S Richard
email: richard@math.nagoya-u.ac.jp

T Umeda
email: umeda@sci.u-hyogo.ac.jp

On some integral operators appearing in scattering theory, and their resolutions

Keywords: 2010 Mathematics Subject Classification: 47G10 integral operators, Hilbert transform, Hankel transform, dilation operator, scattering theory

We discuss a few integral operators and provide expressions for them in terms of smooth functions of some natural self-adjoint operators. These operators appear in the context of scattering theory, but are independent of any perturbation theory. The Hilbert transform, the Hankel transform, and the finite interval Hilbert transform are among the operators considered.

Introduction

Investigations on the wave operators in the context of scattering theory have a long history, and several powerful technics have been developed for the proof of their existence and of their completeness. More recently, properties of these operators in various spaces have been studied, and the importance of these operators for non-linear problems has also been acknowledged. A quick search on MathSciNet shows that the terms wave operator(s) appear in the title of numerous papers, confirming their importance in various fields of mathematics.

For the last decade, the wave operators have also played a key role for the search of index theorems in scattering theory, as a tool linking the scattering part of a physical system to its bound states. For such investigations, a very detailed understanding of the wave operators has been necessary, and it is during such investigations that several integral operators or singular integral operators have appeared, and that their resolutions in terms of smooth functions of natural selfadjoint operators have been provided. The present review paper is an attempt to gather some of the formulas obtained during these investigations.

Singular integral operators are quite familiar to analysts, and refined estimates have often been obtained directly from their kernels. However, for index theorems or for topological properties these kernels are usually not so friendly: They can hardly fit into any algebraic framework. For that reason, we have been looking for representations in which these singular integral operators have a smoother appearance. It turns out that for several integral operators this program has been successful, and suitable representations have been exhibited. On the other hand, let us stress that even though these formulas are necessary for a C * -algebraic approach, it seems that for pure analysis they do not lead to any new refined estimates.

Let us now be more precise for a few of these operators, and refer to the subsequent sections for more information. The Hilbert transform is certainly one of the most famous and ubiquitous singular integral operators. Its explicit form in L 2 (R) is recalled in (2.1). A nice representation of this singular operator does not take place directly in L 2 (R), but by decomposing this Hilbert space into odd and even functions, then it becomes possible to obtain an expression for the Hilbert transform in terms of the operators tanh(πA) and cosh(πA) where A denotes the generator of dilations in L 2 (R + ). Such an expression is provided in Proposition 2.1. Similarly, the Hankel transform whose definition in L 2 (R + ) is recalled in (2.3) is an integral operator whose kernel involves a Bessel function of the first kind. This operator is not invariant under the dilation group, and so does the inversion operator J defined on f ∈ L 2 (R + ) by [Jf ](x) = 1

x f ( 1 x ). However, the product of these two operators is invariant under the dilation group and can be represented by a smooth function of the generator of this group, see Proposition 2.2 for the details.

For several integral operators on R + or on R n , the dilation group plays an important role, as emphasized in Section 2. On the other hand, on a finite interval (a, b) this group does not play any role (and is even not defined on such a space). For singular operators on such an interval, the notion of rescaled energy representation can be introduced, and then tools from natural operators on R can be exploited. This approach is developed in Section 3.

As a conclusion, this short expository paper does not pretend to be exhaustive or self-contained. Its (expected) interest lies on the collection of several integral operators which can be represented by smooth functions of some natural selfadjoint operators. It is not clear to the authors if a general theory will ever be built from these examples, but gathering the known examples in a single place seemed to be a useful preliminary step.

Resolutions involving the dilation group

The importance of the dilation group in scattering theory is well-known, at least since the seminal work of Enss [START_REF] Enss | Geometric methods in scattering theory[END_REF]. In this section we recall a few integral operators which appeared during our investigations on the wave operators. They share the common property of being diagonal in the spectral representation of the generator of the dilation group. Before introducing these operators, we recall the action of this group in L 2 (R n ) and in L 2 (R + ).

In L 2 (R n ) let us set A for the self-adjoint generator of the unitary group of dilations, namely for f ∈ L 2 (R n ) and x ∈ R n :

[e itA f ](x) := e nt/2 f (e t x).

There also exists a representation in L 2 (R + ) which is going to play an important role: its action is given by [e itA f ](x) := e t/2 f (e t x) for f ∈ L 2 (R + ) and x ∈ R + . Note that we use the same notation for these generators independently of the dimension, but this slight abuse of notation will not lead to any confusion.

The Hilbert transform

Let us start by recalling that the Hilbert transform is defined for f in the Schwartz space S(R) by the formula

[Hf ](x) := 1 π P.v. R 1 x -y f (y) dy = - i √ 2π R e ikx k [F f ](k) dk. (2.1)
Here P.v. denotes the principal value, k := k |k| for any k ∈ R * and F f stands for the Fourier transform of f defined by

[F f ](k) := 1 √ 2π R e -ikx f (x) dx. (2.2)
It is well known that this formula extends to a bounded operator in L 2 (R), still denoted by H. In this Hilbert space, if we use the notation X for the canonical self-adjoint operator of multiplication by the variable, and D for the self-adjoint realization of the operator -i d dx , then the Hilbert transform also satisfies the equality H = -i sgn(D), with sgn the usual sign function.

Based on the first expression provided in (2.1) one easily observes that this operator is invariant under the dilation group in L 2 (R). Since this group leaves the odd and even functions invariant, one can further decompose the Hilbert space in order to get its irreducible representations, and a more explicit formula for H. Thus, let us introduce the even / odd representation of L 2 (R). Given any function ρ on R, we write ρ e and ρ o for the even part and the odd part of ρ. We then introduce the unitary map

U : L 2 (R) f → √ 2 f e f o ∈ L 2 (R + ; C 2 ).
Its adjoint is given on h1 h2 ∈ L 2 (R + ; C 2 ) and for x ∈ R by

U * h1 h2 (x) := 1 √ 2 [h 1 (|x|) + sgn(x)h 2 (|x|)] .
A new representation of the Hilbert transform can now be stated. We refer to [START_REF] Kellendonk | On the structure of the wave operators in one dimensional potential scattering[END_REF]Lem. 3] for the initial proof, and to [19, Lem. 2.1] for a presentation corresponding to the one introduced here.

Proposition 2.1. The Hilbert transform H in L 2 (R) satisfies the following equality:

U H U * = -i 0 tanh(πA) -i cosh(πA) -1 tanh(πA) + i cosh(πA) -1 0 .
Note that this representation emphasizes several properties of the Hilbert transform. For example, it makes it clear that the Hilbert transform exchanges even and odd functions. By taking the equality tanh 2 + cosh -2 = 1, one also easily deduces that the norm of H is equal to 1. This latter property can certainly not be directly deduced from the initial definition of the Hilbert transform based on the principle value.

The Hilbert transform appears quite naturally in scattering theory, as emphasized for example in the seminal papers [START_REF] D'ancona | L p -boundedness of the wave operator for the one dimensional Schrödinger operator[END_REF][START_REF] Weder | ,p -continuity of the Schrödinger wave operators on the line[END_REF][START_REF] Yajima | The L p boundedness of wave operators for Schrödinger operators with threshold singularities I, The odd dimensional case[END_REF]. The explicit formula presented in Lemma 2.1 is used for the wave operators in [START_REF] Richard | New formulae for the wave operators for a rank one interaction[END_REF]Thm. 1.2]. A slightly different version also appears in [13, Eq. ( 1)].

