Isabelle Chalendar

Lucas Oger

Jonathan R Partington

Linear isometries of Hol(D)

Keywords: 2020 Mathematics Subject Classification. 47B33, 30H05, 47A10 Fréchet space, holomorphic functions, isometry, Blaschke product, weighted composition operator, spectrum. 1

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Introduction

Let X be a topological space of holomorphic functions on the open unit disc D. For m a holomorphic function on D and φ a holomorphic self-map of D, then the linear operator defined by W m,φ (f ) = m(f • φ), f ∈ X, is called the weighted composition operator with symbols m and φ.

Observe that W m,φ (f ) = M m C φ (f ), where M m (f ) = mf is the multiplication operator with symbol m and C φ (f ) = f • φ is the composition operator with symbol φ. If m is identically 1, then W m,φ = C φ , and if φ is the identity, then W m,φ = M m . Weighted composition operators are fundamental in the study of Banach and Hilbert spaces that embed continuously in Hol(D), the algebra of all holomorphic functions on D. Indeed, the study of the geometry of a space X is centered on the identification of the linear isometries on X, and there is an obvious connection between weighted composition operators and isometries. This connection can be traced back to Banach himself. In [START_REF] Banach | Théorie des opérations linéaires[END_REF], Banach proved that the surjective isometries on C(K), the space of continuous real-valued functions on a compact metric space K, are of the form T : f → m(f • φ), with |m| ≡ 1 and φ a homeomorphism of K onto itself.

Although the characterisation of isometries is an open problem for most Banach spaces of holomorphic functions, there are many spaces for which the isometries are known.

r On the Hardy space H p of the open unit disc D, p ̸ = 2, Forelli has shown that the isometries are certain weighted composition operators ( [START_REF] Forelli | The isometries of H p[END_REF]). r On the Hilbertian Hardy space H 2 on D, there are too many isometries to have a complete characterisation. However, in [START_REF] Chalendar | Weighted composition operators: isometries and asymptotic behaviour[END_REF]Prop. 4.2], Chalendar and Partington showed that provided that the symbol φ is inner, there exist weighted composition operators W m,φ that are isometries. Recall that on H 2 , C φ is isometric if and only if φ is inner and φ(0) = 0 (see [START_REF] Schwartz | Composition Operators on H p[END_REF]), whereas W m,φ is isometric implies that φ is inner but no extra condition on φ is required.

r On the Bergman space A p of D, p ̸ = 2, Kolaski ( [START_REF] Kolaski | Isometries of weighted Bergman spaces[END_REF]) showed that the surjective isometries are weighted composition operators. Zorboska ([21]) initiated the study of isometric weighted composition operators on A 2 of D. Those classes of operators were also considered in [START_REF] Chalendar | Weighted composition operators: isometries and asymptotic behaviour[END_REF] on weighted Bergman spaces. r On the disc algebra A(D), El-Gebeily and Wolfe ( [START_REF] El-Gebeily | Isometries of the disc algebra[END_REF]) showed that the isometries are of two types: either they are weighted composition operators (« Type 1 » isometries), or we have to add an extension operator to these « Type 1 » isometries. r On the Bloch space, the first study of the isometries was made by Cima and Wogen in [START_REF] Cima | On isometries of the Bloch space[END_REF]. They analyzed the isometries on the subspace of the Bloch space of the open unit disc whose elements fix the origin. On this space, they showed that the surjective isometries are normalized compressions of weighted composition operators induced by disc automorphisms, whereas Colonna ([9]) gave a characterisation of the isometric composition operators on the Bloch space.

We will not give an exhaustive list of contributions in this area but we would like to emphasize the fact that the weighted composition operators play a central role in the study of the isometries on several spaces of holomorphic functions on D.

The goal of this paper is to characterise the linear isometries on the set Hol(D) of all holomorphic functions on the open unit disc D. This is a Fréchet space, endowed with the family of seminorms defined by

∥f ∥ ∞,1-1/p := sup |z|≤1-1/p |f (z)| , f ∈ Hol(D), p ∈ N.
Other sequences are possible, and the results we prove will apply to these too. These seminorms are associated with the topology of uniform convergence on all compact subsets of D. An isometry of Hol(D) is simply an isometry for all the seminorms ∥•∥ ∞,1-1/p , thanks to the following lemma.

Lemma 1.1. Let X be a Fréchet space, and (∥•∥ k ) k≥1 an associated increasing family of semi-norms. A linear operator T : X → X is isometric, considering the distance d X defined by

d X (x, y) = ∞ k=1 2 -k min(1, ∥x -y∥ k ), if and only if for all x ∈ X and k ∈ N, ∥T x∥ k = ∥x∥ k . Proof. Assume that for all x ∈ X and k ∈ N, ∥T x∥ k = ∥x∥ k . Then, for all x, y ∈ X, d X (T x, T y) = ∞ k=1 1 2 k min(1, ∥T x -T y∥ k ) = ∞ k=1 1 2 k min(1, ∥T (x -y)∥ k ) = ∞ k=1 1 2 k min(1, ∥x -y∥ k ) = d X (x, y).
Hence, the operator T is isometric on X for the distance d X .

Conversely, if there exist x ∈ X and p ∈ N such that ∥T x∥ p ̸ = ∥x∥ p , consider the smallest such p (that is ∥T x∥ q = ∥x∥ q for all q < p). Assume that ∥T x∥ p < ∥x∥ p , and take y = x/ ∥x∥ p . Recall that for q < p < r, ∥y∥ q ≤ ∥y∥ p = 1 ≤ ∥y∥ r .

Then, since ∥y∥ p > ∥T y∥ p ,

d X (y, 0) = ∞ k=1 1 2 k min(1, ∥y∥ k ) = p-1 k=1 1 2 k ∥y∥ k + 1 2 p ∥y∥ p + ∞ k=p+1 1 2 k > p-1 k=1 1 2 k ∥T y∥ k + 1 2 p ∥T y∥ p + ∞ k=p+1 1 2 k min(1, ∥T y∥ k ) = d X (T y, 0).
Finally, the operator T is not isometric on X, for the distance d X .

If we have the reverse inequality ∥T x∥ p > ∥x∥ p , then by taking z = x/ ∥T x∥ p , since ∥z∥ p < ∥T z∥ p = 1, the same calculations give

d X (T z, 0) = ∞ k=1 1 2 k min(1, ∥T z∥ k ) = p-1 k=1 1 2 k ∥T z∥ k + 1 2 p ∥T z∥ p + ∞ k=p+1 1 2 k > p-1 k=1 1 2 k ∥z∥ k + 1 2 p ∥z∥ p + ∞ k=p+1 1 2 k min(1, ∥z∥ k ) = d X (z, 0),
so T is also not isometric on X for the classical distance. □

A similar result holds for another commonly-used metric on Hol(D).

Lemma 1.2. Let X be a Fréchet space, and (∥•∥ k ) k≥1 the associated increasing family of semi-norms. A linear operator T : X → X is isometric, considering the distance d ′ X defined by

d ′ X (x, y) = ∞ k=1 2 -k ∥x -y∥ k 1 + ∥x -y∥ k ,
if and only if for all x ∈ X and k ∈ N, ∥T x∥ k = ∥x∥ k .

Proof. The « if » implication is clear, and we now prove the « only if » implication. Let a k = ∥x∥ k and b k = ∥T x∥ k . These are increasing sequences and we can clearly assume that not all the a k are 0 and not all the b k are 0, as otherwise x = 0. So let us suppose that

a k > 0 for k ≥ k 1 and similarly b k > 0 for k ≥ k 2 (so that a k = 0 if k < k 1 and b k = 0 if k < k 2 ).
We have that for each λ > 0

∞ k=k 1 1 2 k λa k 1 + λa k = ∞ k=k 2 1 2 k λb k 1 + λb k .
Divide by λ, set λ = 1 + s and write α k = 1/a k and β k = 1/b k . We have

∞ k=k 1 1 2 k 1 α k + 1 + s = ∞ k=k 2 1 2 k 1 β k + 1 + s ,
initially for s > 0 but, because these series define H 1 (C + ) ∩ H 2 (C + ) functions on the right half-plane C + , they define the same function. These are sums of reproducing kernels (see, for example, [START_REF] Partington | Linear operators and linear systems. An analytical approach to control theory[END_REF]Sec. 1.2]), and we may take an inner product with the function e -ws ∈ H ∞ (C + ) for w > 0 to get

∞ k=k 1 1 2 k exp(-w(α k + 1)) = ∞ k=k 2 1 2 k exp(-w(β k + 1)).
Again, these are equal as H ∞ (C + ) functions of w, so on w = iy for y ∈ R they are equal almost everywhere, and they are both almost-periodic functions of y, so that α k = β k for all k (this standard fact can be found in many places: for example, [START_REF] Partington | Linear operators and linear systems. An analytical approach to control theory[END_REF]Sec. 5.1]). This completes the proof. □

The paper is organised as follows. First, in Section 2, we characterise the linear isometries of Hol(D), using only two different seminorms. We prove (Theorem 2.1) that they are trivial weighted composition operators in the following sense: they are defined by

T α,β (f )(z) = αf (βz), f ∈ Hol(D), z ∈ D, |α| = |β| = 1.
We will mainly need the two following results. The first one comes from [2, Proposition 2.1] and gives a necessary and sufficient condition to be a composition operator on Hol(D). The second one is a new characterisation of finite Blaschke products. Then X := {ξ ∈ T : |g(ξ)| = 1} is either finite, or is equal to T. Moreover, if X = T, then g is a finite Blaschke product.

Proof. Note that g(D) ⊂ D. Indeed, by the open mapping theorem, if there exists z 0 ∈ D such that |g(z 0 )| = 1, then there exists some z 1 ∈ D close to z 0 such that |g(z 1 )| > 1, which is impossible.

Consider now g as a continuous map from T to D. Then X = g -1 (T) is a closed subset of T (since g(D) ⊂ D).

