual sense. For instance, if 'a is close to b' and 'b close to c', it is not so true that we should conclude that 'a is close to c' in any case (at least with a too restrictive acceptance of "close"). This exemplifies the weakening of transitivity. In order to avoid undesirable transitivity effects and obtain results at least approximately valid with respect to a numerical semantics, Raiman is obliged to introduce arbitrary limitations on the chaining of rules (by means of control techniques).

The use of fuzzy relations for modeling relations Ne, Cl and Co can solve the above problems, and provide numerical semantics to the symbolic calculus of orders of magnitude performed by the FOG system.

Fuzzy Equalities and Inequalities

In this section, we consider particular fuzzy relations which are of interest in practice in some forms of qualitative reasoning. Approximate equalities and strong inequalities (e.g., "much greater than") are examples of binary fuzzy relations which can be easily handled using fuzzy arithmetic techniques. An approximate equality can be modelled by a fuzzy relation E of the form µ E (x,y) = µ L (|x -y|). For instance:

µ E (x,y) = 1 if |x -y| ≤ δ; µ E (x,y) = 0 if |x -y| ≥ δ + ε; µ E (x,y) = (δ + ε -|x -y|) / ε otherwise (1)
where δ and ε are respectively positive and strictly positive parameters which modulate the approximate equality. Then the approximate equality of quantities a and b (in the sense of E) will be denoted by: a

-b ∈ L (2)
with the following intended meaning: the possible values of the difference a -b are restricted by the fuzzy set L. Here L is a fuzzy set centered in 0, i.e., µ L (d) = µ L (-d), or if we prefer µ E (x,y) = µ E (y,x). See Figure1. It ensures the symmetry of the approximate equality relation.

Classical equality is recovered for δ = 0 and ε → 0. We shall write equivalently (a,b)

∈ E(δ,ε) (⇔ (b,a) ∈ E(δ,ε)). Moreover (a,b) ∈ E(δ,ε) ⇔ (a -b, 0) ∈ E(δ,ε).
Similarly a more or less strong inequality can be modelled, for instance, by a relation I of the form:

µ I (x,y) = 1 if x ≥ y + λ + ρ; µ I (x,y) = 0 if x ≤ y + λ; µ I (x,y) = (x -y -λ) / ρ otherwise (3)
with λ ≥ 0 and ρ > 0. A constraint of the form 'a is slightly / moderately /…/ much larger than b' (in the sense of I) can then be written, with µ I (x,y) = µ K (x -y):

a -b ∈ K (4)
where K is a fuzzy interval such that K = [K,+∞), where µ [K,+∞) (t) = sup s≤t µ K (s)). It means that K identifies itself with the set of values equal or greater than a value restricted by K. See Figure1. According to the values of parameters λ and ρ, the modality which specifies how much a is larger than b may be linguistically expressed by "slightly", "moderately", "much",…, in a given context. When λ = 0 and ρ → 0 the usual inequality relations '>' or '≥' are recovered. We shall write (a,b) ∈ I(λ,ρ) equivalently to a -b ∈ K.

The composition of two fuzzy relations E and I, with µ I (x,y) = µ K (x -y) et µ E (y,z) = µ L (y -z) is defined by: ∀x, ∀z, µ EôI (x,z) = sup y∈Y min(µ K (x -y), µ L (y -z)) = sup s,t min(µ K (s), µ L (t))

x-z=s+t = µ K⊕L (x -z),

where ⊕ denotes the addition extended to fuzzy intervals (e.g., [START_REF] Dubois | Fuzzy numbers: An overview[END_REF]. Indeed arithmetic operations are extended to fuzzy quantities in the following way:

µ K©L (u) = sup s,t min(µ K (s), µ L (t)), u=s * t
where © denotes the extension of the arithmetic operation * to fuzzy sets of the real line.

Note that if K and L have trapezoidal membership functions, i.e., K = (k1, k2, k3, k4) where [k1, k4] is the support of µ K and [k2, k3] is the core of µ K , and L = (l1, l2, l3, l4), with a similar understanding of the components of the 4-tuple, then K ⊕ L = (k1 + l1, k2 + l2, k3 + l3, k4 + l4) et K Á L = (k1 -l4, k2 -l3, k3 -l2, k4 -l1). In the Figure 1

, L = (-δ-ε, -δ, δ, δ + ε), Κ = (λ, λ + ρ, +∞, +∞) and L ⊕ K = (λ -δ -ε, λ + ρ -δ, +∞, +∞).
For more details about fuzzy arithmetic, see [START_REF] Dubois | Fuzzy numbers: An overview[END_REF].

1 -δ-ε -δ 0 δ δ+ε λ λ+ρ L K λ-δ-ε λ+ρ-δ x-y ⊕ L K ⊕ Figure 1
Let us take an example. If we know for instance that: 'a is approximately equal to b' (i.e., a -b ∈ L) and that 'b is much greater than c' (i.e., b -c ∈ K), using (5) we can deduce that a -c ∈ L ⊕ K. This result is represented in Figure 1. Symbolically, this inference can be written:

(a,b) ∈ Ε(δ,ε) and (b,c) ∈ I(λ,ρ) ⇒ (a,c) ∈ I(λ -δ -ε, ρ + ε) (6)
which means that if a and b are approximately equal (in the sense of the approximate equality relation E defined with parameters δ et ε) and b is (more or less) larger than c (in the sense of the fuzzy inequality relation I defined with parameters λ and ρ) then we infer that a is (more or less) larger than c (in the sense of of the fuzzy relation I defined with parameters λδε and ρ + ε). Note that I(λ -δε, ρ + ε) means 'larger than' only if λδε > 0, strictly speaking. Otherwise if λδε ≤ 0, it is at least somewhat possible that a is not greater than c, when (a,c) ∈ I(λ -δε, ρ + ε); and it is fully possible that a is not greater than c, if λ + ρδ ≤ 0. Note also that (a,b) ∈ I(λ,ρ) ⇔ (a -b, 0) ∈ I(λ,ρ).

In the same way, we can justify the following inference rules:

(a,b) ∈ Ε(δ,ε) and (b,c) ∈ E(δ',ε') ⇒ (a,c) ∈ E(δ + δ', ε + ε'), ( 7
) (a,b) ∈ I(λ,ρ) and (b,c) ∈ I(λ',ρ') ⇒ (a,c) ∈ I(λ + λ', ρ + ρ'). ( 8
)
Other rules expressing the behavior of fuzzy relations E and I with respect to arithmetic operations can be easily derived. For instance, we have

(a,b) ∈ Ε(δ,ε) ⇒ (a + c, b + c) ∈ E(δ,ε) (9) (a,b) ∈ I(λ,ρ) ⇒ (a + c, b + c) ∈ I(λ,ρ) (10) (a,b) ∈ Ε(δ,ε) and (c,d) ∈ E(δ',ε') ⇒ (a + c, b + d) ∈ E(δ + δ', ε + ε') (11) (a,b) ∈ I(λ,ρ) and (c,d) ∈ I(λ', ρ' ) ⇒ (a + c, b + d) ∈ I(λ + λ', ρ + ρ' ) (12) (a,b) ∈ Ε(δ,ε) and (c,d) ∈ I(λ,ρ) ⇒ (a + c, b + d) ∈ I(λ -δ -ε,ρ + ε) (13) (a + b, c) ∈ Ε(δ,ε) and (a,b) ∈ I(λ,ρ) ⇒ (a, c -a) ∈ I(λ -δ -ε, ρ + ε) (14A) (a + b, c) ∈ E(δ,ε) and (a,b) ∈ I(λ,ρ) ⇒ (c -b, b) ∈ I(λ -δ -ε, ρ + ε) (14B) (a + b, c) ∈ I(λ,ρ) and (c,a) ∈ I(λ',ρ') ⇒ (b,0) ∈ I(λ + λ', ρ + ρ') (15) (a + b, c) ∈ Ε(δ,ε) and (a,c) ∈ E(δ',ε') ⇒ (0,b) ∈ E(δ + δ', ε + ε') (16) (a + b, c) ∈ E(δ,ε) and (c,a) ∈ I(λ,ρ) ⇒ (b,0) ∈ I(λ -δ -ε, ρ + ε) (17) (a -c, b) ∈ Ε(δ,ε) and (c,a) ∈ E(δ',ε') ⇒ (0,b) ∈ E(δ + δ', ε + ε') (18) (a + b, c + d) ∈ E(δ,ε) and (c,a) ∈ E(δ',ε') ⇒ (b,d) ∈ E(δ + δ', ε + ε') (19) (a + b, c + d) ∈ I(λ,ρ) and (c,a) ∈ E(δ,ε) ⇒ (b,d) ∈ I(λ -δ -ε, ρ + ε) (20) (a + b, c + d) ∈ I(λ,ρ) and (c,a) ∈ I(λ',ρ') ⇒ (b,d) ∈ I(λ + λ', ρ + ρ') (21)
Proof: Let us look at the proof of some of these inference rules.