The Hankel transform

Let us recall that the Hankel transform is a transformation involving a Bessel function of the first kind. More precisely, if J m denotes the Bessel function of the first kind with m ∈ C and (m) > -1, the Hankel transform H m is defined on

f ∈ C ∞ c (R + ) by [H m f ](x) = R+ √ xy J m (xy) f (y) dy. (2.3)
Note that slightly different expressions also exist for the Hankel transform, but this version is suitable for its representation in L 2 (R + ). Let us also introduce the unitary and self-adjoint operator J :

L 2 (R + ) → L 2 (R + ) defined for f ∈ L 2 (R + ) and x ∈ R + by [Jf ](x) = 1 x f 1 x .
It is now easily observed that neither H m nor J are invariant under the dilation group. However, the products JH m and H m J are invariant, and thus have a representation in terms of the generator of dilations. The following formulas have been obtained in [START_REF] Dereziński | On Schrödinger operators with inverse square potentials on the half-line[END_REF]Prop. 4.5] based on an earlier version available in [2, Thm. 6.2]. Note that in the statement the notation Γ is used for the usual Gamma function.

Proposition 2.2. For any m ∈ C with (m) > -1 the map H m continuously extends to a bounded invertible operator on L 2 (R + ) satisfying H m = H -1 m . In addition, the equalities

JH m = Ξ m (A) and H m J = Ξ m (-A), hold with Ξ m (t) := e i ln(2)t Γ( m+1+it 2 ) Γ( m+1-it 2 )
.

Let us mention that the function t → Ξ m (t) has not a very interesting asymptotic behavior for large |t|. However, by taking the asymptotic behavior of the Gamma function into account, one can observe that the product of two such functions has a much better behavior, namely for any m, m ∈ C with (m) > -1 and

(m ) > -1 the map t → Ξ m (-t)Ξ m (t) belongs to C [-∞, ∞] and one has Ξ m (∓∞)Ξ m (±∞) = e ∓i π 2 (m-m )
. Note that such a product of two function Ξ m appears at least in two distinct contexts: For the wave operators of Schrödinger operators with an inverse square potential, see [2, Thm. 6.2], [5, Eq. (4.24)] and also [START_REF] Inoue | Index theorems for Fredholm, semi-Fredholm, and almost periodic operators: all in one example[END_REF][START_REF] Inoue | Topological Levinson's theorem for inverse square potentials: complex, infinite, but not exceptional, to appear[END_REF], and for the wave operators of an Aharonov-Bohm system [START_REF] Pankrashkin | Spectral and scattering theory for the Aharonov-Bohm operators[END_REF]Prop. 11]. Let us also mention that additional formulas in terms of functions of A have been found in [START_REF] Pankrashkin | Spectral and scattering theory for the Aharonov-Bohm operators[END_REF]Thm. 12] as a result of a transformation involving a Bessel function of the first kind and a Hankel function of the first kind.

Remark 2.3. Let us still provide a general scheme for operators in L 2 (R + ) which can be written in terms of the dilation group. If K denotes an integral operator with a kernel K(•, •) satisfying for any x, y, λ ∈ R + the relation

K(λx, λy) = 1 λ K(x, y), (2.4)
then this operator commutes with the dilation group. As a consequence, K can be rewritten as a function of the generator A of the dilation group in L 2 (R + ), and one has K = ϕ(A) with ϕ given by

ϕ(t) = ∞ 0 K(1, y) y -1 2 +it dy.
Note that this expression can be obtained by using the general formula

ϕ(A)f = 1 √ 2π R φ(t) e -itA f
in conjunction with the homogeneity relation (2.4).

A 3 dimensional example

When dealing with scattering theory for Schrödinger operators in L 2 (R 3 ) one more operator commuting with the generator of dilations appears quite naturally, see [START_REF] Kellendonk | On the wave operators and Levinson's theorem for potential scattering in R 3[END_REF]Sec. 3]. Let us set F for the Fourier transform in R 3 defined for f ∈ S(R 3 ) and k ∈ R 3 by

[F f ](k) = 1 (2π) 3/2 R 3 e -ik•x f (x) dx.
Then we can define for f ∈ S(R 3 ) and x = 0 the integral operator

[T f ](x) = -i 1 √ 2π R+ e iκ |x| κ |x| [F f ](κ x) κ 2 dκ (2.5)
where we have again used the notation x := x |x| ∈ S 2 . An easy computation shows that this operator is invariant under the action of the dilation group. It is thus natural to express the operator T in terms of the operator A. In fact, the operator T can be further reduced by decomposing the Hilbert space with respect to the spherical harmonics.

Let us set h := L 2 (R + , r 2 dr) and consider the spherical coordinates (r, ω) ∈ R + × S 2 . For any ∈ N = {0, 1, 2, . . .} and m ∈ Z satisfying -≤ m ≤ , let Y m denote the usual spherical harmonics. Then, by taking into account the completeness of the family

{Y m } ∈N,|m|≤ in L 2 (S 2 , dω), one has the canonical decomposition L 2 (R 3 ) = ∈N,|m|≤ H m , (2.6)
where

H m = {f ∈ L 2 (R 3 ) | f (rω) = g(r)Y m (ω)
a.e. for some g ∈ h}. For fixed ∈ N we denote by H the subspace of L 2 (R 3 ) given by -≤m≤ H m . Let us finally observe that since the dilation group acts only on the radial coordinate, its action is also reduced by the above decomposition. In other terms, this group leaves each subspace H m invariant.

As a final ingredient, let us recall that the Fourier transform F also leaves the subspace H m of L 2 (R 3 ) invariant. More precisely, for any g ∈ C ∞ c (R + ) and for (κ,

ω) ∈ R + × S 2 one has [F (gY m )](κω) = (-i) Y m (ω) R+ r 2 J +1/2 (κr) √ κr g(r) dr , (2.7)
where J ν denotes the Bessel function of the first kind. So, we naturally set F :

C ∞ c (R + ) → h by the relation F (gY m ) = F (g)Y m (it is clear from (2.7
) that this operator does not depend on m). Similarly to the Fourier transform in L 2 (R 3 ), this operator extends to a unitary operator from h to h. Remark 2.4. If V : L 2 (R + , r 2 dr) → L 2 (R + , dr) is the unitary map defined by [Vf ](r) := rf (r) for f ∈ L 2 (R + , r 2 dr), then the equality VF V * = (-i) H +1/2 holds, where the r.h.s. corresponds to the Hankel transform defined in (2.3).

By taking the previous two constructions into account, one readily observes that the operator T is reduced by the decomposition (2.6). As a consequence one can look for a representation of the operator T in terms of the dilation group in each subspace H m . For that purpose, let us define for each ∈ N the operator T acting on any g ∈ C ∞ c (R + ) as

[T g](r) = -i 1 √ 2π R+ e iκ r κ r [F g](κ) κ 2 dκ .
The following statement has been proved in [14, Prop. 3.1].

Proposition 2.5. The operator T extends continuously to the bounded operator

ϕ (A) in H m with ϕ ∈ C [-∞, ∞] given explicitly for every x ∈ R by ϕ (x) = 1 2 e -iπ /2 Γ 1 2 ( + 3/2 + ix) Γ 1 2 ( + 3/2 -ix) Γ 1 2 (3/2 -ix) Γ 1 2 (3/2 + ix) 1+tanh(πx)-i cosh(πx) -1
and satisfying ϕ (-∞) = 0 and ϕ (∞) = 1. Furthermore, the operator T defined in (2.5) extends continuously to the operator ϕ(A) ∈ B L 2 (R 3 ) acting as ϕ (A) on H .

Let us finally observe that in the special case = 0, one simply gets

ϕ 0 (A) = 1 2 1 + tanh(πA) -i cosh(πA) -1 .
This operator appears in particular in the expression for the wave operators in R Remark 2.6. Let us mention that operators similar to the one presented in (2.5) appear quite often in the context of scattering theory, and then such expressions can be reformulated in a way similar to the one presented in Proposition 2.5. For example, such kernels can be exhibited from the asymptotic expansion of the generalized eigenfunctions of the relativistic Schrödinger operators in dimension 2 [25, Thm. 6.2] or in dimension 3 [24, Thm. 10.2].