Assume that X is not finite. Then, the set T\X is an open subset of T, that is a union of an infinite number of open intervals of T. We look at the edges of these intervals, which lie in X. If none of these edges is a limit of points of X, then the set X only consists of isolated points (the edges). But since X is infinite, by the Bolzano-Weierstrass theorem, X must contain a limit point, contradicting the assumption. Hence, one of the edges is indeed a limit point of X, denoted in the following as z 0 . This point satisfies |g(z 0 )| = 1, and there exist two sequences (u n ) ⊂ X and (v n ) ⊂ T\X (it is sufficient to take points from the chosen interval) which tend to z 0 , such that |g(u

n )| = 1 and |g(v n )| < 1.
We set ϕ 1 , ϕ 2 two conformal maps from T to R ∪ {∞} defined by

ϕ 1 (z) = i z -z 0 z + z 0 , ϕ 2 (z) = i z -g(z 0 ) z + g(z 0 ) . They satisfy ϕ 1 (z 0 ) = 0 and ϕ 2 (g(z 0 )) = 0. Denote h = ϕ 2 • g • ϕ -1 1 .
Since ϕ -1 1 is holomorphic near 0, as is ϕ 2 near g(z 0 ), the function h is holomorphic around 0, satisfies h(0) = 0 and there exist two sequences

(a n ), (b n ) ⊂ R (corresponding to (u n ) and (v n ) ⊂ T) such that a n → 0, b n → 0, h(a n ) ∈ R and h(b n ) ̸ ∈ R. However, h ′ (0) = lim n→∞ h(a n ) a n ∈ R.
The map h 1 : z → h(z)/z -h ′ (0) is also holomorphic around 0, and satisfies h 1 (a n ) ∈ R. Then, in a same way, we get h ′ 1 (0) = h ′′ (0) ∈ R. Iterating this reasoning, all the derivatives of h at 0 must be real, so

h(b n ) ∈ R (since b n ∈ R). Impossible.
Finally, we must have T\X = ∅, i.e., X = T, that is g(T) ⊂ T. Hence, g is a finite Blaschke product, using [START_REF] Garcia | Introduction to model spaces and their operators[END_REF][START_REF] Nikolski | Operators, functions, and systems: an easy reading[END_REF]. □

Next, in Section 3, we focus on the composition, multiplication, and weighted composition operators that are similar to a linear isometry of Hol(D). We obtain a complete description of those operators (Theorem 3.2).

In Section 4, we give a characterisation of the linear isometries for a single seminorm ∥•∥ ∞,1-1/p (Theorem 4.2). There are more than for two seminorms, as expected, and the results from the disc algebra, by El-Gebeily and Wolfe ( [START_REF] El-Gebeily | Isometries of the disc algebra[END_REF]), will be in the spotlight in this section. We also need the following lemma.

Lemma 1.5. Let ε > 0, and f ∈ Hol((1+ε)D) such that f (D) = D and f ((1+ε)D) ⊂ (1+ε)D.
Then, f is a rotation.

Proof. Using Theorem 1.4, we know that f is a finite Blaschke product. Moreover, the zeroes of f are in (1 + ε) -1 D. We consider three cases.

(i) If f (z) = e iθ z n , for some integer n ≥ 2, then for z 0 = 1 + aε,

|f (z 0 )| = (1 + aε) n > 1 + ε, for a ∈ (0, 1) sufficiently close to 1. Hence, f ((1 + ε)D) ̸ ⊂ (1 + ε)D. (ii) Assume that f (z) = e iθ z-α 1-αz , for some 0 < |α| < 1 1+ε . Since f is continuous, we have f ((1 + ε)D) ⊂ (1 + ε)D =⇒ f ((1 + ε)D) ⊂ (1 + ε)D.
Hence, for all x ∈ R,

(1 + ε)e ix -α 1 -(1 + ε)αe ix ≤ 1 + ε ⇐⇒ 1 -αe -ix 1+ε 1 -(1 + ε)αe ix ≤ 1.
In particular, for x such that αe -ix = |α|, we obtain

1 - |α| 1 + ε ≤ 1 -(1 + ε) |α| ⇐⇒ 1 1 + ε ≥ 1 + ε.
This last inequality is impossible.

(iii) General case: Let f be a finite Blaschke product which is not a rotation. Assume that f has at least two zeroes (counting the multiplicities), and that 0 is not the only zero of f . Hence, we may write

f (z) = e iθ z n z -α 1 -αz × k i=1 z -α i 1 -α i z , with 0 < |α| , |α i | < 1 1+ε . Once again, since f is continuous, we must have f ((1 + ε)D) ⊂ (1 + ε)D.
Take z 0 = (1 + ε)e -i arg(α) . Then, by (ii),

z -α 1 -αz > 1 + ε. Moreover, e iθ z n 0 > 1. Finally, since |z 0 | > 1 and for all i ∈ {1, • • • , k}, |α i | < 1, we get k i=1 z -α i 1 -α i z > 1.
Thus, |f (z 0 )| > 1 + ε, a contradiction. To conclude, the function f must be a rotation. □ Finally, in Section 5, we compute, using the results of [START_REF] Arendt | Spectral properties of weighted composition operators on Hol(D) induced by rotations[END_REF], the spectra of the operators found in the previous sections. The common property of these weighted composition operators is that the symbol is a rotation. Hence, we have to consider separately the operators with constant weights and those whose weights are finite Blaschke products.

Isometries of Hol(D)

We begin by characterising all the isometric operators of Hol(D). To do this, we first focus on the operators that are isometric for two different seminorms ∥•∥ ∞,p .

Theorem 2.1 (Main theorem).

Let 0 < r 1 < r 2 < 1 be fixed, and T : Hol(D) → Hol(D) a linear and continuous operator such that for all f ∈ Hol(D) and i ∈ {1, 2},

∥T (f )∥ ∞,r i = ∥f ∥ ∞,r i = sup |z|≤r i |f (z)| .
Then there exist two constants α, β ∈ T such that for all f ∈ Hol(D),

T (f )(z) = αf (βz) =: T α,β (f )(z).
Proof. We are going to use Lemma 1.3. Recall that for all n ∈ N 0 , e n (z) = z n .

Step 1: We begin by showing that T e 0 = αe 0 , with

|α| = 1. Indeed, since T is isometric for ∥•∥ ∞,r 1 and ∥•∥ ∞,r 2 , we obtain ∥T e 0 ∥ ∞,r 1 = ∥T e 0 ∥ ∞,r 2 = 1.
Using the maximum modulus principle, the map T e 0 is constant and unimodular.

In the following, consider T := αT , so that T is still isometric for the two seminorms ∥•∥ ∞,r 1 and ∥•∥ ∞,r 2 , and T e 0 = e 0 .

Step 2: Set r = r 1 or r 2 . We show that for all n ≥ 1,

r n T ⊂ ( T e n )(rD) =: K.
Indeed, assume that there exists ξ = r n e iθ such that ξ ̸ ∈ K, which is a compact subset of D. Then δ = d(ξ, K) > 0, and K ⊂ r n D\D(ξ, δ). We set f = e 0 + e -iθ e n . Hence, T f = e 0 + e -iθ T e n . Now, compare the seminorms: we have ∥f ∥ ∞,r = 1 + r n and

T f ∞,r = sup |z|≤r 1 + e -iθ ( T e n )(z) = sup w∈K 1 + e -iθ w ≤ sup |z|≤r n ,|z-r n |≥δ |1 + z| < 1 + r n . This is impossible, since T is isometric for ∥•∥ ∞,r . Therefore, r n T ⊂ K.
Moreover, because T is an isometry, we get K ⊂ r n D.

In the following, we set f n = T e n , and for all 0 < r < 1, we consider g n,r defined by

g n,r (z) = 1 r n f n (rz).
Then g n,r is holomorphic on r -1 D, and satisfies

T ⊂ g n,r (D) ⊂ D. Since g n,r (D) ⊂ D, if we set X = {ξ ∈ T : |g n,r (ξ)| = 1}
, then if X was finite, we would have g n,r (X) also finite. Impossible because T ⊂ g n,r (D). Therefore, X = T by Theorem 1.4. This implies that the maps B 1 = g n,r 1 and B 2 = g n,r 2 are finite Blaschke products. Hence,

f n (z) = r n 1 B 1 (z/r 1 ), |z| < r 1 , r n 2 B 2 (z/r 2 ), |z| < r 2 .
Step 3: Let us write all the terms of the Blaschke products, for |z| < r 1 < r 2 . There exist

numbers α 1 , • • • , α K 1 and β 1 , • • • , β K 2 such that 0 < |α i | , |β j | < 1 and for |z| < r 1 , r n-N 1 1 e iθ 1 z N 1 K 1 i=1 z r 1 -α i 1 -α i z r 1 = r n-N 2 2 e iθ 2 z N 2 K 2 j=1 z r 2 -β j 1 -β j z r 2 (1) ⇐⇒ r n-N 1 1 e iθ 1 z N 1 K 1 i=1 z -r 1 α i r 1 -α i z = r n-N 2 2 e iθ 2 z N 2 K 2 j=1 z -r 2 β j r 2 -β j z (2)
We study the poles and zeroes of the Blaschke products. In (2), the zero 0 has same order on each side of the equation, so N 1 = N 2 =: N . Moreover, the zeroes and poles of the two Blaschke products should coincide, so K 1 = K 2 =: K, and for each i ∈ {1, • • • , K}, there exists j ∈ {1, • • • , K} such that

α i r 1 = β j r 2 and r 1 α i = r 2 β j .
In particular, we obtain

r 1 r 2 = β j α i = α i β j , so |α i | 2 = |β j | 2 ,
(z) = e iθ 1 z N 1 , B 2 (z) = e iθ 2 z N 2 .
Step 4: Putting the formulas in the expression of f n , we get

f n (z) = r n 1 e iθ 1 z N 1 r N 1 1 = r n 2 e iθ 2 z N 2 r N 2 2 , |z| < r 1 ≤ r 2 .
The order of the zero 0 is unique, so we obtain N 1 = N 2 =: N , and

r n-N 1 e iθ 1 z N = r n-N 2 e iθ 2 z N .
Identifying the modulus and argument of the coefficients, we have r n-N 1 = r n-N

2

, so n = N (since r 1 ̸ = r 2 ), and e iθ 1 = e iθ 2 =: δ n , that is

f n (z) = e iθ z n = δ n e n (z), |z| < r 1 .
Using analytic continuation, the last equality is valid for z ∈ D, so we have proved that for all n ∈ N, there exists δ n ∈ T such that T e n = δ n e n .