• The left side of ( 11) can be written under the form, with L' = (-δ'-ε', -δ', +δ', δ' + ε')

sup x,y,z,t min (µ E(δ,ε) (x,y), µ E(δ',ε') (z,t)) u=x+z; v=y+t = sup x,y,z,t min (µ L (x -y), µ L' (z -t)) = µ L⊕L' (u -v). u-v=(x-y)+(z-t)
This last result implies that (a + c, b + d) ∈ E(δ + δ', ε + ε').

• Rule ( 17) can be established noticing that sup x,z min(µ E (x + y, z), µ I (z,x))

= sup x,z min(µ L (x + y -z), µ K (z -x))

= µ L⊕K (y), observing that y=(x+y-z)+(z-x).

Thus, we conclude that (b,0) ∈ I(λ -δε, ρ + ε) when (a + b, c) ∈ E(δ,ε) and (c,a) ∈ I(λ,ρ). Note that it is certain that b > 0 only if λδε > 0.

• Now, considering rule (20), the composition of relations (a + b, c + d) ∈ I(λ,ρ) and (c,a) ∈ E(δ,ε) is given by

sup x,z min(µ I (x + y, z + t), µ E (z,x)) = sup x,z min(µ K ((x + y) -(z + t)), µ L (z -x)) = µ K⊕L (y -t), observing that y-t=[(x+y)-(z+t)]+(z-x).
This means that (b,d) ∈ I(λ -δε, ρ + ε).

The rules ( 9)-( 21) are not independent. ( 9)-( 10) are particular cases of ( 11)-( 12) replacing (c,d) by (c,c), noticing that (c,c) ∈ E(0,0) and (c,c) ∈ I(0,0). Similarly, ( 16) and ( 15) are particular cases of ( 19) and ( 21) respectively for d = 0. Rule (18) can be obtained from ( 16 As it can be seen, arithmetic computations on fuzzy numbers are mirrored at a symbolic level by inference rules which exhibit formal manipulations of the parameters underlying the approximate equality and inequality relations. The fuzzy number-based interpretation of these relations provides a natural semantics, and interfaces the symbolic level with the numerical one. From these inference rules, new conclusions can be symbolically drawn, still keeping the ability to interpret the obtained result numerically. For instance, from (7) and the equivalence (a + b, c)

∈ E(δ,ε) ⇔ (a, c -b) ∈ E(δ,ε), we can conclude for instance that if (a -c, b) ∈ Ε(δ,ε) and (d + b, c) ∈ E(δ',ε') then (a -c, c -d) ∈ E(δ + δ', ε + ε').
Lastly, let us recall that this kind of fuzzy relations (i.e., relations E and I) can offer a useful setting for representing and processing temporal knowledge (Dubois and Prade, 1989b). For instance • Knowing that the date 'b' takes place "much after" the date 'a' is represented by π By the transitivity rule (7), we obtain (a,c) ∈ E(δ + δ', ε + ε'). Since E is symmetric, this last result implies that (c,a) ∈ E(δ + δ', ε + ε'). Rule (20), applied to this last relation and to the relation in (iii), enables us to conclude that

(b,d) ∈ I[λ -(δ + δ') -(ε + ε'), ρ + (ε + ε')].
This means that date b is "much after" date d, where the meaning of "much" is modelled by means of parameters λ' = λ -(δ + δ') -(ε + ε') and ρ' = ρ + (ε + ε'), provided that λ' > 0.

Symbolic reasoning with fuzzy negligibility and closeness relations

Modeling closeness and negligibility

In the previous section, we have seen that some relational order of magnitude information can be expressed in practice by fuzzy relations whose membership functions depend on difference of values. If we prefer, we can also work in terms of the ratio x/y, rather than in terms of the difference x -y, especially when both x and y are positive. Then an approximate equality is modelled by a fuzzy relation µ E (x,y) = µ M (x / y) where M is a fuzzy interval such as µ M (1) = 1, i.e., M represents a "fuzzy one". More or less strong inequality relations can be obtained by sustituting x / y to x -y in definition (3) of µ K , i.e., µ I (x,y) = µ K (x / y). The use of ratios look natural for the modeling of negligibility relations. When µ K and µ L are functions of x / y (instead of x -y), we obtain µ IôE (x,z) = µ K⊗L (x / z) instead of ( 5), where ⊗ denotes the product extended to fuzzy numbers. In the following, we shall omit '⊗' when writing products.

As pointed out by Dubois and Prade (1989a), the idea of relative closeness (Cl) can be captured by the following approximate equality relation between numbers x and y:

µ Cl (x,y) = µ M (x / y) (22)
where the characteristic function µ M is such that µ M (1) = 1 (since x is close to x), µ M (t) = 0 if t ≤ 0 (since two numbers which are close should have the same sign), and µ M (t) = µ M (1 / t) in order to ensure that µ Cl (x,y) = µ Cl (y,x). In other words, M is a fuzzy interval which restricts values around 1 and which is equal to its "inverse", M -1 which is defined by µ M -1 (t) = sup t=1/s µ M (s) = µ M (1 / t). Thus, we have the semantic equivalence

(a,b) ∈ Cl ⇔ (b,a) ∈ Cl. ( 23
)
Strict equality is recovered for M = 1, i.e., µ 1 (x/y) = 1 if x = y and µ 1 (x/y) = 0 otherwise.

Raiman, in his inference system FOG, relates the ideas of closeness and of negligibility in the following way: 'a is close to b' is equivalent to 'a -b is negligible in comparison with b'. In other words, 'a is negligible in comparison with b' if and only if 'a + b is close to b' (see rule 8 of FOG). It leads to define the extent to which x is negligible in comparison with y, by µ Ne (x,y) = µ M ((x + y) / y) = µ M (1 + x / y).

(24)

By definition, if a, b, c > 0 then the following equivalence hold, ∀c:

(a,b) ∈ Cl ⇔(ac,bc) ∈ Cl, (a,b) ∈ Ne ⇔ (ac,bc) ∈ Ne.
The third relation Co is such that (see rule 2 of FOG): a Cl b ⇒ a Co b, which means that comparability is a less restrictive notion than closeness. Note that it is desirable to have not only the fuzzy relation inclusion Cl ⊆ Co but also to express that a and b are no longer comparable (in the sense of Co) when a becomes negligible in comparison with b, or b becomes negligible in comparison to a. This leads to the definition:

((a,b) ∈ Co ⇔ ¬ [(a,b) ∈ Ne or (b,a) ∈ Ne]) i.e., µ Co (x,y) = 1 -max(µ Ne (x,y), µ Ne (y,x)) = min(1 -µ M (1 + x / y), 1 -µ M (1 + y / x)). ( 25
)
However, in the rest of the paper, we only consider rules involving the more basic relations Cl and Ne.