Resolutions in the rescaled energy representation

In the previous section, the dilation group played a special role since all operators were invariant under its action. For many other singular kernels appearing in scattering theory, this is no more true, and in many settings there is no analog of the dilation group. However, a replacement for the operator A can often be found by looking at the rescaled energy representation, see for example [START_REF] Bellissard | Scattering theory for lattice operators in dimension d ≥ 3[END_REF]Sec. 2.4] and [START_REF] Schulz-Baldes | The density of surface states as the total time delay[END_REF]Sec. 3.1]. The main idea in this approach is to rescale the underlying space such that it covers R, and then to use the canonical operators X and D on R. Let us stress that here the energy space corresponds to the underlying space since the following operators are directly defined in the energy representation.

The finite interval Hilbert transform

In this section we consider an analog of the Hilbert transform but localized on a finite interval. More precisely, let us consider the interval Λ := (a, b) ⊂ R. For any f ∈ C ∞ c (Λ) and λ ∈ Λ we consider the operator defined by

[T f ](λ) := 1 π P.v. Λ 1 λ -µ f (µ) dµ. ( 3.1)
This operator corresponds to the Hilbert transform but restricted to a finite interval.

In order to get a better understanding of this operator, let us consider the Hilbert space L 2 (R) and the unitary map

U : L 2 (Λ) → L 2 (R) defined on any f ∈ L 2 (Λ) and for x ∈ R by [Uf ](x) := b -a 2 1 cosh(x) f a + be 2x 1 + e 2x .
The adjoint of this map is given for h ∈ L 2 (R) and λ ∈ Λ by

[U * h](λ) = b -a 2 1 (λ -a)(b -λ) h 1 2 ln λ -a b -λ .
Let us now denote by L the operator of multiplication by the variable in L 2 (Λ) and set ρ(L) for the operator of multiplication in L 2 (Λ) by a function ρ ∈ L ∞ (Λ). Then, a straightforward computation leads to the following expression for its representation in L 2 (R): ρ(X) := U ρ(L) U * is the operator of multiplication by the function x → ρ(x) = ρ a+be 2x 1+e 2x . In particular, by choosing ρ(λ) = λ one obtains that U L U * is the operator ρ(X) = a+be 2X 1+e 2X . Note that the underlying function is strictly increasing on R and takes the asymptotic values ρ(-∞) = a and ρ(∞) = b.

Let us now perform a similar conjugation to the operator T . A straightforward computation leads then to the following equality for any h ∈ C ∞ c (R) and x ∈ R:

[U T U * h](x) = 1 π P.v. R 1 sinh(x -y) h(y) dy .
Thus if we keep denoting by D the self-adjoint operator corresponding to -i d dx in L 2 (R), and if one takes into account the formula Such an operator plays a central role for the wave operator in the context of the Friedrichs-Faddeev model [START_REF] Isozaki | On the wave operators for the Friedrichs-Faddeev model[END_REF]Thm. 2]. Let us also emphasize one of the main interest of such a formula. Recall that X and D satisfy the usual canonical commutation relations in L 2 (R). Obviously, the same property holds for the selfadjoint operators X Λ := U * X U and D Λ := U * D U in L 2 (Λ). More interestingly for us is that for any functions ϕ ∈ L ∞ (R) and ρ ∈ L ∞ (Λ) the operator ϕ(D) ρ(X) in L 2 (R) is unitarily equivalent to the operator ϕ(D Λ ) ρ(L) in L 2 (Λ). In particular, this allows us to define quite naturally isomorphic C * -subalgebras of B L 2 (R) and of B L 2 (Λ) , either generated by functions of D and X, or by functions of D Λ and L. By formula (3.2), one easily infers that the singular operator T defined in (3.1) belongs to such an algebra.

i π P.v.

The finite interval Hilbert transform with weights

The operator considered in this section is associated with a discrete adjacency operator on Z. Once considered in its energy representation, this operator leads to a Hilbert transform on a finite interval multiplied by some weights.

We consider the Hilbert space L 2 (-2, 2) and the weight function β : (-2, 2) → R given by β(λ) := 4 -λ 2 1/4 .

For any f ∈ C ∞ c (-2, 2) and for λ ∈ (-2, 2) we define the operator

[T f ](λ) := 1 2πi P.v. 2 -2 β(λ) 1 λ -µ β(µ) -1 f (µ) dµ. (3.3)
Clearly, this operator has several singularities: on the diagonal but also at ±2. In order to get a better understanding of it, let us introduce the unitary transformation U :

L 2 (-2, 2) → L 2 (R) defined on f ∈ L 2 (-2, 2) by [Uf ](x) := √ 2 1 cosh(x) f 2 tanh(x) .
The action of its adjoint is given on h ∈ L 2 (R) by

[U * h](λ) = √ 2 √ 4 -λ 2 h arctanh(λ/2) .
We also introduce the multiplication operators b ± (X) ∈ B L 2 (R) defined by the real functions b ± (x) := e x/2 ± e -x/2 (e x + e -x ) 1/2 . The function b + is continuous, bounded, non-vanishing, and satisfies lim x→±∞ b + (x) = 1. The functions b -is also continuous, bounded, and satisfies lim x→±∞ b -(x) = ±1.

With these notations, the following statement has been proved in [START_REF] Nguyen | Spectral and scattering theory for the discrete Laplacian in a half-space with a periodic potential on the boundary[END_REF].

Proposition 3.2. One has

U T U * = - 1 2 b + (X) tanh(πD)b + (X) -1 -ib -(X) cosh(πD) -1 b + (X) -1 . (3.4)
Note that a slightly simpler expression is also possible, once a compact error is accepted. More precisely, since the functions appearing in the statement of the previous proposition have limits at ±∞ the operator in the r.h.s. of (3.4) can be rewritten as

- 1 2 tanh(πD) -i tanh(X) cosh(πD) -1 + K (3.5)
with K ∈ K L 2 (R) , see for example [START_REF] Cordes | On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators[END_REF] for a justification of the compactness of the commutators. Note that this expression can also be brought back to the initial representation by a conjugation with the unitary operator U. Let us also mention that the operators obtained above play an important role for the wave operator of discrete Schrödinger operators on Z n . Such operators are currently under investigations in [START_REF] Inoue | Schrödinger wave operators on the discrete half-line[END_REF] and in [START_REF] Nguyen | Spectral and scattering theory for the discrete Laplacian in a half-space with a periodic potential on the boundary[END_REF].

The upside down representation

In this section we deal with a singular kernel which is related to a 1-dimensional Dirac operator. Compared to the operators introduced so far, its specificity comes from its matrix-values. Dirac operators depend also on a parameter m which we choose strictly positive. The following construction takes already place in the energy representation of the Dirac operator, namely on its spectrum. Let us define the set Σ := (-∞, -m) ∪ (m, +∞) and for each λ ∈ Σ the 2 × 2 matrix

B(λ) = 1 √ 2 diag 4 λ -m λ + m , 4 λ + m λ -m .
Clearly, for any λ ∈ Σ the matrix B(λ) is well defined and invertible, but it does not have a limit as λ m or as λ -m. For f ∈ C ∞ c (Σ; C 2 ) we consider the singular operator T defined by The trick for this singular operator is to consider the following unitary transformation which sends the values ±m at ±∞, while any neighbourhood of the points ±∞ is then located near the point 0. More precisely, let us define the unitary operator U : L 2 (Σ; C 2 ) → L 2 (R; C 2 ) given for f ∈ L 2 (Σ; C 2 ) and x ∈ R by Uf (x) := √ 2m e x/2 e x -1 f m e x + 1 e x -1 .

The adjoint of the operator U is provided for h ∈ L 2 (R; C 2 ) and λ ∈ Σ by the expression

U * h (λ) = √ 2m λ + m λ -m 1 λ + m h ln λ + m λ -m .
We shall now compute the kernel of the operator U T U * , and observe that this new kernel has a very simple form.