Step 5: Since T is isometric, for all r ∈ {r 1 , r 2 },

sup |z|=r T (e k + e k+1 + e k+2 )(z) = sup |z|=r |(δ k e k + δ k+1 e k+1 + δ k+2 e k+2 )(z)| = sup |z|=r δ k z k + δ k+1 z k+1 + δ k+2 z k+2 = sup |z|=r z k + z k+1 + z k+2 .
Dividing by z k , we obtain

sup |z|=r δ k + δ k+1 z + δ k+2 z 2 = sup |z|=r 1 + z + z 2 = 1 + r + r 2 .
Factorising by δ k , since |δ k | = 1, we have

sup |z|=r 1 + δ k+1 δ k z + δ k+2 δ k z 2 = sup |z|=r 1 + z + z 2 = 1 + r + r 2 .
Using the triangle inequality, for all z satisfying |z| = r,

1 + δ k+1 δ k z + δ k+2 δ k z 2 ≤ 1 + r + r 2 ,
with equality if and only if

0 = arg(1) ≡ arg δ k+1 δ k z ≡ arg δ k+2 δ k z 2 mod 2π. Denoting θ = arg[z], this gives 0 ≡ arg[δ k+1 ] -arg[δ k ] + θ ≡ arg[δ k+2 ] -arg[δ k ] + 2θ mod 2π.
Finally,

(3) 2θ + 2 arg[δ k+1 ] -2 arg[δ k ] ≡ 0 mod 4π 2θ + arg[δ k+2 ] -arg[δ k ] ≡ 0 mod 2π, so arg[δ k ] -2 arg[δ k+1 ] + arg[δ k+2 ] ≡ 0 mod 2π.
Step Proof. Let T be a linear isometry of Hol(D). Then, in particular, T is a linear isometry for two different seminorms associated with Hol(D), using Lemma 1.1. Theorem 2.1 concludes.

Conversely, let |α| = |β| = 1. Then, for all p ∈ N and f ∈ Hol(D),

∥T α,β (f )∥ ∞,1-1/p = sup |z|≤1-1/p |αf (βz)| = sup |w|≤1-1/p |f (w)| = ∥f ∥ ∞,1-1/p .
Hence, T α,β is a linear isometry of Hol(D). □

Operators similar to an isometry on Hol(D)

In this section, we will focus on linear operators of Hol(D) that are similar to an isometry of Hol(D). Let us recall the concept of similarity. Definition 3.1. Two linear operators T, V ∈ L(Hol(D)) are similar if there exists some U ∈ L(Hol(D)) invertible such that U -1 T U = V .

In the following, we will only consider some classes of operators: composition, multiplication, and weighted composition ones. The goal of this section is to obtain a characterisation of such operators that are similar to a linear isometry of Hol(D). The main theorem is the following. Theorem 3.2. Let T be a continuous operator on Hol(D).

(i) T = C φ is a composition operator similar to a linear isometry of Hol(D) if and only if φ is bijective and elliptic, i.e., with a fixed point on D.

(ii) T = M m is a multiplication operator similar to a linear isometry of Hol(D) if and only if m is constant and unimodular.

(iii) T = W m,φ is a weighted composition operator similar to a linear isometry of Hol(D) if and only if φ is bijective and elliptic with σ p (W m,φ ) ∩ T ̸ = ∅.

It is now time to prove this result. We recall that the linear isometries of Hol(D) are defined by T α,β (f )(z) = αf (βz), f ∈ Hol(D), |α| = |β| = 1. as we proved in Theorem 2.1.

Composition operators -Proof of Theorem 3.2.(i ).

First, assume that φ is bijective and elliptic. Denote by α ∈ D its fixed point. Consider the map ψ defined by

ψ(z) = z -α 1 -αz .
Then, ψ ∈ Hol(D), ψ is bijective, ψ -1 = ψ, and ψ(α) = 0. Hence, φ = ψ • φ • ψ is a bijective self-map of D, with a fixed point at 0. By Schwarz's Lemma ( [START_REF] Abate | Iteration theory of holomorphic maps on taut manifolds[END_REF][START_REF] Rudin | Function theory in the unit ball of C n[END_REF]), there exists β ∈ C such that |β| = 1 and φ(z) = βz. Hence,

C ψ • C φ • C ψ = C φ = T 1,β .
The operator C φ is therefore similar to a linear isometry of Hol(D). Now, consider φ not both bijective and elliptic. If C φ was similar to a linear isometry of Hol(D), then we would write

C φ = U V U -1 ⇐⇒ U -1 C φ U = V,
with V isometric. Iterating this equality n times, and we get

U -1 C n φ U = U -1 C φ [n] U = V n . Thanks to Lemma 1.1, for all n, p ∈ N and f ∈ Hol(D), (U -1 C n φ U )(f ) ∞,1-1/p = ∥V n (f )∥ ∞,p = ∥f ∥ ∞,1-1/p . Since U is invertible, writing g = U (f ), for all p ∈ N and g ∈ Hol(D), (U -1 C n φ )(g) ∞,1-1/p = U -1 (g • φ [n] ) ∞,1-1/p = U -1 (g) ∞,1-1/p .
By the Denjoy-Wolff theory ( [START_REF] Bracci | Continuous semigroups of holomorphic self-maps of the unit disc[END_REF][START_REF] Cowen | Composition operators on spaces of analytic functions[END_REF]), there exists a point ω ∈ D such that the iterates φ [n] of φ converge uniformly on all compact subsets of D to ω. Finally, considering g(z) = z -ω, we have g ∈ Hol(D), and using the continuity of U ,

U -1 (g) ∞,1-1/p = U -1 (φ [n] -ω) ∞,1-1/p ----→ n→+∞ 0.
However, g ̸ ≡ 0, so U -1 (g) ̸ ≡ 0 (because U is invertible). Thus, there exists p ∈ N such that ∥U -1 (g)∥ ∞,1-1/p ̸ = 0, leading to an absurdity. Conclusion: C φ is similar to a linear isometry of Hol(D) if and only if φ is bijective and elliptic. The proof of Theorem 3.2.(i) is then complete.

Multiplication operators -Proof of Theorem 3.2.(ii ).

Assume that m is constant and unimodular. Denote by α this constant. Therefore, for all f ∈ Hol(D), we may write

[M m (f )](z) = m(z)f (z) = αf (z) = [T α,1 (f )](z).
Hence, M m = T α,1 is a linear isometry of Hol(D).

Conversely, let M m be similar to a linear isometry of Hol(D). We may write M m = U V U -1 , with U invertible and V isometric. Iterating this n times, we obtain

M n m = M m n = U V n U -1
, with V n also isometric (by induction). We will consider two cases.

r Assume that for all z ∈ D, |m(z)| < 1. Then, for all p ∈ N, ∥m∥ ∞,1-1/p = c < 1. Hence, ( 4)

∥m n ∥ ∞,1-1/p = c n ---→ n→∞ 0.
Consider 1, the map defined by 1(z) = 1 for all z ∈ D. Then, for all n ∈ N 0 , since V n is isometric and V n U -1 = U -1 M m n , for all p ∈ N, by Lemma 1.1,

U -1 (1) ∞,1-1/p = V n U -1 (1) ∞,1-1/p = U -1 M m n (1) ∞,1-1/p = U -1 (m n ) ∞,1-1/p ----→ n→+∞ 0,
using the continuity of U and ( 4). But U -1 (1) ̸ ≡ 0 (since U is invertible), so there exists p ∈ N such that ∥U -1 (1)∥ ∞,1-1/p ̸ = 0. Impossible.

r Assume that there exists

z 0 ∈ D such that |m(z 0 )| > 1.
Then, for all p ∈ N, n ∈ N 0 and f ∈ Hol(D), we have

∥m n f ∥ ∞,1-1/p = U V n U -1 (f ) ∞,1-1/p .
In particular, for p = p 0 such that

|z 0 | ≤ 1 -1 p 0 and f = 1, U V n U -1 (1) ∞,1-1/p 0 = ∥m n ∥ ∞,1-1/p 0 ---→ n→∞ +∞.
However, U is continuous and V is isometric, so there exist C > 0 and q ∈ N such that Remark 3.3. We have proved that there is no multiplication operator similar to a linear isometry of Hol(D), other than the isometries themselves.

U V n U -1 (1) ∞,1-1/p 0 ≤ C U -1 (1) ∞,1-1/q < +∞.

Weighted composition operators -Proof of Theorem 3.2.(iii ).

To obtain the third and last result of Theorem 3.2, we begin with a simple observation. If W m,φ is similar to a linear isometry of Hol(D), then W m,φ is invertible.

Proof. If the operator W m,φ is similar to a linear isometry of Hol(D), then we have

W m,φ = U V U -1 ,
with U invertible and V isometric. Since all linear isometries of Hol(D) are invertible (Theorem 2.1 and [3, Proposition 2.1]), W m,φ is also invertible, with inverse U V -1 U -1 . □ Therefore, in the following, using [3, Proposition 2.1], we may consider only weighted composition operators with symbol φ bijective, and weight m such that ∀z ∈ D, m(z) ̸ = 0.

1 st case: Assume that φ is not elliptic. Then, by [3, Proposition 2.3, Theorems 6.1 & 7.1], there are only two possiblities: either

σ p (W m,φ ) = C * , or σ p (W m,φ ) = ∅. If σ p (W m,φ ) = C * , consider f ∈ Hol(D)\{0} such that W m,φ (f ) = 2f . Suppose that W m,φ
is similar to a linear isometry of Hol(D). We may write W m,φ = U V U -1 , with U invertible and V isometric. An easy induction gives

W n m,φ = U V n U -1 for n ∈ N, so W n m,φ (f ) = 2 n f = U V n U -1 f. Since U , U -1 and V are continuous, there exists a constant c > 0 such that U V n U -1 f ∞,1/2 ≤ c. But f ̸ ≡ 0, so ∥f ∥ ∞,1/2 > 0, and W n m,φ (f ) ∞,1/2 = ∥2 n f ∥ ∞,1/2 = 2 n ∥f ∥ ∞,1/2 ---→ n→∞ +∞.
We have obtained an absurdity.