Inference rules with Cl and Ne relations

In the following we show that it is possible to develop inference rules where the fuzzy sets which parametrize the closeness and negligibility relations are manipulated at a symbolic level. The use of fuzzy relations can provide, as we shall see, an appropriate numerical semantics to the order of magnitude symbolic calculus performed by the formal system FOG. From now on, we use the notations Cl[M] and Ne[M] in order to express that Cl (resp. Ne) is defined from a membership function µ M by (22) (resp. ( 24)). Using fuzzy arithmetic, the following inference rules can be established (Dubois andPrade, 1989a, 1991):

(a,b) ∈ Ne[M] ⇔ (a + b, b) ∈ Cl[M] (26) (a,b) ∈ Cl[M] ⇔ (a -b, b) ∈ Ne[M] (27) (a,b) ∈ Ne[M] ⇔ (-a,b) ∈ Ne[2 Á M] (28) (a, a + b) ∈ Ne[M] ⇔ (b, a + b) ∈ Cl[2 Á M] (29) (a,b) ∈ Cl[M] and (b,c) ∈ Cl[N] ⇒ (a,c) ∈ Cl[MN] (30) (a,b) ∈ Ne[M] and (b,c) ∈ Ne[N] ⇒ (a,c) ∈ Ne[(M Á 1)(N Á 1) ⊕ 1] (31) (a,b) ∈ Cl[M] and (b,c) ∈ Ne[N] ⇒ (a,c) ∈ Ne[M(N Á 1) ⊕ 1] (32) (a,c) ∈ Cl[M] and (b,a) ∈ Ne[N] ⇒ (a + b, c) ∈ Cl[MN] (33) (a,c) ∈ Cl[M] and (b,c) ∈ Ne[N] ⇒ (a + b, c) ∈ Cl[M ⊕ N Á 1] (34) (a,c) ∈ Ne[M] and (b,c) ∈ Ne[N] ⇒ (a + b, c) ∈ Ne[M ⊕ N Á 1] (35) (a,b) ∈ Cl[M] and (c,d) ∈ Ne[N] ⇒ (a • c, b • d) ∈ Ne[M(N Á 1) ⊕ 1] (36) (a,b) ∈ Ne[M] and (c,d) ∈ Ne[N] ⇒ (a • c, b • d) ∈ Ne[(M Á 1)(N Á 1) ⊕ 1] (37) (a,b) ∈ Cl[M] and (c,d) ∈ Cl[N] ⇒ (a • c, b • d) ∈ Cl[MN] (38) (a + b, c) ∈ Cl[M] and (b,a) ∈ Ne[N] ⇒ (a,c) ∈ Cl[MN].
(39) Most of these rules have counterparts in the FOG system. For instance, the rules (30)-( 32) and ( 39) are the counterparts of the following rules of FOG: The fuzzy relations approach shows that these rules are only "qualitatively valid". The repeated use of some of these rules without control can lead to dubious conclusions. Fuzzy relations enable us to acknowledge that the relation Cl is "weakly transitive" (since for M = N, Cl[M] ⊆ Cl[M 2 ], i.e., 'a may be less close to c' than 'a is close to b'), and that the relation Ne is "strongly transitive" (since for M = N, Ne[M] ⊇ Ne[(M Á 1)(M Á 1) ⊕ 1], i.e., 'a is more negligible in comparison with c' than 'a in comparison with b', and 'b with c').

In order to equip all the rules of FOG with numerical semantics, we propose new inference rules (not studied in Dubois and Prade (1989a)) involving the product of the considered quantities:

(a • b, c • d) ∈ Ne[M] and (c,a) ∈ Ne[N] ⇒ (b,d) ∈ Ne[(M Á 1)(N Á 1) ⊕ 1] (40) (a • b, c • d) ∈ Cl[M] and (c,a) ∈ Cl[N] ⇒ (b,d) ∈ Cl[MN] (41) (a • b, c • d) ∈ Cl[M] and (a,c) ∈ Ne[N] ⇒ (d,b) ∈ Ne[M(N Á 1) ⊕ 1] (42)
Proof:

• The composition of relations (a • b, c • d) ∈ Ne[M] and (c,a) ∈ Ne[N] can be written under the form:

sup x,z min (µ Ne[M] (x • y, z • t), µ Ne[N] (z,x)) = sup x,z min(µ M (1 + [x • y / z • t]), µ N (1 + z / x)) = sup u, v min(µ M (u), µ N (v)), letting u=1+[x•y/z•t] and v=1+z/x (u-1)(v-1)+1=1+y/t = µ [(MÁ1)(NÁ1)⊕1] (1 + y / t).
This implies that (b,d) ∈ Ne[(MÁ 1)(NÁ 1) ⊕ 1].

• In the same way, we can establish rules (41) and (42). Considering the rule (42), the composition of relations (a • b, c • d) ∈ Cl[M] and (a,c) ∈ Ne[N] can be written under the form:

sup x,z min (µ Cl[M] (x • y, z • t), µ Ne[N] (x,z)) = sup x,z min(µ M (x • y / z • t), µ N (1 + x / z)) = sup x,z min(µ M (z • t / x • y), µ N (1 + x / z)) = sup u, v min(µ M (u), µ N (v)), letting u=z•t/x•y and v=1+x/z u(v-1)+1=1+t/y = µ [M(NÁ1)⊕1] (1 + t / y).
This latter result implies that (d,b) ∈ Ne[M(NÁ 1) ⊕ 1]. y

The syntactic approximations of rules ( 40), ( 41) and ( 42) are given by the following FOG's rules: In the following, we show that it is possible to derive other symbolic inference rules which are not explicit in the formal system FOG. First, we start with simple rules whose premises are not compound, namely

(a, a • b) ∈ Ne[M] ⇒ (1 + b, b) ∈ Cl[M] if a ≠ 0 (43) (a, a • b) ∈ Cl[M] ⇒ (1,b) ∈ Cl[M] if a ≠ 0 (44) (a -b, a) ∈ Ne[M] ⇒ (b,a) ∈ Cl[2 Á M] (45) (a -b, a) ∈ Cl[M] ⇒ (b,a) ∈ Ne[2 Á M]. (46)
These rules can be easily proved by using the definitions of relations Cl and Ne given in ( 22) and ( 24) respectively.

More complex rules expressing the behavior of the relations Cl[M] and Ne[N] with respect to arithmetic operations (addition, subtraction and product) either in the premise or in the conclusion, can be derived as well

(a,c) ∈ Ne[M] and (b,a) ∈ Ne[N] ⇒ (a + b, c) ∈ Ne[(M Á 1)N ⊕ 1] (47) (a,b) ∈ Ne[M] and (c,b) ∈ Ne[N] ⇒ (a, b + c) ∈ Ne[(M Á 1)N ⊕ 1] (48) (a • b, c • d) ∈ Ne[M] and (a,c) ∈ Cl[N] ⇒ (b,d) ∈ Ne[(M Á 1)N ⊕ 1] (49) (a + b, c) ∈ Cl[M] and (b,c) ∈ Cl[N] ⇒ (a,c) ∈ Ne[M Á N ⊕ 1] (50) (a + b, c) ∈ Cl[M] and (b,c) ∈ Ne[N] ⇒ (a,c) ∈ Cl[M Á N ⊕ 1] (51) (a + b, c) ∈ Ne[M] and (b,c) ∈ Ne[N] ⇒ (a,c) ∈ Ne[M Á N ⊕ 1] (52) (a -b, c) ∈ Cl[M] and (b,a) ∈ Ne[N] ⇒ (c,a) ∈ Cl[M(2 Á N)] (53) (a -b, c) ∈ Cl[M] and (a,c) ∈ Cl[N] ⇒ (b,c) ∈ Ne[N Á M ⊕ 1] (54) (a -b, c) ∈ Ne[M] and (a,c) ∈ Cl[N] ⇒ (b,c) ∈ Cl[N Á M ⊕ 1] (55) (a -b, c) ∈ Cl[M] and (b,c) ∈ Ne[N] ⇒ (a,c) ∈ Cl[M ⊕ N Á 1]. ( 56
)
Proof: Let us look at the proof of some of these rules:

• Rule (47) can be established noticing that i.e., F = 100(N Á 1), which can be easily computed. This illustrates that fuzzy relations can provide a natural framework for interfacing symbolic information and numerical data. Generally, different results (some less restrictive than others) can be obtained from different chainings of inference rules, as suggested by the above example. The conjunctive combination of these partial results restricting the possible values of a ratio of interest enables us to refine the conclusion. Note that the most restrictive result would be obtained by applying the combination/projection method [START_REF] Zadeh | A theory of approximate reasoning[END_REF], because in this approach all the existing constraints are exploited. Nevertheless, this approach can be computationally untractable in practice, and no explanation can be provided for the obtained results using such a global handling of the constraints.

Example 3: Let us now consider a practical example of a counter-current heat exchanger as described in [START_REF] Mavrovouniotis | Formal order of magnitude reasoning in process engineering[END_REF]. The important parameters in the analysis of the device are the molar-heat KH and the molar-flowrate FH of the hot stream, and the molar-heat KC and the molar-flowrate FC of the cold stream. Four temperature differences are defined: DTH is the temperature drop of the hot stream, DTC is the temperature rise of the cold stream, DT1 is the driving force at the left end of the device, and DT2 is the driving force at the right end of the device. The two following equations hold

(e1) DTH -DT1 -DTC + DT2 = 0 (e2) DTH • KH • FH = DTC • KC • FC.
The first one is a consequence of the definition of the temperature differences, and the second one is the energy balance of the device. Let us take the following assumptions expressed as orders of magnitude relations

(i) (DT2, DT1] ∈ Cl[M] (ii) (DT1, DTH) ∈ Ne[N] (iii) (KC, KH) ∈ Cl[P].
The problem now is to deduce the relation between the parameters FC and FH. Rule ( 32

Conclusion

Fuzzy relations can provide an appropriate semantics for inference rules about relative orders of magnitude. Each rule has a precise interpretation in terms of fuzzy sets. The composition of relations Ne[M] and Cl[N] reduces to simple arithmetic operations on the fuzzy numbers M and N. Moreover, the fuzzy relations calculus enables us to reason about closeness and negligibility in a rigorous way without being obliged to introduce arbitrary limitations on the chaining of rules (by means of control techniques) in order to avoid undesirable transitivity effects for instance. The approach is faithful to the gradual nature of comparators. Another clear advantage of the use of fuzzy relations is that it enables us to interface numbers with symbolic information in a convenient way as hinted in example 1. Thus, it provides an example of computing with the words 'negligible' and ' close to' in a symbolic manner, in full agreement with the numerical semantics underlying these words. As briefly suggested in Section 3, a similar approach based on other fuzzy relations could be also developed in temporal reasoning (where approximate equalities or fuzzy ordering relations can apply to dates), or even in spatial resoning.

A noticeable feature of the approach is the separation of the symbolic processing of the labels of the inferred relations from the expression of their semantics in terms of their membership functions. It is why it is important to always check that the produced conclusions are semantically meaningful, e.g. that the obtained relation of closeness is not too permissive. If a relation Ne[P] of negligibility is rather obtained, we should make sure that it is in agreement with the intuitive semantics underlying this notion (e.g., µ P (2) = 0, otherwise x would be negligible with respect to itself, since µ P (1 + x / x) = µ P (2)). Indeed even if we start with relations having semantically meaningful parameters, it may happen that after several chainings of the rules, the result is no longer a closeness or a negligibilty relation semantically speaking, in spite of its appearance at the syntactic level.

Future works should deal especially with: i) the conception of a complete system of inference rules (for particular types of equations systems, for example, linear equations involving only addition and subtraction operations, and product by a constant); ii) the choice of a set of formal computation rules which enables us to write equations in a suitable way when necessary (since the form of equations can interfere with the possibility of applying inference rules); iii) to implement this system and the set of computation rules in order to obtain automated strategies for simplifying equations involving relative orders of magnitude relations and arithmetic expressions. This should be useful for solving numerical equations, in an approximate way, for physico-chimical systems, or for the qualitative calculus of probabilities [START_REF] Dubois | Semantic considerations on order of magnitude reasoning[END_REF][START_REF] Dubois | Qualitative reasoning with imprecise probabilities[END_REF].

Finally, it should be also emphasized that reasoning with relative orders of magnitude can be seen as a particular type of fuzzy Constraint Satisfaction Problems [START_REF] Dubois | Possibility theory in constraint satisfaction problems: Handling priority, preference and uncertainty[END_REF], which may provide a source of inspiration for the implementation. Thus, there is a balance to find between a minimal complete set of rules and a larger set of rules exhibiting some redundancy, but offering more possibilities of being applicable to a given set of constraints. See [START_REF] Hadj-Ali | Raisonnement sur les ordres de grandeurs relatifs avec des relations floues: Appliqué aux probabilités qualitatives[END_REF][START_REF] Dubois | Granular Computing with Fuzzy Closeness and Negligibility Relations[END_REF] for the study of a reduced set of rules from which all the rules provided in Section 4 can be derived. Moreover it is possible to approximate all these rules using symmetrical fuzzy numbers as parameters (e.g. in terms of MN, if both M and N are symmetrical); a method for establishing the inconsistency of a set of hypotheses w.r.t. a set of equations, is also hinted (see [START_REF] Hadj-Ali | Résolution approchée d'équations par le raisonnement qualitatif flou[END_REF].

  ) by noticing that (a -c, b) ∈ E(δ,ε) ⇔ (a, b + c) ∈ E(δ,ε) and (a,c) ∈ E(δ',ε') ⇔ (c,a) ∈ E(δ',ε'). Besides, rule (17) can be generalized into the rule (a + b, c + d) ∈ E(δ,ε) and (c,a) ∈ I(λ,ρ) ⇒ (b,d) ∈ I(λ -δε, ρ + ε). Then, letting a + b = C, c + d = A, c = A + B, a = C + D, we obtain (C,A) ∈ E(δ,ε) and (A + B, C + D) ∈ I(λ,ρ) ⇒ (-D, -B) ∈ I(λ -δε, ρ + ε) which is nothing but (20), noticing that (-d, -b) ∈ I(λ',ρ') ⇔ (b,d) ∈ I(λ',ρ') always holds. y

  (a,b) (x,y) = µ I (y,x), where I is defined by (3) and where the meaning of "much" is modelled by means of the parameters λ and ρ. π (a,b) is the joint possibility distribution expressing the precedence relations between dates a and b. • Knowing that dates 'a' and 'b' are "approximately equal" is represented by π (a,b) (x,y) = µ E (x,y), where E is defined by (1) and where the meaning of "approximately" is modelled by means of the parameters δ and ε. Let us consider an example of reasoning involving temporal knowledge, where we have the following information about dates a, b, c and d: i) dates a and b almost coincide (in the sense of E(δ,ε)); ii) dates b and c are "approximately equal" (in the sense of E(δ',ε')); iii) date a + b takes place "much after" date c + d (in the sense of I(λ,ρ)). Let us consider the question: What can be said about date b with respect to d? Information in (i)-(iii) can be represented respectively by the following relations: i) (a,b) ∈ E(δ,ε) or (b,a) ∈ E(δ,ε); ii) (b,c) ∈ E(δ',ε'); iii) (a + b, c + d) ∈ I(λ,ρ).