For that purpose, we keep the notations X and D for the canonical self-adjoint operators on L 2 (R), and denote by F the Fourier transform in L 2 (R; C 2 ), namely two copies of the Fourier transform (2.2). One then checks, by a direct computation, that for any measurable function ρ : Σ → M 2 (C) one has U ρ(L) U * = ρ m e X + 1 e X -1 .

Furthermore, for any f = (f 1 , f 2 ) ∈ C ∞ c (R; C 2 ) and x ∈ R, it can be obtained straightforwardly that We refer to [17, Sec. III.D] for the details of the computation, and for the use of this expression in the context of 1-dimensional Dirac operators. Note that in Section IV of this reference the C * -algebraic properties mentioned at the end of Section 3.1 are exploited and the construction leads naturally to some index theorem in scattering theory.

[U T U * h](x) = 1 4π P.v.

Proposition 3 . 1 .

 31 The following equality holdsU T U * = -i tanh πD/2 .(3.2)

[

Proposition 3 . 3 .

 33 y-x)/4) h(y) dy.By summing up the information obtained so far one obtains: For any m > 0 one hasU T U * = i tanh(2πD) + i cosh(2πD) -1 0 0 tanh(2πD) -i cosh(2πD) -1 .

Acknowledgement

S. Richard thanks the Department of Mathematics of the National University of Singapore for its hospitality in February 2019. The authors also thank the referee for suggesting the addition of Remark 2.3. Its content is due to him/her.

* Supported by the grantTopological invariants through scattering theory and noncommutative geometry from Nagoya University, and by JSPS Grant-in-Aid for scientific research (C) no 18K03328, and on leave of absence from Univ. † Supported by JSPS Grant-in-Aid for scientific research (C) no 18K03340.


Fadi Badra
email: badra@univ-paris13.fr

Karima Sedki
email: karima.sedki@univ-paris13.fr

Adrien Ugon
email: adrien.ugon@esiee.fr

On the Role of Similarity in Analogical Transfer

Keywords: Analogical Transfer, Qualitative Modeling, Similarity Measurement

Analogical transfer consists in making the assumption that if two situations are alike in some respect, they may be alike in others. This paper studies analogical transfer from the viewpoint of measurement theory. We show that analogical transfer can be seen as a similaritybased reasoning on "contour lines" in a qualitative similarity model: if a contour line is included in another at a given point, then it may also be included at other points that are not too (analogically) dissimilar.

Introduction

Case-based reasoning (CBR) can be seen as the application of an analogical inference to problem solving. Analogical reasoning is a cognitive process in which some structural pattern identified in a source conceptualization is transferred to a target domain (possibly the same domain) in order to learn a target conceptualization [START_REF] Gust | Analogical Reasoning: A Core of Cognition[END_REF]. Three main subtasks are identified: retrieval, mapping, and transfer. The aim of the retrieval phase is to use some memory to identify to which source situation the new (target) situation can be compared with. The mapping phase produces a mapping from the retrieved source situation to the target situation. The transfer phase uses this mapping to makes some prediction on how to complete the description of the target situation. Analogies in which mapping is followed with a transfer of information or explanations are called predictive analogies in [START_REF] Gust | Analogical Reasoning: A Core of Cognition[END_REF].

Analogical transfer may be used to predict some commonalities between a source situation and a target situation. An example of predictive analogy that transfers a commonality is the k-Nearest Neighbor algorithm [START_REF] Mitchell | Machine Learning[END_REF], an instancebased machine learning algorithm in which the class clspx 0 q of a target instance x 0 is predicted from the class value clspxq of the k instances x that are most similar to x 0 . But, analogical transfer is also used to predict differences between a target situation and the source situations retrieved from memory. Adaptation, i.e., the cognitive ability to envision a target solution that is different from any previously encountered solution, is considered in cognitive psychology [START_REF] Novick | Mathematical problem solving by analogy. Journal of experimental psychology[END_REF][START_REF] Paritosh | Cognitive processes in quantitative estimation: analogical anchors and causal adjustment[END_REF], as one of the core components of analogical transfer. In the following example, taken from [START_REF] Novick | Mathematical problem solving by analogy. Journal of experimental psychology[END_REF], students adapt solution procedures to solve new mathematical problems. They are asked to adapt the solution procedure of a mathematical problem (called the garden problem [START_REF] Novick | Analogical Transfer, Problem Similarity, and Expertise[END_REF]), in which the goal is to calculate the number of plants that two people have in their garden. The solution procedure consists in (i) computing the least common multiple (LCM) of a set of numbers (the LCM of 10, 4, and 5, is 20), (ii) generate the first multiples of the obtained number (20, 40, 60, etc.), (iii) add a left over to each multiple (the left over is 2, so we obtain 22, 42, 62, etc.), and (iv) find the smallest candidate solution that is divisible by a given number (the smallest divisible by 6 is 42). Two other mathematical problems, the marching band problem and the cookies problem, are structurally very similar to the the garden problem, but differ by the value given to three attributes: the goal (g), the set of divisors (d), and the left over (l). Fig. 1 represents the difference between these problems: garden denotes the description of the garden problem, band denotes the description of the marching band problem, and cookies denotes the description of the cookies problem. The cookies problem is solved by adapting the solution procedure of the garden problem. To do so, the student has to assess how to modify the solution procedure to account for the differences observed between the two problems. From the remembering of how the marching band problem has been previously solved by adapting the solution of the garden problem, most students were able to induce that a problem with the same structure as the garden problem may be solved by the same solution procedure, up to a substitution of attribute values. However, more complex differences between problems may require a deeper modification of the solution procedure, such as adding or removing steps. The authors give the example of the seashell problem, for which the number of multipliers is greater than the one of the garden problem. Adapting the solution of the garden problem to solve the seashell problem would require to add some reasoning steps to its solution procedure.

Adaptation is recognized as being part of the case-based reasoning cycle. But surprisingly, a study of the CBR literature shows that the adaptation step is not included in the case-based analogical inference [START_REF] Ontañón | On Knowledge Transfer in Case-Based Inference[END_REF]. The adaptation step is always performed after the analogical inference (i.e., retrieval, mapping, and transfer) has taken place, and only aims at modifying its result. Some adaptation methods such as critique-based adaptation [START_REF] Hammond | CHEF: A Model of Case-based Planning[END_REF], or conservative adaptation [START_REF] Lieber | Application of the Revision Theory to Adaptation in Case-Based Reasoning: The Conservative Adaptation[END_REF] are used to resolve inconsistencies in the reused source case, whereas others such as differential adaptation [START_REF] Fuchs | Differential adaptation: An operational approach to adaptation for solving numerical problems with CBR[END_REF], case-based adaptation [START_REF] Craw | Learning adaptation knowledge to improve case-based reasoning[END_REF] or adaptation by reformulation [START_REF] Lieber | Correct and Complete Retrieval for Case-Based Problem-Solving[END_REF] modify the reused source case in order to fit the requirements on the target case. This singular status of adaptation in CBR research with respect to analogical transfer appears even more striking since many adaptation strategies apply to themselves an analogical process, by reusing adaptation knowledge retrieved from memory, in form of adaptation rules [START_REF] D'aquin | Case base mining for adaptation knowledge acquisition[END_REF], substitutions [START_REF] Mcsherry | Demand-driven discovery of adaptation knowledge[END_REF], or adaptation cases [START_REF] Craw | Learning adaptation knowledge to improve case-based reasoning[END_REF]. One of the reasons why adaptation is left out of the casebased analogical inference is that adaptation essentially consists in reasoning on the differences that exist between two cases and the case-based inference is currently unable to transfer such differences to the target case. The case-based inference typically relies on a measure of similarity between cases, and while the importance of capturing and reusing case differences has long been acknowleged in adaptation research (see e.g., [START_REF] D'aquin | Case base mining for adaptation knowledge acquisition[END_REF][START_REF] Fuchs | Differential adaptation: An operational approach to adaptation for solving numerical problems with CBR[END_REF][START_REF] Hanney | The adaptation knowledge bottleneck: How to ease it by learning from cases[END_REF][START_REF] Jalali | Learning and applying case adaptation rules for classification: An ensemble approach[END_REF][START_REF] Mcsherry | Demand-driven discovery of adaptation knowledge[END_REF], the link between case differences and a theory of similarity remains unclear. What is the role of similarity in analogical transfer?