If σ p (W m,φ ) = ∅ and W m,φ is similar to a linear isometry V of Hol(D), then 1 st conclusion: In order to make W m,φ similar to a linear isometry of Hol(D), the symbol φ must be bijective and elliptic. Now, we consider symbols φ that are elliptic and bijective. Without loss of generality (using the map ψ p.10), we will assume that the fixed point of φ is 0. Using Schwarz's Lemma ( [START_REF] Abate | Iteration theory of holomorphic maps on taut manifolds[END_REF][START_REF] Rudin | Function theory in the unit ball of C n[END_REF]), we can write

∅ = σ p (W m,φ ) = σ p (U V U -1 ) = σ p (V ), since U is invertible. But σ p (V ) ̸ = ∅,
φ(z) = βz, |β| = 1. Definition 3.5. Let φ : z → βz, with |β| = 1.
r The map φ is periodic if there exists N ∈ N such that β N = 1.

r The map φ is aperiodic if for all n ∈ N,

β n ̸ = 1.
First, we will assume that |m(0)| = 1. Indeed, we have the following lemma.

Lemma 3.6. Let φ : D → D be elliptic and bijective such that φ(0) = 0, and m ∈ Hol(D).

If W m,φ is similar to a linear isometry of Hol(D), then |m(0)| = 1.

Proof. If |m(0)| < 1, then considering the map g = U -1 1, with 1(z) = 1, we obtain U -1 V n U g ∞,0 = W n m,φ (g) ∞,0 = |m(0)| n |g(0)| ---→ n→∞ 0.
Using the continuity of U , we get

1 = ∥1∥ ∞,0 = ∥V n U g∥ ∞,0 -→ 0, which is impossible. If |m(0)| > 1, then U V n U -1 1 ∞,0 = W n m,φ (1) ∞,0 = |m(0)| n ---→ n→∞ +∞.
However, there exists p ∈ N and c > 0 such that

U V n U -1 1 ∞,0 ≤ c V n U -1 1 ∞,1-1/p = c U -1 1 ∞,1-1/p < +∞.
This leads to an absurdity. □ Now, we consider separately periodic and aperiodic symbols.

2 nd case: Assume that φ : z → βz is elliptic periodic. Then, there exists N ∈ N such that

W N m,φ = M m N , with m N (z) = N -1 k=0 (m • φ [k] )(z) = N -1 k=0 m(β k z).
If W m,φ is similar to a linear isometry of Hol(D), then so is W N m,φ = M m N . Using Theorem 3.2.(ii) (proved in Section 3.2), the map m N must be constant, and unimodular.

Conversely, if m N is constant and unimodular, thanks to [4, Lemma 3.5], since |m(0)| = 1, we may write

m = exp( m), with m = iθ + N -1 k=1 z k f k (z N ), θ ∈ R, f 1 , • • • , f N -1 ∈ Hol(D).
Moreover, we show that there exists U invertible such that U W m,φ U -1 = V , with V a linear isometry of Hol(D). Indeed, we look at particular U and V :

U = M w , V = T e iθ ,β ,
with w ∈ Hol(D) not vanishing on D, e iθ in the definition of m, and β in the definition of φ.

Note that since w does not vanish on D, we may write w = exp( w).

Then, for all h ∈ Hol(D) and z ∈ D,

(U W m,φ U -1 )(h)(z) = V (h)(z) ⇐⇒ h(βz) w(βz) m(z)w(z) = e iθ h(βz) ⇐⇒ w(z) w(βz) exp N -1 k=1 z k f k (z N ) = 1 ⇐⇒ exp w(z) -w(βz) + N -1 k=1 z k f k (z N ) = 1 ⇐= w(z) -w(βz) + N -1 k=1 z k f k (z N ) = 0. Consider f k (z) = ℓ≥0 a (k)
ℓ z ℓ and w(z) = ℓ≥0 b ℓ z ℓ . We are searching for maps w such that ( 5)

w(z) -w(βz) + N -1 k=1 z k f k (z N ) = ℓ≥0 b ℓ (1 -β ℓ )z ℓ + N -1 k=1 a (k) ℓ z N ℓ+k = N -1 k=1 ℓ≥0 b N ℓ+k (1 -β k ) + a (k) ℓ z N ℓ+k = 0.
Finally, for all ℓ ≥ 0, considering b N ℓ = 0 and

b N ℓ+k = a (k) ℓ β k -1 , 1 ≤ k ≤ N -1,
we obtain the equation [START_REF] Banach | Théorie des opérations linéaires[END_REF]. Hence, for this particular w, we get M w W m,φ M -1 w = T e iθ ,β .

2 nd conclusion: If φ is periodic, then the operator W m,φ is similar to a linear isometry of Hol(D) if and only if the map m N is constant and unimodular, that is if and only if σ p (W m,φ ) ∩ T ̸ = ∅, by [START_REF] Arendt | Spectral properties of weighted composition operators on Hol(D) induced by rotations[END_REF]Theorem 3.4].

3 rd case: Assume that φ is elliptic aperiodic. If W m,φ is similar to a linear isometry of Hol(D), then we may write

U -1 W m,φ U = T α,β ,
for some |α| = 1, |β| = 1. We apply this equality for f = 1. Hence,

(U -1 W m,φ U )(1) = T α,β (1) = α1 ⇐⇒ m(U 1 • φ) = α(U 1). Therefore, α ∈ T is an eigenvalue of W m,φ . Conversely, assume that σ p (W m,φ ) ∩ T ̸ = ∅. Let λ ∈ σ p (W m,φ ) ∩ T, and f ∈ Hol(D) such that W m,φ (f ) = m(f • φ) = λf.
Then, for all z ∈ D, f (z) ̸ = 0. Indeed, assume that there exists z 0 ∈ D such that f (z 0 ) = 0. Since λ ̸ = 0, by induction, for all n ∈ N 0 , we get (f • φ [n] )(z 0 ) = f (β n z 0 ) = 0. Because φ is aperiodic, it implies that f = 0 on {z ∈ C : |z| = |z 0 |}, so f = 0 on D using analytic continuation. Let U = M f . Then, U is invertible, and for all g ∈ Hol(D),

(U -1 W m,φ U )(g) = (U -1 W m,φ )(f g) = U -1 (m(f • φ)(g • φ)) = λU -1 (f (g • φ)) = λ(g • φ) = T λ,β (g).
Hence, W m,φ is similar to a linear isometry of Hol(D).

Final conclusion: A weighted composition operator W m,φ is similar to a linear isometry of Hol(D) if and only if φ is bijective and elliptic with σ p (W m,φ ) ∩ T ̸ = ∅. By [4, Proposition 3.3 and 3.6] a necessary condition to get a nonempty point spectrum is that m does not vanish on D.

Isometries of Hol(D) with a single seminorm

We now consider the isometries for a single seminorm. In this case, we expect to get more operators. This is indeed the case. In order to show it, we will use an analogy with the disc algebra A(D), a case already discussed by El-Gebeily and Wolfe in [START_REF] El-Gebeily | Isometries of the disc algebra[END_REF].

We begin this section with a useful proposition. It will allow us to prove the main theorem of this section. Then there exist a map B 1 , which is constant or a finite Blaschke product, and β ∈ T such that for all f ∈ Hol(D), T (f )(z) = B 1 (z/r)f (βz).

Proof. To begin, let us consider U : Hol(D) → Hol(r -1 D) the operator defined by

U (f )(z) = f (rz), |z| ≤ 1/r.
Then, by setting S = U T U -1 : Hol(r -1 D) → Hol(r -1 D), the operator S is isometric for ∥•∥ ∞,1 , that is, for the disc algebra norm.

Note that the set of all polynomials is dense in Hol(r -1 D) and in A(D) for the norm ∥•∥ ∞,1 . Hence, we can extend S to an operator S : A(D) → A(D) so that S is still isometric for ∥•∥ ∞,1 .

In the following, we will continue to write S even if we consider S.

Step 1: Recall that for all n ∈ N 0 , we denote e n (z) = z n , and f n = Se n . Let K be the set defined in Proposition 4.1. Then for all ξ ∈ K,

ρ(ξ) × f 0 (ξ) = 1, ρ(ξ) × f 1 (ξ) = ϕ(ξ).
Using the first equation, we obtain that for all ξ ∈ K, ρ(ξ) ̸ = 0. We can say more: since ρ is continuous on the compact set K, there exists δ > 0 such that |ρ(ξ)| ≤ δ for ξ ∈ K. We have the same result for f 0 . Furthermore,

ϕ(ξ) = f 1 (ξ) f 0 (ξ) , ξ ∈ K.
We focus on the Lebesgue measure m(K) of the set K relatively to T. We will show that m(K) > 0. Indeed, first, we can say that K is not finite (otherwise, we would have ϕ(K) = T also finite, impossible).

r If the map f 0 is constant, then ϕ is of the form cf 1 , with c ∈ C. Hence, we can extend ϕ to a holomorphic function on r -1 D, which is in particular C 1 on T. Assume that m(K) = 0, we would get m(ϕ(K)) = m(T) = 0, impossible.

r If the map f 0 is not constant, since it is analytic on r -1 D, because the zeroes are isolated, there is only a finite number of zeroes of f 0 that are unimodular. Denote them by ξ 1 , • • • , ξ n . Therefore, the map ϕ can be extended to a C 1 map on T\{ξ

1 , • • • , ξ n }, such that ϕ(K\{ξ 1 , • • • , ξ n }) contains T, except at most n points. Assume that m(K) = 0,
we would get m(T\{n points}) = 0, impossible. Finally, we have proved that m(K) > 0.