  (a,b) ∈ Cl and (b,c) ∈ Cl ⇒ (a,c) ∈ Cl, (a,b) ∈ Ne and (b,c) ∈ Ne ⇒ (a,c) ∈ Ne, (a,b) ∈ Cl and (b,c) ∈ Ne ⇒ (a,c) ∈ Ne, (a + b, c) ∈ Cl and (b,a) ∈ Ne ⇒ (a,c) ∈ Cl.

  (a • b, c • d) ∈ Ne and (c,a) ∈ Ne ⇒ (b,d) ∈ Ne (a • b, c • d) ∈ Cl and (a,c) ∈ Cl ⇒ (b,d) ∈ Cl (a • b, c • d) ∈ Cl and (a,c) ∈ Ne ⇒ (d,b) ∈ Ne.

  sup x,y min(µ Ne[M] (x,z), µ Ne[N] (y,x)) x+y=w = sup x,y min(µ M (1 + x / z), µ N (1 + y / x)) By applying rule (42), we can deduce that (d,b) ∈ Ne[M(N Á 1) ⊕ 1]. The relation (d / b, f / e) ∈ Cl[P] implies that (d • e, b • f) ∈ Cl[P]. Rule (42), based on this latter relation and (d,b) ∈ Ne[Q] with Q = M(N Á 1) ⊕ 1, enables us to conclude that (f,e) ∈ Ne[PM(N Á 1) ⊕ 1]. If we start with classical equalities, i.e., a • b = c • d and d / b = f / e, then letting M = 1 and P = 1, it leads to (f,e) ∈ Ne[N]. Now, if the parameter e = 100, then we can come up with an estimate of the parameter f under the form of a fuzzy number F, namely: µ F (x) = µ Ne[N] (x,100) = µ N (1 + x / 100) = µ [100(NÁ1)] (x)

Example 2 :

 2 (taken from Mavrovouniotis and Stephanopoulos (1988)) (d,b) ∈ Ne[M]; (b,c) ∈ Ne[N]; (e,g) ∈ Ne[P]; a -b -c -d = 0; a • e • f -c • g • h = 0. The question now is to compare the values of f and h. By applying rule (26), we get (d + b, b) ∈ Cl[M]. Substituting a -c to d + b (because a -b -c -d = 0), we obtain (a -c, b) ∈ Cl[M]. Rule (32), based on this latter relation and (b, c) ∈ Ne[N], enables us to deduce that (a -c, c) ∈ Ne[M(N Á 1) ⊕ 1]. This relation implies that (a,c) ∈ Cl[M(N Á 1) ⊕ 1]. Using rule (36), we get (a • e, c • g) ∈ Ne[Q(P Á 1) ⊕ 1], with Q = M(N Á 1) ⊕ 1. Since (a • e / c • g) = h / f, the pair (h,f) should satisfy the same fuzzy relation, i.e., (h,f) ∈ Ne[Q(P Á 1) ⊕ 1]. This example can be solved using other inference rules. For example, the chaining of rules (31), (35) and (27) enables us to obtain (a,c) ∈ Cl[R ⊕ N], with R = (M Á 1)(N Á 1); then using rule (36) we get (a • e, c • g) ∈ Ne[(R ⊕ N)(P Á 1) ⊕ 1] and finally (h,f) ∈ Ne[(R ⊕ N)(P Á 1) ⊕ 1].

  ), based on the relations (i) and (ii) enables us to obtain(DT2, DTH) ∈ Ne[M(N Á 1) ⊕ 1].The following inferences are easily obtained as well(DT1, DTH) ∈ Ne[N] ⇒ (-DT1, DTH) ∈ Ne[2 Á N](by rule (28)) (DT2, DTH) ∈ Ne[M(N Á 1) ⊕ 1] and (-DT1, DTH) ∈ Ne[2 Á N] ⇒ (DT2 -DT1, DTH) ∈ Ne[M(N Á 1) ⊕ 2 Á N] (by rule (35)).In the latter result, we can substitute DTC -DTH to DT2 -DT1 using equation (e1), we get(DTC -DTH, DTH) ∈ Ne[M(N Á 1) ⊕ 2 Á N].Then applying rule (27), we get (DTC, DTH) ∈ Cl[M(N Á 1) ⊕ 2 Á N]. Rule (38), based on this relation and relation (iii) enables us to conclude that (DTC • KC, DTH • KH) ∈ Cl[(M(N Á 1) ⊕ 2 Á N)P] and finally since DTC • KC / DTH • KH = FH / FC, see equation (e2), the pair (FH, FC) should satisfy the same fuzzy relation, i.e., (FH, FC) ∈ Cl[(M(N Á 1) ⊕ 2 Á M)P].

Rule (48) can be obtained similarly noticing that 1 + x y -1 y y + z + 1 = 1 + x y + z and using µ N 1 + z y = µ N y y + z .

• The result of rule ( 50) can be easily obtained noticing also that

Propagation of Relative Orders of Magnitude: Some Examples

In this section, we show that the previous inference rules can be useful for solving equations in an approximate way. Such equations involve arithmetic expressions and order of magnitude relations (expressed in terms of parametrized relations Cl and Ne).

Example 1: The following relations are supposed to be known to hold:

The problem is to compare the orders of magnitude of quantities e and f. The following relations can be easily obtained: Allel HADJ-ALI is a "Chargé de Cours" at the Computer Sciences Institute of Mouloud Mammeri University, Tizi-ouzou, Algeria. He received the Engineer and the "Magister" degrees in computer sciences from Mouloud Mammeri University in 1988 and 1991 respectively. He is currently preparing his "Doctorat d'Etat" degree, with D. Dubois and H. Prade as advisors. His current research interests include fuzzy logic, qualitative and approximate reasoning, constraint satisfaction problems, artificial intelligence and expert systems. He has already contributed papers on fuzzy expert systems and fuzzy qualitative reasoning.

Henri PRADE is a "Directeur de Recherche" at C.N.R.S. since 1988, and works as a Research Advisor at IRIT (Institut de Recherches en Informatique de Toulouse). He received a Doctor-Engineer degree from ENSAE, Toulouse (1977), his "Doctorat d'Etat" (1982) and the"Habilitation a Diriger des Recherches" (1986) both from Toulouse University. He is the co-author, with Didier Dubois, of two monographs on fuzzy sets and possibility theory published by Academic Press (1980) andPlenum Press (1988) respectively. He has contributed a great number of technical papers on uncertainty modelling and applications. He is the co-editor (with Didier Dubois and Ronald Yager) of a volume entitled "Fuzzy Information Engineering: A Guided Tour of Applications" published by Wiley in 1997, and of the "Handbooks of Fuzzy Sets Series" (Kluwer). He is co-editor-in chief of Fuzzy Sets and Systems and a member of the the Editorial Board of several technical journals including IEEE Transactions on Fuzzy Systems, the Artificial Intelligence J., the Inter. J. of Approximate Reasoning, the Inter. J. of Intelligent Systems, the J. of Intelligent


Carlos Arrambide Cruz

Sébastien Marie

Guilhem Arrachart
email: guilhem.arrachart@cea.fr

Stéphane Pellet-Rostaing

Selective extraction and separation of germanium by catechol based resins

come

Introduction

Germanium is a trace component in the Earth's crust and natural waters. It is used as a component of optical fibers, polymerization catalyst, element in infrared optics and electronics, and in many other applications. The wide use of Ge in the manufacture of electronic makes it an element of high technological value [1]. Germanium is recovered as a by-product of zinc or copper and zinc ores, it is also extracted from some coals deposits [START_REF] Depuydt | Germanium-based Technologies[END_REF]. The separation of germanium from acid solutions has received considerable attention because of its economic implications.