In this article, analogical transfer is studied from the viewpoint of a qualitative measurement of similarity. We show that analogical transfer can be seen as a similarity-based reasoning on "contour lines" in a qualitative similarity model: if a contour line is included in another at a given point, then it may also be included at other points that are not too (analogically) dissimilar.

The paper is organized as follows. In the next section, we start with a formulation of analogical transfer as a rule of inference. To understand the role played by similarity in this rule of inference, Sec. 3 introduces qualitative measures of similarity. Sec. 4 shows that these measures can be used to measure analogical mappings. Sec. 5 defines analogical transfer as a reasoning on contour lines. Sec. 6 illustrates the approach on a use case, and Sec. 7 concludes the paper.

Analogical Transfer from a Logical Perspective

Analogical transfer is stated from a logical perspective in [START_REF] Davis | A Logical Approach to Reasoning by Analogy[END_REF] as the following hypothetical rule of inference, referred to as the "analogical jump" (AJ):

P psq P ptq Qpsq Qptq (AJ)
The properties P and Q may represent commonalities. For example, Bob's car and John's car share the property P of being a 1982 Mustang GLX V6 hatchbacks, and Bob's car has the property Q of having a price of 3500 $. The inference is that the price of John's car should also be around 3 500 $. But P and Q may also represent differences. For example, in Fig. 2, adapted from [START_REF] Tversky | Features of Similarity[END_REF], the pair (a, a 1 ) can be mapped to the pair (b, b 1 ) because b 1 differs from b in the same way that a 1 differs from a. Both pairs instanciate the two relational predicates R 1 =" differs in profile shape from round to sharp " and R 2 =" has the same eyebrow as ". If we know that d 1 has a sharp profile (i.e., R 1 pd, d 1 q holds), then assuming d 1 is to d what a 1 is to a, applying the rule (AJ) enables to make the hypothesis that d 1 has the same eyebrow (curved) as d, i.e., that R 2 pd, d 1 q holds:

R 1 pa, a 1 q R 2 pa, a 1 q R 1 pd, d 1 q R 2 pd, d 1 q
The authors of [START_REF] Davis | A Logical Approach to Reasoning by Analogy[END_REF] claim that a sufficient condition for the inference (AJ) to be justified should: (i) be weaker than a generalization rule @xP pxq ñ Qpxq (otherwise the inference is simply deductive), (ii) on the contrary, be stronger than an enumerative induction that consists in applying the rule P ñ Q to t only because it was seen to hold on one example s, (iii) be in proportion to the amount of similarity between sources and targets, but similarity degrees are highly subjective and context-dependent. One criteria that seems satisfying is when a functional dependency holds between P and Q.

To be able to make an analogical jump, instance-based learning approaches apply a similarity principle, that states that the more similar two causes are, the more plausible it is that they have the same effect. In numerical settings, this property can be translated in the geometrical property that locally, a bounded similarity on problems results in a bounded similarity on solutions [START_REF] Hüllermeier | Possibilistic instance-based learning[END_REF]. The inference schema (AJ) is applied on a neighborhood N pxq of the retrieved instance x:

x P N pxq clspxq " c x 0 P N pxq clspx 0 q " c Another example of predictive analogy is the analogical proportion-based inference, used in analogical classification, which, as shown e.g., in [START_REF] Bounhas | Analogy-based classifiers for nominal or numerical data[END_REF], also consists in making an analogical jump.

Qualitative Similarity

Qualitative Similarity. A qualitative representation of similarity consists in dropping numerical values in similarity assessment, which is justified by the fact that actual values of distance between two cases are less significant than the ability to compare two distances. Let U be a finite, non-empty set, called the universe.

An element of the square product U ˆU is denoted pa, bq, or simply ab. Following [START_REF] Yao | Qualitative Similarity[END_REF], a qualitative similarity relation ą is defined as a strict weak order on UˆU. For a, b, c, d P U, a strict ordering ab ą cd for a similarity relation ą is interpreted as "a and b are more similar than c and d", and represents a strict inequality. A strict weak order is a binary relation with the following two properties, for a, b, c, d, e, f P U: asymetry (ab ą cd ñ pcd ą abq), and negative transitivity ( pab ą cdq ^ pcd ą ef q ñ pab ą ef q). The axioms of asymetry and negative transitivity define a form of rationality of a similarity comparison [START_REF] Robert | Measurement Theory[END_REF]. For a strict weak order ą, let " be a binary relation, called the indifference relation, defined as: ab " cd ô pab ą cdq ^ pcd ą abq.

An element ab " cd is interpreted as "a and b are as similar as c and d". The indifference relation " is an equivalence relation: it is reflexive, transitive, and symmetric. Each equivalence relation defines a partition, which is a collection of disjoint subsets of a set whose union is the whole set. The subset of " that contains a pair ab P U ˆU is written rabs " " tuv P U ˆU | ab " uvu, and is called the equivalence class of ab. The relation ľ represents a (non-strict) inequality, and is defined as ab ľ cd iff ab ą cd or ab " cd. Because ą is strict weak, there holds that for any a, b, c, d P U, exactly one of ab ą cd, cd ą ab, or ab " cd hold. A direct consequence of the negative transitivity of ą is that an (in)equality ab ľ cd (resp., ab ą cd, ab " cd) between two pairs ab and cd implies an (in)equality between the two equivalence classes rabs " and rcds " :

rabs " ľ rcds " ô @uv P rabs " , u 1 v 1 P rcds " , uv ľ u 1 v 1 ô ab ľ cd
Scaling. A natural way to discretize a set is to apply a discrete-valued function. Such a function should preserve the underlying structure of the set. The set pU ˆU, ľq is a relational structure, i.e., a set along with a collection of finitary operations and binary relations. A scale is an homomorphism between two relational structures [START_REF] Robert | Measurement Theory[END_REF].

Definition 1. A variation is a scale υ : pU ˆU, ľq ÝÑ pV, ľq.

A homomorphism is a mapping that preserves all relations and operations. Therefore, for a, b, c, d P U,

ab ľ cd ô υpabq ľ υpcdq
Variations can be used to describe a wide range of similarity relations, from the most specific one (υpabq " ab) to the most general (υpabq " 1). In this paper, we will restrict the discussion to variations that are constructed by scaling the set of pairs of values taken by a given feature 3 . A feature is a variable on the set U, and can be modeled as a function ϕ : U ÝÑ X . In the following, the term feature denotes either a binary variable (X " t0, 1u), a nominal variable (X is Table 1. Some examples of scales o : X ˆX ÝÑ V defined on the square product of a feature space X .

Feature space Examples

All " pxyq "

# 1 if x " y 0 if x ‰ y 1pxyq " # 1 if x " y " 1 0 otherwise Idpxyq " xy Ordinal ĺpxyq " # 1 if x ĺ y 0 otherwise Continuous ´pxyq " y ´x Signpxyq " $ ' & ' % `if x ă y o if x " y ´if x ą y d p pxyq " }y ´x} p Sets ∆pXY q " |XzY | `|Y zX| AP pXY q " pXzY , Y zXq spXY q " pXzY , X X Y , Y zXq
an enumerated set, like td, m, uu), a quantity (X is a set of values on an ordinal, interval, or ratio scale), or a variable that takes its value in the powerset PpAq of a set A. The term feature space denotes the set X of values taken by a feature.