Step 2: By [16, Theorem 1.1] and [11, Proposition 1], there exist a function φ ∈ H ∞ of the form φ = h 1 /h 2 , with h 1 , h 2 ∈ A(D), and h 2 = 0 on a certain subset L ⊂ T; also, another map

ψ ∈ A(D) such that ∥φ∥ ∞,1 = ∥ψ∥ ∞,1 = 1, φ(K) = T, ψ = 0 on L and |ψ| = 1 on K satisfying S(f ) = ψ(f • φ), f ∈ A(D).
Recall that at the beginning, S is defined for f ∈ Hol(r -1 D). Thus, we obtain φ ∈ Hol(r -1 D), that is L = ∅. Hence, the map ψ does not vanish on T. In addition, by setting X = {ξ ∈ T : |φ(ξ)| = 1}, if X was finite, we would get φ(X) finite. This is impossible since φ(K) = T, so T ⊂ φ(D). By Theorem 1.4, φ is a finite Blaschke product. Now, assume that ψ is not constant. If ψ is not a finite Blaschke product, note that since ∥ψ∥ ∞,1 = 1, we have ψ(D) ⊂ D. By Theorem 1.4, the set X = {ξ ∈ T : |ψ(ξ)| = 1} is finite. Choose z 0 ∈ T\X such that φ -1 (φ(z 0 )) ⊂ T\X; this is possible since X is finite. Set

g(z) = φ(z 0 )z + 1 2 .
Then ∥g∥ ∞,1 = 1, and |g(φ(z

))| = 1 ⇐⇒ φ(z) = φ(z 0 ) ⇐⇒ z ∈ φ -1 (φ(z 0 )) ⊂ T\X. Thus, |g(φ(z))| = 1 =⇒ |ψ(z)| < 1 =⇒ |S(g)(z)| = |ψ(z)| |g(φ(z))| < 1. Therefore, ∥S(g)∥ ∞,1 < 1, which is impossible because S is isometric for ∥•∥ ∞,1 .
To conclude, X = T (by Theorem 1.4), and ψ is a finite Blaschke product.

Finally, S(f

)(z) = B 1 (z)f (B 2 (z))
, with B 1 and B 2 of the wanted form. We have only left to go back to T via the invertible operator U , and that gives

T (f )(z) = (U -1 S)(U f )(z) = U -1 (B 1 (z) × (U f )(B 2 (z))) = U -1 (B 1 (z) × f (rB 2 (z)) = B 1 (z/r)f (rB 2 (z/r)).
We finish the proof by noting that since the equality must hold for all f ∈ Hol(D), if we denote ϕ(z) = rB 2 (z/r), we must have ϕ(D) ⊂ D. Moreover, ϕ(rD) = rD, so by a renormalisation and Lemma 1.5, ϕ is a rotation. Hence, there exists |β| = 1 such that T (f )(z) = B 1 (z/r)f (βz). □

Spectral study of the isometries

In this section, we make a spectral study of the isometries of Hol(D), endowed with two or a single seminorm. We will consider separately the two cases. 5.1. Spectra of the isometries for two seminorms.

First, we focus on the isometries of Hol(D), that is the operators

T α,β : f -→ αf (β•).
Following the results of [START_REF] Arendt | Spectral properties of weighted composition operators on Hol(D) induced by rotations[END_REF], the results depend on the nature of the number β (cf. Definition 3.5). 5.2. Spectra of the isometries for one seminorm. Now, let 0 < r < 1. Using the results of Section 4, we want to find the spectra of the operators defined by T (f )(z) = B(z/r)f (βz), with B constant or a finite Blaschke product, and β ∈ T.

1 st case: If B is a constant map of modulus one, denote by ξ ∈ T the constant. Then we obtain T = T ξ,β , so the spectral study has already been done previously. Thus, the spectrum of T contains a set which is dense in the circle centered at 0 and of radius |B(0)|.

Lemma 1 . 3 .Theorem 1 . 4 .

 1314 Let T : Hol(D) → Hol(D) be linear and continuous. Then T is a composition operator if and only if for all n ∈ N 0 , T e n = (T e 1 ) n , with e n (z) = z n . Let ε > 0 and g ∈ Hol((1 + ε)D) such that g(D) ⊂ D.

6 :Corollary 2 . 2 .

 622 We show by induction that for all n ∈ N 0 , arg[δ n ] ≡ n arg[δ 1 ] mod 2π. r n = 0 or 1: It is immediate (see Step 1 for n = 0). r n = 2: We only need that arg[δ 0 ] -2 arg[δ 1 ] + arg[δ 2 ] ≡ 0 mod 2π and arg[δ 0 ] = 0. r Assume that the formula is valid for n ≥ 2. Using (3), arg[δ n-1 ] -2 arg[δ n ] + arg[δ n+1 ] ≡ 0 mod 2π. Using induction hypothesis, we have (n-1) arg[δ 1 ]-2n arg[δ 1 ]+arg[δ n+1 ] ≡ 0 mod 2π. After rearranging the terms, we obtain arg[δ n+1 ] ≡ (n + 1) arg[δ 1 ] mod 2π. Conclusion: For all n ∈ N 0 , T (e n ) = δ n 1 e n . Hence, T (e n ) = ( T (e 1 )) n , so T is a composition operator, with symbol φ = T (e 1 ) = δ 1 e 1 , by Lemma 1.3. Finally, multiplying by α, for all f ∈ Hol(D), (T f )(z) = αf (δ 1 z). □ We obtain the following corollary. The only linear operators T : Hol(D) → Hol(D) that are isometries of Hol(D) are those of the form T α,β , with |α| = |β| = 1.

  The two cases lead to an absurdity. Ergo, for all z ∈ D, |m(z)| ≤ 1, and there exists z 0 ∈ D such that |m(z 0 )| = 1. Using the maximum modulus principle, we have shown that m is constant, of modulus one.Conclusion: M m is similar to a linear isometry of Hol(D) if and only if m is constant and unimodular. The proof of Theorem 3.2.(ii) is then complete.

Lemma 3 . 4 .

 34 Let φ : D → D holomorphic, and m ∈ Hol(D).

  using Theorem 2.1 and [4, Theorems 3.4 & 3.7].

Proposition 4 . 1 (

 41 [START_REF] El-Gebeily | Isometries of the disc algebra[END_REF]). Let S : A(D) → A(D) an isometry. Then there exists a closed subset K ⊂ T, and two maps ρ : K → T continuous, ϕ : K → T continuous and onto, such that for all f ∈ A(D) and ξ ∈ K, ρ(ξ) × (Sf )(ξ) = f (ϕ(ξ)).

Theorem 4 . 2 .

 42 Let 0 < r < 1 fixed, and T : Hol(D) → Hol(D) a linear continuous operator such that for all f ∈ Hol(D), ∥T (f )∥ ∞,r = ∥f ∥ ∞,r .

r

  If β is periodic, then σ(T α,β ) = σ p (T α,β ) = {αβ k : k ∈ N 0 }. r If β is aperiodic, then σ p (T α,β ) = {αβ k : k ∈ N 0 } ⊂ σ(T α,β ) ⊂ T,and if we consider for τ > 2 the set of all Diophantine numbers of order τ , defined byD(τ ) = {ξ ∈ R : ∃γ > 0, ∀p ∈ Z, ∀q ∈ N, |p/q -ξ| ≥ γq -τ },we obtain a little improvement: for β = e 2iπξ , ξ ∈ D(τ ), σ(T α,β ) ⊂ {e 2iπx : x ̸ ∈ Q} ∪ {1}.

2 4 ,

 4 nd case: If B is a finite Blaschke product, then since B is vanishing on D, we obtain σ p (T ) = ∅ (see[START_REF] Arendt | Spectral properties of weighted composition operators on Hol(D) induced by rotations[END_REF] Proposition 3.3 and 3.6]). Now, to compute the spectrum of T , consider once again the two subcases depending on the nature of β.r If β is periodic, then there exists N ∈ N, N ≥ 2 such that β N = 1 and β k ̸ = 1 for k < N . It follows that for all f ∈ Hol(D),T N (f )(z) = m N (z)f (z), m N (z) = Proposition 3.2], we have σ(T ) = {λ ∈ C : λ N ∈ m N (D)}. r If β is aperiodic, then by [4, Corollary 2.2], {β n B(0) : n ∈ N 0 } ⊂ σ(T ).Moreover, using [4, Proposition 7.2], we have σ(T ) ⊂ D(0, M ), with M defined by Re it /r) dt .Note that if m(z) = B(z/r), we can writem(z) = z N m(z), with m(z) = e iθ r N K j=1 z -rα j r -α j z , with α 1 , • • • , α K ̸ = 0.Since m is well defined in D, the equation (7.2) of[START_REF] Arendt | Spectral properties of weighted composition operators on Hol(D) induced by rotations[END_REF] is also valid for M R with R → 1, that is for M 1 = M . Hence, .Therefore, if B has d zeroes (counting the multiplicities), then {β n B(0) : n ∈ N 0 } ⊂ σ(T ) ⊂ D(0, r -d ).

  and r 1 = r 2 . Impossible. Ergo, the Blaschke products B 1 and B 2 are of the form B 1

Acknowledgments: This research is partly supported by the Bézout Labex, funded by ANR, reference ANR-10-LABX-58. The first two authors wish to thank E. Abakoumov for stimulating questions and suggestions.


Luis Palacios Medinacelli

Florian Noyrit
email: florian.noyrit@cea.fr

Chokri Mraidha
email: chokri.mraidha@cea.fr

A Methodology for Knowledge Integration and Acquisition in Model-Based Systems Engineering

Keywords: Domain Specific Ontologies, Knowledge Integration, MBSE, Semantic Interoperability

In Model-Based System Engineering (MBSE) systems are represented as models using a predefined metalanguage such as SysML that hides some of the complexity behind the specification of a system, and provides experts with a rich syntax to define, share and constrain these models. Even though current MBSE design tools are sophisticated and support expressive meta-languages, these tools have limited capabilities on the detection of semantic errors, the integration of expert knowledge, or the ability to formalize the knowledge from the expert's design. Our work addresses these limitations by annotating the SysML models with domain-specific ontologies. Enabling this interaction not only makes the ontology's semantics available to the tooling environment, but the UML model specified by the designer can be translated into Ontology Web Language format (OWL), generating a system specification in terms of the ontology. These "annotated models" are suitable for reasoning tasks, like consistency check and instance checking. We particularly ensure the annotated model is a consistent extension of the domain-specific ontology, thus formalizing the expert's knowledge as a subontology. This extended ontology can be re-used and shared to evaluate and constrain further models, or be itself evaluated by semantic-compatible tools. In this article we present our approach for an OWL-MBSE integration, and show its feasibility via an implementation in the UAVs domain.

INTRODUCTION

Complex systems design, like Autonomous or Cyber-Physical Systems, require specific design and engineering techniques to ensure the resulting systems comply with their specifications. Among these techniques Model-Based System Engineering (MBSE) provides good practices and formalized syntax that make the engineering process systematic.