Hydrometallurgy appears to be the most attractive process for the recovery of germanium. This includes, the liquid-liquid extraction [START_REF] Cote | [END_REF][4][5][6], flotation [7,8] and the resin [9][10][11] separation methods. Polyols are known to form complexes with germanium [12]; therefore polymeric solids containing hydroxy groups, such as N-methylglucamine [9,13] and catechol derivatives [14][15][16][17] have been used for the preconcentration and separation of germanium. Catechol forms a negatively charged complex with germanium in solution [17][18][19][20][21] and that complexation with germanium and catechol forms a symmetric structure in three dimensions which is a more stable complex compared with the silicate ion [15,16]. In most of the recovery system using catechol, the germanium ion is complexed with catechol in a homogeneous system, and the resulting anionic germanium complex is then captured using an anion-exchange resin or membrane [15,17]. The predominant oxidation state of germanium is Ge(IV) however the Ge4+ ion is easily hydrolysed and is rarely existing in aqueous solutions [13]. Indeed, germanium is mainly found in the form of germanic acid, Ge(OH)4, and Ge-oxoanions in aqueous solutions. At pH lower than 4, germanium predominantly exist in non-complexed form and at higher pH the formation of a catechol complex is proposed according to Eq. (1)and illustrated in Fig. 1. [17,21] The selectivity of Ge towards Zn and Cu is important when considering the Ge recovery from zinc or copper and zinc ores. From the waste from the industry of semiconductor, optical fibers and solar panels, Ge is contained along with large concentration of silicate ion which implies a high selectivity against silicate ion using the hydrometallurgy process. In this process, the germanium-containing waste is dissolved into an alkaline or acid solution. The formo-phenolic resins, which result from the alkaline polycondensation of formaldehyde with a phenolic compound, are one of the best known types of chelating polymers [22,23]. The ion-exchange properties are provided by the presence of the phenolic OH groups, therefore a wide variety of phenolic-formaldehyde polymers that contain a variety of active chelating groups have been reported in the literature and used to sorb various heavy metals [24][25][26], lanthanide metal ions [27][28][29][30][31] and radionucleides [32][33][34][35][36]. Interestingly, to our knowledge, in the literature there is no example of the use of such resins for the selective extraction of Ge. Due to the selective complexation of germanium ions with catechol, and the potential of oxine for the extraction of germanium, we decide to explore the possibility to use formo-phenolic resins based on catechol and/or oxine (8-quinolinol/8-hydroxyquinoline) for the separation of Ge from Zn and Si. Also other phenolic precursor such as phenol and resorcinol have been explored.

Experimental

Resins synthesis

Catechol (C), 8-quinolinol or 8-hydroxyquinoline (Q), phenol (P), resorcinol (R), and formaldehyde (F) (37% aqueous solution) were obtained from Aldrich or Acros and used without further purification. The resins were synthesized by alkaline polycondensation of formaldehyde with phenolic compounds according to procedure adapted from the literature [37,38]. Typically, a phenolic compound/ formaldehyde/NaOH/H2O molar ratio of 1:2.5:1.5:80 were used with the objective to obtain a resol type resin resulting in insoluble and infusible polymeric materials. The homopolymerization of 8hydroxyquinoline using formaldehyde as a crosslinker in basic conditions gave a large amount of watersoluble materials. To overcome this drawback, the resins were prepared through a polycondensation reaction between 8-hydroxyquinoline and catechol using formaldehyde as crosslinker in the presence of an alkaline hydroxide (NaOH) as a catalyst.

The phenolic or mixture of phenolic precursors (1 mmol) were dissolved in an alkaline hydroxide solution (1.5 mmol of NaOH from a 6M solution). Water is added to reach the amount desired (80 mmol of H2O). The solution was stirred at room temperature, then the formaldehyde (2.5 mmol) was added using a syringe pump at 20 mL/h. During formaldehyde addition, the solution was maintained at 0 or 25 °C. After the addition, the mixture was stirred for 24 h at room temperature. After stirring, the mixture was heated in an oven in air at 100 °C for 96 h. After curing, the polymers were crushed by ball-milling and submitted to a washing step with 1M HCl/ H2O /1 M NaOH/ H2O cycles to remove the unreacted compounds and the oligomers which have not been well crosslinked. Finally, the resin was dried in air at 80 °C for 24 h. A list of synthesized resins is summarized in Table 1 and an example of their synthesis is illustrated in Fig. SI-1 (see supporting information).

Resin characterization

The ball-milling was performed on a Retsch mixer mill MM 200 using a Zr ball under strong horizontal stirring at 50 Hz for 10 min. The statistical study of the size of the particles was carried out using Fiji software from the treatment of optical microscopy images obtained from a Zeiss Axio Imager microscope. From the analysis of images, different treatments can be performed on the particles (thresholding, erosion/dilatation, etc.) to estimate the size and polydispersity of the particles.

FTIR spectra were recorded with a Perkin Elmer 100 spectrometer between 400 and 4000 cm -1 using an ATR crystal with 4 cm -1 resolution.

The 13 C solid-state MAS NMR spectra were recorded with a Varian VNMRS 300 solid spectrometer with MAS probe using 3.2mm outer diameter rotors with a spinning rate of 12 kHz (typical contact time: 2-3 ms, typical recycle delay: 5 s).

Thermal analysis (TGA) was performed with a Mettler Toledo Instrument under air flow with a heating rate of 10 °C/min from 25 to 950 °C. The moisture regain was determined from the TGA analysis after an heating after in air from 25 to 100 °C at 10 °C/min followed by an isothermal treatment of 30 min at 100 °C. The corresponding weight loss giving the percentage of water in the resin (see Table S1 supporting information).

For determination of total ion-exchange capacity, 0.1 g of resin in the H-form was equilibrated for 24 h with 20 mL of 0.1 N NaOH solution containing 5% NaCl. The amount of NaOH consumed in the H-Na exchange was determined by titrating the remaining NaOH in the supernatant with 0.1M HCl solution. Assuming a complete crosslinking and if all the OH groups are accessible in the polymer, the theoretical expected H-Na ion-exchange capacity of the polymer would be 8.9 meq/g and 15.6 meq/g for a hydroxybenzene and dihydroxybenzene resin respectively [37]. For the various resins around 50% of the phenolic group can undergo ion-exchange process. The ion-exchange capacities are summarized in Table S1 (see supporting information) and are close to those obtained in the literature for such resins [28].

Elemental analysis (Fig. SI-2 see supporting information) were carried out with an Elementar Vario Micro Cube instrument. The morphology of the resins was observed using an environmental scanning electron microscope (model FEI QUANTA 200 ESEM FEG). XEDS analyses were performed using a Bruker AXS X-Flash 5010 detector coupled with SEM to quantitatively determine the elementary atomic percentages.

Quantification of cation concentration in solution was conducted by inductively coupled plasma/atomic emission spectroscopy (ICP/AES) using a SPECTRO ARCOS spectrometer. The wavelengths for the measurements for the Ge (164.919, 206.866, 209.426, 219.871 and 265.118 nm), Zn (202.613, 213.856 and 334.502 nm), Si (152.672, 251.612 and 288.158 nm) and Cu (224.700, 324.754 and 327.396 nm) were chosen to avoid any spectral interference between the elements.

Batch contact with cation solutions

The metallic solutions were obtained after dilution with Milli-Q® water of from 1000 mg/L of ICP standard (in 1% HNO3) of germanium (Ge), and /or silicon (Si), zinc (Zn), and copper (Cu). For all solutions, the pH was measured by a pH meter (Metrohm 780 pH Meter) and found to be approximately 6. The concentrations were initially set at 10 ppm of Ge, Si and Zn in an attempt to avoid saturation of the functional sites.