Proposition 1. Given a feature ϕ : U Ñ X , a scale o : X ˆX ÝÑ V, and a strict weak order ą on V, the function υ ϕ o : UˆU ÝÑ V defined by υ ϕ o pabq " opϕpaq, ϕpbqq is a variation iff ab ą cd ô ϕpaqϕpbq ą X ˆX ϕpcqϕpdq, where ą X ˆX is the (strict weak) order induced on X ˆX by the scale o ( i.e., opxyq ą opztq ô xy ą X ˆX zt).

Proof. (Sketch) For two pairs ab and cd of U ˆU, ϕpaqϕpbq P X ˆX , and ϕpcqϕpdq P X ˆX . The relation ą on V is strict weak, so exactly one of ϕpaqϕpbq ą X ˆX ϕpcqϕpdq, ϕpaqϕpbq ă X ˆX ϕpcqϕpdq, or ϕpaqϕpbq " X ˆX ϕpcqϕpdq holds.

Examples. Tab. 1 gives some examples of scales o defined on the set X ˆX . The variation υ eyebrow " can be defined from the nominal feature ϕ " eyebrow : U ÝÑ X and the scale ": X ˆX ÝÑ t0, 1u as follows:

υ eyebrow " pabq " # 1 if eyebrowpaq " eyebrowpbq 0 if eyebrowpaq ‰ eyebrowpbq
The indicator function 1 P of the class "sharing the property P " can be defined from the feature ϕ " P and the scale 1 : X ˆX ÝÑ t0, 1u as follows:

1 P pabq " υ P 1 pabq " # 1 if both P paq and P pbq hold 0 otherwise

The variation υ color Id pabq " pcolorpaq, colorpbqq can be defined from the nominal feature ϕ " color : U ÝÑ X with X " tr, g, bu, and the scale Idpxyq " xy. The variation υ age d1 pxyq " |agepyq ´agepxq| can be defined from the numerical feature ϕ " age : U ÝÑ N and the scale d 1 : N ˆN ÝÑ N where d 1 is the Manhattan distance. This function returns a natural number representing the (absolute) difference in value of the property age. Let ϕ 1 , ϕ 2 , . . . , ϕ n be n numerical features, and v be the function such that vpaq " pϕ 1 paq, ϕ 2 paq, . . . , ϕ n paqq P R n for a P U. The variation Ý Ñ ab " υ v ´pabq " vpbq ´vpaq gives the difference between two vectors. Let ľ aÑ be the restriction of ľ to tau ˆU. An inequality ab ľ aÑ ac is interpreted as "b is at least as similar to a as c is to a". If the order ľ aÑ is a complete preorder on tau ˆU, there exists a function u : tau ˆU ÝÑ R, called a utility function, such that for all b, c P U, ab ą ac ô upabq ą upacq. The utility function can be defined for example using a distance measure }.} p on U by taking u " υ v dp where υ v dp pabq " } Ý Ñ ab} p gives the distance between a and b. Consider the function ϕ " a Ñ : U ÝÑ N `that gives for each p P U the size of the principal ideal Ó ap " tab P tau ˆU | ap ľ abu. The variation υ aÑ d1 pbcq " |sizepÓ acq ´sizepÓ abq| gives a numerical measure of distance between two elements b and c when a is chosen as a frame of reference. In particular, if b " a, υ aÑ d1 pacq " |sizepÓ acq ´1| gives a distance from a to c in the sense of the nearest neighbors: υ aÑ d1 pacq " k iff c is the k´Nearest Neighbor of a. The scale AP pXY q " pXzY , Y zXq represents the difference from a set X to a set Y by "what is lost" from X to Y and "what is gained" from X to Y . The scale ∆pXY q " |XzY | `|Y zX| measures the symmetric difference between X and Y . If the sets X " pX 1 , X 2 , . . . , X n q and Y " pY 1 , Y 2 , . . . , Y n q are elements of a cartesian product A 1 ˆA2 ˆ. . . A n , the scale ∆ is given by ∆pXY q " ř i p|X i zY i | `|Y i zX i |q. The function spX, Y q " pXzY , X X Y , Y zXq corresponds to what Amos Tversky called a matching function in [START_REF] Tversky | Features of Similarity[END_REF].

Mapping

Analogical Equalities. An equivalence class of a variation can be seen as a contour line in the underlying similarity model. An analogical equality ab " υ cd holds if d differs from c as b differs from a according to the scale υ. The equivalence class rabs υ " tcd P U ˆU | υpabq " υ υpcdqu of a pair ab groups a set of pairs of U ˆU which are " equally different ", i.e., which differ from each other as much as b differs from a. Mapping a pair ab to a pair cd consists in establishing an analogical equality, i.e., finding a variation υ for which the two pairs belong to the same contour line for υ:

Find υ such that ab " υ cd (Mapping)
If " υ is the identity relation (ab " υ cd ô υpabq " υpcdq), then each value P V defines an equivalence class denoted by pυ : q " tab P U ˆU | υpabq " u Sharing a difference defines a contour line. In Fig. 2, the pair aa 1 can be mapped to the pair bb 1 since υ eyebrow " paa 1 q " υ eyebrow " pbb 1 q " 1, so they belong to the same equivalence class pυ eyebrow " : 1q for the variation υ eyebrow " . Similarly, a contour line of the variation υ v ´groups in the same equivalence class the pairs ab and cd when their difference is represented by the same vector (i.e., such that Ý Ñ ab " Ý Ñ cd). Sharing a commonality (like having a property P ) also defines a contour line. Let s P U be an element that has a property P (i.e., such that P psq holds). The variation 1 P can be used to map the pair ss to all pairs st P rsss 1 P " p1 P : 1q (and thus s to t). An analogical proportion also defines a contour line. The analogical proportion X : Y :: Z : T is defined [START_REF] Prade | Reasoning with Logical Proportions[END_REF] by XzY " ZzT and Y zX " T zZ, which means that the analogical proportion X : Y :: Z : T holds for two pairs XY and ZT iff the two pairs take the same value for the variation AP pXY q " pXzY , Y zXq. Similarly, in preference theory, the pairs of transactions satisfying the same conditional preference rules constitute a contour line. Suppose U denotes a set of objects called transactions. Let I be a set of distinct literals (called items), and ϕ : U ÝÑ PpIq the function that associates to each transaction a set of items. A transaction t is said to be preferred to u according to a conditional preference rule π : i `ą i ´| C iff pi `, C, i ´q Ď spϕptq, ϕpuqq with spXY q " pXzY , X X Y , Y zXq [START_REF] De Amo | Contextual preference mining for user profile construction[END_REF]. As a result, two transactions that take the same value for this variation satisfy the same conditional preference rules.

Analogical Inequalities. The strict weak relation ą on V defines a linear order which acts as a contour scale on U ˆU. Inequalities define a linear order on contour lines (since ab ľ cd ô rabs " ľ rcds " ). For example, in Fig. 2, two faces having the same eyebrow may be more similar than two faces having different eyebrows, i.e., 1 ľ 0 ô pυ eyebrow " : 1q ľ pυ eyebrow " : 0q. Similarly, by taking the usual order ĺ on N, one can define a contour scale: n ĺ m ô pυ age d1 : nq ľ pυ age d1 : mq. This scale expresses that " the lower the age difference, the more similar ". By structuring the set PpAq ˆPpAq with the inclusion relation Ď defined by pA, Bq Ď pA 1 , B 1 q iff A Ď A 1 and B Ď B 1 , a contour scale can be defined on analogical proportions: Ď k ô pAP : q ľ pAP : kq. Such a scale formalizes that all things equal, the less properties are lost or gained when going from a set X to a set Y , the more similar X and Y are. For example, if X " ta, bu, Y " tb, du, Z " ta, cu, and T " tcu, then ZT ľ XY since AP pZT q " ttau, tuu " Ď AP pX, Y q " ttau, tduu " k. The relation rabs " ľ rcds " is noted a : b ! c : d in [START_REF] Prade | Analogical inequalities[END_REF]. According to Tversky, a matching function should enforce a monotony property. This property is equivalent to the definition of a contour scale. By structuring the set PpAq ˆPpAq ˆPpAq with the inclusion relation Ď defined by pA, B, Cq Ď pA 1 , B 1 , C 1 q iff A Ď A 1 , B Ě B 1 , and C Ď C 1 a contour scale on conditional preference rules can be defined as: Ď k ô ps : q ľ ps : kq.