It notably helps in sharing the same interpretation of the models among experts. Among the existing modeling languages available for complex system design we consider SysML 1 , a UML 2 profile for system engineering. These languages have a very rich expressiveness. The sophistication of current modeling tools is correlated to this high expressiveness, thus requiring high expertise in the tool's specific representation language. Moreover, new modeling projects have to be built from scratch, rebuilding structures and descriptions that are common to specific domains (e.g. Autonomous Systems), where knowledge reuse is available in a limited way (e.g. model libraries). Despite the formal syntax of MBSE, its semantics are expressed in natural language, which poses a limitation for interpretation, sharing and reuse of these models. Moreover, because different stakeholders and different projects may define the same systems differently, semantic interoperability and the construction of knowledge bases from these system models remains challenging. The use of formal semantics is essential for explicit, shareable, and reusable knowledge representation. [START_REF] Yang | Ontology-based systems engineering: A state-of-the-art review[END_REF] There exists extensive research [START_REF] Yang | Ontology-based systems engineering: A state-of-the-art review[END_REF][START_REF] Atkinson | A detailed comparison of uml and owl[END_REF][START_REF] Berardi | Reasoning on uml class diagrams[END_REF][START_REF] Parreiras | Marrying model-driven engineering and ontology technologies: The twouse approach[END_REF]) on the benefits and potential of the interaction and integration of MBSE and ontologies. These vary from providing reference concepts and the definition of homogeneous terminologies, to allowing for machine readable axioms and integration, exchange and reuse of knowledge, among others. It is also noted in these works the challenges for a successful integration, among which we find: the (un)availability of the required ontologies along with their documentation; the different levels of abstraction and viewpoints, where the domain and scope of the ontology needs considerable effort to be aligned to a new application domain; the different interpretations of a given language construct depending on the viewpoint; and the gap between theoretically focused approaches and real-world applications.

To enable the interaction between different viewpoints and representations, and to enable the application and reuse of ontological knowledge within MBSE, in this article we present a formal approach to integrate and extract knowledge, expressed as an ontology, into/from MBSE tools.

Our approach emphasizes in the following challenges : 1) The integration of standardized and domain-specific languages, 2) The definition of the components a bidirectional mapping (OWL-UML) should consider, guided by the expressivity of specific description logic languages, 3) The ability to capture complex system structures and describe them as complex ontological definitions 4) the re-use of the captured knowledge.

In this paper we present the formalization of our approach, to integrate standardized domain-specific ontologies into MBSE tooling environments, and show its feasibility via an implementation in the UAVs (Unmanned Aerial Vehicles) domain.

The rest of this paper is organized as follows: in section 2 we present the works related to ontologies and MBSE. In section 3 we present our approach. In section 4 we present an implementation of the approach, motivated by the UAVs use case. Finally, we present our conclusions in section 5 and the further works in section ??.

RELATED WORKS

In this section we present some of the most relevant works regarding ontologies, UML/SysML and MBSE technologies. These works present the motivations, state of the art, challenges, and envisaged benefits from the interaction of the aforementioned technologies. A recurrent issue addressed by works combining UML and ontologies is the problem of semantic heterogeneity in distributed and delocalized companies, where problems of misunderstanding and information exchange may arise, due to different viewpoints, for which applications are developed [START_REF] Parreiras | Marrying model-driven engineering and ontology technologies: The twouse approach[END_REF][START_REF] Elasri | Semantic integration process of business components to support information system designers[END_REF]. There is also the risk of loss of information when exchanging between heterogeneous systems. In these works, the use of ontologies as models is proposed, to trace relevant and shared information related to the knowledge domain in question.

The work in [START_REF] Atkinson | A detailed comparison of uml and owl[END_REF] evidences that there is a lack of a complete mapping between the constructs of the two languages. Although, within the terminology of an ontology, there might be specific concepts and relations that suit specific UML constructs, allowing for a more complete mapping.

The work in [START_REF] Berardi | Reasoning on uml class diagrams[END_REF] explores the expressivity and reasoning complexity in UML diagrams, and the work in [START_REF] Parreiras | Marrying model-driven engineering and ontology technologies: The twouse approach[END_REF] aims to provide an integrated approach for UML class-based modeling and ontology modeling. There are several areas of system engineering (SE) knowledge where research between ontologies and SE has been conducted [START_REF] Yang | Ontology-based systems engineering: A state-of-the-art review[END_REF] : System Fundamentals, System engineering Standards, Generic Life Cycle Stages, Representing Systems with Models, Engineered System Contexts and System Engineering Management. The works considered in [START_REF] Yang | Ontology-based systems engineering: A state-of-the-art review[END_REF] summarize the causes for the difficulties in developing systems on budget and on time, and the considerable resource waste dealing with the correction of mistakes, into four reasons: 1) the implicit nature of SE, 2) the limitations of best-practice standards and meta-models, 3) the absence of a widely accepted and consistent terminology, and 4) inefficient collaborations due to the misunderstanding and misinterpretation.

Ontologies can improve system design, by facilitating communication among stakeholders having different concerns when designing, for example, a Cyber-Physical-System. Common, interdependent properties can be harmonized and synchronized, to manage inconsistencies (et al., 2016). Thus ontologies can ensure that multiple systems share a common terminology, which is the essence of knowledge sharing and reuse. Formal definitions for the different properties and processes of SE would be a significant contribution towards improving accuracy and precision in the implementation of SE [START_REF] Mezhuyev | Ontology based development of domain specific languages for systems engineering[END_REF]. And, by using a predefined ontology, it is possible to reduce the number of misinterpretations within projects [START_REF] Hallberg | Ontology for systems development[END_REF].

APPROACH

Let us first introduce some preliminary notions required to define our approach. In the following we assume the reader is familiar with OWL 3 ontologies and Description Logics DLs. DL is a family of FOL languages. Thanks to a carefully bounded expressiv-ity, some of them can provide tractable reasoning services, such as consistency or instance checking.

In our work we consider the description logics ALC H I D . The rationale behind this choice lies in that we aimed to at least provide ALC ′ s expressivity, plus enabling the extension of an ontology's taxonomy, and the use of datatypes. Next, we need to recall the corresponding syntax definitions of ALC H I D .

For further details the reader may refer to [START_REF] Baader | Appendix: description logic terminology. The Description logic handbook: Theory, implementation, and applications[END_REF] and [START_REF] Baader | Introduction to description logic[END_REF].

Let N C , N R , N D , and N I be pairwise disjoint (nonempty) sets of concept names, object property names, data property names, and individual names, respectively. We denote by

N - R the set of inverses of all r ∈ N R . A role is an element of N R ∪ N - R ∪ N D . Concepts are defined as follows: every φ ∈ N C is a concept, if o 1 , o 2 , . . . o n ∈ N I then {o 1 , o 2 , . . . o n } is a concept. If φ, φ 1 and φ 2 are concepts, then (φ 1 ⊔ φ 2 ),(φ 1 ⊓ φ 2 )
, and (¬φ) are concepts (called conjunction, disjunction, and negation, respectively). If r ∈ N R ∪ N - R then ∃r.φ, ∀r.φ are concepts. If D is a datatype and s ∈ N D , then ∃s.D, ∀s.D are concepts as well. We write ⊤ and ⊥ to abbreviate the concepts (φ ⊔ ¬φ) and (φ ⊓ ¬φ) , respectively. We eliminate parentheses as usual.

Axioms. An axiom in ALC H I has one of the To denote our concepts, roles and individuals, we adhere to the notation in [START_REF] Baader | The description logic handbook: Theory, implementation and applications[END_REF], where predicates starting in uppercase, like Device or SensorDevice, denote concepts; predicates starting in lower case, like sensingPart, denote roles. Individuals are denoted by terms in lowercase, and possibly carrying a subscript number. Individuals are asserted as instances of a class, as parameters of unary predicates, like Device(d 1 ), or as taking part in a role, like hasPart(d 1 , p 1 ); inverse roles carry the superscript " -", as in hasPart -.

Approach Definitions

In the following let O up , O d be two ontologies, where we define the union ontology as:

O C = {C 1 ,C 2 ...,C n }
O d → O up = ⟨IC, IR⟩ where: IC = {⟨C d 1 ,C up 1 ⟩, ⟨C d 2 ,C up 2 ⟩, . . . , ⟨C d n ,C up n ⟩} IR = {⟨r d 1 , r up 1 ⟩, ⟨r d 2 ,
U O d ,O up = O d ∪ O up ∪ { n i=1 C d i ⊑ C up i } ∪ { m j=1 r d j ⊑ r up j }
Where each ⟨C d i ,C

U O Drone ,O CORA = O Drone ∪ O CORA ∪ {SensorDevice ⊑ Device, Motor ⊑ ActuatorDevice}∪ {hasPart ⊑ part -, isConnectedTo ⊑ interactsWith}
O d → O up is consistent if: U O d ,O up |= ⊤ i.e. U O d ,O up is satisfiable.
The intended meaning of the concepts and relations from the ontology, have to be preserved throughout the transformation into UML and back to OWL. Because of the variety of the constructs in both formalisms, their correlation is not only not evident, but case-dependent. Moreover, there are specific sets of constructs and diagrams in UML for describing specific systems and entities. The UML specification 4 defines the UML meta-model. The root concepts : Element and Relationship, provide the basis for all other modeling concepts in UML. In the rest of this article, we prefix elements of the UML language with "uml::" (like uml::Class or uml::Component) and properties of those elements are separated by "::" as in uml::Element::ownedElement. For our work, the elements in the UML language, are considered as UML constructs, that can be specialized and/or instantiated by the system's designer to specify his model. We target these constructs to define a mapping between OWL and UML. Definition 4 (A mapping function µ O→UML ). Let

σ UML = {U 1 , U 2 , . . . U n }
be the set of names of all constructs in the UML metamodel (i.e. UML language), and let O be an ontology, with the set of concept names O C , and the set of role names O r .