Other experiments were conducted with a ten-fold higher concentration of competitors metals with respect to germanium: (i) 10 ppm of Ge with 100 ppm of Si; (ii) 10 ppm of Ge with 100 ppm of Zn; (iii) 10 ppm of Ge with 100 ppm of Zn and 100 ppm of Cu. The choice of concentrations was chosen to best represent the characteristic values of the effluents as well as allow their characterization through the ICP analyses after the extractions experiments. Even if the concentrations are not representative of those encountered in the mineral deposits, they make it possible to give trends on the extraction and selectivity performances in the context of germanium recovery. Batch contacts were generally run for 24 h. The mixtures were shaken (rotary mixer speed rate: 1 rotation/s). The solution to solid ratio was typically 1000 (1 g/L), although it was varied from 400 (2.5 g/L) to 100 (10 g/L) in certain experiments to determine the effect on the extraction efficiencies. After shaking the mixture for the specified time, they were filtered through a 0.2 μm filter and analysed by ICP to determine the residual metal concentration.

The distribution coefficient Kd (mL/g) was calculated using the following Eq. ( 1) in which Ci is the initial concentration of the metal ion in solution, Cf is the residual metal ion concentration, and V/m is the solution volume to solid-mass ratio (V=volume of the treated solution and m=mass of the resin used): Kd = ((Ci-Cf)/Ci)×V/m. The adsorption efficiency E (%) was calculated according to the equation E=((Ci-Cf)/Ci)×100.

The separation factor SF Ge/M was determined according to the equation SF Ge/M=EGe/EM. Kinetics experiments were run in the same fashion as for the batch contacts, except that aliquots were removed at specified time intervals, filtered through a 0.2 μm filter, and subjected to ICP analysis. Kinetics experiments revealed that equilibrium was achieved in 1-3 h, indicating all experimental results represent true equilibrium conditions.

Results and discussion

Resin synthesis and characterizations

The influence of the preparation of polymers bearing various phenolic compounds as monomers was preliminary studied. Resol resins type were prepared using an excess of formaldehyde compared to the phenolic compound in basic conditions. The molar ratio was considered to obtain solubilisation of the mixture before the addition of formaldehyde and adjusted from previous synthesis of formophenolic resins [32]. The co-polymerisation results in an uncontrolled crosslinking through a methylene linkage between the phenolic precursors. Following this procedure various resins were synthesized by varying the nature of the phenolic precursors (Table 1).

After conditioning of the insoluble resins (three cycles 1M HCl/ H2O/1M NaOH/H2O), we noticed a decrease in the mass which can be attributed to an elimination of the partially soluble oligomeric species. We measured the highest percent loss of mass resin when 8-hydroxyquinoline was used whether during the homopolymerization or heteropolymerization with 25% of catechol with respectively 70% of loss in comparison to the values obtained for the other one that are around 35-40%. The homopolymerization of 8-hydroxyquinoline using formaldehyde as crosslinker in basic conditions gave a large amount of water-soluble materials which are not applicable in solid-liquid extraction. This is due to the fact that the 8-hydroxyquinoline have two reactive sites (an ortho-and a para-position on its phenol ring) which are difficult to polymerize.

Table 1 List of polymeric resins synthesized by alkaline polycondensation of formaldehyde and phenolic derivatives: Catechol (C), 8-hydroxyQuinoline (Q), Phenol (P) and Resorcinol (R).

To overcome this drawback, the resins were prepared by polycondensation reaction between the 8hydroxyquinoline and catechol using formaldehyde as crosslinker in the presence of alkaline hydroxide as catalyst (NaOH). After the washing step infusible, insoluble, amorphous, crosslinked polymer are obtained which are applicable in solid-liquid extraction.

Each resins were crushed thanks to a ball-milling treatment. The ball-milling of the polymers allows the reduction of the particle size from 500 μm to sizes smaller than 100 μm with mainly particles size about 10 μm (Fig. SI-3 see supporting information). This treatment makes it possible to reduce the size of the particles which is conventionally included between 250-850 μm for ion-exchanger material and allow to increase the surface contact area of these resins. The crushed resins were examined using SEM and showed particles with smooth glass-like surfaces that lack pore structure (Fig. SI-4 see supporting information). Scratches on the surface shows a rough surface inside the particles which may be due to a local swelling.

The typical structure of the polymers is depicted with the IR spectra of the polymers I-V in Fig. SI-5 (see supporting information). We observed the characteristics band of the precursors such as the broad band at 3300 cm -1 indicative of O-H bond stretching from the ring hydroxyls. The band at 2985 and 2900 cm -1 arise from the C-H stretching of the methylene linkages between the phenolic rings and the ring skeletal vibrations appears at around 1600, 1460 and at 815 cm -1 with the aromatic C-H out-ofplane deformation band which is indicative of the ring substitution pattern. Characteristic signature of the 8-hydroxyquinoline is observed at 1500 and at 780 cm -1 , this two band increase with the increase of 8-hydroxyquinoline ratio in the polymeric resin. The 3500-3000 cm -1 region appear to be different in the resin V, this difference can be attributed to the hydroxy groups of 8-hydroxyquinoline intramolecularly hydrogen bonded to the nitrogen and maybe the presence of methylol hydroxyl group. Also, the different bands of this polymeric resin (V) are narrower reflecting the fact that the resin is less well crosslinked than the one with a catechol matrix.

Elemental analysis for the resins R1 to R5 were carried out and the progressive increase in the nitrogen content from 0 to 8% with the increase in the percentage of 8-hydroxyquinoline suggest the incorporation of this group into the polymer matrix (Fig. SI-2). This result has been correlated with 13 C MAS solid state NMR with the typical displacement of the aromatic carbons at 125 ppm and phenolic carbons at 145 ppm and bridging methylene carbons at around 30 ppm (Fig. 2). As illustrated in the Fig. 2, the introduction of 8-hydroxyquinoline in the catechol resin (III) slightly modifies the characteristic region of aromatic carbons with a displacement around 148 ppm for the phenolic carbons and the apparition of a shoulder around 138 ppm characteristic of aromatic carbons of the 8hydroxyquinoline. The moisture regain was determined from the thermogravimetric analysis after a heating after in air at 100 °C. The weight loss corresponding to the percentage of water in the resin are summarized in Table SI-1 (see supporting information). A moisture content of about 10% was estimated for the various polymers which is in agreement with the values obtained for such resins [28]. To facilitate comparison between the thermograms, they are plotted after the heat treatment allowing the evaluation of the moisture regain. The thermogravimetry curves along with the derivative thermogravimetry (DTG) curves of two resins are shown in Fig. 3a andb for the resin I and III, respectively. Thermal analysis shows clearly the mass loss of both polymers by air which indicate different behaviour with one stage decomposition for polymer I and two stages decomposition for polymer III (Fig. 3a). The thermal degradation takes place over a wide temperature range and finally both resins were completely degraded resulting from their thermooxidation. Derivative thermogravimetry curves (Fig. 3b) shows a mass reduction through one endothermic peak at 500 °C for polymer I and two endothermic peaks at around 500 °C and 600 °C for polymer III. The degradation process of the catechol resin I is mainly obtained in one stage while for the resin III incorporating the 8-hydroxyquinoline in the polymer matrix the degradation process can be divided into two stages. It appears that the decomposition takes place over a wide temperature range with endothermic peak centred at 500 and 600 °C corresponding to the catechol and 8-hydroxyquinoline pattern of the resin respectively.

Batch contact with cation solutions

The pH is an important parameter affecting the affinity of a resin for a particular ion and also affecting the germanium-catechol chelate formation, a pH of 6 was considered in our study since it has been shown that germanium is poorly extracted at pH < 4 and pH > 9 [17,21]. The distribution coefficient (Kd) and adsorption efficiency (E) for each cation were calculated after submersing the resins in different aqueous solutions for 24 h. Kinetics experiments (Fig. SI-6 see supporting information) revealed that equilibrium was achieved in only 1-3 h, confirming the systems were in true equilibrium. Inductively coupled plasma (ICP) was used to determine the initial and residual metal concentrations.