a b c d X 1 1 0 0 Y 0 0 1 0 Z 0 1 1 0 T 1 0 1 1
Analogical Dissimilarity. Analogical dissimilarity measures the distance between two contour lines. Analogical dissimilarity is given by a scale ad on V, which is coherent with analogy, i.e., adpab, cdq " 0 ô ab " υ cd. As a consequence, if ab " υ a 1 b 1 and cd " υ c 1 d 1 , then adpab, cdq " adpa 1 b 1 , c 1 d 1 q. If " υ is the identity relation, the previous observation implies that adpab, cdq gives the same value for all ab P pυ : q and cd P pυ : 1 q taken in two different contour lines. Therefore, we will write: adp , 1 q " adpab, cdq for ab P pυ : q and cd P pυ : 1 q The variation υ aÑ d1 pbcq " |sizepÓ acq ´sizepÓ abq| gives a numerical measure of distance between b and c with respect to a. The analogical dissimilarity is given by adpab, cdq " υ The analogical dissimilarity between two vectors [START_REF] Miclet | Analogical Dissimilarity: Definition, Algorithms and Two Experiments in Machine Learning[END_REF] gives the difference between two contour lines:

adpab, cdq " υ υ v dp pab, cdq " } Ý Ñ ab ´Ý Ñ cd} p i.e., adp Ý Ñ u , Ý Ñ v q " } Ý Ñ v ´Ý Ñ u } p
If the feature ϕ : U ÝÑ PpAq associates to each element of U a subset of a set A, the analogical dissimilarity between analogical proportions is given by a symmetric difference:

adpab, cdq " υ υ ϕ AP ∆ pab, cdq " |U zW | `|W zU | `|V zQ| `|QzV | with U " XzY, V " Y zX, W " ZzT, and Q " T zZ
where X " ϕpaq, Y " ϕpbq, Z " ϕpcq, and T " ϕpdq To illustrate, take A " ta, b, c, du, and a, b, c, d P U such that X " ϕpaq " ta, bu, Y " ϕpbq " tcu, Z " ϕpcq " tb, cu, and T " ϕpdq " ta, c, du (Tab. 2). The scale AP gives AP pXY q " U V with

-U " XzY " ta, bu is what is lost from X to Y -V " Y zX " tcu is what is gained from X to Y and AP pZT q " W Q with -W " ZzT " tbu is what is lost from Z to T -Q " T zZ " ta, du is what is gained from Z to T
The analogical dissimilarity between ab and cd is adpab, cdq " 1 `0 `1 `2 " 4. If the feature ϕ : U ÝÑ PpIq associates to each transaction a set of items, the analogical dissimilarity between two pairs of transactions is given by the symmetric difference:

adpab, cdq " υ υ ϕ s ∆ pab, cdq " |U zW | `|W zU | `|RzS| `|SzR| `|V zQ| `|QzV | with U " XzY, V " Y zX, R " X X Y, S " Z X T, W " ZzT, and Q " T zZ
where X " ϕpaq, Y " ϕpbq, Z " ϕpcq, and T " ϕpdq For example, if I " ta, b, c, du, and a, b, c, d P U are defined as above by X " ϕpaq " ta, bu, Y " ϕpbq " tcu, Z " ϕpcq " tb, cu, and T " ϕpdq " ta, c, du, then

-R " X X Y " H is what is preserved X to Y -S " Z X T " tcu is what is preserved from Z to T
The analogical dissimilarity between ab and cd is adpab, cdq " 1`0`0`1`1`2 " 5. Therefore, we can say that the distance between the two conditional preference rules π

1 : i `1 ą i ´1 | C 1 and π 2 : i `2 ą i ´2 | C 2 is adpπ 1 , π 2 q " ˇˇi `1 ∆ i `2ˇ`ˇi ´1 ∆ i ´2ˇ`| C 1 ∆ C 2 |
For example, the distance between π 1 : c ą ab | H and π 2 : ad ą b | c is 5. Such measure could be useful e.g., to predict preferences in a transductive way [START_REF] Pirlot | Completing Preferences by Means of Analogical Proportions[END_REF].

Analogical Transfer

A situation of special interest is when two contour lines intersect. Let Υ " tυ 1 , υ 2 , . . . , υ n u be a set of n variations with υ i : U ˆU ÝÑ V i , and n ľ 2. For υ i , υ j P Υ, the inclusion of two contour lines at a point ab is called a co-variation and is written υ i ab ñ υ j . We say that the variation υ i co-varies with the variation υ j at ab. The notion of co-variation at a point ab P U ˆU can be extended to a subset S Ď U ˆU by stating that: If each " υi is the identity relation, then a co-variation is a functional dependency [START_REF] Badra | Reasoning with Co-variations[END_REF] between the two values " υ i pabq and k " υ j pabq, and the co-variation υ i ab ñ υ j can be written pυ i : q ab ñ pυ j : kq. For example, the co-variation pυ profile Id : pround, sharpqq taa 1 ,bb 1 u ñ pυ eyebrow " : 1q represents the fact that when the profile changes from round to sharp, the eyebrow stays the same.

υ i S ñ υ j iff υ i ab ñ υ j for all ab P S
Different semantics can be given to a co-variation, depending on how it is interpreted with respect to the inclusion of equivalence classes rabs υi Ď rabs υj . Such an inclusion of equivalence classes is known as the similarity principle, and states that the equivalence relations " υi and " υj are locally co-monotone, i.e., for all a, b, c, d P U, ab " υi cd ñ ab " υj cd. This principle expresses that if a and b are as similar as c and d for υ i , then a and b are also as similar as c and d for υ j . Therefore, different semantics can be given to a co-variation depending on which subset R Ď rabs υi of the equivalence class rabs υi the similarity principle is to be verified (Fig. 3):

-Bayesian semantics: R " rabs υi , i.e., the co-variation is verified on the whole equivalence class ; -Semi-bayesian semantics: R " Ş kPtiuYD rabs υ k , i.e., the co-variation is verified all things equal, but only in the subset of rabs υi where a set of variations υ k with k P D take the same value [START_REF] Žabkar | Extracting qualitative relations from categorical data[END_REF] ; -Ceteris paribus semantics: R " Ş k‰j rabs υ k , i.e., the co-variation is verified all things equal, in the subset of rabs υi where all other variations take the same value.

Performing an analogical transfer consists in taking the similarity principle as a local inference rule, in order to make the hypothesis that if a contour line rabs υi is included in a contour line rabs υj at a point ab, then rabs υi should also be included in rabs υj at (at least some of) other points cd of rabs υi (i.e., in some points cd for which adpab, cdq " 0), and even maybe at points that are not too dissimilar (i.e., in some points cd for which adpab, cdq ĺ k):

υ i ab ñ υ j , adpab, cdq ĺ k υ i cd ñ υ j (Transfer)
If each " υi is the identity relation, then (Transfer) gives: pυ i : 0 q ab ñ pυ j : γq, cd P pυ i : q such that adp 0 , q ĺ k pυ i : q cd ñ pυ j : γq