We define the mapping:

µ O→UML = {µ C , µ OP , µ DP } with: µ C = {⟨C 1 , U 1 ⟩, ⟨C 2 , U 2 ⟩, . . . , ⟨C m , U m ⟩} µ OP = {⟨r 1 , U 1 ⟩, ⟨r 2 , U 2 ⟩, . . . , ⟨r n , U n ⟩} µ DP = {⟨r 1 , U 1 ⟩, ⟨r 2 , U 2 ⟩, . . . , ⟨r p , U p ⟩}
where : We can also see that conditions 1), 2) and 3) from definition 4 are satisfied.

1)∀⟨x, y⟩ ∈ µ C | x ∈ O C , y ∈ σ UML , ⟨z, y⟩ ̸ ∈ {µ OP , µ DP } 2)∀⟨x, y⟩ ∈ µ OP | x ∈ O r , y ∈ σ UML , ⟨z, y⟩ ̸ ∈ {µ C , µ DP } 3)∀⟨x, y⟩ ∈ µ DP | x ∈ O r , y ∈ σ UML , ⟨z, y⟩ ̸ ∈ {µ C , µ OP } Example 3 (Mapping µ O→UML ).
Definition 4 allows single ontology concepts (and relations), to be mapped to multiple UML constructs, and vice-versa. Note that the mapping µ O→UML does not map names in O to actual UML models elements, but to its meta-language. The mapping µ O→UML allows to select a relevant subset of names in O, and defines the elements in a UML model that can be typed by these names. This is achieved by creating UML stereotypes for each entry in the mapping. Stereotypes are a UML mechanism to extend its language, that allows to specialize an element, and are defined in a UML profile. Each stereotype can be applied to a restricted set of elements. Thus, the mapping is not applied to the UML model automatically. It is the system designer who, in the end, defines which ontology concepts correspond to (or generalise) elements in his model (i.e. by applying the stereotypes). The annotated UML model might further specialize the classes and relations from the ontology. It might represent single concepts as composite structures or as properties of classes. Thus an "inverse" mapping, from an annotated UML model to O, will indeed depend on µ O→UML , but we can not simply "inverse it" to get a meaningful representation of the UML model. To consider the multiple representations that a UML model can have, the mapping from UML to OWL needs to distinguish instances from concepts. It also needs to take into account the specialization mechanisms in UML, and it is desirable that complex structures and complex concept definitions are handled.

Definition 5 (A mapping function µ UML→O ). Let M be a UML model, and let σ M = {M 1 , M 2 , . . . M n } be the set of names of all elements in M . Then the mapping from UML model M to ontology O is defined by:

µ UML→O = {µ ′ ⊑ , µ ′ ≡ , µ ′ OP , µ ′ DP , µ ′ C(i) , µ ′ r(a,b) } with: µ ′ ⊑ = {⟨M 1 ,C 1 ⟩, ⟨M 2 ,C 2 ⟩, . . . , ⟨M i ,C i ⟩} µ ′ ≡ = {⟨M 1 ,C 1 ⟩, ⟨M 2 ,C 2 ⟩, . . . , ⟨M j ,C j ⟩} µ ′ OP = {⟨M 1 , r 1 ⟩, ⟨M 2 , r 2 ⟩, . . . , ⟨M k , r k ⟩} µ ′ DP = {⟨M 1 , r 1 ⟩, ⟨M 2 , r 2 ⟩, . . . , ⟨M l , r l ⟩} µ ′ C(i) = {⟨M 1 ,C(i) 1 ⟩, ⟨M 2 ,C(i) 2 ⟩, . . . , ⟨M m ,C(i) m ⟩} µ ′ r(a,b) = {⟨M 1 , r(a, b) 1 ⟩, ⟨M 2 , r(a, b) 2 ⟩, . . . ,
O UML = M µ UML→O
, recursively defined by:

∀⟨A, B⟩ ∈ µ ⊑ ⇔ {A ⊑ B} ∈ O UML ∀⟨A, B⟩ ∈ µ ≡ ⇔ {A ≡ B} ∈ O UML ∀⟨A, B⟩ ∈ µ OP ⇔ {A ⊑ B} ∈ O UML ∀⟨A, B⟩ ∈ µ DP ⇔ {A ⊑ B} ∈ O UML ∀⟨A,C(i)⟩ ∈ µ C(i) ⇔ {C(i)} ∈ O UML ∀⟨A, r(a, b)⟩ ∈ µ r(a,b) ⇔ {r(a, b)} ∈ O UML
Example 4 (Definition and Application of µ UML→O ). Consider a UML model M ′ , a simplified version of the model in figure 1, containing the following elements:

σ M ′ = {DroneSystem 1 , BatteryType 1 , MotorType 1 , PropellerType 1 , D subSystem1,Camera1, attachedTo, is 1 , s 1 .ownedElement.m 1 , m 1 , p 1 , s 1 , s 1 .ownedElement.p 1 }
, we define the mapping µ UML→O as:

µ ⊑ = {⟨PropellerType1, Propeller⟩, ⟨MotorType1, Motor⟩, ⟨DroneSystem1, Device⟩} µ ≡ = {⟨DroneSystem 1 , hasPart.PropellerType1 ⊔hasPart.MotorType1⊔ hasPart.BatteryType1⟩} µ OP = {⟨attachedTo, connectedTo⟩} µ DP = {} µ C(i) = {⟨m 1 , MotorType1(m 1 )⟩, ⟨p 1 , PropellerType1(p 1 )⟩, ⟨s 1 , DroneSystem1(s 1 )⟩} µ r(a,b) = {⟨is 1 , attachedTo(m 1 , p 1 )⟩, ⟨s 1 :: ownedElement :: m 1 , hasPart(s 1 , m 1 )⟩, ⟨s 1 :: ownedElement :: p 1 , hasPart(s 1 , p 1 )⟩}
Then, the application of µ UML→O to model M yields:

O UML = { PropellerType1 ⊑ Propeller, MotorType1 ⊑ Motor, DroneSystem1 ⊑ Device, DroneSystem1 ≡ hasPart.PropellerType1⊓ hasPart.PropellerType1, attachedTo ⊑ connectedTo, MotorType1(m 1 ), PropellerType1(p 1 ), DroneSystem1(s 1 ), attachedTo(m 1 , p 1 ), hasPart(s 1 , m 1 ), hasPart(s 1 , p 1 )} Definition 7 (A System Specification T-Box)
. Given an UML model M and the mapping µ UML→O , the System Specification T-Box T UML is the result of applying the restricted mapping:

µ T = {µ ⊑ , µ ≡ , µ OP , µ DP } to M.
T UML = M µ T Example 5 (System Specification T-Box). Given the UML model M ′ and the mapping µ T , the application of µ T to M yields:

T UML = { PropellerType1 ⊑ Propeller, MotorType1 ⊑ Motor, DroneSystem1 ⊑ Device, DroneSystem1 ≡ hasPart.PropellerType1⊓
hasPart.PropellerType1, attachedTo ⊑ connectedTo} Definition 8 (A UML System Instance). Given an UML model M and the mapping µ UML→O , a system instance A-Box A UML , is the result of applying the restricted mapping:

µ A = {µ C(i) , µ r(a,b) } to M. A UML = M µ A
Example 6 (System Instance). Given the UML model M and the mapping µ A , the application of µ A to M yields:

A UML = {MotorType1(m 1 ), PropellerType1(p 1 ), DroneSystem1(s 1 ), attachedTo(m 1 , p 1 ), hasPart(s 1 , m 1 ), hasPart(s 1 , p 1 )} Definition 9 (Consistent System Specification).

Given an ontology O, an UML model M and a mapping µ UML→O , a consistent system specification is a T-Box T UML s.t.

O ∪ T UML |= ⊤

Definition 10 (System Specification Model). Given an ontology O, an UML model M, a mapping µ UML→O , and a consistent system specification T UML ,a system specification model for the consistent

specification T UML is an A-Box A UML s.t. O ∪ T UML ∪ A UML |= ⊤

IMPLEMENTATION

The process of integration of ontologies and an UML system model is illustrated in Figure 2. In the implementation we target Papyrus5 (an open-source MBSE tool) and use UML Profiles to make the terminology of the ontology available to the system's designer.

The workflow is divided into four tasks, represented by the circles in the diagram. The inputs and outputs of these processes are depicted as color-coded files with extension .oml (blue) or .uml (yellow). In the following, we first explain the workflow and next, we detail the construction of the mappings.

In Figure 2 on the top left we have two ontologies:

CORA.owl = O 1 and ODrone.owl = O 2 . ODrone.owl is the Domain Specific Ontology [START_REF] Medinacelli | Augmenting model-based systems engineering with knowledge[END_REF]. This is the specific vocabulary used in the target domain of interest. Our implementation is encompassed within the autonomous systems domain, and specifically the UAVs subdomain. The relevant definitions in the domain specific ontology need to be mapped to UML. Specific ODrone concepts can be asserted as specializations of UML classes, and ODrone object and data properties can be asserted as specializations of UML associations.

Task 1 outputs a consistent integration of O 2 into O 1 , namely Union O 2 ,O 1 (see section 3). The union ontology Union O 2 ,O 1 consistently integrates the domain specific ontology ODrone into the upper and standardized ontology CORA. Thus providing a standardized domain specific ontology Union O 2 ,O 1 . Task 2 maps concepts and roles from Union O 2 ,O 1 to a UML profile thanks to the mapping specification µ O→UML , making the ontology semantics available to the system designer in the MBSE tooling.

Indeed, the profile can now be used to model a system in task 3. Thus it is the system designer who applies the mapping µ O→UML to a specific UML model M. Once the annotated model is ready, the application of the "inverse" mapping µ UML→O takes place in task 4. This process yields O UML , an ODrone/OWL com- pliant representation of the UML system, from where a system specification T UML and a system instance A UML can be obtained. The translated model O UML is suitable to be analyzed by reasoners. At this stage, constraints expressed as queries (SPARQL), rules (SWRL) or complex concept definitions (DL's/OWL) can be evaluated over the annotated UML model, in the same manner as in [START_REF] Medinacelli | Augmenting model-based systems engineering with knowledge[END_REF].

Mappings

To demonstrate the feasibility of the approach, we have implemented a mapping between ODrone and UML targeting: 1) Class Diagrams and 2) Composite Structure Diagrams. In the literature [START_REF] Mkhinini | Combining uml and ontology: An exploratory survey[END_REF], it is common to target Class Diagrams constructs due to its evident similarity with ontology constructs (e.g. classes, relations). Furthermore, Composite Structure diagrams allow further specifying the relations between the parts of a system. Using constructs in these two diagrams, the designer can describe the entities that play a role in his design, as well as specific composite structures that carry interconnected parts.