Preliminary screening tests were performed to evaluate the efficiency of the resins towards the extraction of Ge, Si, Zn. The suitability of the resin is associated with an extraction efficiency value which is as high as possible and with the highest possible separation factor SF Ge/M with M=Si or Zn. The first experiments were conducted with 1 g/L (10 mg in 10 mL; V/m=1000) of resin I, II and III, the resin was submersed in pure water doped with 10 ppm of each metals (pH 6). The results illustrated in Fig. 4 indicate that the Ge is preferably extract in comparison to Zn and Si. After extraction, the presence of germanium in the resins has been highlighted through X-EDS analyses (Fig. SI-7, see supporting information). The catechol resin (polymer I) is able to selectively extract the Ge towards Si with separating factor SF Ge/Si of 28, when 8-hydroxyquinoline is introduced, such as in the resin II (25%) and resin III (50%), the extraction of Si is increased and therefore the SF Ge/Si decrease up to 1.6. An interesting point is that the introduction of the 8-hydroxyquinoline makes it possible to have a more pronounced selectivity towards Zn which reaches a SF Ge/Zn of 16. It appear that the selective separation of Ge towards Si and Ge towards Zn can be proceed thanks to the catechol resin (resin I) and catechol resin with 8-hydroxyquinoline (resin II and III) respectively. The V/m ratio was 1000.

By increasing the amount of resins (Fig. 5) the efficiency was increased with the possibility to quantitatively extract the germanium with separation factors SF Ge/Si higher than 100 using a concentration of 10 g/L of resin I and a SF Ge/Zn higher than 20 using a concentration of 10 g/L of resin III (distributions coefficients Kd are illustrated in Fig. SI-8 see supporting information). It appear that the best compromise for the separation of germanium towards zinc is the use of the resin II or III with a concentration of 2.5 g/L allowing a SF Ge/Zn higher than 80. One limitation is when Cu is introduced as a competitive metal (Fig. 6c), indeed the extraction of the Cu increase significantly with the concentration of the resins. However it possible to selectively extract the Ge with respect to Zn and Cu with SF Ge/Zn > 300 and SF Ge/Cu > 10 when a concentration of resin is 1 g/L. The preorganization of OH functions in the phenolic precursors was investigated with the implementation of resins based on phenol (VI) or a resorcinol (VII). The results (Table SI-2 see supporting information) indicate that the Ge is poorly extract around 7 and 16% for the phenol and resorcinol resin respectively with a 10 g/L concentration of resin. It appear that the catechol (1,2-dihydroxybenzene) shows better performance than resorcinol (1,3dihydroxybenzene) and phenol (1-hydroxybenzene) suggesting an appropriate preorganization of the OH for the Ge extraction in the catechol.

Preliminary experiments have been performed to study the recuperation of germanium from the resin. After extraction, 25 mg of resin I or resin III were contacted with 10 mL of a 1M HCl solution to evaluate the stripping efficiency. It appear that in each case more than 80% of the germanium extracted by the resin can be stripped. The modifications of desorption conditions to obtain a complete stripping of the germanium could make it possible to envisage the resins reutilization.

Conclusion

The investigation on the behaviour of germanium extraction by phenolic polymers demonstrate that the resins with a catechol moiety are effective for the selective recovery of germanium. The selective sorption of germanium was studied in the presence of elements such as silicon, zinc or copper which are likely to be competitive cations in solutions from urban mines or mine deposits. The incorporation of 8-hydroxyquinoline in the catechol matrix produces changes in the selectivities properties of the resins. The selectivity Ge/Si or Ge/Zn is found to be influenced by the nature of the phenolic resins. Indeed, the catechol-formaldehyde resin shows the best performance for the separation of Ge towards Si with high separation factor while the catechol 8-hydroxyquinoline-based resins are not efficient for the Ge/Si separation but they show a selectivity between Ge and Zn. Preliminary experiments indicate that the recuperation of germanium is possible using an acidic eluting solution.

Fig. 1 .

 1 Fig. 1. Schematic structure of the germanium-catechol complex.

Fig. 2 .

 2 Fig. 2. 13 C solid state NMR spectra of Polymers I and III.

Fig. 3 .

 3 Fig. 3. Thermal analysis data of the resins I and III: (a) TG and (b) DTG curves at 10 °C/min under an air atmosphere.

Fig. 4 .

 4 Fig. 4. Extraction of Ge, Si and Zn by resin I, II or III (at 1 g/L) from mixed cation solutions with each cation at nominal 10 mg/L (pH=6).The V/m ratio was 1000.

Fig. 5 .

 5 Fig. 5. Extraction of Ge, Si and Zn by resin (a) I, (b) II or (c) III at various concentration (1-10 g/L) from mixed cation solutions with each cation at nominal 10 mg/L (pH=6).A relevant comparison with similar or close sorption experiment is difficult. Indeed, sorption properties depends mainly on the nature of the ion-exchanger used and on the nature of the solution used. In a previous work, the selectivity of germanium ion against silicate ion was determined of approximately 15 for germanium extraction using a process based on a catechol complexation and permeation through an anion-exchange membrane with an initial Ge and Si concentration of 58 and 188 ppm respectively[15]. Similarly, Torralvo et al. described the sorption of the catechol -germanium complex onto anionic resins with a Ge/Zn selectivity of about 125 with an initial Ge and Zn concentration of 36 and 2 ppm respectively[17,19]. Therefore, compared with the literature, this work obtained competitive selectivity values with respect to the extraction condition studied. From the preliminary extraction experiments it could be identified that the resins likely to have the best performance for the separation Ge/Si are the resins containing only a catechol group (resin I) while introduction of with 8-hydroxyquinoline (resin II and III) allow the separation Ge/Zn.A second set of experiments have been performed with two different aqueous solutions: i) a mixture of 100 ppm of Si and 10 ppm of Ge for the resin I or ii) a mixture of 100 ppm of Zn and 10 ppm of Ge with or without Cu (100 ppm) for the resin III, in order to investigate the extraction of the Ge with respect to higher concentration of Si or Zn and Cu. The results confirm the previously observed tendencies even with a 10 times greater amount of competitors (Si or Zn) than Ge. No extraction of Si by the resin I or Zn by the resin III are observed while the Ge can be quantitatively extracted (Fig.6aand b, distributions coefficients Kd are illustrated in Fig. SI-9 see supporting information). One limitation is when Cu is introduced as a competitive metal (Fig.6c), indeed the extraction of the Cu increase significantly with the concentration of the resins. However it possible to selectively extract the Ge with respect to Zn and Cu with SF Ge/Zn > 300 and SF Ge/Cu > 10 when a concentration of resin is 1 g/L. The preorganization of OH functions in the phenolic precursors was investigated with the implementation of resins based on phenol (VI) or a resorcinol (VII). The results (Table SI-2 see supporting information) indicate that the Ge is poorly extract around 7 and 16% for the phenol and resorcinol resin respectively with a 10 g/L concentration of resin.

Fig. 6 .

 6 Fig. 6. Extraction efficiencies of cations by resin I and III at various concentration (1-10 g/L) from mixed cation solutions with (a) Ge at 10 mg/L and Si 100 mg/L or (b) Ge at 10 mg/L and Zn 100 mg/L or (c) Ge at 10 mg/L, Zn 100 mg/L and Cu 100 mg/L (pH=6).

Acknowledgments

The authors acknowledge C. Rey, B. Baus Lagarde and P. Gaveau for their technical support regarding the TGA, ICP/AES and solid state NMR analyses. We gratefully acknowledge the financial support for this project provided by the Montpellier University and the Labex Project CheMISyst (ANR-10-LABX-05-01).

Appendix A. Supplementary material

Supplementary data associated with this article can be found, in the online version.