The analogical "jump" (AJ) is a the expression of an analogical transfer where υ i " 1 P , and υ j " 1 Q : p1 P : 1q ss ñ p1 Q : 1q, st P p1 P : 1q, with adp1, 1q " 0 p1 P : 1q st ñ p1 Q : 1q By applying the co-variation 1 P st ñ 1 Q , the state of sharing the property Q is transferred to st. The co-variation is a functional dependency, which is consistent with the observation of [START_REF] Davis | A Logical Approach to Reasoning by Analogy[END_REF] that a sufficient condition for an analogical jump (AJ) to be justified is that there exists a functional dependency holding between P and Q. Differences may also be used in the transfer. In Fig. 2, if we know that d 1 has a sharp profile, we can make the hypothesis that d A fortiori reasoning is a type of inference that exploits the co-monotony of two partial orders to estimate the value of an attribute. In [START_REF] Derrac | Inducing semantic relations from conceptual spaces: A data-driven approach to plausible reasoning[END_REF], the authors give the following example: "If whiskey is stronger than beer, and buying beer is illegal under the age of 18, then we can plausibly derive that buying whiskey is illegal under the age of 18". This inference consists in applying the co-variation pυ degree ĺ : 1q bw ñ pυ legal age ĺ : 1q, where b stands for beer, w stands for whiskey, and degree and legal age are two numerical features that associate to alcohols their degree and minimum legal age of consumption. Suppose now that we want to estimate the minimum legal age for cider (denoted by c). If we know that cider is less strong than beer (degreepcq ĺ degreepbq), then one can use analogical transfer to infer make the hypothesis that the minimum legal age for cider is less than the one for beer (18): Analogical classifiers and instance-based-learning methods apply the similarity principle to the nearest analogical neighbors [START_REF] Hug | Analogical classifiers: A theoretical perspective[END_REF]. In instance-based learning, the relation " belonging to the same class " is transferred from an element x to an element x 0 whenever x 0 is found to be in the neighborhood of x. For example, in the k-Nearest Neighbor algorithm, pυ xÑ dp : nq is the equivalence relation that relates the pairs xy that are the n th most distant from x according to }.} p , and pυ cls " : 1q is the equivalence relation that relates all elements that share the same class. The rule (Transfer) writes: Method. Among the attributes ϕ 1 , ϕ 2 , . . . , ϕ 18 , the quantity of α waves on the occipital lobe is given by the numerical attribute ϕ 1 " α.occipital. The quantity of left (resp., right) eye movements is given by the numerical attribute ϕ 2 " left eye (resp., ϕ 3 " right eye). The muscle tonus is given by the numerical attribute ϕ 4 " tonus. For each attribute, the square product of its feature space is discretized using the scale Sign (for a real value ľ 0), such that: is successfully applied to cd using the confidence measure P pcd P Bq " |AXB| |A| . Similarly, we estimate the probability that the reverse rule B ñ A is verified using the confidence measure P pcd P Aq " |AXB| |B| . These two indicators are used to test the two hypothesis H 0 and H 1 . Results. First results (Tab. 3) show that whatever the discretization, the presence in polysomnographies of property A is not discriminant to predict the sleep transition. However, the sleep transition from wakefulness (W ) to light sleep stage N 1 can be characterized by a signal having the property A. When signals are discretized with " 0, a very large proportion (97%) of such sleep transitions are associated with a polysomnography in which the form described by the expert is verified.

υ ϕi

Conclusion

There is an intimate link between the qualitative measurement of similarity and computational analogy. Some essential notions of formal models of analogy, such as analogical equalities/inequalities, or analogical dissimilarity, and the related inferences (mapping and transfer) can be formulated as operations on ordinal similarity relations. Qualitative similarity measurement thus provides a bridge between qualitative models and analogical methods. One direct outcome is that it should allow to inject more psychologically grounded similarity measures in learning methods.

Fig. 1 .

 1 Fig.1. The differences between the values of the three attributes goal (g), set of divisors (d), and left over (l) in the descriptions of some mathematical problems.

Fig. 2 .

 2 Fig. 2. Six faces.

υ aÑ d 1 d1

 1 pab, cdq for a, b, c, d P U i.e., adpn, mq " |n ´m| for n, m P N

Fig. 3 .

 3 Fig. 3. The co-variation pυi : αq ab ñ pυj : γq can be interpreted with different semantics on the set Υ " tυi, υj, υ k , υmu. Hatched areas on the figure represent the resulting domains of validity of the co-variation. a) Bayesian semantics: the co-variation is verified on the whole equivalence class pυi : αq b) Semi-bayesian semantics: the co-variation is verified only on a subset of the equivalence class where some other variations (here, υm) keep the same value c) Ceteris paribus semantics: the co-variation is verified on the subset of the equivalence class where all other variations keep the same value.

1

 1 has the same eyebrow as d (curved), since d has a round profile and d 1 has a sharp profile:

Table 2 .

 2 Four subsets X, Y , Z, and T .

  In this experiment, an expert formulated some knowledge the he/she uses in the interpretation of polysomnographies, and we want to check to which extent this knowledge fits the actual data. The knowledge acquired from the expert expresses the fact that the transition between wakefulness (stage W ) and light sleep stage N 1 is characterized by a decrease of α waves on the occipital lobe, and/or a decrease of the quantity of eye movements (left or right) and/or a decrease of muscle tonus.Materials.Ten subjects showing no clinical sign of sleep disorders spent three consecutive nights at the hospital to monitor their sleep with a polysomnography. Afterwards, an experienced sleep expert scored all the curves to generate a hypnogram, which is the succession of sleep stages during the night by 30second epochs. Our work focused on sleep transitions, which are more difficult to describe since they are not similar to any given sleep stage, but curves are composed of waveforms and sleep events varying from one state to another, which makes their study of interest. Let U be the set of epochs of a given patient. Each epoch e i is described by 18 numerical attributes ϕ 1 , ϕ

	pυ xÑ dp : 0q	xx ñ pυ cls " : 1q, xx0 P pυ xÑ dp : nq with adp0, nq ĺ k
	pυ xÑ dp : nq	xx 0 ñ pυ cls " : 1q
	6 Experiments

2 , . . . , ϕ 18 , described by a categorical attribute s : U ÝÑ tW, N 1 , N 2 , N 3 , Ru giving the associated sleep stage: W (Wakefulness), N 1 , N 2 (light sleep), N 3 (deep sleep), and R (REM, Rapid Eye Movements Sleep).

  `if ϕ i pyq ą ϕ i pxq ` {2 ´if ϕ i pyq ă ϕ i pxq ´ {2 o if ϕ i pyq P rϕ i pxq ´ {2, ϕ i pxq ` {2sThe expert knowledge is formalized by the two rules A ñ B and B ñ A with: Id : pW, N 1 qqThe goal is to test two hypothesis (labelled H 0 and H 1 hereafter) on the set of pairs of epochs E " tpe n´2 , e n q | e n´2 , e n P Uu Ď U ˆU that are distant of one minute, for every patient and every night of the dataset.H 0 The rule A ñ B is verified H 1 The rule B ñ A is verifiedAnalogical dissimilarity is defined for the rule A ñ B by the scale adpab, cdq " 1 A pab, cdq " 1 if Apabq and Apcdq, and 0 otherwise. Similarly, analogical dissimilarity is defined for the rule B ñ A by the scale adpab, cdq " 1 B pab, cdq " 1 if Bpabq and Bpcdq, and 0 otherwise. As a result, for two pairs ab, cd P A, if we know that ab P B, we estimate the probability P pcd P Bq that the transfer rule A

	A : pυ α.occipital Sign	: ´q _ pυ left eye Sign	: ´q _ pυ right eye Sign	: ´q _ pυ tonus Sign :	´q
	B : pυ s
		$
		'
		&
	Sign pxyq "
		'
		%

ab ñ B, cd P A such that adpab, cdq " 0 A cd ñ B

Table 3 .

 3 Confidence indicators for the rules A ñ B and B ñ A according to the value given to the discretization parameter .

	Rule	Indicator			Value for
				0	0.1 0.5	1	5	10 50 100
	A ñ B |AXB| |A|	(in %)	2.5 2.8 3.2 3.8 4.8 6.2 12 19.5
	B ñ A |AXB| |B|	(in %)	97.4 94.6 89.5 84.5 62.3 45 7.8 1.5

The latter are a very common way to represent differences between two objects, although not the only one. Variations could for example represent complex rewriting rules, such as term reduction relations.