In the ODrone implementation, we aim to describe the physical components of UAV's design, that is, the parts of the system and how these are interconnected. UML as well as ODrone, consider a "composition relation" named part. The ODrone:hasPart object property's meaning is analogous to the uml::AggregationKind::composite property. Both describe composition of complex structures by their parts. This correspondence between ODrone and UML, is taken into account by algorithm 1. It shows how to implement a mapping that considers case-specific correspondences (between the ontology and UML) and allows for the description of complex composed UML structures using the ontology concepts and relations. The output of algorithm 1, is the restricted mapping µ T with which we can obtain a system specification T UML . Whereas algorithm 2, constructs the mapping µ A , which can be applied to a UML model M to obtain a system instance A UML .

These two algorithms are presented next.

Recall:

µ T = {µ ⊑ , µ ≡ , µ OP , µ DP }
In algorithm 1 we first (1) initialize the sets µ ⊑ , µ ≡ , µ OP , µ DP to be empty.

Then for each uml::Element in the model M (2) we verify if it is a (3) uml::Class. Regardless of whether the uml::Class has a stereotype (4), we add the pair ⟨uml :: Class, owl : T hing⟩ to µ ⊑ , thus every uml::Class is mapped as subclass of owl:Thing. Line (5) handles the case of specialized classes, and in (9) we state that each stereotyped class is subsumed by its stereotype. In line (11) we introduce the set CCD E = {} (Complex Concept Description for E), this set will be incrementally constructed to hold all stereotyped relations from the class (12-16), and all its uml::CompositeAggregation (19-23) in the form of a complex concept definitions. Note in line ( 22) that each uml::CompositeAggregation is mapped to odrone:hasPart. Thus relating a uml::Structure with a domain specific relation, existing only in ODrone. Line (26) states that the pair ⟨E,CCD E ⟩ belongs to µ ≡ , effectively providing a complex concept definition for E, in terms of ODrone. Finally, in (29) a subsumption relation is added to µ OP for each stereotyped uml::Relationship. Note that this specific mapping allows only for one stereotype per relation.

Let us now introduce the algorithm 2 to construct the mapping µ A . Recall:

µ A = {µ C(i) , µ r(a,b) }
Algorithm 2 constructs the sets composing µ A , and aggregates them into the output µ A . Algorithm 2 is more straightforward, since we target only two UML In algorithm 2 we first (1) initialize the sets µ C(i) , µ r(a,b) to be empty. Then for each uml::Element in the model M (2) we verify if it is a (3) uml::InstanceSpecification. The way UML assigns a class to an instance is through its classifiers. Lines (4-7) state that for every classifier C of an instance E, the pair ⟨E,C(E)⟩ belongs to the set µ C(i) . Note that this process allows for multiple classifiers for the same instance, which is intended in OWL. In the case where an instance has no classifier (for example an anonymous individual which just participates in a relation) These artifacts can be reused by new designs, or exploited by external tools and services that are compatible with ODrone and CORA, thus effectively allowing the system designer to extend the ontology via UML.

CONCLUSIONS

Our approach tackled two main problems: the availability and integration of domain specific ontologies into MBSE; and the capture and formalization of knowledge from the expert's design.

We have provided an end-to-end solution for the integration of formal vocabularies into MBSE tooling, effectively enabling the system designer to describe its system in terms of the ontology. This integration provides the context for other viewpoints and stakeholders to interact. We have formally defined the components a mapping from the terminology of an ontology, to UML and back should consider (encoded in OWL/UML format). This specification aims to ensure that the application of the mappings is consistent w.r.t. the domain specific ontology. Furthermore, by clearly defining a system specification as a T-Box, and a system instance as an A-Box, we separate these two aspects of system design, and we enable logic-based techniques (like reasoning or SAT solving) to evaluate and generate models for these artifacts.

As further work we aim to enable the integration of different ontologies, different system design tools (eg. safety, simulation, etc.), and to explore model generation for the obtained specifications.

  following forms: 1) φ ⊑ ψ (called concept inclusion axiom), where φ and ψ are concepts; (2) r ⊑ s (called role inclusion axiom), where either r, s ∈ N R ∪ N - R or r, s ∈ N D (3) φ(a) (called concept membership axiom), where φ is a concept and a ∈ N I ; (4) r(a, b) (resp., d(a, b)) (called role membership axiom), where r ∈ N R ∪ N - R (resp., d ∈ N D ) and a, v ∈ N I (resp., a ∈ N I and v is a data value). A Knowledge Base L is a finite set of axioms. Terminological axioms A general concept inclusion (GCI) has the form C ⊑ D where C and D are concepts. We write C ≡ D when C ⊑ D and D ⊑ C. A T-Box is a finite set of GCIs. Assertional axioms A concept assertion is a statement of the form C(a) where a ∈ N I and C is a concept. A role assertion is a statement of the form r(a, b) where a, b ∈ N I and r is a role. An A-Box is a finite set of assertional axioms. A DL knowledge base (KB) is a pair (T , A) for T-Box T and A-Box A. Since knowledge bases encode ontologies, in the rest of this paper we refer to a DL knowledge base and an ontology indistinctly.

  is the set of concepts names in ontology O, and O r = {r 1 , r 2 ..., r m } is the set of object and data property names in O. Definition 1 (Integration Specification). Given two ontologies O up , O d , an Integration Specification of O d into O up is a pair, of sets of pairs, of the form:

  r up 2 ⟩, . . . , ⟨r d m , r up m ⟩} , for m, n ≥ 0 , and each C d i ∈ O C d Example 1 (Upper and Domain-Specific Ontologies). Let O CORA and O Drone be two ontologies, with O C CORA = {Device, ActuatorDevice} O r CORA = {part, robotPart, interactsWith} O C Drone = {SensorDevice,Camera, Motor, Propeller} O r Drone = {hasPart, isConnectedTo} An Integration Specification of O Drone into O CORA can be defined as: O Drone → O CORA = ⟨{⟨SensorDevice, Device⟩, ⟨Motor, ActuatorDevice⟩}, {⟨hasPart, part -⟩, ⟨isConnectedTo, interactsWith⟩}⟩ Definition 2 (Union Ontology). Given two ontologies O d , O up and an integration specification O d → O up ,

  up i ⟩ ∈ IC with n = |IC| and, each ⟨r d j , r up j ⟩ ∈ IR with m = |IR|. Example 2 (Union Ontology). Given the integration specification O Drone → O CORA defined in example 1, the union ontology of O CORA and O Drone is:

Definition 3 (

 3 Consistent Integration). Given two ontologies O d , O up , an integration specification O d → O up and the union ontology U O d ,O up , we say that

  where: µ C = {⟨Device, uml :: Class⟩, ⟨Device, uml :: Component⟩} µ OP = {⟨hasPart, uml :: Element :: ownedElement⟩, ⟨isConnectedTo, uml :: Association⟩} µ DP = {} 4 https://www.omg.org/spec/UML/2.0/

Figure 1 :

 1 Figure 1: On the left, the UML Class Diagram for model M in example 4, on the right the Composite Structure Diagram for DroneSystem1.

Figure 2 :

 2 Figure 2: The workflow and resources for the implementation of the approach.

Algorithm 2

 2 Construction of µ A Input: M, µ O→UML Output: µ A 1: µ C(i) = µ r(a,b) = {} 2: for each uml::Element E ∈ M do for 18: return µ A = {µ C(i) , µ r(a,b) } line (9) assigns to instance E the most general concept in CORA, i.e. cora : Entity. If it is the case the element E is an instance specification of a relationship (11), we obtain (12) the source a and (13) target b of the relation, and state that ⟨E, E(a, b)⟩ ∈ µ r(a,b) . Note that only stereotyped relations are captured. Once µ A and µ T are constructed, we can apply them to an UML model M, to automatically obtain T UML and A UML .

  and where ⟨M i , x⟩ and ⟨M i , y⟩ can not occur, ifx ̸ = y.

	Definition 6 (The application of a mapping function
	µ UML→O ). Given an ontology O, a mapping µ UML→O and an UML model M, the application of µ UML→O to
	M yields the set of axioms O UML , written as :

⟨M n , r(a, b) n ⟩} with i, j, k, l, m, n ≥ 0. Where each C i is a (possibly complex) concept in ALC H I D , each r i is a role in O,

  Algorithm 1 Construction of µ T Input: M, µ O→UML Output: µ T 1: µ ⊑ = µ ≡ = µ OP = µ DP = {} 2: for each uml::Element E ∈ M do ⟨E, S⟩ ∈ µ OP 32: end if 33: return µ T = {µ ⊑ , µ ≡ , µ OP , {}}elements, both specializations of the same top element uml::InstanceSpecification.

	3:	if E isA uml::Class then
	4:	⟨E, owl : T hing⟩ ∈ µ ⊑
	5:	if E specializaionOf B ∈ M then
	6:	⟨E, B⟩ ∈ µ ⊑
	7:	end if
	8:	for each E.AppliedStereotypes S ∈ M do
	9:	⟨E, S⟩ ∈ µ ⊑
	10:	end for
	11:	CCD E = {}
	12:	for each E.getRelationships R ∈ M do
	13:	if R isA uml::Association and
		R.AppliedStereotypes ̸ = / 0 then
	14:	b= R.target
	15:	S= R.AppliedStereotype
	16:	CCD E = CCD E ⊓ {∃S.b}
	17:	end if
	18:	end for
	19:	for each E.getOwnedElements OE ∈ M do
	20:	if OE isA uml::CompositeAggregation
		then
	21:	b= OE.getType
	22:	S= odrone:hasPart
	23:	CCD E = CCD E ⊓ {∃S.b}
	24:	end if
	25:	end for
	26:	⟨E,CCD E ⟩ ∈ µ ≡
	27:	end if
	28: end for
	29: if	E	isA	uml::Relationship	and
	|E.AppliedStereotypes|= 1 then
	30:	S= R.AppliedStereotype
	31:

https://www.w3.org/TR/owl2-overview/

https://www.eclipse.org/papyrus/index.php